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confronting nuclear physics , we should highlight the great importance of the equation of state ( eos ) , for it being significantly important to study the structure of nuclei , the reaction dynamics of heavy - ion collisions , and many issues in astrophysics @xcite . the nuclear eos consists usually of two ingredients : the energy density for symmetric matter and the density dependence of the symmetry energy . for the former , the saturation properties are quite clear nowadays , though its high - density behavior remains to be revealed in more details . however , the density dependence of the symmetry energy is still poorly known especially at high densities @xcite , and even the trend of the density dependence of the symmetry energy can be predicted to be contrary . while most relativistic theories @xcite and some non - relativistic theories @xcite predict that the symmetry energy increases continuously at all densities , many other non - relativistic theories ( for instance , see @xcite ) , in contrast , predict that the symmetry energy first increases , then decreases above certain supra - saturation densities , and even in some predictions @xcite becomes negative at high densities , referred as the super - soft symmetry energy . therefore , the experimental extraction is of necessity . recently , by analyzing the fopi / gsi data on the @xmath0 radio in relativistic heavy - ion collisions @xcite , the evidence for a super - soft symmetry energy was found @xcite . this finding can result in many consequences , while a direct challenge is how to stabilize a normal neutron star with the super - soft symmetry energy . conventionally , a mechanical instability may occur if the symmetry energy starts decreasing quickly above the certain supra - saturation density @xcite . to solve this problem , one possible way is to take into account the hadron - quark phase transition which lifts up the pressure in pure quark matter @xcite , while the transition is expected to occur at much higher densities within a narrow region of parameters . instead , one may consider the possible correction to the gravity . though the gravitational force was first discovered in the history , it is still the most poorly characterized , compared to three other fundamental forces that can be favorably unified within the gauge theory . for the further grand unification of four forces , the correction to the conventional gravity seems necessary . the light u - boson , which is proposed beyond the standard model , can play the role in deviating from the inverse square law of the gravity due to the yukawa - type coupling , see refs . @xcite and references therein . this light u - boson was used as the interaction propagator of the mev dark matter and was used to account for the bright 511 kev @xmath1-ray from the galactic bulge @xcite . as a consequence of its weak coupling to baryons , the stable neutron star can be obtained in the presence of the super - soft symmetry energy @xcite . in addition , it is noted that through the reanalysis of the fopi / gsi data with a different dynamical model another group extracted a contrary density dependent trend of the symmetry energy at high densities @xcite . the solution of the controversy is still in progress . in pursuit of the covariance in addressing neutron stars bound by the strong gravity , the relativistic models are favorable to obtain the eos , though the fraction , arisen from the relativistic effect of fast particles in the compact core of neutron stars , is just moderate . apart from the non - relativistic models to obtain the eos of neutron stars in ref . @xcite , we will adopt the relativistic mean - field ( rmf ) models in this work . the rmf theory which is based on the dirac equations for nucleons with the potentials given by the meson exchanges achieved great success in the past few decades @xcite . the original lagrangian of the rmf model was first proposed by walecka more than 30 years ago @xcite . the walecka model and its improved versions were characteristic of the cancellation between the big attractive scalar field and the big repulsive vector field . to soften the eos obtained with the simple walecka model , the proper medium effects were accounted with the inclusion of the nonlinear self - interactions of the @xmath2 meson proposed by boguta et . a few successful nonlinear rmf models , such as nl1 @xcite , nl2 @xcite , nl - sh @xcite , nl3 @xcite , and etc . , had been obtained by fitting saturation properties and ground - state properties of a few spherical nuclei . later on , an extension to include the self - interaction of @xmath3 meson was implemented to obtain rmf potentials which were required to be consistent with the dirac - brueckner self - energies @xcite . in this direction , besides the early model tm1 @xcite , there were recent versions pk1 @xcite and fsugold @xcite . although various rmf models reproduce successfully the saturation properties of nuclear matter and structural properties of finite nuclei , the corresponding eos s may behave quite differently at high densities especially in isospin - asymmetric nuclear matter . it was reported in the literature @xcite that the light u - boson can significantly modify the eos in isospin - asymmetric matter . however , the further systematic work to analyze the effect of the light u - boson on various nuclear eos s is still absent . in this work , we will investigate in detail the effect of light u - boson on the eos and properties of neutron stars with various rmf models . in particular , we will address the difference of the effects induced by the u - boson in various rmf models . the paper is organized as follows . in sec . [ rmf ] , we present briefly the formalism based on the lagrangian of the relativistic mean - field models . in sec . [ results ] , numerical results and discussions are presented . at last , a summary is given in sec . [ summary ] . in the rmf approach , the nucleon - nucleon interaction is usually described via the exchange of three mesons : the isoscalar meson @xmath4 , which provides the medium - range attraction between the nucleons , the isoscalar - vector meson @xmath5 , which offers the short - range repulsion , and the isovector - vector meson @xmath6 , which accounts for the isospin dependence of the nuclear force . the relativistic lagrangian can be written as : & = & [ i_^-m+g_-g _ _ ^-g___3 b_0^ ] + & & - f_f^+ m_^2_^ - b _ b^ + m_^2 b_0 b_0^ + & & + ( _ ^-m_^2 ^ 2 ) + u_eff ( , , b_0)+l_u , [ eq : lag1 ] where @xmath7,@xmath6 are the fields of the nucleon , scalar , vector , and neutral isovector - vector mesons , with their masses @xmath8 , and @xmath9 , respectively . @xmath10 are the corresponding meson - nucleon couplings . @xmath11 and @xmath12 are the strength tensors of @xmath5 and @xmath13 mesons respectively , @xmath14 the self - interacting terms of @xmath2 , @xmath5 mesons and the isoscalar - isovector coupling are given generally as u_eff(,^ , b_0^)&=&-g_2 ^ 3-g_3 ^ 4 + c_3(_^)^2 + & & + 4_vg^2_g_^2 _ ^b_0b_0^. [ eq : u ] here , the isoscalar - isovector coupling term is introduced to modify the density dependence of the symmetry energy @xcite . in addition , we include in lagrangian @xmath15 for the u - boson that is beyond the standard model . a very light u - boson can be utilized to interpret the deviation from the newton s gravitational potential which is usually characterized in the form @xcite : v(r)=-(1+e^-r/ ) [ grav]where @xmath16 is the universal gravitational constant , @xmath17 is a dimensionless strength parameter with @xmath18 and @xmath19 being the boson - nucleon coupling constant and baryon mass , respectively , and @xmath20 is the length scale with @xmath21 being the boson mass . according to the conventional view , the yukawa - type correction to the newtonian gravity resides at the matter part rather than the geometric part . thus , following the form of the vector meson , @xmath15 is written as : _ u&=&-g_u_u^-u _ u^ + m_u^2u_u^ , with @xmath22 the field of u - boson . @xmath23 is the strength tensor of u - boson , @xmath24 with the standard euler - lagrange formala , we can deduce from the lagrangian the equations of motion for the nucleon and mesons . they are given as follows : = 0 ( _ t^2-^2+m_^2)&=&g_-g_2 ^ 2-g_3 ^ 3 , + ( _ t^2-^2+m_^2)_&=&g__- c_3_^3 + & & -8_v g_^2g_^2b_0 b_0^ _ , + ( _ t^2-^2+m_^2)b_0&=&g _ _ _ 3 + & & -8_v g_^2g_^2 _ ^b_0 , + ( _ t^2-^2+m_u^2)u_&=&g_u_. in the mean - field approximation , all derivative terms drop out and the expectation values of space - like components of vector fields vanish ( only zero components survive ) due to translational invariance and rotational symmetry of the nuclear matter . in addition , only the third component of isovector fields survives because of the charge conservation . in the mean - field approximation , after the dirac field of nucleons is quantized @xcite , the fields of mesons and u - boson , which are replaced by their classical expectation values , obey following equations : m_^2&=&g__s - g_2 ^ 2-g_3 ^ 3 , + m_^2_0&=&g__b - c_3_0 ^ 3 - 8_v g_^2g_^2b_0 ^ 2_0 , + m_^2b_0&=&g__3 - 8_v g_^2g_^2 _ 0 ^ 2b_0 , + m_u^2u_0&=&g_u_b , where @xmath25 and @xmath26 are the scalar and baryon densities , respectively , and @xmath27 is the difference between the proton and neutron densities , namely , @xmath28 . the set of coupled equations can be solved self - consistently using the iteration method . with these mean - field quantities , the resulting energy density @xmath29 and pressure @xmath30 are written as : & = & _ i = p , n^k_f_i d^3k e^*_i+ m_^2_0 ^ 2+_b^2 + & & + m_^2_0 ^ 2+m_^2 b_0 ^ 2 + g_2 ^ 3+g_3 ^ 4 + & & + c_3_0 ^ 4 + 12_vg^2_g_^2 _ 0 ^ 2 b_0 ^ 2,[eq : e ] + p&=&_i = p , n^k_f_i d^3k + m_^2_0 ^ 2 + _ b^2 + & & -m_^2_0 ^ 2+m_^2 b_0 ^ 2 -g_2 ^ 3-g_3 ^ 4 + & & + c_3_0 ^ 4 + 4_vg^2_g_^2 _ 0 ^ 2 b_0 ^ 2 , [ eq : p]with @xmath31 . given above is the formalism for nuclear matter without considering the @xmath32 equilibrium . for asymmetric nuclear matter at @xmath32 equilibrium , the chemical equilibrium and charge neutrality conditions need to be additionally considered , which are written as : _ n&=&_p+_e , + _ e&=&_p , + _ b&=&_n+_p , where @xmath33 are the chemical potential of neutron , proton and electron , respectively , and @xmath34 is the number density of electrons . in neutron star matter , the eos is obtained by adding in eqs.([eq : e ] ) and ( [ eq : p ] ) the contribution of the free electron gas . the neutron star properties are obtained from solving the tolman - oppenheimer - volkoff ( tov ) equation @xcite : @xmath35[m(r)+4\pi r^3 p(r ) ] } { r(r-2m(r))},\label{eq : tov1}\\ m(r)&=&4\pi\int^r_0d\!\tilde{r}\tilde{r}^2 \vep(\tilde{r}),\label{eq : tov2}\end{aligned}\ ] ] where @xmath36 is the radial coordinate from the center of the star , @xmath37 and @xmath38 are the pressure and energy density at position @xmath36 , respectively , and @xmath39 is the mass contained in the sphere of the radius @xmath36 . note that here we use units for which the gravitation constant is @xmath40 . the radius @xmath41 and mass @xmath42 of a neutron star are obtained from the condition @xmath43 . because the neutron star matter , consisting of neutrons , protons , and electrons ( npe ) at @xmath44 equilibrium in this work , undergoes a phase transition from the homogeneous matter to the inhomogeneous matter at the low density region , the rmf eos obtained from the homogeneous matter does not apply to the low density region . for a thorough description of neutron stars , we thus adopt the empirical low - density eos in the literature @xcite . among a number of nonlinear rmf parametrizations , we select several typical best - fit parameter sets , for instance nl1 @xcite , nl - sh @xcite , nl3 @xcite , tm1 @xcite and fsugold @xcite , to investigate the effects of the u - boson on the eos of isospin - asymmetric nuclear matter and properties of neutron stars . the nonlinear rmf models usually include the nonlinear self - interactions of the @xmath2 meson to simulate appropriate medium dependence of the strong interaction . this is typical in rmf parameter sets nl1 , nl - sh and nl3 . in addition to the nonlinear @xmath2 meson self - interactions , in tm1 and fsugold the nonlinear self - interaction of the @xmath3 meson is also included . parameters and saturation properties of these parameter sets are listed in table [ t : t1 ] . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] ( upper panel ) and pressure p ( lower panel ) as a function of density with various rmf parameter sets , nl3 , nl1 , nl - sh , tm1 , and fsugold in npe matter at @xmath32 equilibrium . [ f : rho],width=321,height=321 ] in fig . [ f : rho ] , the energy density and pressure of npe matter at @xmath32 equilibrium are shown as a function of nucleon density for various models without the inclusion of the u - boson . it is seen that the eos with parameter sets tm1 and fsugold is clearly softer than that with the nl1 , nl - sh and nl3 with the increase of the density . the softening stems from the inclusion of the nonlinear self - interaction of the @xmath5 meson that lowers the repulsion provided by the @xmath3 meson at high densities , while the excess softening with the fsugold as compared to that with the tm1 can be attributed dominately to the larger parameter @xmath45 in fsugold . equilibrium with various rmf models . [ ep0],width=321,height=321 ] shown in fig . [ ep0 ] is the correlation between the pressure and the energy density given in fig . [ f : rho ] . this correlation is usually regarded as the eos that is used as the input of the tolman - oppenheimer - volkoff ( tov ) equation @xcite for the evaluation of the neutron star properties . once again , we see the large deviations in the eos with different rmf models especially at high densities . in the following , it is thus interesting to see how the u - boson affects the eos produced by various rmf models that differs largely at high densities . in units of @xmath46 . [ epnl],width=321,height=385 ] but for the rmf models tm1 and fsugold.[eptf],width=321,height=321 ] in the rmf approximation , the contribution of the u - boson in a linear form is just decided by the ratio of the coupling constant to its mass , i.e. , @xmath47 , as seen in eqs.([eq : e ] ) and ( [ eq : p ] ) . in figs . [ epnl ] and [ eptf ] , the eos s with various models are depicted for a set of ratios @xmath48 . it is shown in figs . [ epnl ] and [ eptf ] that the inclusion of the u - boson stiffens the eos . this is physically obvious since the vector form of the u - boson provides an excess repulsion in addition to the vector mesons , whereas an interestingly large difference appears for different types of models . as shown in figs . [ epnl ] and [ eptf ] , the eos s with the tm1 and fsugold acquires a much more apparent stiffening than that with the nl1 , nl - sh and nl3 by including the u - boson . this phenomenon can be understood by the inherent feature of these models . in models nl1 , nl - sh and nl3 , the repulsion is quadratic in the density because the nonlinear self - interaction of the @xmath5 meson is not considered . with the increase of the density , the repulsion provided by the @xmath3 meson dominates the attraction provided by the @xmath2 meson . the cancellation between the repulsion and attraction in the pressure ( see eq.([eq : p ] ) is not prominent at high densities so that the u - boson plays a similar role in the energy density and pressure . thus , these eos s are just moderately modified by the u - boson , as shown in fig . [ epnl ] . for models tm1 and fsugold that feature a clearly softer eos at high densities , the cancellation between the repulsion and attraction becomes significant and thus sharpens the importance of the u - boson in the pressure . comparing to the addition of the big repulsion and attraction in the energy density , the u - boson just plays a marginal role in modifying the energy density . thus , the u - boson can modify appreciably the correlation between the pressure and energy density in the high - density region in favorably softened models , for instance , the tm1 and fsugold , as shown in fig . [ eptf ] . because in tm1 and fsugold the nonlinear term of the @xmath5 meson plays a decisive role in softening the eos , the larger the parameter @xmath45 , the more apparent the modification , as shown comparatively in the upper and lower panels of fig . [ eptf ] . but to exhibit the difference between the cases with and without the modification to the symmetry energy . left panels represent the results with the nl3 and nl3@xmath49 , and right panels are the results with the tm1 and tm1@xmath49 . different density dependencies of the symmetry energy are drawn in the insets of upper panels , while given in the insets of lower panels are the eos of two cases in the absence of the u - boson . [ epnl3],width=321,height=321 ] in addition , it is interesting to examine whether the significant difference in the u - boson - induced modification to the eos can be created by softening the symmetry energy . the symmetry energy is softened by including the isoscalar - isovector coupling term in rmf models ( see eq.([eq : u ] ) ) . in fig . [ epnl3 ] , we depict the eos without ( upper panels ) and with ( lower panels ) the softening of the symmetry energy in nl3 and tm1 . however , no visible difference in two cases with the nl3 is observed , and with the tm1 the difference is not significant . this observation seems to show a contrast with that in ref . @xcite where the fluffy eos due to the super - soft symmetry energy can be lifted up by the u - boson to support a normal neutron star . in deed , the magnitude of the modification to the eos caused by the u - boson relies on the softness of the eos . as long as the eos is modified significantly by softening the symmetry energy , the stiffening role of the u - boson in the eos can be considerably enhanced accordingly . given that the stiff eos with the nl3 is little modified by softening the symmetry energy , as shown in the inset of the left lower panel in fig . [ epnl3 ] , the softening of the symmetry energy can scarcely affect the role of the u - boson . for models with a softer eos , the situation can turn out to be different when the eos is modified appreciably by softening the symmetry energy . indeed , the vital role of the u - boson in the eos of the non - relativistic mdi model with a super - soft symmetry energy @xcite is a typical case that the role of the u - boson can be largely amplified due to the softening of the symmetry energy . in rmf models , for instance , the tm1 whose eos is softer than that with the nl3 , the softening of the symmetry energy can also result in some visible difference in the eos and thereby the role of the u - boson , as shown in right panels of fig . [ epnl3 ] . .[mr],width=321,height=302 ] next , we turn to the consequences in hydrostatic neutron stars with the eos modified by the u - boson . using eqs.([eq : tov1 ] ) and ( [ eq : tov2 ] ) , the mass and radius of hydrostatic neutron stars can be obtained with the given eos . in fig . [ mr ] , the mass - radius ( m - r ) relation of neutron stars is depicted with different ratio parameter @xmath48 for the u - boson in various models . with the inclusion of the u - boson , we can see that both the maximum mass and radius of neutron stars increase significantly . it is clearly seen that the star maximum mass with the soft eos is modified more significantly by the u - boson . this is consistent with the corresponding modification to the high - density eos caused by the u - boson , as shown in figs . [ epnl ] and [ eptf ] . the consistency is established on the fact that the maximum mass of neutron stars is dominated by the high - density behavior of the eos . in the past , a few neutron stars with large masses around @xmath50 had been observed @xcite . though it can have improvements in experimental aspects , the observation of neutron stars with large masses is not so scarce . recently , the mass of the lmxb 4u1608 - 52 is measured to be 1.74@xmath51 @xcite , and most recently a @xmath50 neutron star j1614 - 2230 was measured through the shapiro delay @xcite . note that the model fsugold which is well consistent with the nuclear laboratory constraints just produces a maximum mass about 1.7@xmath51 for the neutron star without hyperons , whereas the hyperonization can further reduce the maximum mass to a value below @xmath52 . in this case , the role of the u - boson is constructive in increasing the maximum mass of neutron stars , either as the eos is softened by the creation of new degrees of freedom , or the eos is too soft to obtain a large maximum mass . on the other hand , the radius of neutron stars is primarily determined by the eos in the lower density region of @xmath53 to @xmath54 , see refs.@xcite and references therein . because the symmetry energy in this density region offers the most important ingredient of the pressure in pure neutron matter , the density dependence of the symmetry energy plays a crucial role in determining the radius of neutron stars . while in the present case the pressure in the lower density region is increased appreciably by the u - boson , it is not surprising that the sensitive variation of the neutron star radius is obtained accordingly . this is similar to the non - relativistic case in ref . in fact , the radius of neutron stars relies sensitively on the stiffness of the eos . thus , the stiffening of the eos caused by the u - boson gives rise to a significant increase of the radius . concretely , we can see from fig . [ mr ] that the larger rise of the radius comes up with the more apparent stiffening role of the u - boson in softer models . it is known that the radius of neutron stars extracted from the observation can have a wide range due to the uncertainties of the distance measurement and theoretical models used for the spectrum analyses @xcite . a more precise extraction of the neutron star radius , probably through the coincident measurements , thus becomes very significant , because it can test the non - newtonian gravity due to its promising sensitivity to the star radius . ( left panel ) and the @xmath55 ( right panel).[mr100],width=321,height=264 ] to stress the role of the u - boson in the maximum mass and radius of neutron stars , we depict in fig . [ mr100 ] the m - r relation for various models with and without the u - boson . here , for the case with the inclusion of the u - boson , the calculation is performed with @xmath56 . it is seen clearly that the large difference in maximum masses with various types of models can be reduced largely by the u - boson with suitable parameter @xmath57 . we can see once again that the reduction of the difference is mainly attributed to the role of the u - boson in the models featuring much softer eos s . interestingly , we see that the uncertainty of the radius for a canonical neutron star ( with the mass @xmath52 ) can also be reduced by the u - boson . in view of interesting and significant roles of the u - boson , we may say that the task to look for the u - boson and further confirm the non - newtonian gravity is also confronted . the recent experimental constraints on the relationship between parameters @xmath58 ( @xmath18 ) and @xmath59 ( @xmath21 ) can be found in ref . @xcite . to recover the stability of neutron stars using the eos constrained by the fopi / gsi data @xcite , the ratio @xmath60 was found to be needed @xcite . in this work , the effect of the u - boson is investigated within the parameter region @xmath61 . to avoid the visible effect beyond low energy constraints in finite nuclei , with these values of the ratio parameter we may estimate that the mass of the u - boson should be of order below @xmath62 with the coupling strength being almost or at least three orders less than the fine - structure constant , while these estimated orders can be compatible with parameter regions allowed by a few experimental constraints , see ref . we expect that more precision experiments will be performed to better determine or exclude the parameter regions for the non - newtonian gravity . at last , it is interesting to discuss the relevance between the parameters of the non - newtonian gravity touched upon in this work and the solution to the dark matter problem . in order to explain the flatness of the rotational curve of galactic spirals , one needs to assume the non - luminous dark matter being the additional gravitational source . alternatively , the newtonian gravity that was well tested in the solar system may be assumed to fail at the large distance scales of galaxies , and hence the newtonian gravity should be modified to be the non - newtonian one @xcite . the yukawa - type modification to the newtonian gravity due to the boson exchange may possibly be considered as a candidate to solve the dark matter problem . in this work , the vector coupling of the u - boson that is restrained by the u(1 ) symmetry produces a repulsion other than the anticipated attraction . we may thus suppose to solve the dark matter problem through the introduction of light scalar bosons . however , since the flatness of the rotational curve requires a supplemental force roughly linear inversely in the distance from the center of the galaxy , even if the light scalar boson is assumed to provide the needed attraction in one region , the exponential suppression factor of the yukawa - type potential ( see eq.([grav ] ) ) actually inhibits the reproduction of the rotational curve in other regions . in deed , in addition to the introduction of the light scalar boson , more considerations are necessary to solve the dark matter problem @xcite . on the other hand , we may explore the constraints from the effect of the u - boson on the dark matter . however , the coupling of the u - boson with the dark matter candidates should be assumed to be much stronger than that with the normal particles to explain the @xmath63 @xmath1-ray observation while simultaneously compatible with the low - energy constraints @xcite . to sum up , we are presently not able to restrain the parameters of the non - newtonian gravity originated from the u - boson exchange in this work directly by using the effect of the u - boson on the dark matter and/or the solution to the dark matter problem with the modified newtonian dynamics . nonetheless , this deserves further exploration . for instance , the further first - principle understanding of the underlying origin of the difference in the u - boson couplings to normal and dark matter particles may open possibility to extract constraints on the parameters of the non - newtonian gravity . we have studied in this work the effects of the u - boson in rmf models on the equation of state and subsequently the consequence in neutron stars . all rmf models are chosen to have similarly nice reproduction of saturation properties and ground - state properties of finite nuclei , whereas they can give rise to a significantly large difference in eos s at high densities and mass - radius relations of neutron stars . interestingly , we find that the u - boson in models with much softer eos plays a much more significant role in increasing the maximum mass of neutron stars . the distinction can be attributed analytically to the different modification caused by the u - boson in soft and stiff models to the pressure . thus , the inclusion of the u - boson may allow the existence of the non - nucleonic degrees of freedom in the interior of large mass neutron stars initiated with the favorably soft eos of normal nuclear matter . in addition , it is worth notifying that the radius of canonical neutron stars in all models can be sensitively modified by the u - boson due to its stiffening role in the eos . meanwhile , the difference in the mass - radius relations predicted by various models can favorably be reduced by increasing the coupling strength between the u - boson and baryons . at last , constraints on the parameters of the non - newtonian gravity are discussed . presently , we have not found the direct relevance between the parameters of the non - newtonian gravity originated from the u - boson exchange and its effect on the dark matter concerning the dark matter problem . together with the future coincident measurements and more precise extraction of the mass and radius of neutron stars , the sensitive role of the u - boson in the m - r relation may be helpfully used to test the physics beyond the standard model and consequently the existence of the non - newtonian gravity in the dense neutron star . authors thank professors de - hua wen , lie - wen chen and bao - an li for useful discussions . the work was supported in part by the srtp grant of the educational ministry of china , the national natural science foundation of china under grant no . 10975033 , the china jiangsu provincial natural science foundation under grant no.bk2009261 , and the china major state basic research development program under contract no . 2007cb815004 . 99 j. m. lattimer and m. prakash , phy . rep . * 333 * , 121 ( 2000 ) ; astrophys . j. * 550 * , 426 ( 2001 ) ; science * 304 * , 536 ( 2004 ) ; phy . rep . * 442 * , 109 ( 2007 ) . c. j. horowitz and j. piekarewicz , phy . lett . * 86 * , 5647 ( 2001 ) . d. j. nice , i. h. stairs , and l. e. kasian , 2008 , in aip conf . 983 , 40 years of pulsars : millisecond pulsars , magnetars and more , ed . c. bassa , z.wang , a. cumming , and v. m. kaspi ( aip : new york ) , 453 | we investigate the effects of the light vector u - boson that couples weakly to nucleons in relativistic mean - field models on the equation of state and subsequently the consequence in neutron stars .
it is analyzed that the u - boson can lead to a much clearer rise of the neutron star maximum mass in models with the much softer equation of state .
the inclusion of the u - boson may thus allow the existence of the non - nucleonic degrees of freedom in the interior of large mass neutron stars initiated with the favorably soft eos of normal nuclear matter .
in addition , the sensitive role of the u - boson in the neutron star radius and its relation to the test of the non - newtonian gravity that is herein addressed by the light u - boson are discussed . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
automatic object recognition and image classification are important and challenging tasks . this paper is inspired by the remarkable recent work of poggio , serre , and their colleagues @xcite , on rapid object categorization using a feedforward architecture closely modeled on the human visual system . the main directions it departs from that work are twofold . first , trading - off biological accuracy for computational efficiency , our results exploit more engineering - motivated mathematical tools such as wavelet and grouplet transforms @xcite , allowing faster computation and limiting ad - hoc parameters . second , the approach is generalized by adding a degree of _ feedback _ ( another known component of human perception ) , yielding significant performance and robustness improvement in multiple - object scenes . in experiments , the resulting scale- and translation - invariant algorithm achieves or exceeds state - of - the - art performance in object recognition , but also in texture and satellite image classification , and in language identification . as in @xcite , the algorithm is hierarchical . in addition , motivated in part by the relative uniformity of cortical anatomy @xcite , the two layers of the hierarchy are made to be computationally similar , as shown in fig . [ fig : algo ] . layer one performs a wavelet transform @xcite in the @xmath0 unit followed by a local maximum operation in the @xmath1 unit . the transform in the @xmath2 unit in layer two is similar to the grouplet transform @xcite , and is followed by a global maximum operation in the @xmath3 unit . + @xmath4 * wavelet transform*. the frequency and orientation tuning of cells in visual cortex v1 can be interpreted as performing a wavelet transform of the retinal image @xcite . let us denote @xmath5 a gray - level image of size @xmath6 . a translation - invariant wavelet transform is performed on the ] where @xmath7 denotes the orientation ( horizontal , vertical , diagonal ) , @xmath8 is a wavelet function and @xmath9 are the wavelet coefficients . scale invariance is achieved by a normalization @xmath10 where @xmath11 is the image energy within the support of the wavelet @xmath12 . one can verify that @xmath13 where @xmath14 and @xmath15 are the coefficients of @xmath5 and of its @xmath16-time zoomed version @xmath17 . the normalization also makes the recognition invariant to global linear illumination change . + @xmath18 * local maximum * limited translation invariance is achieved at this stage by keeping the local maximum of @xmath14 coefficients in a subsampling procedure : @xmath19 the maximum being taken at each scale @xmath20 and orientation @xmath21 within a spatial neighborhood of size proportional to @xmath22 . the resulting @xmath23 map at scale @xmath20 and orientation @xmath21 is thus of size @xmath24 . @xmath25 * grouplet - like transform*. cells in visual cortex v2 and v4 have larger receptive fields comparing to those in v1 and are tuned to geometrically more complex stimuli such as contours and corners @xcite . the geometrical grouplets recently proposed by mallat @xcite imitate this mechanism by grouping and re - transforming the wavelet coefficients . the procedure in @xmath2 is similar to the grouplet transform . instead of grouping the wavelet coefficients with a multi - scale geometrically adaptive association field and then re - transforming them with haar - like functions as in @xcite , responses of @xmath2 are obtained via inner products between @xmath23 coefficients and sliding patch functions of different sizes : @xmath26 where @xmath27 of support size @xmath28 are patch functions that group the 3 wavelet orientations in a square of size @xmath29 . while the grouplet functions are adaptively chosen to fit the geometry in the image @xcite , the patch functions @xmath27 , @xmath30 are learned with a simple random sampling as in @xcite : each patch is extracted at a random scale and a random position from the @xmath23 coefficients of a randomly selected training image , the rationale being that patterns that appear with high probability are likely to be learned . + @xmath31 * global maximum*. a global maximum operation in space and in scale is applied on @xmath32 and the resulting @xmath33 coefficients @xmath34 are thus invariant to image translation and scale change . + * classification * the classification uses @xmath33 coefficients as features and thus inherits the translation and scale invariance . while various classifiers such as svms can be used , a simple but robust nearest neighbor classifier will be applied in the experiments . structures that appear with a high probability are likely to be learned as patch functions through random sampling . however , they are not necessarily salient and neither are the resulting @xmath33 features . this suggests active selection of the learned patches . for example , lowe and mutch have constructed sparse patches by retaining one salient direction at each position @xcite . a simple patch selection is proposed here by sorting the variances of the @xmath33 coefficients of the _ training _ images . a small @xmath35 variance implies that the corresponding patch @xmath27 is not salient . [ fig : c2:variance]-a plots the variance of the @xmath33 coefficients of the motorcycle and the background images in the caltech5 database ( see fig . [ fig : caltech5 ] ) , the @xmath32 patches being learned from the same images . out of the 1000 patches , 200 salient ones whose resulting @xmath33 have non - negligible variance are selected . other patches usually correspond to nonsalient structures such as a common background and are therefore excluded . [ fig : c2:variance]-b and c show that after patch selection the 200 @xmath33 coefficients are mainly positioned around the object , as opposed to the 1000 @xmath33 coefficients spreading over all the image prior to patch selection . the recognition using these salient patches is not only more robust but also 5 times faster . [ cols="^,^,^ " , ] inspired by the biologically motivated work of @xcite , we have described a wavelet - based algorithm which can compete with the state - of - the - art methods for fast and robust object recognition , texture and satellite image classification , language recognition and sound classification . a feedback procedure has been introduced to improve recognition performance in multiple - object scenes . potential applications also include video archiving ( semantic video analysis ) , video surveillance , high - throughput drug development , texture retrieval , and robotic learning by imitation . to further improve and extend the algorithm , a key aspect will be a more refined use of feedback between different levels . such feedback will naturally involve stability and convergence questions , which will in turn both guide the design of the algorithm and shape its performance . in addition , contrary to the nervous system , the algorithm need not be constrained by information transmission delays between different levels . preliminary ideas in this direction are briefly discussed in the appendix . + * appendix * the first step towards introducing a dynamic systems perspective aimed at further development of feedback mechanisms is simply to rewrite the algorithm in terms of differential equations , which puts it in a form more suitable to subsequent analysis of stability and convergence . with spike amplitude equal to @xmath46 @xmath47 the function @xmath48 is discussed later in this section and in a companion paper . the dynamics of @xmath49 can be modified in turn so that states corresponding to each object appear in sequence according to the state @xmath50 where , componentwise , @xmath53 where @xmath54 is active and and @xmath55 otherwise . note that @xmath50 smoothly transitions between @xmath56 and @xmath46 according to the attended object . the positive gain @xmath57 is chosen such that @xmath58 , where @xmath59 is the spike duration , itself a fraction of the interspike period . techniques for globally stable spike - based clustering are described in a companion paper , based on modified fitzhugh - nagumo neural oscillators @xcite , similar to @xcite , @xmath60\nonumber\end{aligned}\ ] ] where @xmath61 is the membrane potential of the oscillator , @xmath62 is an internal state variable representing gate voltage , @xmath63 represents the external current input , and @xmath64 , @xmath65 and @xmath66 are strictly positive constants . using a diagonal metric transformation @xmath67 , one easily shows , similarly to @xcite , that one of the most immediate additional feedback mechanisms to be explored is that of generalized diffusive connections ( @xcite , section 3.1.2 ) . in a feedback hierarchy , these correspond to achieving consensus between multiple processes of different dimensions . similarly to @xcite , composite variables for dynamic tracking can be used at every level , based on both top - down an bottom - up information . this allows one to implicitly introduce time - derivatives of signals in the differential equations , without having to measure or compute these terms explicitly . 99 k. chen and d.l . wang , `` a dynamically coupled neural oscillator network for image segmentation . '' , _ neural networks _ , 15 , 423 - 439 . r. fergus and p. perona and a. zisserman , `` object class recognition by unsupervised scale - invariant learning '' , _ cvrp _ , vol.2 , pp.264 - 271 , 2003 . r. fitzhugh , `` impulses and physiological states in theoretical models of nerve membrane '' , _ biophysical journal _ , vol.1 , pp.445 - 466 , 1961 . j. hawkins , s.blakeslee , _ on intelligence _ , times books , 2004 . r.m.haralick , k.shanmugam , i.dinstein , `` textural features for image classification '' , _ ieee trans . on sys man cy , smc-3 , ( 6 ) : 610 - 621 _ , 1973 . k. kim , k. jung , s. park , and h. kim , `` support vector machines for texture classification '' , _ ieee trans . pami _ , vol.24 , no.11 , pp.1542 - 1550 , 2002 . s. lazebnik , c. schmid and j. ponce , `` a sparse texture representation using local affine regions '' , _ ieee transactions on pattern analysis and machine intelligence _ , vol . 27 , no . 8 , pp . 1265 - 1278 , 2005 . x. liu and d. wang , `` texture classification using spectral histograms '' , _ ieee trans . pami _ , vol.12 , pp.661 - 670 , 2003 . w. lohmiller , and j.j.e . slotine , `` global convergence rates of nonlinear diffusion for time - varying images '' , _ scale - space theories in computer vision , lecture notes in computer science _ , vol.1682 , springer verlag ( 1999 ) . s. lu and c. tan , `` script and language identification in noisy and degraded document images '' , _ ieee trans . pami _ , vol.30 , no.1 pp.14 - 24 , 2008 . b.luo , j-f.aujol , y.gousseau , s.ladjal , `` indexing of satellite images with different resolutions by wavelet features '' , _ ieee trans image proc _ , accepted , 2008 . s. mallat , `` geometrical grouplets '' , _ acha _ , to appear , 2008 . s. mallat , _ a wavelet tour of signal processing _ , academic press , 2nd edition , 1999 . v. mountcastle , `` an organizing principle for cerebral function : the unit model and the distributed system '' , the mindful brain , mit press , 1978 . j. nagumo . and s. arimoto , and s. yoshizawa , `` an active pulse transmission line simulating nerve axon , '' _ proceedings of the ire _ , 50(10 ) , pp.2061 - 2070,1962 . pham and j.j.e . slotine , `` stable concurrent synchronization in dynamic system networks , '' _ neural networks _ , 20(1 ) , 2007 . t. randen and j. husoy , `` filtering for texture classification : a comparative study '' , _ ieee trans on image proc _ , vol.21 , no.4 , pp.291 - 310 , 1999 . j. mutch and d. lowe , `` multiclass object recognition with sparse , localized features '' , _ cvpr 06 _ , pp.11 - 18 . rao and d.h . ballard , `` predictive coding in the visual cortex : a functional interpretation of some extra - classical receptive - field effects '' , _ nature neuroscience _ , 2 , 1 , 79 , 1999 . t. serre , l. wolf , s. bileschi , m. riesenhuber and t. poggio , `` robust object recognition with cortex - like mechanisms '' , _ ieee trans . pami _ , vol.29 , no.3 , pp.411 - 426 , 2007 . von melchner , l. , pallas , s.l . and sur , m , `` visual behavior mediated by retinal projections directed to the auditory pathway '' , _ nature _ , 404 , 2000 . d. walther , t. serre , t. poggio and c. koch , `` modeling feature sharing between object detection and top - down attention '' , vss , may 2005 . w. wang and j.j.e . slotine , `` on partial contraction analysis for coupled nonlinear oscillators , '' _ biological cybernetics _ , 92(1 ) , 2005 . m. weber and m. welling and p. perona , `` unsupervised learning of models for recognition '' , _ eccv _ , pp.18 - 32 , 2000 . | we investigate a biologically motivated approach to fast visual classification , directly inspired by the recent work @xcite . specifically , trading - off biological accuracy for computational efficiency , we explore using wavelet and grouplet - like transforms to parallel the tuning of visual cortex v1 and v2 cells , alternated with max operations to achieve scale and translation invariance .
a feature selection procedure is applied during learning to accelerate recognition .
we introduce a simple attention - like feedback mechanism , significantly improving recognition and robustness in multiple - object scenes . in experiments ,
the proposed algorithm achieves or exceeds state - of - the - art success rate on object recognition , texture and satellite image classification , language identification and sound classification . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in our current understanding of pulsar magnetospheres and radiation mechanisms , strongly magnetized rotating neutron stars play a central role . the underlying plasma processes like particle acceleration , pair creation and pulsed emission profiles throughout the whole electromagnetic spectrum strongly depend on the peculiar magnetic field geometry and strength adopted or extracted from numerical simulations of the magnetosphere . for instance radio emission is believed to emanate from the polar caps , therefore in regions of strong gravity where curvature and frame - dragging effects are considerable due to the high compacity of neutron stars @xmath4 for typical models with its mass @xmath5 , its radius @xmath6 and the schwarzschild radius given by @xmath1 , @xmath7 being the gravitational constant and @xmath8 the speed of light . detailed quantitative analysis of radio pulse polarization and pair cascade dynamics could greatly benefit from a better quantitative description of the electromagnetic field around the polar caps . although there exists an extensive literature about flat space - time electrodynamics , only little work has been done to include general - relativistic effects . the first general solution for an oblique rotator in flat vacuum space - time was found by @xcite with closed analytical formulas . this solution is often quoted to explain the magnetic dipole radiation losses . to be truly exact , we emphasize that the poynting flux @xmath9 derived from his solution does not strictly coincide with the point dipole losses @xmath10 but depends on the ratio @xmath11 , where @xmath12 is the light cylinder radius and @xmath3 the rotation rate of the neutron star . it is only equal to the textbook equation for dipole losses in the limit of vanishing radius @xmath13 . the distinction is meaningful at least for checking results emanating from numerical computations . indeed , because of limited computer resources , we are often forced to take ratios @xmath14 not completely negligible compared to unity . therefore the computed spin - down luminosity can significantly deviate from the point dipole losses . moreover , @xcite showed in the case of an aligned rotator that the electric field induced by frame - dragging effects could be as high as the one induced by the stellar rotation itself . these results were extended to an oblique rotator a few years later by thanks to a formalism developed earlier by @xcite . it is therefore crucial to treat maxwell equations in the general - relativistic framework in order to analyse quantitatively acceleration and radiation in the vicinity of the neutron star . this led @xcite to seek for an approximate solution of maxwell equations in a curved space - time either described by the schwarzschild metric or by the kerr metric , using a linearised approach employing the newman - penrose formalism . he computed the structure of the electromagnetic waves propagating in vacuum and launched by a rotating dipole . he also gave an expression for the poynting flux @xmath15 depending on the ratio @xmath11 . the exact analytical solution for the static magnetic dipole in schwarzschild space - time was given by @xcite and extended to multipoles by . @xcite also studied the influence of space - time curvature and frame dragging effects on the electric field around the polar caps of a pulsar and confirmed the earlier claims of an increase in its strength . @xcite computed the electric field for an aligned rotator in vacuum in the schwarzschild metric . the aligned rotator has also been investigated by @xcite with special emphasize to particle acceleration in vacuum . @xcite and @xcite took a similar approach to study the acceleration of particles around polar caps . @xcite computed the electromagnetic field in the exterior of a slowly rotating neutron star in the slow rotation metric as well as inside the star and investigated the impact of oscillations . they gave approximate analytical expressions for the external electromagnetic field close to the neutron star . @xcite extended the previous work by solving numerically the equations for the oblique rotator in vacuum in general relativity . they retrieve @xcite results close to the surface and the deutsch solution for distances larger than the light cylinder @xmath16 . it is the purpose of this paper to elucidate quantitatively and accurately some aspects of general - relativistic effects on the electrodynamics close to the neutron star . our goal is to derive a general formalism to compute the solution of maxwell equations in curved space - time for any multipole component of the magnetic field . consequently , we use a 3 + 1 formalism of electrodynamics in curved space - time as presented in [ sec : modele ] . next we show how to solve for the electromagnetic field for an aligned rotator in [ sec : aligne ] . this method is easily extended to a perpendicular rotator as explained in [ sec : orthogonal ] . because maxwell equations in vacuum are linear , the most general solution for an oblique rotator will be a linear superposition of the weighted aligned and perpendicular rotator . conclusions and future possible work are drawn in [ sec : conclusion ] . the covariant form to describe the gravitational and electromagnetic field in general relativity is the natural way to write them down in a frame independent way . nevertheless , it is more intuitive to split space - time into an absolute space and a universal time , similar to our all day three dimensional space , rather than to use the full four dimensional formalism . another important advantage of a 3 + 1 split is a straightforward transcription of flat space techniques for scalar , vector and tensor fields to curved spaces . we start with a description of the special foliation used for the metric . next we derive maxwell equations in this foliation and conclude on some words about force - free electrodynamics which will be treated in another work but for completeness we give the useful expressions already in this paper . we therefore split the four dimensional space - time into a 3 + 1 foliation such that the metric @xmath17 can be expressed as @xmath18 where @xmath19 , @xmath20 is the time coordinate or universal time and @xmath21 some associated space coordinates . we use the landau - lifschitz convention for the metric signature given by @xmath22 @xcite . @xmath23 is the lapse function , @xmath24 the shift vector and @xmath25 the spatial metric of absolute space . by convention , latin letters from @xmath26 to @xmath27 are used for the components of vectors in absolute space ( in the range @xmath28 ) whereas latin letters starting from @xmath29 are used for four dimensional vectors and tensors ( in the range @xmath30 ) . our derivation of the 3 + 1 equations follow the method outlined by @xcite . a fiducial observer ( fido ) is defined by its 4-velocity @xmath31 such that @xmath32 this vector is orthogonal to the hyper - surface of constant time coordinate @xmath33 . its proper time @xmath34 is measured according to @xmath35 the relation between the determinants of the space - time metric @xmath36 and the pure spatial metric @xmath37 is given by @xmath38 for a slowly rotating neutron star , the lapse function is @xmath39 and the shift vector @xmath40 we use spherical coordinates @xmath41 and an orthonormal spatial basis @xmath42 . the spin @xmath26 is related to the angular momentum @xmath43 by @xmath44 . it follows that @xmath26 has units of a length and should satisfy @xmath45 . introducing the moment of inertia @xmath46 , we also have @xmath47 . for the remainder of the paper , it is also convenient to introduce the relative rotation of the neutron star according to @xmath48 in the special case of a homogeneous and uniform neutron star interior with spherical symmetry , the moment of inertia is @xmath49 thus the spin parameter can be expressed as @xmath50 we adopt this simplification for the neutron star interior in order to compute the spin parameter @xmath26 . let @xmath51 and @xmath52 be the electromagnetic tensor and its dual respectively , see appendix [ app : metric ] . it is useful to introduce the following spatial vectors @xmath53 such that [ eq : bdeh ] @xmath54 @xmath55 is the vacuum permittivity and @xmath56 the vacuum permeability . @xmath57 is the fully antisymmetric spatial tensor and @xmath58 the three dimensional levi - civita symbol . the contravariant analog is @xmath59 . relations eq . ( [ eq : bdeh ] ) can be inverted such that @xmath60 these three dimensional vectors can be recast into @xmath61 these expressions are also easily inverted such that @xmath62 all these antisymmetric tensors are summarized in appendix [ app : metric ] . with these definitions of the spatial vectors , maxwell equations take a more traditional form in the curved three dimensional space . the system reads @xmath63 the source terms @xmath64 are given by @xmath65 @xmath66 being the 4-current density . the above differential operators should be understood as defined in a three dimensional curved space , the absolute space with associated spatial metric @xmath25 , such that @xmath67 the special case of a diagonal spatial metric is given in appendix [ app : operateur ] . the three dimensional vector fields are not independent , they are related by two important constitutive relations , namely [ eq : constitutive ] @xmath68 the curvature of absolute space is taken into account by the lapse function factor @xmath23 in the first term on the right - hand side and the frame dragging effect is included in the second term , the cross - product between the shift vector @xmath69 and the fields . we see that @xmath70 are the fundamental fields , actually those measured by a fido , see below . the source terms have not yet been specified . having in mind to apply the above equations to the pulsar magnetosphere , we give the expressions for the current in the limit of a force - free plasma , neglecting inertia and pressure . the force - free condition in covariant form reads @xmath71 and in the 3 + 1 formalism it becomes @xmath72 which implies @xmath73 and therefore also @xmath74 . as in the special relativistic case , the current density is found to be , see the derivation for instance in @xcite @xmath75 because @xmath76 and @xmath77 , @xmath78 and @xmath79 can be interpreted as the magnetic and electric field respectively as measured by the fido . moreover @xmath80 thus @xmath81 is the electric charge density as measured by this same observer . using the projection tensor defined by @xmath82 its electric current density @xmath83 is given by @xmath84 maxwell equations ( [ eq : maxwell1])-([eq : maxwell4 ] ) , the constitutive relations ( [ eq : constitutivee]),([eq : constitutiveh ] ) and the prescription for the source terms set the background system to be solved for any prescribed metric . in the next section , we show how to solve this system in a simple way by introducing a vector spherical harmonic basis in curved space as summarized in appendix [ app : hsv ] . for the remainder of this paper , we will only focus on the vacuum field solutions , leaving the force - free case for future work . note that we choose to keep all physical constants in the formulas because this helps to check easier the consistency with dimensionality of the equations . the system to be solved being linear , we treat separately the aligned and the perpendicular case , the general oblique configuration being a weighted linear superposition of both solutions . we first address the simple static and rotating aligned dipole magnetic field before investigating the interesting perpendicular rotator as a special case of an oblique rotator . we start with a non rotating neutron star , setting the spin to zero , @xmath85 , therefore @xmath86 , followed by a simplification of the constitutive relations . the electric field vanishes , thus @xmath87 whereas @xmath88 . as a consequence , the magnetic field satisfies the static ( @xmath89 ) maxwell equations given by @xmath90 far from the neutron star , we expect to retrieve the flat space - time expression for the dipole magnetic field with magnetic moment @xmath91 or , written explicitly , @xmath92 = - \frac{\mu_0\,\mu}{4\,\pi } \ , \sqrt{\frac{8\,\pi}{3 } } \ , \mathrm{re } \left [ { \mathbf{\nabla } \times}\frac{\mathbf \phi_{1,0}}{r^2}\right]\ ] ] in curved space - time , the meaning of a dipole field needs to be explicitly defined . we take as a definition for the dipolar magnetic field the one which is expressed only with the first vector spherical harmonic @xmath93 corresponding to the mode @xmath94 according to its flat space - time expression eq . ( [ eq : bdipoleplat ] ) . this is valid for a symmetry around the @xmath95-axis because @xmath96 . the perpendicular case or more generally the oblique rotator would include the mode @xmath97 for the dipolar field . this will be done in section [ sec : orthogonal ] . thus we expand the magnetic field according to the divergencelessness prescription and look for a separable solution with the prescription @xmath98\ ] ] with the boundary condition @xmath99 @xmath93 is a vector spherical harmonic , see for instance @xcite . the vector spherical harmonics being proper functions of the curl linear differential operator insure that such separable solutions do indeed exist . these linear algebra properties are absolutely fundamental and make vector spherical harmonics extremely useful to solve linear partial differential equations involving vector fields . note that @xmath100 is the unique unknown in this simple problem and depends only on the radial coordinate @xmath101 . ( [ eq : static1 ] ) is automatically satisfied by construction whereas inserting the expansion eq . ( [ eq : bstatic1 ] ) into eq . ( [ eq : static2 ] ) following the property eq . ( [ eq : rotrotphi ] ) of appendix [ app : hsv ] ( for @xmath102 ) leads to a second order linear ordinary differential equation for the scalar function @xmath103 such that @xmath104 the exact solution to this boundary problem which asymptotes to the flat dipole at large distances as prescribed by eq . ( [ eq : bstatic2 ] ) is given by @xmath105\ ] ] which corresponds to the solution shown in @xcite . the non vanishing magnetic field components are [ eq : magneticstatic ] @xmath106 \ , \frac{\mu\,\cos\vartheta}{r_s^3 } \\ \label{eq : magneticstatict } b^{\hat \vartheta } & = 3 \ , \frac{\mu_0}{4\,\pi } \ , \left [ 2 \ , \sqrt { 1 - \frac{r_s}{r } } \ , { \rm ln } \left ( 1 - \frac{r_s}{r } \right ) + \frac{r_s}{r } \ , \frac{2\,r - r_s}{\sqrt{r\,(r - r_s ) } } \right ] \ , \frac{\mu\,\sin\vartheta}{r_s^3}\end{aligned}\ ] ] corrections to first order compared to flat space - time are @xmath107 \\ b^{\hat \vartheta } & = \frac{\mu_0}{4\,\pi } \ , \frac{\mu\,\sin\vartheta}{r^3 } \ , \left [ 1 + \frac{r_s}{r } + o\left ( \frac{r_s}{r } \right ) \right]\end{aligned}\ ] ] this first example shows how easy it is to compute the solution once the expansion onto vector spherical harmonics has been performed and knowing their properties and action on linear differential operators . next we consider the more useful case of a rotating magnetic dipole with magnetic moment aligned to the rotation axis . now the situation becomes much more involved . first , rotation induces an electric field and secondly frame dragging effects mix electric and magnetic fields through the constitutive relations eq . ( [ eq : constitutive ] ) . to demonstrate how our formalism works , we decided to split the task in two steps . first we neglect frame dragging effects and look solely for the induced electric field . in a second stage , we add frame dragging . frame dragging effects could become important and should be included . nevertheless , before dealing with the most general expression including frame dragging , we think it is educational to introduce the reasoning by hand and work out a low order expansion explicitly without any frame dragging effect . this would be acceptable for sufficiently low rotation and we can in the first stage neglect the shift vector setting @xmath108 as in the previous paragraph . maxwell equations then become @xmath109 these equations are particularly straightforward to solve because it represents a decoupled system of two unknown vector fields , one for @xmath110 and one for @xmath78 . from the flat space - time solution , we know that the electric field will be quadrupolar which means only one mode is present namely @xmath111 for the axisymmetric case . thus we expand both fields according to @xmath112 \\ \label{eq : tournantb1 } \mathbf b & = \mathrm{re } \left [ { \mathbf{\nabla } \times } ( f_{1,0}^b \ , \mathbf \phi_{1,0 } ) \right ] \end{aligned}\ ] ] this expansion insure automatically and analytically the divergencelessness nature of both @xmath110 and @xmath78 . moreover , these expressions lead as in the previous static regime to a separable solution for both the electric and magnetic field . straightforward calculations show that @xmath103 again satisfies eq . ( [ eq : laplacef10 ] ) whereas @xmath113 has to be solution of another second order linear differential equation given by @xmath114 it is obtained by inserting the expansion eq . ( [ eq : tournantd2 ] ) into eq . ( [ eq : tournant2 ] ) following the property eq . ( [ eq : rotrotphi ] ) of appendix [ app : hsv ] but now for @xmath115 . the exact solution of this homogeneous linear differential equation and vanishing at infinity reads @xmath116\ ] ] where @xmath117 is a constant to be determined from the boundary conditions at the surface of the neutron star . we now discuss this inner boundary condition in more details . inside a perfectly conducting star , the rotation of the plasma induces an electric field @xmath118 which satisfies @xmath119 this implies an electric field as measured by a fido given by @xmath120 for this fido , the electromagnetic field symbolized by @xmath70 has to verify the jump conditions across an interface as in flat space - time . in other words , the magnetic field component normal to the surface and the electric field components lying in the plane of the interface are continuous functions . more explicitly , the normal component @xmath121 and the tangential components @xmath122 have to be continuous across the stellar surface . by construction , it can be verified by projection of eq . ( [ eq : tournantd2 ] ) onto @xmath123 that the component @xmath124 remains zero in the exterior vacuum space , as it is inside the star . note that this remark is consistent with the projection of eq . ( [ eq : cld ] ) onto @xmath123 . for the other tangential component , by projection of eq . ( [ eq : cld ] ) onto @xmath125 we have to enforce the condition @xmath126 this has to be compared with the projection of eq . ( [ eq : tournantd2 ] ) onto @xmath125 and given by @xmath127 in order to deduce the constant of integration @xmath117 in eq . ( [ eq : dipoleschwarzf20 ] ) , eq . ( [ eq : limitefd20 ] ) and ( [ eq : limited20 ] ) should be compared at the stellar surface setting @xmath128 . @xmath121 is known from the static dipole solution and given by eq . ( [ eq : magneticstaticr ] ) . by direct calculation from eq . ( [ eq : dipoleschwarzf20 ] ) we arrive at @xmath129\ ] ] the constant @xmath117 then follows immediately from the above condition . we get @xmath130 where @xmath131^{-1}\end{aligned}\ ] ] the magnetic field remains the same as for the static dipole and the electric field yields @xmath132 \ , ( 3\,\cos^2\vartheta - 1 ) \\ d^{\hat \vartheta } & = 6 \ , \frac{\varepsilon_0 \ , \mu_0 \ , \mu}{4\,\pi } \ , \frac{r}{r_s^3 } \ , \frac{\tilde{\omega}_r}{\alpha_r^2 } \ , \alpha \ , c_1 \ , c_2 \ , \left [ \left ( 1 - 2\,\frac{r}{r_s } \right ) \ , \ln\alpha^2 - 2 - \frac{r_s^2}{6\,r^2\,\alpha^2 } \right ] \ , \cos\vartheta \ , \sin\vartheta \\ d^{\hat \varphi } & = 0\end{aligned}\ ] ] so far , we did not include any frame dragging effect symbolized by the cross product in the constitutive relations eq . ( [ eq : constitutivee ] ) and ( [ eq : constitutiveh ] ) . now we proceed to the inclusion of the frame dragging effect to look for more accurate solutions taking explicitly into account the rotation of the neutron star . because these constitutive relations and the vacuum maxwell equations are linear , we use a power series expansion of the unknown vector fields with respect to a small adimensionalized parameter @xmath133 which is related to the neutron star spin such that @xmath134 . any vector field @xmath135 is expanded into @xmath136 @xmath137 is the static field for the non rotating star . thus , to zero - th order , the electric field vanishes , @xmath138 . they are at least first order in @xmath3 . the shift vector is a quantity of first order so we write it as @xmath139 . from the constitutive relations , we get the @xmath140-th order of the auxiliary electric field for @xmath141 as @xmath142 and for the @xmath140-th order of the auxiliary magnetic field @xmath143 moreover , for any order , we have the constraints @xmath144 as a consequence , we obtain a hierarchical set of partial differential equations for the fields @xmath145 such that @xmath146 for @xmath141 . the initialisation for @xmath147 corresponds to the static dipole with @xmath148 given by eq . ( [ eq : dipoleschwarzf20 ] ) , therefore @xmath138 as expected . we immediately conclude that the first perturbation in magnetic field corresponds to a second order term symbolized by @xmath149 ( @xmath150 ) . we look for the first order perturbation in electric field corresponding to an electric quadrupole with @xmath111 such that @xmath151 from now on , we suppress the real part symbol , it should be understood that the physical fields correspond to the real parts of the expressions derived below . inserting this expansion into eq . ( [ eq : hierarchy1 ] ) with @xmath152 , the function @xmath113 is solution of the following second order inhomogeneous linear ordinary differential equation @xmath153\ ] ] the right hand side is obtained from the property eq . ( [ eq : rotbetarotvhs ] ) . to solve this equation , we use standard techniques . first we look for the general solution to the homogeneous equation which is nothing else than eq . ( [ eq : laplacef20 ] ) with the subsequent solution eq . ( [ eq : dipoleschwarzf20 ] ) , which we write here as @xmath154 . next a peculiar solution of the inhomogeneous eq . ( [ eq : laplacesourcef20 ] ) and vanishing at infinity is given by @xmath155\ ] ] in order to satisfy the boundary condition on the star , we also need the following expression @xmath156 in order to compute @xmath157 in eq . ( [ eq : limited20 ] ) from @xmath158 . the constant @xmath117 will be determined from the inner boundary condition , now taking the frame dragging effect into account because of the presence of the peculiar solution @xmath159 , it has to be set to @xmath160\ ] ] putting all pieces together , the full solution @xmath158 reads @xmath161 \\ - 2 \ , \frac{\varepsilon_0 \ , \mu_0 \ , \mu}{4\,\pi } \ , \sqrt{\frac{6\,\pi}{5 } } \ , \frac{a\,c}{r_s^2\,r } \ , \left [ \ln\alpha^2 + \frac{r_s}{r } \right]\end{gathered}\ ] ] taking the value of the constant @xmath117 into account , we finally get @xmath162 - 2 \ , \frac{\omega\,r^3}{r_s^3 } \ , \left ( \ln\alpha^2 + \frac{r_s}{r } \right ) \right\}\end{gathered}\ ] ] if we separate the frame dragging effect @xmath163 from the pure rotation @xmath3 , we get @xmath164 - \right . \\ \left . 2 \ , \frac{\omega\,r^4}{r_s^5 } \ , \left ( \frac{r_s^2}{r^2 } \ , \left ( \ln\alpha^2 + \frac{r_s}{r } \right ) + \frac{c_2\,r_s^2}{3\,\alpha_r^2\,r^2 } \ , \left ( \ln\alpha_r^2 + \frac{r_s}{r } \right ) \ , \times \right . \right . \\ \left.\left.\left [ \left ( 3 - 4\,\frac{r}{r_s } \right ) \ , \ln\alpha^2 + \frac{r_s^2}{6\,r^2 } + \frac{r_s}{r } - 4 \right ] \right ) \right\}\end{gathered}\ ] ] the components of the electric field are then @xmath165 with the radial derivative given explicitly by @xmath166 \right . \\ & \left . - \frac{\omega\,r^3}{r_s^5 } \left ( \frac{6\,c_2\,r_s^2}{\alpha_r^2\,r^2 } \ , \left ( \ln\alpha_r^2 + \frac{r_s}{r } \right ) \ , \left [ \left ( 1 - 2 \ , \frac{r}{r_s } \right ) \ , { \rm ln } \alpha^2 - 2 - \frac{r_s^2}{6\,\alpha^2\,r^2 } \right ] + 3 \ , \frac{r_s^4}{\alpha^2\,r^4 } \right ) \right\ } \nonumber\end{aligned}\ ] ] these expressions are the same as equations ( 124)-(125)-(126 ) in @xcite specialized to the aligned rotator . in the newtonian limit we find @xmath167 as it should be . the properties of the vector spherical harmonics in curved space allow us to derive in a systematic way the relations between the expansion coefficients of @xmath145 . because of the axisymmetry of the problem , there are no toroidal components of neither the magnetic nor the electric part . therefore , all coefficients with @xmath168 vanish . thus we expand both fields according to @xmath169 the superscript @xmath170 denotes the order of the expansion in the spin parameter , related to the frame dragging effect . injecting those expressions into eqs . ( [ eq : hierarchy1 ] ) and ( [ eq : hierarchy2 ] ) , we get for @xmath171 according to eq . ( [ eq : rotbetarotvhs ] ) in appendix [ app : hsv ] @xmath172\end{gathered}\ ] ] @xmath173\end{gathered}\ ] ] it is understood that @xmath174 . the very important fact about this hierarchical system of second order linear partial differential equations relating the @xmath175 to the @xmath175 is its _ uncoupled _ nature . indeed the coefficients @xmath175 and @xmath176 are related to the immediately lowest order expansion coefficients @xmath177 and @xmath178 . consequently , we can find the solution to any order by simply computing more coefficients . the recurrence starts with the static aligned magnetic dipole which is the zero - th order approximation of the solution , with subscript @xmath179 . as already noted in the previous paragraph , the electric field is a first order effect at least . for concreteness , let us work out the approximate solution to third order , i.e. including to vector spherical harmonic functions in the expansion of both the electric and the magnetic field . at first glance , this seems a rather high degree of accuracy for such solution in contrast to the first order expansion of the background metric . nevertheless , we have in mind to use such results as a benchmark to test forthcoming general relativistic electromagnetic solvers in free space and in the force - free approximation in order to extend the code presented in @xcite . this justifies our wish to reach a high degree of accuracy for the numerical solutions even if the metric is only first order accurate in the spin parameter @xmath26 . the electric field is a consequence of the rotation of the star , thus to zero - th order , there is only a magnetic field , i.e. @xmath180 and @xmath181 , the dipole in schwarzschild space - time given by eq . ( [ eq : dipoleschwarzf10 ] ) , all other @xmath182 with @xmath183 being equal to zero . the initialisation with @xmath180 implies that there are no first order corrections to the magnetic field because eq . ( [ eq : fl0bk ] ) has a vanishing right hand side . it is a linear homogeneous second order partial differential equation with zero boundary conditions on the star and at infinity . therefore the solution vanishes identically leading to @xmath184 . the first correction comes from the coefficients @xmath185 which have to satisfy eq . ( [ eq : fl0dk ] ) . there is only one inhomogeneous equation corresponding to @xmath115 with the right hand side including @xmath186 . if written explicitly , we retrieve eq . ( [ eq : laplacesourcef20 ] ) namely @xmath187 with its subsequent solution . the next order includes a perturbation in the magnetic field . indeed , the coefficients @xmath188 have to satisfy eq . ( [ eq : fl0bk ] ) with source terms emanating only from @xmath189 , supplemented with vanishing boundary conditions . the two equations are inhomogeneous , namely @xmath190 finally , this perturbed magnetic field will feed back to the electric field to third order with the non - vanishing coefficients satisfying @xmath191 \\ \label{eq : f40d3 } \partial_r(\alpha^2\,\partial_r(r\,f_{4,0}^{d(3 ) } ) ) - \frac{20}{r } \ , f_{4,0}^{d(3 ) } & = 4 \ , \sqrt{\frac{15}{7 } } \ , \varepsilon_0 \ , \omega \ , f_{3,0}^{b(2)}\end{aligned}\ ] ] but with boundary conditions at the stellar surface according to eq . ( [ eq : cld ] ) . equations ( [ eq : f10b2])-([eq : f20d3 ] ) show the hierarchical set we are led to in order to improve the solution step by step by including an increasing number of multipoles of order @xmath192 in accordance with the degree of approximation desired in the spin parameter . some of these equations can be solved analytically with source terms , but we were unable to write down simple expressions for the solution with appropriate boundary conditions except for the very few first coefficients . finding closed expression is a cumbersome task . eventually , we decided to solve the above set of equations numerically by spectral methods . we expand the solutions into rational chebyshev functions as defined in @xcite . see below for the details . our starting point is to use a finite number of multipolar coefficients in the expansion of both the fields , @xmath193 terms for @xmath194 and @xmath195 terms for @xmath196 , writing [ eq : dbdvlptalign ] @xmath197 the order in the spin parameter , previously labelled as @xmath170 , has disappeared in the numerical solution , we do not perturb anymore according to @xmath26 . each of the coefficient @xmath198 and @xmath199 has to satisfy the differential equation which is given for the magnetic field by @xmath200\end{gathered}\ ] ] and for the electric field by @xmath201\end{gathered}\ ] ] the boundary conditions at infinity enforce vanishing coefficients whereas on the neutron star surface , we have to impose continuity of the tangential @xmath110 and normal @xmath78 components on the stellar surface . introducing the expansions eq . ( [ eq : dbdvlptalign ] ) into eq . ( [ eq : cld ] ) , then projecting along @xmath125 and using the useful identities for frame - dragging presented in appendix [ sec : framedragging ] we get the relation between the coefficients of @xmath110 and @xmath78 as @xmath202 = \\ \varepsilon_0 \ , r \ , \tilde{\omega } \ , \left [ \sqrt{l\,(l+1 ) } \ , ( 1 - j_{l,0}^2 - j_{l+1,0}^2 ) \ , f_{l,0}^{b } - \right . \\ \left . \sqrt{(l-2)\,(l-1 ) } \ , j_{l,0 } \ , j_{l-1,0 } \ , f_{l-2,0}^{b } - \sqrt{(l+2)\,(l+3 ) } \ , j_{l+1,0 } \ , j_{l+2,0 } \ , f_{l+2,0}^{b}\right ] \end{gathered}\ ] ] where quantities have to be evaluated on the neutron star surface , at @xmath128 . some recurrence formulas are very useful to deal with spherical harmonics . the three recurrences used to impose the boundary conditions are @xmath203 with the constants given by @xmath204 let us write down explicitly the equations for the three first coefficients in @xmath196 and @xmath194 . the system of partial differential equations reads @xmath205 \\ \partial_r(\alpha^2\,\partial_r(r\,f_{5,0}^{b } ) ) - \frac{30}{r } \ , f_{5,0}^{b } & = \frac{2}{\sqrt{11 } } \ , \frac{\omega}{\varepsilon_0 \ , c^2 } \ , \left [ - 5 \ , \sqrt{6 } \ , f_{4,0}^{d } + 9 \ , \sqrt{\frac{35}{13 } } \ , f_{6,0}^{d } \right ] \\ \label{eq : f20d } \partial_r(\alpha^2\,\partial_r(r\,f_{2,0}^{d } ) ) - \frac{6}{r } \ , f_{2,0}^{d } & = \frac{6}{\sqrt{5 } } \ , \varepsilon_0 \ , \omega \ , \left [ f_{1,0}^{b } - 3 \sqrt{\frac{2}{7 } } \ , f_{3,0}^{b } \right ] \\ \label{eq : f40d } \partial_r(\alpha^2\,\partial_r(r\,f_{4,0}^{d } ) ) - \frac{20}{r } \ , f_{4,0}^{d } & = 2 \ , \sqrt{3 } \ , \varepsilon_0 \ , \omega \ , \left [ 2 \ , \sqrt{\frac{5}{7 } } \ , f_{3,0}^{b } - 5 \ , \sqrt{\frac{2}{11 } } \ , f_{5,0}^{b } \right ] \\ \partial_r(\alpha^2\,\partial_r(r\,f_{6,0}^{d } ) ) - \frac{42}{r } \ , f_{6,0}^{d } & = 18 \ , \sqrt{\frac{35}{143 } } \ , \varepsilon_0 \ , \omega \ , f_{5,0}^{b } \end{aligned}\ ] ] the associated boundary conditions are [ eq : clfd ] @xmath206 \\ \alpha^2 \ , \left [ \frac{2}{3}\,\sqrt{\frac{5}{7 } } \ , \partial_r(r\,f_{4,0}^{d } ) - \sqrt{\frac{6}{35 } } \ , \partial_r(r\,f_{2,0}^{d } ) \right ] & = \varepsilon_0 \ , r \ , \tilde{\omega } \ , \left [ \frac{44}{15\,\sqrt{3 } } \ , f_{3,0}^{b } - \frac{2}{5 } \ , \sqrt{\frac{6}{7 } } \ , f_{1,0}^{b } - \frac{20}{3}\,\sqrt{\frac{10}{231 } } \ , f_{5,0}^{b } \right ] \\ \alpha^2 \ , \left [ \sqrt{\frac{42}{143 } } \ , \partial_r(r\,f_{6,0}^{d } ) - \frac{2}{3}\,\sqrt{\frac{5}{11 } } \ , \partial_r(r\,f_{4,0}^{d } ) \right ] & = \varepsilon_0 \ , r \ , \tilde{\omega } \ , \left [ \frac{58}{39 } \ , \sqrt{\frac{10}{3 } } \ , f_{5,0}^{b } - \frac{40}{3\,\sqrt{231 } } \ , f_{3,0}^{b } \right]\end{aligned}\ ] ] again where quantities have to be evaluated on the neutron star surface , at @xmath128 . we emphasize that the magnetic field at the neutron star surface is exactly matched to the expression for the general - relativistic static dipole , eq . ( [ eq : dipoleschwarzf10 ] ) . all other multipole fields @xmath199 with @xmath207 vanish at @xmath128 by our definition . the computation of the electromagnetic field in vacuum has been reduced to a system of linear ordinary differential equations of second order . moreover , it is a boundary value problem to be solved in a semi - infinite interval , from @xmath128 to @xmath208 . several different techniques exist to treat such a system . we choose to employ spectral methods , expanding the unknown functions onto special basis functions . according to @xcite , dealing with rational chebyshev functions @xmath209 is a judicious choice for the interval @xmath210 . these functions are defined by @xmath211 where @xmath212 are the chebyshev polynomials of order @xmath140 and @xmath213 . @xmath214 is a scaling parameter which should reproduce the characteristic length of the problem . we choose @xmath215 . any radial function @xmath216 is therefore expanded into a finite number of @xmath217 terms such that @xmath218 the inner and outer boundary conditions for the magnetic field coefficients @xmath219 are expressed as @xmath220 actually , all the @xmath221 vanish except for the dipole @xmath222 , recall that we strictly impose a dipolar magnetic field on the neutron star surface . the outer boundary conditions for the electric field coefficients @xmath223 are the same as eq . ( [ eq : clf1 ] ) . the inner boundary conditions are different because we enforce conditions on the derivative , see eq . ( [ eq : clfd ] ) , but it remains a relation involving linear terms in the expansion coefficients . we use what @xcite calls the boundary - bordering method leading to a linear algebra system to be solved . for more details on spectral and pseudo - spectral methods , see for instance also @xcite . technically , we use mathematica 9 to compute the matrix coefficients and invert the system to find the expansion coefficients . for the subsequent numerical applications , we normalize the magnetic moment of the neutron star to unity , @xmath224 . in order to demonstrate the accuracy of our spectral algorithm to solve the system of linear ordinary differential equations , we begin with the static aligned dipole . the neutron star radius is set to @xmath225 although it is irrelevant for the static dipole case because there is no rotation and no scaling with @xmath226 . the schwarzschild radius is chosen with increasing value compared to the stellar radius , we take @xmath227 . the absolute value of the expansion coefficients of the function @xmath228 are shown in fig . [ fig : fb10_align ] with a number of collocations points @xmath229 . for any value of the compactness parameter @xmath230 , we get the prescribed 15 digits of accuracy , although that for compactness close to unity , we need more coefficients . indeed , for very low compactness @xmath231 , red curve with full circles , less than ten coefficients are required to get full accuracy . the same remark holds for any @xmath232 . for the typical compactness of a neutron star , @xmath233 , magenta curve with full triangles , we need almost 25 coefficients to achieve the required accuracy . nevertheless , the spectral convergence of our computation is clearly identified by the exponential decrease of the magnitude of the highest - order coefficients within an accuracy of 15 digits . the other question to address is the efficiency of our spectral algorithm to reproduce the exact solution depending on the strength of deviation from the flat space - time metric . in order to check the correctness of these coefficients , we have to compare them with the expansion coefficients of the analytical solution given by eq . ( [ eq : dipoleschwarzf10 ] ) . for the static aligned dipole . @xmath140 corresponds to the order of the k - th rational chebyshev function and the numbers in the legend depict the ratio @xmath234.,scaledwidth=50.0% ] to do this , we define the absolute error between the analytical @xmath235 and the numerical @xmath236 solution by @xmath237 this error is plotted in fig . [ fig : fb10_align_erreur ] and shows a perfect match between both solutions , within the numerical accuracy . we reach 15 digits of significance for the relevant coefficients , those which are not zero numerically . this explains the decreasing number of significant digits when the coefficients are close to zero . they are meaningless . for the static aligned dipole . @xmath140 corresponds to the order of the k - th rational chebyshev function and the numbers in the legend depict the ratio @xmath234.,scaledwidth=50.0% ] this first example demonstrates the very high accuracy obtainable by our spectral method . next , we pursue with the rotating aligned dipole . rotation combined with frame dragging effects will produce higher order multipole coefficients which to first order depend linearly on the spin parameter @xmath26 . we already gave an approximate analytical solution to the lowest order , i.e. the induced electric field without taking into account the perturbation in the magnetic field . nevertheless with our numerical integration procedure , we are able to give solutions to any order in the multipole moments @xmath192 . we therefore proceed in an increasing order of complexity . starting with only the two functions @xmath228 and @xmath238 to the lowest approximation , corresponding to the magnetic dipole and to the electric quadrupole , we then successively add the couple @xmath239 and conclude with two more functions @xmath240 . consequently , we can quantitatively estimate the contribution to the electromagnetic field from higher multipoles other than dipole and quadrupole . we performed different sets of calculation by combining slow and fast rotation @xmath241 with low and high compactness @xmath242 with normalized magnetic moment @xmath224 . we start with a very slowly rotating dipole for which @xmath243 and a low compactness @xmath244 in order to look for small perturbations of the electric field induced by frame dragging effects . we can therefore compare the approximate analytical expressions with the more accurate numerical one . the absolute value of the rational chebyshev coefficients of the lowest order approximation are shown in fig . [ fig : f_align_rot_1_r1000_rs2000 ] for @xmath228 and @xmath238 . spectral convergence is achieved as expected . the discrepancy between the analytical solution and the numerical computation are small , less than @xmath245 , the absolute error between both sets of coefficients is close to zero as can be seen in fig . [ fig : erreur_align_rot_1_r1000_rs2000 ] . in red circles and @xmath238 in blue squares , for the aligned rotating dipole . the parameters are @xmath244 and @xmath243.,scaledwidth=50.0% ] in red circles and for the electric field @xmath246 in blue squares . the parameters are @xmath244 and @xmath243.,scaledwidth=50.0% ] in a second set of calculations , we increased the frame - dragging effects by taking @xmath247 and @xmath244 . the absolute value of the rational chebyshev coefficients of the lowest order approximation are shown in fig . [ fig : f_align_rot_1_r10_rs2000 ] for @xmath228 and @xmath238 . spectral convergence is achieved as expected . here also the absolute discrepancy between both sets of coefficients is close to zero as can be seen in fig . [ fig : erreur_align_rot_1_r10_rs2000 ] . in red circles and @xmath238 in blue squares , for the aligned rotating dipole . the parameters are @xmath244 and @xmath247.,scaledwidth=50.0% ] in red circles and for the electric field @xmath246 in blue squares . the parameters are @xmath244 and @xmath247.,scaledwidth=50.0% ] in a third set of calculations , we increased the compactness by taking @xmath243 and @xmath248 . these values are typical for radio pulsars . the absolute value of the rational chebyshev coefficients of the lowest order approximation are shown in fig . [ fig : f_align_rot_1_r1000_rs2 ] for @xmath228 and @xmath238 . spectral convergence is achieved as expected . here also the absolute discrepancy between both sets of coefficients is close to zero as can be seen in fig . [ fig : erreur_align_rot_1_r1000_rs2 ] . in red circles and @xmath238 in blue squares , for the aligned rotating dipole . the parameters are @xmath248 and @xmath243.,scaledwidth=50.0% ] in red circles and for the electric field @xmath246 in blue squares . the parameters are @xmath248 and @xmath243.,scaledwidth=50.0% ] in a last set of calculations , we increased the rotation frequency by taking @xmath247 and @xmath248 . these values are typical for millisecond pulsars . the absolute value of the rational chebyshev coefficients of the lowest order approximation are shown in fig . [ fig : f_align_rot_1_r10_rs2 ] for @xmath228 and @xmath238 . spectral convergence is achieved as expected . the absolute discrepancy is shown in fig . [ fig : erreur_align_rot_1_r10_rs2 ] in red circles and @xmath238 in blue squares , for the aligned rotating dipole . the parameters are @xmath248 and @xmath247.,scaledwidth=50.0% ] in red circles and for the electric field @xmath246 in blue squares . the parameters are @xmath248 and @xmath247.,scaledwidth=50.0% ] next , we go on in this section about the aligned rotator by computing higher order multipoles @xmath249 to demonstrate that they are several orders of magnetic less than the magnetic dipolar and electric quadrupolar moment . results are shown in fig . [ fig : f_align_rot_2_r1000_rs2000 ] for two more multipoles with a slowly rotating non compact star . the same for a rapidly rotating neutron star is shown in fig . [ fig : f_align_rot_2_r10_rs2 ] . and @xmath250 for the aligned rotating dipole . the parameters are @xmath244 and @xmath243.,scaledwidth=50.0% ] and @xmath250 for the aligned rotating dipole . the parameters are @xmath248 and @xmath247.,scaledwidth=50.0% ] we conclude this section by computing even higher order multipoles @xmath251 to demonstrate that they are also several orders of magnetic less than the lower multipolar moments . for a total of 6 multipoles , we get the coefficients represented in fig . [ fig : f_align_rot_3_r1000_rs2000 ] for the slowly rotating non compact star and for a rapidly rotating neutron star in fig . [ fig : f_align_rot_3_r10_rs2 ] . and @xmath252 for the aligned rotating dipole.,scaledwidth=50.0% ] and @xmath252 for the aligned rotating dipole.,scaledwidth=50.0% ] the dipolar magnetic field as well as the electric quadrupolar field are not significantly affected by the higher multipolar fields . indeed , we show the discrepancy in the expansion coefficients in fig . [ fig : erreur_multipole_r10_rs2 ] . we first compare the dipole magnetic field quadrupole electric field expansion versus a dipole plus octupole @xmath253 expansion of the magnetic field and a quadrupole plus @xmath254 electric fields , denoted by @xmath255 and @xmath256 . the same can be performed with a threefold expansion for both fields and denoted by @xmath257 and @xmath258 . higher order multipoles can also be compared by inspection of @xmath259 and @xmath260 . comparison with the lowest order expansion is not possible because this approximate solution does not contain neither @xmath261 nor @xmath262 . we conclude from the plots in fig . [ fig : erreur_multipole_r10_rs2 ] that the discrepancy in the expansion coefficients is not relevant . in other words , adding higher multipoles will not significantly perturb the lower expansion coefficients . for an almost non rotating and non compact star , the discrepancies are shown in fig . [ fig : erreur_multipole_r1000_rs2000 ] . they are weaker than in the previous case . [ cols="^,^ " , ] in this paper , we showed how to look for a systematic solution to the stationary maxwell equations in the background space - time of a slowly rotating neutron star following the 3 + 1 foliation and an expansion of the unknown electromagnetic field onto vector spherical harmonics . we obtained numerical solutions of high accuracy for the aligned rotator and less accurate for the orthogonal rotator . we hope that these results will serve as a benchmark for general - relativistic codes solving the electromagnetic field in a static background metric like for instance pulsar and black hole magnetospheres . the orthogonal rotator could still benefit from some improvements by replacing the asymptotic spherical hankel functions by more precise functions which take into account at least to first order the perturbation in the metric induced by the presence of the mass @xmath5 . this would lead to more rapid convergence of the solution but those analytical solutions do not exist . a next step would be to solve the time dependent maxwell equations instead of looking for solutions to the boundary value problem , especially difficult to handle with high accuracy for an orthogonal rotator . this could improve the estimate of the magnetic dipole losses in a curved space - time . a further step to this work will be to include a force - free plasma surrounding the neutron star in order to compute the pulsar force - free magnetosphere in the general - relativistic case . the same technique could be useful for the black hole magnetosphere . however , because of the non linearity implied by the force - free current , it is impossible to solve the system semi - analytically as we did here . computing the force - free magnetosphere requires numerical simulations . this is the subject of a forthcoming paper in which we will describe a time dependent pseudo - spectral code using the vector spherical harmonics expansion to solve maxwell equations in a curved vacuum space - time . i would like to thank ric gourgoulhon and serguei komissarov for helpful discussions . j. l. , cohen j. m. , 1970 , ap&ss , 9 , 146 j. p. , 2001 , chebyshev and fourier spectral methods . springer - verlag c. , hussaini m. y. , quarteroni a. , zang t. a. , 2006 , spectral methods j. m. , kearney m. w. , 1980 , ap&ss , 70 , 295 j. m. , kegeles l. s. , 1974a , physical review d , 10 , 1070 j. m. , kegeles l. s. , 1974b , physics letters a , 47 , 261 j. m. , kegeles l. s. , 1975 , physics letters a , 54 , 5 j. m. , toton e. t. , 1974 , annals of physics , 87 , 244 a. j. , 1955 , annales dastrophysique , 18 , 1 v. l. , ozernoy l. m. , 1964 , zh . teor . fiz . , 47 , 1030 y. , matsunaga n. , okita t. , 2004 , mnras , 348 , 1388 s. s. , 2011 , mnras , 418 , l94 k. , kojima y. , 2000 , progress of theoretical physics , 104 , 1117 l. , lifchitz e. , 1989 , _ thorie des champs_. editions mir moscou a. , harding a. k. , 1997 , apj , 485 , 735 a. g. , tsygan a. i. , 1992 , mnras , 255 , 61 j. , 2012 , mnras , 424 , 605 j. a. , 1974 , physical review d , 10 , 3166 j. , 1976 , general relativity and gravitation , 7 , 459 l. , ahmedov b. j. , miller j. c. , 2001 , mnras , 322 , 723 l. , j. ahmedov b. , 2004 , mnras , 352 , 1161 n. , shibata s. , 2003 , apj , 584 , 427 s. , 1995 , apj , 449 , 224 o. , rezzolla l. , 2002 , mnras , 331 , 376 we give the explicit expressions for the metric and the electromagnetic field tensor using the landau - lifschitz signature @xmath22 @xcite . the metric decomposed into time and space components reads then @xmath263 where @xmath23 is the lapse function , @xmath24 the shift vector and @xmath264 . the spatial metric is simply given by @xmath265 and the inverse metric by @xmath266 the contravariant components of the electromagnetic field tensor expressed with the fields @xmath267 are @xmath268 and its dual expressed with the fields @xmath269 are @xmath270 the covariant components of the electromagnetic field tensor expressed with the fields @xmath269 are @xmath271 and for its dual with respect to @xmath267 are @xmath272 the metric of a slowly rotating neutron star remains very close to the usual flat space , except for the radial direction . indeed the spatial metric is diagonal such that @xmath273 for any scalar field @xmath274 , the gradient and laplacian are respectively @xmath275 the physical components of a vector are depicted by hatted indexes . the differential vector operators are then for the divergence and the curl @xmath276 \\ ( { \mathbf{\nabla } \times}\mathbf b)^{\hat \vartheta } & = \frac{1}{r\,\sin\vartheta } \ , \partial_\varphi\,b^{\hat r } - \frac{\alpha}{r } \ , \partial_r(r\,b^{\hat \varphi } ) \\ ( { \mathbf{\nabla } \times}\mathbf b)^{\hat \varphi } & = \frac{\alpha}{r } \ , \partial_r(r\,b^{\hat \vartheta } ) - \frac{1}{r } \ , \partial_\vartheta\,b^{\hat r } \end{aligned}\ ] ] these expressions are very similar to their flat space equivalent , except for the replacement of the radial derivative @xmath277 by @xmath278 in each term . the transverse part of the spatial metric @xmath25 with @xmath279 is exactly the same as for the flat space . because the flat space vector spherical harmonics ( vsh ) lie only in this transverse sub - space , it is straightforward to extend these vsh to the special metric eq . ( [ eq : metric3d ] ) as shown in the next paragraph . we generalize the vector spherical harmonics ( vsh ) introduced in @xcite to a three - dimensional curved space . the three sets of vector spherical harmonics we use are defined by the vector spherical harmonics share some useful properties with respect to their spatial derivatives . first , assume a 3d scalar field @xmath283 expanded onto the scalar spherical harmonics such that @xmath284 then , its gradient expanded onto the vsh becomes @xmath285 the action of the same gradient on the vsh gives the divergence of any vector field @xmath281 as @xmath286 and for the curl @xmath287\end{gathered}\ ] ] for each component taken individually , we find for the divergence finally , define the radial differential operator @xmath290 by @xmath291 = \frac{\alpha}{r } \ , \frac{d}{dr } \left ( \alpha \ , \frac{d}{dr } ( r\,f ) \right ) - \frac{l(l+1)}{r^2 } \ , f = \frac{\alpha}{r^2 } \ , \frac{d}{dr } \left ( \alpha \ , r^2 \ , \frac{df}{dr } \right ) - \frac{l(l+1)}{r^2 } \ , f\ ] ] the vsh noted @xmath292 are eigenvectors in the linear algebra meaning , of the vector laplacian operator @xmath293 . indeed , it is straightforward to show that @xmath294 = \mathcal{d}_l[f ] \ , \mathbf{\phi}_{l , m}\ ] ] it is thus an extension of the properties of the scalar spherical harmonics to the realm of 3d vectors . another useful relation is @xmath295 \ , \mathbf{\phi}_{l , m}\ ] ] where we introduced the operator @xmath296 \equiv \left [ \frac{1}{r } \ , \frac{\partial}{\partial r } \left ( \alpha^2\,\frac{\partial}{\partial r}(r\,f ) \right ) - \frac{l(l+1)}{r^2 } \ , f \right]\ ] ] finding the components of @xmath281 is therefore equivalent to finding the expansion coefficients of three scalar fields onto the scalar spherical harmonics . this procedure works for any vector field . however , the magnetic field being divergencelessness , only two of the three components are independent . it is therefore judicious to deal properly with those kind of fields by analytically enforcing the condition on the divergence as explained below . any divergencelessness vector field is efficiently developed onto the vector spherical harmonic _ orthonormal basis_. this will be the case for the magnetic field and the electric field in our algorithm . assume that the vector field @xmath135 is divergencelessness . it is helpful to introduce two scalar functions @xmath300 and @xmath301 such that the decomposition immediately implies the property of divergencelessness field . this is achieved by writing @xmath302 + g_{l , m}(r , t ) \ , \mathbf{\phi}_{l , m } \right)\ ] ] this expression automatically and _ analytically _ enforces the condition @xmath303 . let us quickly draw the way to compute these functions . the transformation from the spherical components to the functions @xmath304 is given by in our 3 + 1 formulation of maxwell equations in curved space , the frame dragging effects are included in the constitutive relations eq . ( [ eq : constitutivee ] ) , ( [ eq : constitutiveh ] ) , i.e. the cross product of two divergencelessness vector fields @xmath308 and @xmath194 or @xmath308 and @xmath196 . from the definition of the vsh and the shift vector we get @xmath309 and therefore for the curl @xmath310 the second useful set of identities involves @xmath311\ ] ] applying straightforward algebra using the vsh definitions and the scalar harmonics eigenfunction properties , we get @xmath312\end{gathered}\ ] ] we give explicit expressions for the first few modes @xmath313 for the aligned rotator @xmath96 | pulsars are thought to be highly magnetized rotating neutron stars accelerating charged particles along magnetic field lines in their magnetosphere and visible as pulsed emission from the radio wavelength up to high energy x - rays and gamma - rays .
being highly compact objects with compactness close to @xmath0 , where @xmath1 is the schwarzschild radius and @xmath2 the mass and radius of the neutron star , general - relativistic effects become important close to their surface .
this is especially true for the polar caps where radio emission is supposed to emanate from , leading to well defined signatures such as linear and circular polarization . in this paper
, we derive a general formalism to extend to general relativity the deutsch field solution valid in vacuum space .
thanks to a vector spherical harmonic expansion of the electromagnetic field , we are able to express the solution to any order in the spin parameter @xmath3 of the compact object .
we hope this analysis to serve as a benchmark to test numerical codes used to compute black hole and neutron star magnetospheres .
[ firstpage ] stars : neutron - stars : magnetic fields - general relativity - methods : analytical - methods : numerical |
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the purpose of this paper is to give , in the @xmath0-scenario of uniform hyperbolicity , a characterization of those invariant measures that satisfy pesin s entropy formula in terms of their physical - like properties . our main result works , for @xmath0 anosov diffeomorphisms , as ledrappier - young characterization @xcite of the measures @xmath2 that satisfy pesin s entropy formula ( which holds in the @xmath3 context but not in the general @xmath0 context ) , by substituting the property of absolute continuity of the unstable conditional decomposition of @xmath2 , by the weak pseudo - physical property of its ergodic decomposition . pesin theory @xcite gives relevant tools and results of the modern differentiable ergodic theory . it works for @xmath3 ( or at least @xmath0 plus hlder ) dynamical systems . for instance , for @xmath3 hyperbolic systems , pesin s entropy formula computes exactly the metric entropy of a diffeomorphism in terms of the mean value of the sum of its positive lyaypunov exponents . in the @xmath3 scenario , pesin s entropy formula holds if and only if the invariant measure has absolutely continuous conditional decomposition along the unstable manifolds . . through the properties of absolute continuity of invariant measures , and mainly through the absolute continuity of the holonomy along the invariant foliations , pesin theory gives the tools to construct physically significant invariant measures for @xmath0-plus hlder systems . among these measures , the so called sinai - ruelle - bowen ( srb ) measures @xcite , have particular relevance to describe the asymptotic statistics of lebesgue - positive set of orbits , not only for @xmath0 plus hlder uniform and non - uniform hyperbolic systems , but also for @xmath0 plus hlder partially hyperbolic systems @xcite . precisely , one of the most relevant properties of ergodic srb measures for @xmath0 plus hlder hyperbolic systems , is that they are _ physical ; _ namely , their basins of statistical attraction have positive lebesgue measure , even for lebesgue non - preserving systems . _ _ in particular , for transitive anosov @xmath0 plus hlder systems , the theorem of pesin - sinai ( see for instance @xcite ) states that there exists a unique physical measure : it is the unique invariant probability measure that satisfies pesin s entropy formula , and so , the only one with absolutely continuous conditional measures along the unstable foliation . besides , its basin of statistical attraction covers lebesgue all the orbits . in other words , for a @xmath0 plus hlder anosov system , the definition of physical measure , srb measure , and measure that satisfies pesin s entropy formula , are equivalent . nevertheless , in the @xmath0-scenario , the above results do not work , because the theorems of pesin theory that ensure the absolute continuity of unstable conditional measures , and of the holonomies along invariant foliations , fail . even the existence of the unstable manifolds , along which one could construct the conditional unstable measures , fails in the @xmath0 context @xcite . in the particular case of @xmath0-anosov diffeomorphisms , invariant @xmath4 foliations with @xmath0 leaves do exist ( see for instance @xcite ) , but the holonomies along the invariant foliations are not necessarily absolutely continuous @xcite . as a consequence , for @xmath0 systems , if one defined srb measures by the existence of their absolutely continuous unstable conditional measures , one would lack the hope to construct them . nevertheless , one can still define srb or srb - like measures , if one forgets for a while the properties of absolute continuity , and focus the attention of the properties of statistical attraction . in other words , one can try to look directly at their physical properties , dodging the lack of conditional absolute continuity . for that reason , in the @xmath0-scenario , we look for the lebesgue abundance of points in their statistical basins ( or , more precisely , in the @xmath5-approach of their statistical basins ) . this search was used in @xcite to construct the srb - like or pseudo - physical measures for @xmath4 dynamical systems on a compact riemannian manifold . the notion of srb - like or pseudo - physical measures , even in a non differentiable context , translate to the space of probability measures the concept of statistical attraction defined by ilyashenko @xcite in the ambient manifold . in @xcite it is proved that @xmath0 _ generically , _ transitive and uniformly hyperbolic systems do have a unique measure satisfying pesin s entropy formula . besides this measure is physical and its basin of statistical attraction covers lebesgue almost all the orbits . in the general @xmath0-scenario with non uniform hyperbolicity , pesin s entropy formula was first proved considering systems that preserve a smooth measure ( we call a measure @xmath2 smooth if @xmath6 , where @xmath7 is the lebesgue measure ) . in fact , for @xmath0 generic diffeomorphisms that preserve a smooth measure @xmath2 , @xcite proved that @xmath2 satisfies pesin s entropy formula . later , in @xcite , this formula was also proved for any @xmath0 partially hyperbolic system that preserves a smooth measure . _ _ if no smooth measure is preserved , in @xcite is proved that the pseudo - physical or srb - like measures still exist and satisfy pesin s entropy formula , provided that the system is @xmath0 partially hyperbolic . recently , and also for @xmath0 partially hyperbolic systems , @xcite derived a proof of shub s entropy conjecture @xcite from their method of construction of measures that satisfy pesin s entropy formula . in this paper we focus on @xmath0 anosov systems to search for a converse of the result in @xcite . namely , our purpose is to characterize all the invariant measures that satisfy pesin s entropy formula . first , we need to generalize the concept of pseudo - physical or srb - like measure . so , we define the _ weak pseudo physical _ measures @xmath2 , by taking into account only the @xmath5-approach of its basin of statistical attraction up to time @xmath8 , which we denote by @xmath9 , and the exponential rate of the variation of the lebesgue measure of @xmath9 when @xmath10 ( definition [ definitionweakpseudophysicalmeasure ] ) . in theorem [ propositionweaklypseudophysicalmeasures ] we study general properties of the weak pseudo physical measures , which do always exist . we prove that for any @xmath0 anosov diffeomorphism , the weak pseudo - physical measures satisfy pesin s entropy formula ( part a ) of theorem [ maintheo ] ) . besides , we prove a converse result , to conclude that the set of invariant measures that satisfy pesin s entropy formula is the closed convex hull of the weak pseudo - physical measures ( theorem [ maintheo ] ) . _ _ so , theorem [ maintheo ] characterizes all the measures that satisfy pesin s entropy formula in terms of the statistical properties that define the weak pseudo - physical notion . nevertheless , as far as we know , no example is still known of a @xmath11 anosov diffeomorphism for which weak pseudo - physical measures are not physical . in other words , there are not known examples of @xmath0-anosov systems such that an ergodic measure satisfies pesin s entropy formula and is non physical . the proof of theorem [ maintheo ] is based on the construction of local @xmath0 pseudo - unstable foliations , which approach the local @xmath4 unstable foliation , and allow us to apply a fubini decomposition of the lebesgue measure of the @xmath5-basin @xmath9 , for any ergodic measure @xmath2 . the pseudo - unstable foliations are constructed via hadamard graphs whose future iterates have bounded dispersion . this method was introduced by ma in @xcite to prove pesin s entropy formula in the @xmath0 plus hlder context . much later , it was applied also to @xmath0 systems in @xcite and @xcite . as said above , in theorem [ maintheo ] of this paper , we prove that the weak pseudo - physical condition for the ergodic components of an invariant measure , is necessary and sufficient to satisfy pesin s entropy formula . the sufficient condition is just a corollary of the results in @xcite . on the contrary , the proof of the necessary condition is new , although it is also strongly based on ma s method to construct , via hadamard graphs , the @xmath0-pseudo unstable foliations . as a subproduct of the proof of theorem [ maintheo ] , we also obtain an equality for any ergodic measure of a @xmath0-anosov diffeomorphism , even for measures that do not satisfy pesin s entropy formula . this equality , which is stated in theorem [ maintheorem3 ] , considers the exponential rate @xmath12 according to which the lebesgue measure of the @xmath5-basin @xmath13 of each ergodic measure @xmath2 varies with time @xmath8 . theorem [ maintheorem3 ] equals the exponential rate @xmath14 with the difference @xmath15 where @xmath16 is the metric entropy and @xmath17 are the positive lyapunov exponents . in the particular case of ergodic measures @xmath2 satisfying pesin s entropy formula , the exponential rate @xmath14 is null , and conversely . let @xmath18 be a compact , connected , riemannian @xmath0-manifold without boundary and let @xmath19 be continuous . _ _ [ definitionempiricproba ] ( empiric probability ) for each @xmath20 , the _ empiric probability _ @xmath21 along the finite piece of the future orbit of @xmath22 up to time @xmath8 , is defined by @xmath23 where @xmath24 denotes the dirac delta probability measure supported on the point @xmath25 . _ _ _ _ we denote by @xmath26 the space of all the borel probability measures on @xmath18 , endowed with the weak@xmath1 topology . we denote by @xmath27 the space of @xmath28-invariant borel probability measures . it is well known that @xmath26 and @xmath29 are nonempty , weak@xmath1 compact , metrizable , sequentially compact and convex topological spaces . we fix and choose a metric @xmath30 in @xmath26 that induces the weak@xmath1 topology . _ _ [ definitionbasinofattraction ] ( basin and pseudo basin of attraction of a measure . ) let @xmath31 and @xmath32 . we construct the following measurable sets in the manifold @xmath18 : @xmath33 @xmath34 @xmath35 _ _ we call @xmath36 the _ basin of attraction _ of @xmath2 . we call @xmath37 the _ @xmath5-pseudo basin of attraction _ of @xmath2 . we call @xmath9 the _ @xmath5-pseudo basin of @xmath2 up to time @xmath8 . _ _ _ _ _ _ _ in the sequel we denote by @xmath7 the lebesgue measure of @xmath18 , renormalized to be a probability measure . _ _ [ definitionweakpseudophysicalmeasure ] ( physical , pseudo - physical and weak pseudo - physical measures ) _ _ let @xmath38 . we call @xmath2 _ physical _ if @xmath39 . _ _ we call @xmath2 _ pseudo - physical _ if @xmath40 for all @xmath32 . _ _ we call @xmath2 _ weak pseudo - physical _ if @xmath41 we denote @xmath42 _ _ on the one hand , it is standard to check that for continuous mappings @xmath19 the physical measures , if they exist , are @xmath28-invariant . also , pseudo - physical are @xmath28-invariant ( see @xcite ) , page 153 , and as proved in theorem 1.3 of @xcite , the set of pseudo - physical measures is never empty , weak@xmath1 compact and independent of the chosen metric @xmath30 that induces the weak@xmath1 topology of @xmath26 . besides , it is immediate to check that physical measures , if they exist , are particular cases of the always existing pseudo - physical measures . on the other hand , in this paper we will generalize the previous results that hold for pseudo - physical measures , by proving the following properties also for weak pseudo - physical measures : [ propositionweaklypseudophysicalmeasures ] let @xmath19 be a continuous map . then : \a ) weak pseudo - physical measures are @xmath28-invariant . \b ) physical measures and pseudo - physical measures are particular cases of weak pseudo - physical measures . \c ) weak pseudo - physical measures do always exist . \d ) the set @xmath43 of weak pseudo - physical measures does not depend on the choice of the metric @xmath30 that induces the weak@xmath1 topology on @xmath26 . \e ) @xmath43 is weak@xmath1-compact , hence sequentially compact . \f ) @xmath44 for lebesgue almost all @xmath45 . \g ) if the weak pseudo - physical measure @xmath2 is unique , then it is physical and its basin of attraction @xmath36 covers lebesgue a.e . @xmath45 . _ _ [ remarkpseudophysicalnoergodica ] weak pseudo - physical measures _ are not necessaritly ergodic _ ( see example 5.4 of @xcite ) . _ _ _ _ now , let @xmath46 be a @xmath0 diffeomorphism on @xmath18 . [ definitionanosov ] ( anosov diffeomorphisms ) the diffeomorphism @xmath28 is called _ anosov _ if there exists a riemannian metric of @xmath18 and asplitting @xmath47 which is continuous and non trivial ( i.e. @xmath48 ) , and a constant @xmath49 , such that @xmath50 we call @xmath51 and @xmath52 the _ stable and unstable subbundles _ respectively . we call @xmath53 the _ ( uniform ) hyperbolicity constant . _ _ _ _ _ _ _ _ [ remarkanosov ] _ we observe that the condition of continuity of the unstable and stable subbundles is redundant in definition [ definitionanosov ] . besides , since the manifold is connected , from the continuity of @xmath54 and @xmath51 we deduce that they are uniformly transversal sub - bundles and @xmath55 and @xmath56 are constants . _ _ from inequalities ( [ eqnl11 ] ) , for any anosov diffeomorphism @xmath28 and for any regular point @xmath45 , the minimum lyapunov exponent along @xmath57 is not smaller than @xmath58 , and the maximum lyapunov exponent along @xmath51 is not larger than @xmath59 . thus , for any regular point @xmath45 all the lyapunov exponents along @xmath60 are strictly negative and bounded away from zero , and all the lyapunov exponents along @xmath57 are strictly positive and bounded away from zero . _ _ [ definitionpesinformula ] ( pesin s entropy formula ) let @xmath46 . let @xmath61 . we say that @xmath2 _ satisfies pesin s entropy formula _ if @xmath62 where @xmath16 is the metric entropy of @xmath28 with respect to @xmath2 ; for @xmath2-a.e . @xmath45 the lyapunov exponents of the orbit of @xmath22 are denoted by @xmath63 and @xmath64 . _ _ _ _ recall that for any @xmath0- anosov diffeomorphism @xmath28 , the set of measures that satisfy pesin s entropy formula is nonempty ( see for example theorems 4.2.3 and 4.5.6 of @xcite ) . the main purpose of this paper is to prove the following result : [ maintheo ] for @xmath0 anosov diffeomorphisms , the set of ergodic weak pseudo - physical measures is nonempty , and the set of invariant probability measures that satisfy pesin s entropy formula is its closed convex hull . _ _ the following is an equivalent restatement of theorem [ maintheo ] : _ _ \a ) all the weak pseudo - physical measures satisfy pesin s entropy formula . \b ) any invariant probability measure @xmath2 satisfies pesin s entropy formula if and only if its ergodic components @xmath65 are weak pseudo - physical @xmath2-a.e . _ from theorem [ maintheo ] , we obtain the following consequence : [ corollaryc2-bonattidiazviana ] if @xmath66 is anosov , then for lebesgue - almost all @xmath45 any convergent subsequence of the empirical probabilities @xmath21 converges to a measure that satisfies pesin s entropy formula . from assertion f ) of theorem [ propositionweaklypseudophysicalmeasures ] , for lebesgue - almost all @xmath45 any convergent subsequence of @xmath67 converges to a weak pseudo - physical measure @xmath2 . thus , applying part a ) of theorem [ maintheo ] @xmath2 satisfies pesin s entropy formula . the arguments to prove theorem [ maintheo ] are based in the following more general result , which we will prove along the paper : [ maintheorem3 ] if @xmath68 is anosov , if @xmath54 denotes its unstable sub - bundle , and if @xmath2 is an ergodic probability measure for @xmath28 , then the @xmath5-pseudo basin @xmath13 of @xmath2 up to time @xmath8 satisfies the following equality : _ @xmath69 _ in section [ sectionpropertiesweaklypseudophysicalmeasures ] we prove theorem [ propositionweaklypseudophysicalmeasures ] , which states the general properties of weak pseudo physical measures for any continuous map @xmath19 . in section [ sectionsufficientcondition ] , for anosov diffeomorphisms , we prove part a ) of theorem [ maintheo ] and also the first part of b ) precisely , we prove that the weak pseudo - physical property of the ergodic components is a sufficient condition to satisfy pesin s entropy formula . in section [ sectionnecessarycondition ] , for anosov diffeomorphisms , we prove the converse statement in part b ) of theorem [ maintheo ] . namely , the weak pseudo - physical property of the ergodic components is also a necessary condition to satisfy pesin s entropy formula . through the proof of theorem [ maintheo ] , we obtain some stronger intermediate results that hold for any ergodic measure . finally , at the end of section [ sectionnecessarycondition ] , we join those intermediate results to prove theorem [ maintheorem3 ] . the purpose of this section is to prove theorem [ propositionweaklypseudophysicalmeasures ] . along this section , we assume that @xmath28 is only a continuous map from a compact riemannian manifold @xmath18 into itself . let us divide the proof of theorem [ propositionweaklypseudophysicalmeasures ] into its assertions a ) to f ) : theorem [ propositionweaklypseudophysicalmeasures ] a ) _ any weak pseudo - physical measure @xmath2 is @xmath28-invariant . _ _ _ from equality ( [ eqnl12 ] ) , for any fixed value of @xmath32 there exists @xmath70 such that @xmath71 . thus , there exists @xmath72 such that @xmath73 since @xmath74 and @xmath26 is sequentially compact , it is not restrictive to assume that @xmath75 is weak@xmath1 convergent . denote by @xmath76 its limit . we assert that @xmath76 is @xmath28-invariant . in fact , consider the operator @xmath77 defined by @xmath78 for any borel measurable set @xmath79 . then @xmath80 for all @xmath25 ; hence @xmath81 for all @xmath82 . it is well known that @xmath83 is continuous . thus , taking limit in the weak@xmath1 topology , we obtain : @xmath84 since @xmath85 we deduce that the total variation of the signed measure @xmath86 is @xmath87 thus @xmath88 hence @xmath89 , or equivalently @xmath76 is @xmath28-invariant . from ( [ eqnl13 ] ) @xmath90 . we have proved that for all @xmath32 there exists @xmath91 such that @xmath92 . since @xmath29 is sequentially compact , we deduce that @xmath61 , as wanted . theorem [ propositionweaklypseudophysicalmeasures ] b ) _ any physical or pseudo - physical measure is weak pseudo - physical . _ _ _ trivially any physical measure is pseudo - physical . so , it is only left to prove that any pseudo - physical measure @xmath2 is weak pseudo - physical . consider @xmath93 . from equality ( [ eqnl15 ] ) , there exists @xmath70 such that @xmath94 . therefore , from ( [ eqnl16 ] ) @xmath95 since the latter assertions holds for all @xmath93 , we have proved that @xmath96 as @xmath2 is pseudo - physical , we deduce : @xmath97 now , assume by contradiction , that @xmath2 is not weak pseudo - physical . taking into account that @xmath98 , from the contrary of equality ( [ eqnl12 ] ) , we deduce that there exist @xmath32 and @xmath99 such that @xmath100 therefore , there exists @xmath101 such that @xmath102 for all @xmath103 from where we deduce that @xmath104 finally , applying borell - cantelli lemma , we conclude that @xmath105 contradicting inequality ( [ eqnl17 ] ) . theorem [ propositionweaklypseudophysicalmeasures ] c ) _ weak pseudo - physical measures do exist . _ _ _ in theorem 1.3 of @xcite , it is proved for any continuous map @xmath28 on a compact manifold , that the pseudo - physical measures ( which in that paper are also called srb - like or observable ) do exist . since any pseudo - physical measure is weak pseudo - physical , these latter measures always exist . theorem [ propositionweaklypseudophysicalmeasures ] d ) _ the set @xmath43 of weak pseudo - physical measures does not depend on the choice of the metric in @xmath26 that induces the weak@xmath1 topology . _ _ take two metrics @xmath106 and @xmath107 , both inducing the weak@xmath1 topology on @xmath26 . we assume that @xmath108 is weak pseudo - physical according to @xmath106 , and let us prove that it is also weak pseudo - physical according to @xmath107 . since both metric induce the same topology , for any @xmath32 there exists @xmath109 such that @xmath110 in the notation of equality ( [ eqnl16 ] ) , add a subindex 1 or 2 to denote the sets @xmath111 and @xmath112 , according to which metric ( @xmath106 and @xmath107 , respectively ) is used to define them . so , from assertion ( [ eqnl19 ] ) we have : @xmath113 from where @xmath114 since we are assuming that @xmath2 is weak pseudo - physical according to @xmath115 , from equality ( [ eqnl12 ] ) we know that @xmath116 then , @xmath117 as @xmath32 was arbitrarily chosen , the latter inequality holds for all @xmath32 . but the limit in the latter inequality is non positive because @xmath7 is a probability measure . we conclude that @xmath118 ending the proof that @xmath2 is also weak pseudo - physical with respect to the metric @xmath107 . theorem [ propositionweaklypseudophysicalmeasures ] e ) _ the set @xmath119 of weak pseudo - physical measures is weak@xmath1-compact . _ _ _ since @xmath120 and @xmath26 is weak@xmath1-compact , it is enough to prove that @xmath43 is weak@xmath1-closed . assume @xmath121 and @xmath38 such that @xmath122 we will prove that @xmath123 . for any given @xmath32 , choose and fix @xmath124 such that @xmath125 . thus , from equality ( [ eqnl16 ] ) and the triangle property , we obtain : @xmath126 from where @xmath127 since @xmath121 , we can apply equality ( [ eqnl12 ] ) to @xmath128 , which joint with inequality ( [ eqnl18 ] ) implies : @xmath129 finally , since @xmath7 is a probability measure , we deduce that the above limsup equals 0 , concluding that @xmath123 as wanted . theorem [ propositionweaklypseudophysicalmeasures ] f ) _ @xmath44 for lebesgue almost all @xmath45 . _ _ _ theorem 1.5 of @xcite , states that the distance between @xmath21 and the set of pseudo - physical measures converges to zero with @xmath10 for lebesgue almost all @xmath45 . since the pseudo - physical measures are contained in @xmath43 , we trivially deduce the wanted equality . theorem [ propositionweaklypseudophysicalmeasures ] g ) _ if the weak pseudo - physical measure @xmath2 is unique , then it is physical and its basin of attraction @xmath36 covers lebesgue a.e . _ _ _ it is an immediate consequence of part f ) . in the sequel we assume that the map @xmath130 is anosov . the purpose of this section is to deduce , as an immediate consequence from previous known results , part a ) of theorem [ maintheo ] , and the sufficient condition to satisfy pesin s entropy formula in part b ) of theorem [ maintheo ] . namely , we will deduce that if all the ergodic components of an @xmath28-invariant measure @xmath2 are weak pseudo - physical , then @xmath2 satisfies pesin s entropy formula . recall definition [ definitionbasinofattraction ] , which defines the @xmath5-pseudo basin @xmath9 up to time @xmath8 of a probability measure @xmath131 . we will apply the following result : @xcite [ theoremcce1 ] let @xmath18 be a compact riemannian manifold of finite dimension . let @xmath46 be anosov with hyperbolic splitting @xmath47 , where @xmath51 and @xmath54 are the stable and unstable sub - bundles respectively . then , the following inequality holds for any @xmath28-invariant @xmath38 : @xmath132 proof . see proposition 2.1 in @xcite . _ [ remarkruelle ] _ for the non negative lyapunov exponents , we adopt the notation @xmath133 as in definition [ definitionpesinformula ] . for any @xmath46 , margulis and ruelle inequality @xcite states : @xmath134 thus , pesin s entropy formula holds for an invariant measure @xmath2 , if and only if the following inequality holds : @xmath135 besides , from definition [ definitionanosov ] , and from the formula of the integral of the volume form along the unstable sub - bundle @xmath54 , we obtain the following equality for anosov diffeomorphisms : @xmath136 joining the above assertions , we conclude : _ _ _ let @xmath68 be anosov , and @xmath54 be its unstable sub - bundle . then , any @xmath28-invariant probability measure @xmath2 satisfies pesin s entropy formula if and only if @xmath137 _ we are ready to deduce part a ) of theorem [ maintheo ] , which is indeed a corollary of theorem [ theoremcce1 ] : part a ) of theorem [ maintheo ] : _ if @xmath68 is anosov and if @xmath2 is a weak pseudo - physical @xmath28-invariant measure , then @xmath2 satisfies pesin s entropy formula . therefore , the set of invariant probability measures that satisfy pesin s entropy formula is nonempty . _ _ _ by contradiction , assume that @xmath2 does not satisfy pesin s entropy formula . according to remark [ remarkruelle ] , inequality ( [ eqnl21 ] ) does not hold : @xmath138 therefore , applying inequality ( [ eqnl20 ] ) of theorem [ theoremcce1 ] , we conclude that there exists @xmath32 such that @xmath139 so , equality ( [ eqnl12 ] ) does not hold ; hence @xmath2 is not weak pseudo - physical , contradicting the hypothesis . we have proved that all the weak pseudo - physical measures for @xmath28 satisfy pesin s entropy formula . from part c ) of theorem [ propositionweaklypseudophysicalmeasures ] , weak pseudo - physical measures do exist . so , the set of measures that satisfy pesin s entropy formula is nonempty . we now recall the following well known result ( see for instance theorems 4.3.7 and 4.5.6 of @xcite ) : [ theoremconvexhull ] let @xmath68 be anosov . an @xmath28-invariant measure @xmath2 satisfies pesin s entropy formula if and only if its ergodic components @xmath65 satisfy it for @xmath2-a.e . @xmath45 . on the one hand , we recall that any anosov @xmath0 diffeomorphism @xmath28 is expansive , and for any expansive homeomorphism @xmath28 on @xmath140 the metric entropy @xmath16 depends upper semi - continuously on the @xmath28-invariant measure @xmath2 ( see for instance theorem 4.5.6 of @xcite ) . so , we can apply the theorem of the affinity of the entropy function ( see theorem 4.3.7 of @xcite ) , which states that @xmath141 where the measures @xmath142 for @xmath143 are the ergodic components of @xmath2 . on the other hand , the ergodic decomposition theorem states that @xmath144 joining equalities ( [ eqnl22 ] ) and ( [ eqnl23 ] ) , and taking into account margulis and ruelle inequality ( [ eqnruelleinequality ] ) , we deduce that @xmath145 if and only if @xmath146 ending the proof of theorem [ theoremconvexhull ] . as a consequence we obtain : part b ) of theorem [ maintheo ] , sufficient condition : _ if @xmath68 is anosov and if @xmath2 is an invariant measure whose ergodic components @xmath65 are weak pseudo - physical for @xmath2-a.e . @xmath45 , then @xmath2 satisfies pesin s entropy formula . _ _ _ from the hypothesis , and applying part a ) of theorem [ maintheo ] , we deduce that the ergodic components @xmath65 of @xmath2 satisfy pesin s entropy formula for @xmath2-a.e . so , from theorem [ theoremconvexhull ] , the measure @xmath2 also satisfies this formula . in this section we will prove the necessary condition to satisfy pesin s entropy formula , as stated in part b ) of theorem [ maintheo ] . precisely , we will prove that if @xmath46 is anosov , and if the @xmath28-invariant measure @xmath2 satisfies pesin s entropy formula , then the ergodic components of @xmath2 are weak pseudo - physical . we will also prove the equality of theorem [ maintheorem3 ] for any ergodic measure @xmath2 . recall that any anosov diffeomorphism @xmath28 is expansive ( see for instance lemma 3.4 in @xcite ) . namely , there exists a constant @xmath147 , which is called the expansivity constant , such that @xmath148 metric entropy for expansive systems . recall the following result , which follows from kolmorgorov - sinai theorem in the case of expansive homeomorphisms ( see for instance , proposition 2.5 of @xcite , or also theorem 3.2.18 and lemma 4.5.4 of @xcite ) : _ if @xmath150 is a finite partition whose pieces are borel measurable sets and have diameter smaller than the expansivity constant @xmath153 , then @xmath154 generates the borel @xmath155-algebra , and for any @xmath28-invariant measure @xmath2 , the metric entropy @xmath16 can be computed by : _ @xmath156 @xmath157 note : in ( [ eqn39 ] ) at right , @xmath158 denotes the number of elements of the finite set @xmath159 . _ _ rectangles . recall the definition of rectangle @xmath160 in the manifold @xmath18 for the anosov diffeomorphism @xmath28 . ( see @xcite , page 78 . ) in particular a rectangle @xmath160 is proper if @xmath161 . for any proper rectangle @xmath160 and any @xmath162 denote @xmath163 @xmath164 where @xmath165 are the stable and unstable submanifolds of the point @xmath22 . \(ii ) there exists a constant @xmath169 such that , if @xmath170 is a local embedded @xmath0-submanifold , with dimension equal to the unstable dimension , and such that @xmath171 intersects transversally the local stable manifolds @xmath172 for all @xmath162 , then @xmath173 where @xmath174 denotes the lebesgue measure along @xmath171 . let @xmath179 be a markov partition of the manifold @xmath18 , let @xmath45 and denote by @xmath180 the rectangle of @xmath150 that contains @xmath22 . let @xmath181 be a positive natural number . the _ dynamical rectangle _ @xmath182 that contains @xmath22 is defined by @xmath183 _ _ [ lemmaentropia ] let @xmath28 be an anosov diffeomorphism on a compact manifold . then , for any finite markov partition @xmath150 , there exists a constant @xmath185 satisfying the following inequality for any @xmath28-invariant probability measure @xmath2 , for any @xmath186 , for any borel measurable set @xmath187 such that @xmath188 , and for any natural number @xmath189 : @xmath190 denote @xmath191 since @xmath192 we have @xmath193 . if @xmath194 then inequality ( [ eqnl40 ] ) holds trivially as a consequence of ( [ eqn39 ] ) . so , let us prove lemma [ lemmaentropia ] in the case @xmath195 by definition : @xmath196 @xmath197 @xmath198 @xmath199 construct the probability measures @xmath200 and @xmath201 defined by the following equalities for all borelian set @xmath79 : @xmath202 we obtain @xmath203 @xmath204 applying inequality ( [ eqn39 ] ) : @xmath205 @xmath206 taking into account that @xmath207 and that @xmath208 is strictly increasing for @xmath209 and strictly decreasing for @xmath210 , we obtain : @xmath211 @xmath212 @xmath213 where @xmath214 therefore , to end the proof of lemma [ lemmaentropia ] it is enough to show that @xmath215 . in fact , for a markov partition @xmath150 , any rectangle @xmath216 is obtained as a connected component of the intersection @xmath217 for some pair of rectangles @xmath218 . fixing @xmath219 , the maximum number of connected components of the intersections of @xmath220 with the rectangle @xmath221 of the partition , is upper bounded the following quotient @xmath222 @xmath223 where @xmath224 thus , denoting @xmath225 , we have @xmath226 we conclude that @xmath227 which implies @xmath228 ending the proof of lemma [ lemmaentropia ] . in the following lemma we will construct a local @xmath0-foliation , whose leaves are _ pseudo - unstable manifolds _ @xmath5- approaching ( in the @xmath0-topology ) the true local unstable manifolds of any rectangle of a given markov partition . _ _ [ lemmafoliation ] let @xmath68 be anosov . denote the stable and unstable subbundles by @xmath51 and @xmath229 , respectively . denote the expansivity constant by @xmath147 . then , for all @xmath32 there exist @xmath230 and @xmath231 such that , for any finite markov partition @xmath175 into rectangles with diameter smaller than @xmath232 , there exists a finite family @xmath233 of local foliations @xmath234 , each one defined in an open neighborhood of each rectangle @xmath176 , satisfying the following properties for all @xmath235 , for all @xmath236 and for all @xmath237 : \d ) there exist a point @xmath241 and an open subset @xmath242 , in the topology of the stable submanifold @xmath243 , such that _ @xmath244 _ where _ @xmath245 _ denotes the lebesgue measure along the submanifold @xmath243 ; and besides , if @xmath246 , then _ @xmath247 _ where _ @xmath248 _ denotes the lebesgue measure along the submanifold @xmath249 . _ _ _ _ _ _ _ _ proposition 3.6 of @xcite states the existence of @xmath250 and the local @xmath0-foliation @xmath234 satisfying ( a ) , ( b ) and ( c ) . so , it is enough to prove that if @xmath32 is small enough , then any local @xmath0-foliation @xmath234 defined in a neighborhood of the rectangle @xmath176 and satisfying ( a ) and ( b ) , also satisfies ( d ) for some constant @xmath251 . in fact , choose and fix any point @xmath252 in the interior of the rectangle @xmath176 . from the definition of rectangle , for each @xmath253 there exists a unique point in the transversal intersection @xmath254 by continuity of the transversal intersection between @xmath0-manifolds , there exists @xmath255 such that the following assertion holds : if @xmath256 and if @xmath234 is any local foliation whose leaves have dimension @xmath55 , are @xmath0 , and are @xmath257-near the unstable local leaves of @xmath176 in the @xmath0-topology , then for each @xmath253 the intersection @xmath258 is transversal and contains a single point . from the definition of the dynamical rectangle @xmath182 and from the properties of the markov partition , there exists a rectangle @xmath269 of the partition such that local stable manifold @xmath270 for all @xmath271 . so , we deduce @xmath272 for all @xmath267 and for all @xmath273 in other words , the pseudo - unstable @xmath274 submanifold @xmath275 intersects transversally all the local stable submanifolds of the rectangle @xmath221 where it is contained . thus , applying inequality ( [ eqnl25 ] ) we have @xmath276 for any probability measure @xmath2 recall equality ( [ eqnl16 ] ) , defining the measurable set @xmath9 which we called the @xmath5-pseudo basin of @xmath2 up to time @xmath8 . we will end the proof of theorem [ maintheo ] , by applying the following key result which bounds from below the lebesgue measure of the set @xmath9 for any ergodic measure @xmath2 : [ theorem2 ] let @xmath18 be a compact riemannian manifold of finite dimension . let @xmath46 be anosov with hyperbolic splitting @xmath47 , where @xmath51 and @xmath54 are the stable and unstable sub - bundles respectively . let @xmath2 be an ergodic measure . then : @xmath278 we notice that the limit at left in equality ( [ eqnl20-b ] ) does not depend on the choice of metric @xmath30 that induces the weak@xmath1 topology in the space @xmath26 of borel probability measures . in fact , to prove the latter assertion it is enough to argue as in the proof of part ( d ) of theorem [ propositionweaklypseudophysicalmeasures ] in section [ sectionpropertiesweaklypseudophysicalmeasures ] . so , to prove theorem [ theorem2 ] we choose and fix the following metric in @xmath26 : @xmath279 where @xmath280 is any fixed countable family of real continuous functions @xmath281)$ ] that is dense in @xmath282)$ ] . note that , according to the metric @xmath30 , the balls are convex . in other words , if a finite number of probability measures belong to the ball with centre @xmath2 and radius @xmath32 , then any convex combination of those measures also belongs to it . fix any real value @xmath32 . the real funcion @xmath285 is continuous because @xmath28 is of class @xmath0 and the sub - bundle @xmath54 is continuous . thus , from the definition of the weak@xmath1 topology in the space @xmath26 of probability measures , we deduce that there exists @xmath286 such that @xmath287 in particular , for @xmath288 we deduce : @xmath289 since @xmath2 is an ergodic probability measure , we have @xmath290 for @xmath2-a.e . so , for the fixed value of @xmath255 as above , and for @xmath2-a.e . @xmath45 , there exists @xmath291 such that @xmath292 for any natural value of @xmath293 , define the set @xmath294 since @xmath295 and @xmath296 , there exists @xmath297 such that @xmath298 in the sequel , we fix such a value of @xmath293 . from the definition of the metrizable weak@xmath1-topology in the space @xmath26 of borel - probability measures , it is standard to check that the dirac delta probability @xmath299 depends uniformly continuously on the point @xmath45 . since the empiric probability @xmath21 is a convex combination of dirac delta measures , and the balls in @xmath26 are convex , we deduce that there exists @xmath300 such that , for any pair of points @xmath301 and for any natural value of @xmath189 , the following assertion holds : for the fixed value of @xmath303 at the beginning , we construct the real numbers @xmath304 ( where @xmath305 is expansivity constant ) , and @xmath251 , as in lemma [ lemmafoliation ] . we consider any markov partition @xmath175 with diameter smaller than @xmath306 and , for each rectangle @xmath176 , we construct the @xmath0-foliation @xmath234 that satisfies the properties ( a ) to ( d ) of lemma [ lemmafoliation ] . from equality ( [ eqnl31 ] ) , assertion ( [ eqnl35 ] ) , and the triangle property of the metric , we deduce the following assertion for all @xmath307 : @xmath308 @xmath309 recalling equality ( [ eqnl16 ] ) , from the above assertion we deduce that @xmath310 for all @xmath311 . since the rectangle @xmath182 is any piece of the partition @xmath312 that intersects @xmath313 , we deduce the following statement for all @xmath307 : @xmath314 therefore @xmath315 besides , joining assertion ( [ eqnl30 ] ) and inequality ( [ eqnl37 ] ) , we deduce the following property for all @xmath307 : @xmath316 @xmath317 now , for any @xmath307 , let us compute @xmath318 for any rectangle @xmath216 such that @xmath319 . since @xmath320 , to compute @xmath318 we will use the fubini decomposition of the lebesgue measure along the local pseudo - unstable @xmath0-foliation @xmath234 . applying part ( d ) of lemma [ lemmafoliation ] consider the point @xmath241 and the submanifold @xmath321 . taking the fubini decomposition of @xmath7 we obtain : @xmath322 where @xmath323 is a local @xmath0-diffeomorhism that parameterizes the neighborhood of @xmath176 and trivializes the @xmath0-foliation @xmath234 . therefore , @xmath324 is continuous and bounded away from zero by a constant , say @xmath325 . since @xmath326 is an open subset of @xmath263 in the topology of this local stable manifold , we obtain : @xmath327 changing variables @xmath328 in the integral at right , we obtain : @xmath329 @xmath330 since @xmath331 , we can apply inequality at left of part c ) of lemma [ lemmafoliation ] : @xmath332 @xmath333 since @xmath334 and @xmath319 , we can apply inequality ( [ eqnl38 ] ) : @xmath335 @xmath336 @xmath337 from part ( d ) of lemma [ lemmafoliation ] we know that @xmath338 for all @xmath339 , and besides @xmath340 . thus , we have proved the following inequality for all @xmath307 , and for all @xmath216 such that @xmath319 : @xmath341 joining the above inequality with inequality ( [ eqnl36 ] ) , we deduce , for all @xmath307 : @xmath342 therefore , @xmath343 @xmath344 finally , applying lemma [ lemmaentropia ] , we deduce that @xmath343 @xmath345 so , from equality ( [ eqnmetricentropy ] ) , we conclude @xmath346 ending the proof of theorem [ theorem2 ] . part b ) of theorem [ maintheo ] , necessary condition : _ if @xmath68 is anosov and if @xmath2 is an invariant measure satisfying pesin s entropy formula , then its ergodic components @xmath65 are weak pseudo - physical for @xmath2-a.e . _ _ _ so @xmath350 is increasing with @xmath32 . thus @xmath351 but since @xmath7 is a probability measure , we conclude that @xmath352 applying definition [ definitionweakpseudophysicalmeasure ] , we deduce that @xmath2 is weak pseudo - physical . we have proved that any ergodic measure that satisfies pesin s entropy formula is weak pseudo - physical . now let us consider a non ergodic measure @xmath2 that satisfies pesin s entropy formula . from theorem [ theoremconvexhull ] we know that its ergodic components @xmath65 also satisfy that formula for @xmath2-a.e . we conclude that the ergodic components @xmath65 of @xmath2 are weak pseudo - physical for @xmath2-a.e . @xmath45 , as wanted . | we consider @xmath0 anosov diffeomorphisms on a compact riemannian manifold . we define the weak pseudo - physical measures , which include the physical measures when these latter exist .
we prove that ergodic weak pseudo - physical measures do exist , and that the set of invariant probability measures that satisfy pesin s entropy formula is the weak@xmath1-closed convex hull of the ergodic weak pseudo - physical measures . in brief , we give in the @xmath0-scenario of uniform hyperbolicity , a characterization of pesin s entropy formula in terms of physical - like properties .
2010 _ math . subj . class . : _
primary 37a35 , 37d20 .
secondary 37d35 ; 37a05
. _ keywords and phrases : _ anosov diffeomorphisms ; pesin s entropy formula ; physical measures . |
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the optics community has taken a lot of effort into the development of surface modeling methods for optical design @xcite . unfortunately , none of those methods are developed to improve manufacturability @xcite or to reduce the difficulty of fabrication of optical surfaces . for example , zernike polynomials are widely applied to describe surface errors and express wavefront data in the field of optical fabrication and testing . because there is a theoretical relationship between zernike coefficients and seidel aberrations often observed in optical tests @xcite . however , basically , the standard 36-term zernike polynomial set does not aim to or help to optimize the process of optical fabrication or polishing . when it comes to ultra - precision fabrication of optical surfaces , ion beam technologies play a key role in improving surface precision to extreme . as one of the most precise methods , ion beam figuring ( ibf ) @xcite is employed for finishing lithography optics and telescope mirrors . ibf is realized with a dwell time algorithm akin to that used in computer controlled optical surfacing ( ccos ) . longer time the ion beam dwelling at a point results in more material removal around the point , thus it allows correction of surface figures through controlled variations of scanning velocity . in particular , one - dimensional ibf @xcite , or sometimes called ion beam profiling @xcite , applies to the fabrication of elongated synchrotron optics such as aspherical x - ray mirrors . compared with the conventional ibf , one - dimensional ibf is generally implemented by a simpler algorithm meanwhile the long - rectangular - shaped ion beam adopted in one - dimensional case leads to higher output efficiency under the condition for ensuring similar finishing precision . inspired by zernike polynomials bridging the relationship between surface errors and optical aberrations , we propose a surface modeling method on the basis of double fourier series @xcite to narrow the gap between optical fabrication and surface characterization . the fourier series decomposition of an optical surface produces a set of wave surfaces with a sinusoidal profile , which by and large are of different periods , amplitudes , and propagation directions . conceptually , we can sequentially fabricate the decomposed wave surfaces by linear scanning with a linear ion source and the superposition of those fabricated wave surfaces finally build up a surface that approximates to the desired optical surface . moreover , the power spectral density ( psd ) analysis is the statistical analysis of the spatial - distributed wave surfaces described as components of double fourier series . as to the characterization of optical surfaces , the psd function is typically utilized @xcite , especially for evaluating errors in the mid - spatial frequency ( msf ) range . so , we realized , a surface descriptor based on fourier series can build a relationship between the technique of optical fabrication with linear ion source and the psd - based surface characterization method . this paper introduces the basic theory of surface modeling and decomposition , then illustrates the principle of optical fabrication with linear ion source , and finally demonstrates two applications , i.e. , fabrication of large - aperture beam sampling gratings ( bsgs ) and removal of errors in the msf range . a continuous smoothing surface @xmath0 can be described as @xmath1 where @xmath0 is periodic by @xmath2 in x direction and @xmath3 in y direction ; @xmath4 , @xmath5 ; @xmath6 is a complex number . in particular , @xmath0 is a real function that is conjugate symmetric thus the imaginary part can be omitted . after adding a proper piston to every component , @xmath7 , we get a slightly elevated surface , @xmath8 , that is suitable for indicating the spatial distribution of material removals and is displayed as @xmath9 where @xmath10 represents the piston or extra material removal ; note that the arbitrary component @xmath11 . since ion etching is a subtractive process rather than an additive process , the desired material removal should be nonnegative thus it requires that @xmath12 . continued by equation [ eq : removal_surface1 ] , we put forward a new formula to decrease the number of scan strokes and improve the reachability of optical surfacing , which is represented as @xmath13 where @xmath14 and @xmath15 not equal to @xmath16 simultaneously ; @xmath17 denotes a small positive quantity and represents the compensated material removal . since there is an optimum removal in ion - beam figuring @xcite , @xmath17 usually is not infinitesimal in order to make sure that the scan speed is properly limited . the component @xmath18 , indicating the decomposed distribution of material removals , is shown as a sinusoidal wave surface , where the propagation direction is @xmath19 or @xmath20 and the period is @xmath21 . @xmath22 where @xmath23 represents a material removal map by m times n , which is the discrete representation of @xmath24 ; two - dimensional fast fourier transform ( denoted as fft2 in this paper ) algorithm is used for calculating the matrix of fourier coefficients , @xmath25 . to exclude the dc component , @xmath26 , it needs to assign that : @xmath27 . the material removal map can be effectively approximated by the symmetric rectangular partial sum of double fourier series @xcite . the ( @xmath28)-th symmetric rectangular partial sum is defined as @xmath29 then we have the truncated errors @xmath30 to evaluate the feasibility of the approximation , we use psd function to analyze the truncated errors referring to as surface errors . the psd analysis of the truncated errors can be implemented by the two - dimensional fft algorithm , which is displayed as follows : @xmath31 reactive ion etching ( rie ) is a useful method in the fabrication of diffraction optics . recently we fabricated large - aperture bsgs by the rie process , where the ion beam emitted from linear ion source scans over bsgs along a linear guide meanwhile the rotary leaf scans along the elongated footprint of ion beam , thus the spatial distribution of etch depths is finely adjusted for improving the diffractive uniformity of bsg @xcite . now we try to exploit more applications by combining optical fabrication technique with linear ion source and the aforementioned surface modeling method . fig . [ fig : optsurf ] illustrates the basic principle of optical fabrication with linear ion source and also serves as a schematic of the fabrication of bsgs . the two shutters can collimate the ion beam emitted from the linear ion source and adjust the width of ion beam spot ( see fig . [ fig : optsurf ] ) projected on the workpiece or bsg substrate . the workpiece mounted on a workbench spins about its center normal line and moves in the transverse horizontal direction in accompany with the workbench . that means the ion beam having an elongated rectangular footprint can scan over the workpiece along the arbitrary direction on the workpiece surface . linear ion sources , or ion sources with a large - aperture rectangular grid , are suitable for ion figuring of optical surfaces with mainly a one - dimensional profile such as synchrotron reflective mirrors . fortunately , we only need to fabricate wave surfaces with a one - dimensional sinusoidal profile with the help of surface decomposition . a set of decomposed wave surfaces linearly build up a superposed surface approximating to the desired surface . moreover , we have compared two approach in our previous work @xcite to calculate the dwell time distribution in corresponding with a sinusoidal profile . in the fabrication of large - aperture bsgs , the spatial distribution of etch depths should be finely adjusted to improve the diffractive uniformity of bsg . the mesh graph on the top right of fig . [ fig : reduce ] is a typical etch depth map , which describes the desired etch depths spatially distributed on the bsg substrate . the tolerance analysis indicates that the rms of etch depth differences should be within 0.4 nm , which means the rms of truncated errors , @xmath32 , is not greater than 0.4 nm . as is shown on the top left of fig . [ fig : reduce ] , the psd in the low - spatial frequency range is significantly higher than that in the mid / high - spatial frequency range , which indicates that the first several components or decomposed wave surfaces mainly contribute to the surface form . after psd analysis , as is shown in fig . [ fig : decomp ] , the first 5 sinusoidal wave surfaces including @xmath33 are selected and the summation ( @xmath34 ) of the 5 components is well approximated to the desired etch depth map . the etch depth differences , obtained by removing @xmath34 from the desired etch depth map , is shown on the lower part of fig . [ fig : reduce ] , where the peak - to - valley value is 0.19 nm and the average of etch depth differences is 15.95 nm . ( see fig . [ fig : decomp ] ) from the etch depth map . ] ) and their summation ( @xmath35 ) . ] for this example , to fabricate a bsg of high diffractive uniformity , it needs to carry out at least 6 times of linear scanning of ion beam . to start with removing a uniform layer of thickness of 15.95 nm , we then fabricate the 5 wave surfaces sequentially by linear scanning of ion beam . as is shown in fig . [ fig : decomp ] , for the 5 sinusoidal wave surfaces , the rotary directions are ( 0 , 1 ) , ( 1 , -1 ) , ( 1 , 0 ) , ( 1 , 1 ) , and ( 2 , 0 ) ; the periods are 400 mm , 283 mm , 400 mm , 283 mm , and 100 mm . in the process of ion beam etching , the bsg should be rotated according to the rotary directions and the width of ion beam spot should be less than a half of the period of wave surface . as is demonstrated above , optical fabrication with linear ion source can be used to finely adjust etch depths of large - aperture diffractive optics based on the concept of surface decomposition . we prepared a large - aperture optical surface ( @xmath36 ) polished by ccos with small tools . the measured surface map is shown on the upper right of fig . [ fig : errors ] . and more msf errors are found in the horizontal direction than those in the vertical direction , which is attributed to the raster scan path adopted in the small - tool polishing process . the corresponding psd plots are arranged on the left of fig . [ fig : errors ] , where the solid curve and the dotted curve represent the psd functions along the horizontal and the vertical , respectively . as is shown on the upper left of fig . [ fig : errors ] , there is a high peak at the solid psd curve in the spatial frequency of around @xmath37 . in general , a peak at the psd curve indicates more errors existing in the spatial frequency range around the peak . to remove errors in a specific range of spatial frequencies , we should target the errors displayed as sinusoidal wave surfaces and then remove them one by one via optical surfacing with linear ion source . given that @xmath38 ; @xmath39 ; @xmath40 and provided that @xmath41 ( only the errors distributed along the horizontal are of interest ) , then we have @xmath42 , which refers to @xmath43 . actually , to remove the peak totally , it requires a set of sinusoidal wave surfaces . fig . [ fig : waves ] shows a set of sinusoidal wave surfaces , @xmath44 , stacking up to remove errors in the spatial frequency range around @xmath37 . the summation , @xmath45 , represents the to - be - etched material removal , which is shown on the right of fig . [ fig : waves ] . after reducing the to - be - etched material removal from the original surface map , we obtain a surface map with less errors in the frequency range nearby @xmath37 ( shown on the bottom right of fig . [ fig : errors ] ) and the peak disappears in the corresponding psd plot . , summing up to remove errors in the spatial frequency range around @xmath37 . ] the simulation result indicates that , theoretically , one - dimensional polishing is able to get rid of msf errors spatially distributed vertical to the scanning direction . it also suggests that ultra - precision optical surfacing with linear ion source is a feasible and preferred approach to reduce errors in msf range . because optical surfacing with linear ion source is a superposition of multiple one - dimensional polishing processes and there is no discrete stitching of polishing spots in the course of polishing @xcite . as illustrated in fig . [ fig : optsurf ] , the aperture of optical element is less than the length of rectangular beam spot thus the ion beam can fully cover and scan over the optical surface without stitching along the length direction . we propose a surface modeling method based on double fourier series and the concept of surface decomposition makes it feasible to fabricate an ultraprecise optical surface with linear ion source . also , we demonstrate two applications . the first application shows that optical fabrication with linear ion source can be applied to direct fabricating a bsg of high diffractive uniformity without post - processing ( for example , chemical mechanical polishing ) for correcting the spatial distribution of diffraction efficiency . the second application presents a new approach to remove surface errors in msf range and we think the basic concept can be realized with other advanced polishing tools . moreover , optical fabrication with linear ion source will significantly improve the working efficiency compared with the conventional ibf , though small round beam has a higher reachability in the fabrication of complex surfaces . youth innovation promotion association of the chinese academy of sciences . we thank mr . zhentong liu for the help in drawing the schematic of optical fabrication with linear ion source . | we present a concept of surface decomposition extended from double fourier series to nonnegative sinusoidal wave surfaces , on the basis of which linear ion sources apply to the ultra - precision fabrication of complex surfaces and diffractive optics .
it is the first time that we have a surface descriptor for building a relationship between the fabrication process of optical surfaces and the surface characterization based on psd analysis , which akin to zernike polynomials used for mapping the relationship between surface errors and seidel aberrations . also , we demonstrate that the one - dimensional scanning of linear ion source is applicable to the removal of surface errors caused by small - tool polishing in raster scan mode as well as the fabrication of beam sampling grating of high diffractive uniformity without a post - processing procedure .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in a previous paper @xcite , we presented a comprehensive analysis on the lhc signatures of the type ii seesaw model of neutrino masses in the nondegenerate case of the triplet scalars . in this companion paper , another important signature the pair and associated production of the neutral scalars is explored in great detail . this is correlated to the pair production of the standard model ( sm ) higgs boson , @xmath15 , which has attracted lots of theoretical and experimental interest @xcite since its discovery @xcite , because the pair production can be used to gain information on the electroweak symmetry breaking sector @xcite . since any new ingredients in the scalar sector can potentially alter the production and decay properties of the higgs boson , a thorough examination of the properties offers a diagnostic tool to physics effects beyond the sm . the higgs boson pair production has been well studied for collider phenomenology in the framework of the sm and beyond @xcite , and extensively studied in various new physics models @xcite , as well as in the effective field theory approach of anomalous couplings @xcite and effective operators @xcite . the pair production of the sm higgs boson proceeds dominantly through the gluon fusion process @xcite , and has a cross section at the @xmath16 lhc ( lhc14 ) of about @xmath17 at leading order @xcite . at next - to - leading order @xcite and to @xmath18 at next - to - next - to - leading order @xcite . ] it can be utilized to measure the higgs trilinear coupling . a series of studies have surveyed its observability in the @xmath3 , @xmath4 , @xmath19 , @xmath20 , and @xmath21 signal channels @xcite . for the theoretical and experimental status of the higgs trilinear coupling and pair production at the lhc , see refs . @xcite . in summary , at the @xmath16 lhc with an integrated luminosity of @xmath6 ( lhc14@3000 ) , the trilinear coupling could be measured at an accuracy of @xmath22 @xcite , and thus leaves potential space for new physics . as we pointed out in ref . @xcite , in the negative scenario of the type ii seesaw model where the doubly charged scalars @xmath23 are the heaviest and the neutral ones @xmath0 the lightest , i.e. , @xmath1 , the associated @xmath11 production gives the same signals as the sm higgs pair production while enjoying a larger cross section . the leading production channel is the drell - yan process @xmath24 , with a typical cross section @xmath25-@xmath26 in the mass region @xmath27-@xmath28 . additionally , there exists a sizable enhancement from the cascade decays of the heavier charged scalars , which also gives some indirect evidence for these particles . the purpose of this paper is to examine the importance of the @xmath11 production with an emphasis on the contribution from cascade decays and to explore their observability . the paper is organized as follows . in sec . [ decay ] , we summarize the relevant part of the type ii seesaw and explore the decay properties of @xmath29 in the negative scenario . sections [ eh ] and [ signal ] contain our systematical analysis of the impact of cascade decays on the @xmath0 production in the three signal channels , @xmath3 , @xmath4 , and @xmath5 . we discuss the observability of the signals and estimate the required integrated luminosity for a certain mass reach and significance . discussions and conclusions are presented in sec . [ dis ] . in most cases , we will follow the notations and conventions in ref . @xcite . the type ii seesaw and its various experimental constraints have been reviewed in our previous work @xcite . here we recall the most relevant content that is necessary for our study of the decay properties of the scalars in this section and of their detection at the lhc in later sections . the type ii seesaw model introduces an extra scalar triplet @xmath30 of hypercharge two @xcite on top of the sm higgs doublet @xmath31 of hypercharge unity . writing @xmath30 in matrix form , the most general scalar potential is @xmath32 as in the sm , @xmath33 is assumed to trigger spontaneous symmetry breaking , while @xmath34 sets the mass scale of the new scalars . the vacuum expectation value ( vev ) @xmath35 of @xmath31 then induces via the @xmath36 term a vev @xmath37 for @xmath30 . the components of equal charge ( and also of identical @xmath38 in the case of neutral components ) in @xmath30 and @xmath31 then mix into physical scalars @xmath39 ; @xmath40 ; @xmath41 and would - be goldstone bosons @xmath42 , with the mixing angles specified by ( see , for instance , refs . @xcite ) @xmath43 where an auxiliary parameter is introduced for convenience , @xmath44 to a good approximation , the sm - like higgs boson @xmath15 has the mass @xmath45 , the new neutral scalars @xmath29 have an equal mass @xmath46 , and the new scalars of various charges are equidistant in squared masses : @xmath47 there are thus two scenarios of spectra , positive or negative , according to the sign of @xmath48 . for convenience , we define @xmath49 . and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] and @xmath23 versus @xmath50 at some benchmark points of @xmath51 and @xmath52 : @xmath53 , from the upper to the lower panels . [ brhp],title="fig : " ] in the rest of this section , we discuss the decay properties of the new scalars in the negative scenario with an emphasis on @xmath54 and @xmath40 . the explicit expressions for the relevant decay widths can be found in refs . it has been shown that @xmath0 decays dominantly into neutrinos for @xmath55 @xcite , resulting in totally invisible final states . we will restrict ourselves to @xmath56 in this work , where @xmath0 dominantly decays into visible particles . before we detail their decay properties , we give a brief account of the cascade decays of the charged scalars . the branching ratios of the cascade decays are controlled by the three parameters , @xmath52 , @xmath51 , and @xmath50 . the cascade decays dominate in the moderate region of @xmath52 and for @xmath51 not too small , where a minimum value of @xmath57 appears around @xmath58 @xcite . in fig . [ brhp ] , the branching ratios of @xmath59 and @xmath23 are shown as a function of @xmath50 at some benchmark points of @xmath52 and @xmath51 . basically speaking , in the mass region @xmath60-@xmath28 , the cascade decays are dominant for a relatively large mass splitting @xmath51 ( as shown in the middle panel of fig . [ brhp ] ) or a relatively small @xmath52 ( in the lower panel ) . at tree level , @xmath54 can decay to @xmath61 , @xmath62 , @xmath63 , @xmath64 , and @xmath65 . it can also decay to @xmath66 , @xmath67 , and @xmath68 through radiative effects . similarly , @xmath69 , @xmath62 , @xmath70 at tree level , and it has the same decay modes as @xmath54 at the loop level . since we have chosen @xmath56 , the neutrino mode can be safely neglected for both @xmath54 and @xmath40 . previous work usually concentrated on the decoupling region where the neutral scalars @xmath0 are much heaver than the light @xmath38-even higgs @xmath15 and the scalar self - couplings @xmath71 are taken to be zero for simplicity @xcite . in this case , the mixing angle @xmath72 , and the @xmath73 coupling [ being proportional to @xmath74 tends to vanish . as a consequence , the @xmath75-pair mode is absent and the dominant channels are @xmath76 , @xmath64 for a heavy @xmath54 . in contrast , we take into account the effect of scalar self - interactions and focus on the nondecoupling regime , i.e. , @xmath0 are not much heavier than @xmath15 . for illustration , we choose the benchmark values @xmath77 , @xmath78 ; then , @xmath48 is determined by eq . ( [ massrelation ] ) upon specifying @xmath79 . in the range @xmath80-@xmath81 would not change the branching ratios significantly . ] to investigate the effect of the scalar self - interactions , we note the following features in the decays of @xmath54 . 1 ) the decay widths of @xmath82 differ from those of @xmath15 only by a factor of @xmath83 , which leads to similar behavior for @xmath54 and @xmath15 . 2 ) the only free parameter for the mixing between @xmath54 and @xmath15 is @xmath84 , because [ as shown in eq . ( [ mixangles ] ) ] the impact of @xmath85 is suppressed by a small @xmath52 and a relatively large mass difference between @xmath50 and @xmath86 while @xmath87 is fixed by @xmath86 . 3 ) @xmath84 enters the @xmath73 and @xmath88 couplings and thus affects the decays @xmath89 . 4 ) the @xmath90 coupling simplifies for @xmath91 such that the only free parameter in the decay @xmath92 is again @xmath84 . as a consequence of these features , we shall choose @xmath84 as a free parameter and vary it in the range @xmath93 $ ] , and fix the couplings @xmath94 which are involved in loop - induced decays . and @xmath95 as a function of @xmath96 for various values of @xmath84 . [ brh0tt],title="fig : " ] and @xmath95 as a function of @xmath96 for various values of @xmath84 . [ brh0tt],title="fig : " ] we first examine the branching ratios of @xmath97 . br(@xmath98 ) and br(@xmath95 ) are plotted in fig . [ brh0tt ] for different mass regions of @xmath54 . for light fermions @xmath99 and gluons is similar , so we only present br(@xmath98 ) in fig . [ brh0tt ] . ] it is clear that the variation of br(@xmath98 ) is more dramatic for @xmath100 . the maximum of br(@xmath98 ) appears at @xmath101 . obviously , br(@xmath98 ) is a nonmonotonic function of @xmath84 , while br(@xmath95 ) monotonically increases with @xmath84 . as will be discussed later , this different behavior in the two mass regions is due mainly to a zero in the @xmath88 coupling . as a function of @xmath96 in the mass region @xmath27-@xmath28 . right : branching ratios of @xmath102 as a function of @xmath96 in the mass region @xmath103-@xmath104 . [ brh0ww],title="fig : " ] as a function of @xmath96 in the mass region @xmath27-@xmath28 . right : branching ratios of @xmath102 as a function of @xmath96 in the mass region @xmath103-@xmath104 . [ brh0ww],title="fig : " ] as a function of @xmath96 in the mass region @xmath27-@xmath28 . right : branching ratios of @xmath102 as a function of @xmath96 in the mass region @xmath103-@xmath104 . [ brh0ww],title="fig : " ] as a function of @xmath96 in the mass region @xmath27-@xmath28 . right : branching ratios of @xmath102 as a function of @xmath96 in the mass region @xmath103-@xmath104 . [ brh0ww],title="fig : " ] now we study the bosonic decays @xmath105 . in the left panel of fig . [ brh0ww ] , we present the branching ratios of @xmath89 in the mass region @xmath27-@xmath28 . for most values of @xmath84 , br(@xmath106 ) increases with @xmath96 when @xmath107 , and varying @xmath84 for @xmath100 changes it considerably . @xmath84 has a strong impact on br(@xmath106 ) in the mass region @xmath108 where the decay channel dominates overwhelmingly for @xmath109 but becomes negligible for @xmath84 approaching about @xmath110 . however , once the @xmath92 channel is opened , @xmath106 is suppressed significantly independent of @xmath84 . the decay @xmath111 can not dominate when @xmath107 . in the mass region @xmath108 , it is complementary with the @xmath112 channel , so their behavior is just opposite . more interestingly , there is a zero point for the @xmath88 coupling , which is proportional to @xmath113 . according to eq . ( [ mixangles ] ) , one obtains the corresponding @xmath50 at the zero : @xmath114 note that the above relation only holds for @xmath115 , since we are working in the scenario where @xmath116 . the existence of the zero coupling explains the presence of the nodes in br(@xmath111 ) for @xmath117 . as a function of @xmath96 for various sets of @xmath118 values . [ brh0aa],title="fig : " ] as a function of @xmath96 for various sets of @xmath118 values . [ brh0aa],title="fig : " ] in the right panel of fig . [ brh0ww ] , br(@xmath119 ) are shown in the mass region @xmath103-@xmath104 . when @xmath120 , the dependence on @xmath84 is simple : a larger @xmath84 corresponds to a smaller br(@xmath92 ) and a larger br(@xmath111 ) . it is clear that @xmath84 has a more significant impact in the mass region @xmath121 , and varying @xmath84 could change br(@xmath111 ) from @xmath122 to @xmath123 . once @xmath96 exceeds @xmath124 , the evolution of br(@xmath119 ) becomes smooth with the increase of @xmath96 . there also exists a zero point for the @xmath90 coupling , which can be obtained as for the @xmath64 channel : @xmath125 which is valid for @xmath126 . finally , we investigate the loop - induced decays , @xmath127 . in addition to the usual contributions from the top quark and @xmath75 boson , the new charged scalars @xmath59 and @xmath23 also contribute to the decays . these new terms involve the @xmath128 and @xmath129 couplings , which are proportional to @xmath130 , \nonumber \\ h^0h^{++}h^{--}&:&(\lambda_2\sin\alpha\cos\theta_0-\lambda_4\cos\alpha\sin\theta_0).\end{aligned}\ ] ] one therefore has to consider the scalar self - couplings @xmath85 . for simplicity , we set @xmath131 and vary them from @xmath132 to @xmath133 . in fig . [ brh0aa ] , we display br(@xmath134 ) and br(@xmath135 ) versus @xmath96 for some typical sets of @xmath118 values . the evolution of both branching ratios crosses 3 orders of magnitude in this parameter region . the resulting enhancement compared with @xmath136 in the sm looks significant : the maximal enhancement can be achieved at the level of @xmath137 for the @xmath134 channel at @xmath138 , and of @xmath139 for the @xmath135 channel at @xmath140 . as a function of @xmath141 for various values of @xmath84 . [ bra0tt],title="fig : " ] as a function of @xmath141 for various values of @xmath84 . [ bra0tt],title="fig : " ] similar to @xmath54 , the decay widths of @xmath142 differ from those of @xmath15 by a factor of @xmath143 with @xmath144 being given in eq . ( [ mixangles ] ) . the only vertex which involves @xmath71 is the @xmath145 coupling proportional to @xmath146 . as a consequence , one can only choose @xmath84 as a free parameter to illustrate the influence of scalar interactions . in this section , we also vary @xmath84 from @xmath147 to @xmath148 and take the same benchmark values for @xmath52 and @xmath51 as for the @xmath54 decays . in the left panel of fig . [ bra0tt ] , we present br(@xmath149 ) as a function of @xmath141 . on the @xmath150 channels is similar to the @xmath151 mode . ] for a fixed value of @xmath84 , br(@xmath149 ) decreases as @xmath141 increases . the dependence of br(@xmath149 ) on @xmath84 is simple : the larger @xmath84 is , the larger br(@xmath149 ) is . and br(@xmath149 ) can be dominant with @xmath152 as long as @xmath153 is not fully opened . the right panel of fig . [ bra0tt ] shows br(@xmath154 ) , which is very similar to br(@xmath95 ) . as a function of @xmath141 for various values of @xmath84 .. [ bra0zh],title="fig : " ] as a function of @xmath141 for various values of @xmath84 .. [ bra0zh],title="fig : " ] and @xmath155 as a function of @xmath141 for various values of @xmath84 . [ bra0aa],title="fig : " ] and @xmath155 as a function of @xmath141 for various values of @xmath84 . [ bra0aa],title="fig : " ] we then study the most important decay @xmath153 . in fig . [ bra0zh ] , we present br(@xmath153 ) as a function of @xmath141 in the low - mass region ( @xmath27-@xmath28 ) and high - mass region ( @xmath156-@xmath104 ) , respectively . the evolution of br(@xmath153 ) with @xmath141 and @xmath84 is just opposite to that of @xmath157 in the low- ( high- ) mass region . the variation of br(@xmath153 ) with @xmath84 is dramatic below the @xmath70 threshold . in particular , near the @xmath70 threshold br(@xmath158 for @xmath159 , while br(@xmath153 ) tends to vanish for @xmath152 , which corresponds to the zero point of the @xmath145 coupling : @xmath160 with @xmath161 . br(@xmath153 ) is totally dominant in the mass region between the @xmath70 and @xmath162 thresholds , and becomes comparable to br(@xmath154 ) when @xmath163 . at last , we study the one - loop - induced decays , @xmath164 . these two channels can only be induced by the top quark in the loop since the @xmath165 , @xmath166 , and @xmath167 couplings are absent in the @xmath38-conserving case . in fig . [ bra0aa ] , both br(@xmath168 ) and br(@xmath155 ) are displayed . for @xmath141 below the @xmath70 threshold , the variation in @xmath84 of br(@xmath168 ) increases as @xmath141 increases . br(@xmath169 ) could reach @xmath170 for @xmath171 and @xmath152 , which is much smaller than the maximum of br(@xmath172 ) . the variation in @xmath84 of br(@xmath155 ) is slightly steeper , with a maximum of @xmath173 at @xmath174 and @xmath152 . as a function of @xmath175 at the benchmark point in eq . ( [ bp ] ) . [ brh0],title="fig : " ] as a function of @xmath175 at the benchmark point in eq . ( [ bp ] ) . [ brh0],title="fig : " ] in the above , we have discussed the decay channels of @xmath54 and @xmath40 separately . we have shown that the scalar self - interactions have a large impact on their branching ratios . in sec . [ signal ] , we will explore their lhc signatures . for this purpose , we choose the following benchmark values : @xmath176 the reason that we set relatively small values of @xmath52 and @xmath51 is to obtain large cascade decays of charged scalars as well as a large enhancement of neutral scalar production . in fig . [ brh0 ] , we display all relevant branching ratios versus @xmath175 for this benchmark model , which is to be simulated in sec . [ signal ] for the lhc in the @xmath3 , @xmath4 , and @xmath19 signal channels . we pointed out in ref . @xcite the importance of the associated @xmath11 production in the nondegenerate case . to estimate the number of signal events , we simulated the signal channel @xmath4 at @xmath177 . we found that , with a much higher production cross section than the sm higgs pair ( @xmath65 ) production , a @xmath178 excess in that signal channel is achievable for lhc14@300 . in the present work , we are interested in the observability of the associated @xmath11 production in the nondecoupling mass regime @xmath179-@xmath180 . in fig . [ cs ] we first show the production cross sections for a pair of various scalars at lhc14 versus @xmath50 with a degenerate spectrum . as before , we incorporate the next - to - leading - order ( nlo ) qcd effects by multiplying a @xmath181-factor of @xmath182 in all @xmath183 production channels @xcite . the @xmath65 production through gluon - gluon fusion at nlo ( @xmath184 ) is also indicated ( black dashed line ) for comparison . one can see that the cross section for @xmath11 is about @xmath25-@xmath26 in the mass region @xmath27-@xmath28 , which is much larger than the @xmath65 production for most of the mass region and thus leads to great discovery potential . for a degenerate spectrum . the black dashed line is for the sm @xmath65 production . [ cs],title="fig : " ] for a degenerate spectrum . the black dashed line is for the sm @xmath65 production . [ cs],title="fig : " ] in general , the new scalars are nondegenerate for a nonzero @xmath48 . in the positive scenario where @xmath23 are the lightest , the cascade decays of @xmath39 and @xmath0 can strengthen the observability of @xmath23 @xcite . for the same reason , in the negative scenario where @xmath0 are the lightest , the charged scalars contribute instead to the production of @xmath0 through the cascade decays like @xmath185 . in this work , we study these contributions in the same way as was done for the positive scenario in refs . @xcite . we define the reference cross section @xmath186 for the standard drell - yan process @xmath187 which is independent of the cascade decay parameters @xmath52 and @xmath51 . a detailed study on the @xmath4 signal for this process with @xmath188 can be found in ref . @xcite . besides the above direct production , neutral scalars can also be produced from cascade decays of charged scalars . these extra production channels include @xmath189 , @xmath190 , @xmath191 , and @xmath192 followed by cascade decays of charged scalars . we consider first the associated @xmath189 production followed by cascade decays of @xmath39 , @xmath193 resulting in three final states classified by a pair of neutral scalars : @xmath194 , @xmath195 , and @xmath196 . noting that the last two originate only from cascade decays , any detection of such production channels would be a hint of charged scalars being involved . using the fact that @xmath197 as well as the narrow width approximation , we calculate the production cross sections for these three final states : @xmath198\times \mbox{br}(h^\pm \to a^0 w^ * ) , \\ w^-\to h^- h^0)]\times \mbox{br}(h^\pm \to h^0 w^ * ) , \\ a^0a^0:z_1&=&[\sigma(pp\to w^+\to h^+ a^0)+\sigma(pp\to w^-\to h^- a^0)]\times \mbox{br}(h^\pm \to a^0 w^*).\end{aligned}\ ] ] the factor 2 in @xmath199 accounts for the equal contribution from the process with @xmath54 and @xmath40 interchanged . the relations @xmath200 actually hold true for all of the four production channels , since for a given channel the same branching ratios ( such as for @xmath185 ) are involved , @xmath201 where @xmath202 , and @xmath203 refer to the cross sections for @xmath11 , @xmath195 , and @xmath196 production with the subscript @xmath204 denoting the production channels @xmath189 , @xmath205 , @xmath191 , and @xmath192 , respectively . the relations imply that we may concentrate on the cross section of @xmath11 production . naively , one would expect the next important channel to be @xmath190 since it only involves two cascade decays : @xmath206 but as already mentioned in ref . @xcite , a smaller coupling and destructive interference between the @xmath207 and @xmath208 exchange make the cross section of @xmath190 production an order of magnitude smaller than that of @xmath11 even for a degenerate spectrum . considering further suppression due to cascade decays , @xmath209 is not important for the enhancement of @xmath11 production and can be safely neglected in the numerical analysis . at lhc14 and with @xmath78 , @xmath210 . left : the red solid ( dashed ) line corresponds to @xmath186 ( @xmath211 ) . right : the red line corresponds to @xmath212 from cascade decays @xmath213 , and the green line to @xmath11 from cascade decays ( @xmath214 ) . the shaded regions are filled by scanning over @xmath51 and @xmath37 . [ csx],title="fig : " ] at lhc14 and with @xmath78 , @xmath210 . left : the red solid ( dashed ) line corresponds to @xmath186 ( @xmath211 ) . right : the red line corresponds to @xmath212 from cascade decays @xmath213 , and the green line to @xmath11 from cascade decays ( @xmath214 ) . the shaded regions are filled by scanning over @xmath51 and @xmath37 . [ csx],title="fig : " ] the contribution from @xmath191 is more important despite the fact that it involves three cascade decays : @xmath215\times\\ \nonumber & & \mbox{br}(h^{\pm\pm}\to h^\pm w^*)\mbox{br}(h^{\pm}\to h^0 w^*)\mbox{br}(h^{\pm}\to a^0 w^*).\end{aligned}\ ] ] as shown in fig . [ cs ] , @xmath216 is the largest for a degenerate mass spectrum . when cascade decays are dominant , the phase - space suppression of heavy charged scalars will be important . so we expect that the @xmath11 production receives considerable enhancement from @xmath191 when the mass splitting is small and cascade decays are dominant . finally , the last mechanism is @xmath192 , which involves four cascade decays : @xmath217 this mechanism is also promising since the cross section of @xmath192 production is slightly larger than @xmath218 production for a degenerate mass spectrum . the phase - space suppression of @xmath219 is more severe than that of @xmath220 , because a pair of the heaviest @xmath23 are produced . summing over all four of the above channels yields the contribution to the @xmath11 production from cascade decays , @xmath221 and the total production cross section of @xmath11 is then @xmath222 . using eq . ( [ xxx ] ) , the total cross sections for the pair production @xmath212 , @xmath223 , @xmath224 , are given by @xmath225 since the enhancement from cascade decays depends on a not severely suppressed phase space and a larger branching ratio of cascade decays , we choose to work with a relatively smaller mass splitting and triplet vev as shown in eq . ( [ bp ] ) . figure [ csx ] displays the cross sections of the @xmath11 , @xmath195 , and @xmath196 production as a function of @xmath50 . as can be seen from the figure , the production of @xmath11 can be enhanced by a factor of 3 , while the @xmath212 production at the maximal enhancement can reach the level of @xmath186 . this could make the detection of neutral scalar pair productions very promising in the negative scenario . in this section we investigate the signatures of neutral scalar production at the lhc . from previous studies on the sm @xmath65 production , we already know that the most promising signal is @xmath3 , and @xmath4 is next to it , while both semileptonic and dileptonic decays of @xmath75 s in the @xmath19 channel are challenging . in this work we analyze all three of the signals@xmath3 , @xmath4 , and @xmath226 ( @xmath227 for collider identification)as well as their backgrounds based on the benchmark model presented in eq . ( [ bp ] ) . in sec . [ eh ] we discussed the drell - yan production of @xmath11 and the enhanced pair and associated production of neutral scalars @xmath0 due to cascade decays of charged scalars @xmath228 . we are now ready to incorporate the branching ratios of @xmath0 decays for a specific signal channel . for instance , the cross sections for the @xmath3 signal channel can be written as @xmath229,\\ s(b\bar{b}\gamma\gamma ) & = & x\times\left[\mbox{br}(h^0\to b\bar{b})\mbox{br}(a^0\to\gamma\gamma ) + \mbox{br}(h^0\to\gamma\gamma)\mbox{br}(a^0\to b\bar{b})\right]\\\nonumber & & + 2y\hspace{-0.25em}\times\mbox{br}(h^0\to b\bar{b})\mbox{br}(h^0\to\gamma\gamma ) + 2z\hspace{-0.35em}\times\mbox{br}(a^0\to b\bar{b})\mbox{br}(a^0\to\gamma\gamma).\end{aligned}\ ] ] here @xmath230 denotes the signal from the direct production @xmath24 alone , and @xmath231 includes contributions from cascade decays . @xmath232 has a similar expression as @xmath233 , while @xmath234 is simpler since the decay mode @xmath235 is absent . , @xmath4 , and @xmath236 signal channels at lhc14 . the red solid ( dashed ) line corresponds to the signal from @xmath186 ( @xmath211 ) , the green ( blue ) solid line corresponds to the signal from @xmath237 , and the purple dashed line shows the total cross section @xmath231 for the signal . the sm @xmath65 cross section is shown for comparison . the lower right panel shows the enhancement factor @xmath238 in the three signal channels . [ sgn],title="fig : " ] , @xmath4 , and @xmath236 signal channels at lhc14 . the red solid ( dashed ) line corresponds to the signal from @xmath186 ( @xmath211 ) , the green ( blue ) solid line corresponds to the signal from @xmath237 , and the purple dashed line shows the total cross section @xmath231 for the signal . the sm @xmath65 cross section is shown for comparison . the lower right panel shows the enhancement factor @xmath238 in the three signal channels . [ sgn],title="fig : " ] , @xmath4 , and @xmath236 signal channels at lhc14 . the red solid ( dashed ) line corresponds to the signal from @xmath186 ( @xmath211 ) , the green ( blue ) solid line corresponds to the signal from @xmath237 , and the purple dashed line shows the total cross section @xmath231 for the signal . the sm @xmath65 cross section is shown for comparison . the lower right panel shows the enhancement factor @xmath238 in the three signal channels . [ sgn],title="fig : " ] , @xmath4 , and @xmath236 signal channels at lhc14 . the red solid ( dashed ) line corresponds to the signal from @xmath186 ( @xmath211 ) , the green ( blue ) solid line corresponds to the signal from @xmath237 , and the purple dashed line shows the total cross section @xmath231 for the signal . the sm @xmath65 cross section is shown for comparison . the lower right panel shows the enhancement factor @xmath238 in the three signal channels . [ sgn],title="fig : " ] the theoretical cross sections for the @xmath3 , @xmath4 , and @xmath236 signal channels are plotted in fig . [ sgn ] . the cross section @xmath239 is larger than that of the sm @xmath65 production until @xmath240 ; taking into account cascade enhancement pushes the corresponding @xmath50 further to @xmath241 . @xmath242 is always larger than that of @xmath65 in the mass region @xmath27-@xmath14 , and interestingly , it keeps about the same value when @xmath243 . the signal from @xmath195 is comparable with @xmath230 in these three channels only for @xmath243 , while in contrast the signal from @xmath196 becomes dominant for the @xmath3 and @xmath4 channels when @xmath244 . therefore , we have a chance to probe the @xmath196 pair production in these two channels . also shown in fig . [ sgn ] is the enhancement factor @xmath238 for the three signal channels at the benchmark point ( [ bp ] ) as a function of @xmath79 , which will help us understand the simulation results . in our simulation , the parton - level signal and background events are generated with * madgraph5 * @xcite . we perform parton shower and fast detector simulations with * pythia * @xcite and * delphes3 * @xcite . finally , * madanalysis5 * @xcite is responsible for data analysis and plotting . we take a flat @xmath245-tagging efficiency of 70% , and mistagging rates of 10% for @xmath246 jets and 1% for light - flavor jets , respectively . jet reconstruction is done using the anti-@xmath247 algorithm with a radius parameter of @xmath248 . we further assume a photon identification efficiency of @xmath249 and a jet - faking - photon rate of @xmath250 @xcite . the main sm backgrounds to the signal are as follows : @xmath251 among them , @xmath3 and @xmath70 are irreducible , while @xmath252 is reducible and can be reduced by vetoing the additional @xmath253 s with @xmath254 and @xmath255 . in addition , there exist many reducible sources of fake @xmath3 : @xmath256 where @xmath257 stands for a final - state @xmath258 misidentified as @xmath259 . the remaining fake sources are subdominant and are thus not included in our simulation . the qcd corrections to the backgrounds are included by a multiplicative @xmath181-factor of 1.10 and 1.33 for the leading cross sections of @xmath252 and @xmath70 at lhc14 @xcite , respectively . the cross section of the @xmath3 background has been normalized to include fake sources and does not take nlo corrections into account . , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] , and @xmath260 for the signal @xmath3 and its backgrounds before applying any cuts at lhc14 . [ fig : bbaa],title="fig : " ] the distributions of some kinematical variables before applying any cuts are shown in fig . [ fig : bbaa ] , where we assume @xmath261 . in our analysis , we require that the final states include exactly one @xmath245-jet pair and one @xmath262 pair and satisfy the following basic cuts : @xmath263 where @xmath264 is the particle separation , with @xmath265 and @xmath266 being the separation in the azimuthal angle and rapidity , respectively . here we employ a tighter @xmath267 cut than is usually applied to suppress the qcd - electroweak @xmath3 background . the @xmath245-jet pair and @xmath262 pair are then required to fall in the following windows on the invariant masses and fulfill the @xmath268 cut criteria : @xmath269 [ ! htbp ] .evolution of signal and background cross sections ( in @xmath270 ) at lhc14 for the @xmath3 signal channel upon imposing the cuts one by one . for the cascade - enhanced signal only the cross section passing all the cuts is shown . the last two columns assume an integrated luminosity of @xmath6 . [ tab : bbaacut ] [ cols="^,^,^,^,^,^,^",options="header " , ] with both @xmath75 s decaying leptonically , the final state appears as @xmath5 . the dominant sm backgrounds are as follows : @xmath271 as before , the qcd correction is included by a multiplicative @xmath181-factor of 1.35 for the @xmath272 production @xcite . we pick up the events that include exactly one @xmath245-jet pair and one opposite - sign lepton pair and filter them with the basic cuts : @xmath273 the separation and invariant mass of the @xmath245-jet pair are required to fulfill @xmath274 for the lepton pair , we reconstruct the transverse cluster mass @xmath275 : @xmath276 the distributions of @xmath275 , @xmath277 , and @xmath278 are shown in fig . [ fig : bbll2v ] . the peak of @xmath275 is always lower than @xmath50 by about @xmath279-@xmath280 , and the lepton separation @xmath277 in the signal is much smaller than in the @xmath272 background . accordingly , we set a wide window on @xmath281 while tightening up the cuts on @xmath277 and @xmath278 : @xmath282 we find that @xmath281 is least efficient around @xmath283 , where the peak of @xmath281 for the @xmath272 background is around @xmath284 . the very tight cuts on @xmath277 and @xmath278 are sufficient to suppress the background by 1 or 2 orders of magnitude , while keeping the number of signal events as large as possible . we further combine the @xmath245-jet pair and the lepton pair into a cluster and construct the transverse cluster mass : @xmath285 of the @xmath3 channel versus @xmath50 reachable at lhc14@300 ( red region ) and lhc14@3000 ( green ) . right : required luminosity to reach a @xmath286 ( red region ) and @xmath7 ( green ) significance in the @xmath3 channel versus @xmath50 at lhc14 . the solid line corresponds to the signal from @xmath186 alone , and the dashed line corresponds to the total signal including cascade enhancement . [ bbaa_sen],title="fig : " ] of the @xmath3 channel versus @xmath50 reachable at lhc14@300 ( red region ) and lhc14@3000 ( green ) . right : required luminosity to reach a @xmath286 ( red region ) and @xmath7 ( green ) significance in the @xmath3 channel versus @xmath50 at lhc14 . the solid line corresponds to the signal from @xmath186 alone , and the dashed line corresponds to the total signal including cascade enhancement . [ bbaa_sen],title="fig : " ] which is an analog of @xmath287 in the previous subsection . the distribution of @xmath288 is displayed in fig . [ fig : bbll2v ] , which is very similar to that of @xmath287 in the @xmath3 channel . although it looks from the @xmath288 distributions ( before any cuts are made ) that the @xmath272 background has a large overlap with the signal , the cuts on @xmath275 , @xmath277 , and @xmath278 actually modify them remarkably , so that a further cut on @xmath288 could improve the significance efficiently . we apply a cut on @xmath288 as we did with @xmath287 , as well as one on @xmath260 : @xmath289 the results following the cutflow are summarized in table [ tab : bbllcut ] . for @xmath188 , the final significance is 4.41 ( 17.7 ) without ( with ) cascade enhancement . with cascade enhancement this should be enough to discover the neutral scalars . the signal channel is more promising for @xmath290 due to a slightly larger cross section and higher cut efficiencies . the final significance is 7.78 ( 23.2 ) , which is also better than the @xmath3 and @xmath4 channels with the same mass . finally , for @xmath291 , the significance becomes 3.56 ( 10.1 ) . therefore , for our benchmark model , the only promising signal for such heavy neutral scalars ( @xmath292 ) comes from the @xmath19 channel . , but for the @xmath4 channel . [ bbtata_sen],title="fig : " ] , but for the @xmath4 channel . [ bbtata_sen],title="fig : " ] based on our elaborate analysis of signal channels in secs . [ sec : bbgg][sec : bbww ] , we examine the observability of the neutral scalars @xmath293 in the mass region @xmath294 by adopting essentially the same cuts as before . in the left panel of figs . [ bbaa_sen ] , [ bbtata_sen ] , and [ bbll2v_sen ] we present the significance @xmath295 as a function of @xmath50 in the three signal channels @xmath3 , @xmath4 , and @xmath5 that is reachable for lhc14@300 and lhc14@3000 , respectively . the required luminosity to achieve a @xmath286 and @xmath7 significance is displayed in the right panel of the figures . as was done in our previous analysis , the effect of cascade enhancement is included by a factor @xmath238 in the final results . as shown in figs . [ bbaa_sen ] and [ bbtata_sen ] , both the @xmath3 and @xmath4 channel are typically sensitive to the low - mass region ( @xmath296 ) . in the absence of cascade enhancement , the @xmath286 significance would never be reached for @xmath297 in the @xmath3 ( @xmath4 ) channel for lhc14@300 . however , a cascade enhancement of @xmath298 ( as can be seen from fig . [ sgn ] ) in this mass region can greatly improve the observability , pushing the @xmath286 mass limit up to @xmath299 in the @xmath3 ( @xmath4 ) channel . moreover , with cascade enhancement , one has a good chance to reach a @xmath7 significance if @xmath300 . in other words , the cascade enhancement significantly reduces the required luminosity . for instance , to achieve a @xmath286 and @xmath7 significance in the @xmath3 ( @xmath4 ) channel with @xmath188 , the required luminosity is as low as @xmath301 and @xmath302 at lhc14 , respectively . the @xmath4 channel is more promising , thanks to a relatively larger production rate . at the future lhc14 with @xmath6 data , the heavier mass region can also be probed . with a maximal cascade enhancement , the @xmath286 and @xmath7 mass reach is pushed to @xmath13 and @xmath303 , respectively , in the @xmath3 channel , which should be compared to @xmath304 and @xmath305 in the absence of enhancement . for the @xmath4 channel , the enhancement factor @xmath238 can reach about @xmath306 above the @xmath75-pair threshold , upshifting the @xmath286 and @xmath7 mass reach to @xmath307 and @xmath308 , respectively , from @xmath309 and @xmath284 without the enhancement . , but for the @xmath5 channel . [ bbll2v_sen],title="fig : " ] , but for the @xmath5 channel . [ bbll2v_sen],title="fig : " ] the @xmath5 channel shown in fig . [ bbll2v_sen ] is more special , compared with @xmath3 and @xmath4 . it is relatively more sensitive to a higher mass between @xmath9-@xmath10 , where the decay mode @xmath106 dominates , while its observability deteriorates for @xmath310 due to phase - space suppression in the decay . the cascade enhancement @xmath238 at our benchmark point ( [ bp ] ) is typically @xmath311-@xmath312 in the mass region @xmath27-@xmath14 , and decreases as @xmath50 increases . for lhc14@300 , the @xmath286 and @xmath7 mass reach is , respectively , @xmath313 and @xmath314 with maximal cascade enhancement . these limits would just increase by @xmath315-@xmath316 for lhc14@3000 if there were no cascade enhancement , while with cascade enhancement the @xmath7 limit , for instance , is pushed up to @xmath14 . finally , a @xmath286 or @xmath7 reach in the mass region @xmath9-@xmath10 requires an integrated luminosity of @xmath317 ( @xmath318 ) or @xmath319 ( @xmath320 ) with ( without ) cascade enhancement . in this paper , we have systematically investigated the lhc phenomenology of neutral scalar pair production in the negative scenario of the type ii seesaw model . to achieve this goal , we first examined the decay properties of the neutral scalars @xmath321 and found that the scalar self - couplings @xmath71 have a great impact on the branching ratios of @xmath0 . the coupling @xmath84 is important for tree - level decays of @xmath54 and @xmath40 , while one - loop - induced decays of @xmath54 further depend on @xmath322 and @xmath323 . we found that the decay @xmath106 could dominate for @xmath324 with @xmath109 , while it can be neglected once @xmath96 is above the light scalar pair threshold @xmath325 . moreover , the branching ratios of the decays @xmath127 can cross 3 orders of magnitude when varying the couplings @xmath71 , and there exist zero points for the @xmath88 , @xmath90 , and @xmath145 couplings . the cross section of the drell - yan process @xmath24 for @xmath326 is much larger than that of the sm higgs pair production driven by gluon fusion . in this paper , we studied the contributions to @xmath0 production from cascade decays of the charged scalars @xmath59 and @xmath23 . there are actually three different states for the neutral scalar pair : @xmath11 , @xmath195 , and @xmath196 . here , @xmath195 and @xmath196 can only arise from cascade decays of charged scalars , and their production rates always stay the same to a good approximation . further , for a fixed value of @xmath50 , cascade enhancement is determined by the variables @xmath52 and @xmath51 . by tuning these two variables , the associated production rate of @xmath11 can be maximally enhanced by about a factor of 3 , while those of the @xmath195 and @xmath196 pair production can reach the value of @xmath11 production through the pure drell - yan process . we implemented detailed collider simulations of the associated @xmath11 production for three typical signal channels ( @xmath3 , @xmath4 , and @xmath19 with both @xmath75 s decaying leptonically ) . the enhancement from cascade decays of charged scalars is quantified by a multiplicative factor @xmath238 . due mainly to a larger production rate , all three channels are more promising than the sm higgs pair case . if there were no cascade enhancement , the @xmath7 mass reach of the @xmath3 , @xmath4 , and @xmath5 channels would be , respectively , @xmath8 , @xmath9 , and @xmath10 for lhc14@3000 . the cascade enhancement pushes these limits up to @xmath12 , @xmath13 , and @xmath14 . the @xmath3 and @xmath4 channels are more promising in the mass region below about @xmath284 , and the required luminosities for @xmath7 significance are @xmath327 and @xmath328 , respectively , at our benchmark point . compared with these two channels , the @xmath5 channel is more advantageous in the relatively higher mass region @xmath9-@xmath14 , and the required luminosity for @xmath7 significance is about @xmath319 with maximal cascade enhancement . needless to say , for the purpose of a full investigation on the impact of heavy neutral scalars on the sm higgs pair production , more sophisticated simulations are necessary . we hope that this work may shed some light on further studies in both the phenomenological and experimental communities . this work was supported in part by the grants no . nsfc-11025525 , no . nsfc-11575089 and by the cas center for excellence in particle physics ( ccepp ) . 000 z. l. han , r. ding , and y. liao , phys . d * 91 * , 093006 ( 2015 ) [ arxiv:1502.05242 [ hep - 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ph ] ] ; j. alwall _ et al . _ , jhep * 1407 * , 079 ( 2014 ) [ arxiv:1405.0301 [ hep - ph ] ] . t. sjostrand , s. mrenna , and p. z. skands , jhep * 0605 * , 026 ( 2006 ) [ hep - ph/0603175 ] . s. ovyn , x. rouby , and v. lemaitre , arxiv:0903.2225 [ hep - ph ] ; j. de favereau _ et al . _ [ delphes 3 collaboration ] , jhep * 1402 * , 057 ( 2014 ) [ arxiv:1307.6346 [ hep - ex ] ] . e. conte , b. fuks , and g. serret , comput . commun . * 184 * , 222 ( 2013 ) [ arxiv:1206.1599 [ hep - ph ] ] . g. aad _ et al . _ [ atlas collaboration ] , arxiv:0901.0512 [ hep - ex ] . s. dittmaier _ et al . _ [ lhc higgs cross section working group collaboration ] , arxiv:1101.0593 [ hep - ph ] . g. cowan , k. cranmer , e. gross , and o. vitells , eur . j. c * 71 * , 1554 ( 2011 ) [ erratum - ibid . c * 73 * , 2501 ( 2013 ) ] [ arxiv:1007.1727 [ physics.data-an ] ] . j. m. campbell and r. k. ellis , phys . d * 62 * , 114012 ( 2000 ) [ hep - ph/0006304 ] . m. cacciari , s. frixione , m. l. mangano , p. nason , and g. ridolfi , jhep * 0809 * , 127 ( 2008 ) [ arxiv:0804.2800 [ hep - ph ] ] ; p. brnreuther , m. czakon , and a. mitov , phys . lett . * 109 * , 132001 ( 2012 ) [ arxiv:1204.5201 [ hep - ph ] ] . | this is a sequel to our previous work on lhc phenomenology of the type ii seesaw model in the nondegenerate case . in this work
, we further study the pair and associated production of the neutral scalars @xmath0 .
we restrict ourselves to the so - called negative scenario characterized by the mass order @xmath1 , in which the @xmath0 production receives significant enhancement from cascade decays of the charged scalars @xmath2 .
we consider three important signal channels@xmath3 , @xmath4 , @xmath5and perform detailed simulations .
we find that at the 14 tev lhc with an integrated luminosity of @xmath6 , a @xmath7 mass reach of @xmath8 , @xmath9 , and @xmath10 , respectively , is possible in the three channels from the pure drell - yan @xmath11 production , while the cascade - decay - enhanced @xmath0 production can push the mass limit further to @xmath12 , @xmath13 , and @xmath14 .
the neutral scalars in the negative scenario are thus accessible at lhc run ii . |
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jones ( 1976 ) studied the propagation of light in a moving dielectric and showed by experiment that a rotating medium induces a rotation of the polarisation of the transmitted light . player ( 1976 ) confirmed that this observation could be accounted for through an application of maxwells equations in a moving medium . more recently padgett _ et al . _ ( 2006 ) reasoned that the rotation of the medium turns a transmitted image by the same angle as the polarisation . this is in contrast to the faraday effect ( faraday 1846 ) , where a static magnetic field in a dielectric medium , parallel to the propagation of light , causes a rotation of the polarisation but not a rotation of a transmitted image . rotation of the plane of polarisation and image rotation in a rotating medium may be attributed respectively to the spin and orbital angular momentum of light ( allen _ et al . _ 1999 , 2003 ) . the first theoretical treatment of this problem was published by fermi ( 1923 ) , who considered plane waves and a non - dispersive medium . the theoretical analysis of player ( 1976 ) was also restricted to the propagation of plane waves , but took the dispersion of the medium into account . player assumed that the dielectric response does not depend on the motion of the medium . in our treatment we follow his assumption although a more careful analysis by nienhuis _ et al . _ ( 1992 ) showed that there will be an effect of the motion on the refractive index for a dispersive medium near to an absorption resonance ( see also baranova & zeldovich ( 1979 ) for a discussion on the effect of the coriolis force on the refractive index ) . in contrast to player we allow for more general electromagnetic fields that can carry orbital angular momentum ( oam ) . this leads to an additional term in our wave equation , which corresponds to a fresnel drag term familiar from analysis of uniform motion . for a rotating medium , however , this drag leads to a rotational shift of the image . the propagation of light in a rotating medium thus involves both spin angular momentum ( sam ) and oam . we solve the wave equation for circularly polarised bessel beams and consider two different superpositions of such bessel beams to quantify the effects of both polarisation and image rotation . for rotation of the polarisation we examine a superposition of left- and right - circularly polarised bessel beams carrying the same amount of oam . for image rotation we consider a superposition of bessel beams with the same circular polarisation but opposite oam values . such a superposition creates an intensity pattern with lobes or ` petals ' . in both cases the constituent bessel beams propagate differently in the medium , which leads to a change in their relative phase . this is the origin of the rotation of both the polarisation and the transmitted image . for both phenomena we derive an expression for the angle per unit length of dielectric through which the image or the polarisation is rotated . the significance of the total angular momentum can be most easily seen in the wave equation for the propagation of light in a rotating medium . we derive this wave equation in section [ sec : waveeq ] . in the remaining sections we calculate the rotation of polarisation ( section [ sec : polrot ] ) and the image rotation ( section [ sec : imgrot ] ) and reveal their common form . the wave equation for a general electric displacment @xmath0 in a rigid dielectric medium rotating with angular velocity @xmath1 is given by : @xmath2 \left [ \mathbf{\omega } \times \dot{\mathbf{d } } - ( \mathbf{v } \cdot \nabla ) \dot{\mathbf{d } } \right].\ ] ] an analoguous wave equation can be derived for the magnetic induction @xmath3 . compared to the form derived by player ( 1976 ) , who considered the special case of a plane wave propagating along the direction of @xmath1 , these wave equations contain an additional term @xmath4 ( \mathbf{v } \cdot \nabla ) \dot{\mathbf{d } } $ ] . this term is responsible for the fresnel drag effect which modifies the speed of light in a moving medium ( mccrea 1954 ; barton 1999 ; rindler 2001 ) . in the following we will derive this wave equation for the electric displacement . our analysis starts with the same considerations as player ( 1976 ) , by introducing a rest frame and a moving frame . in the rest frame the dielectric medium rotates with an angular velocity @xmath5 and in the moving frame the medium is at rest . we restrict our analysis to small velocities with @xmath6 and use maxwell s equations in both reference frames ( landau & lifshitz 1975 ) . for the medium at rest we assume the following constitutive relations : [ eq : constitutive ] @xmath7 where we have used primes to denote the fields and their frequency @xmath8 in the moving frame . the fields in the moving frame can be expressed in the rest frame by a lorentz transformation ( stratton 1941 ; jackson 1998 ) , which gives to first order in @xmath9 : [ eq : transforms ] @xmath10 where we have set @xmath11 and work with units in which @xmath12 . the two constitutive relation in ( [ eq : constitutive ] ) in the rest frame are thus given by @xmath13 the dielectric constant is still given as a function of the frequency in the moving frame . we also assume that the dielectric constant depends only on the frequency and is otherwise independent of the state of motion of the medium . on combining these two equations we can express @xmath0 and @xmath3 with the two other fields @xmath14 and @xmath15 to the first order in @xmath16 : [ eq : dbfirstorder ] @xmath17 \mathbf{v } \times \mathbf{h } , \label{eq : dfirstorder}\\ \mathbf{b } & = \mathbf{h } - [ \epsilon(\omega ' ) - 1 ] \mathbf{v } \times \mathbf{e}. \label{eq : bfirstorder}\end{aligned}\ ] ] after taking the curl of ( [ eq : dfirstorder ] ) we can use the maxwell equation @xmath18 and express @xmath19 , with the help of ( [ eq : bfirstorder ] ) , in terms of @xmath20 and @xmath21 . if we assume @xmath22 to be constant ( see [ app : acceleration ] ) , as in player s paper ( player , 1976 ) this yields @xmath23 \mathbf{v } \times \dot{\mathbf{e } } + [ \epsilon(\omega ' ) -1 ] \nabla \times ( \mathbf{v } \times \mathbf{h}).\ ] ] it follows from ( [ eq : dfirstorder ] ) that @xmath24 , to the first order in @xmath22 , and so we can rewrite ( [ eq : vsteady ] ) as : @xmath25 \mathbf{v } \times \dot{\mathbf{d } } + [ \epsilon(\omega ' ) -1 ] \nabla \times ( \mathbf{v } \times \mathbf{h}).\ ] ] we can now take the curl of ( [ eq : curld ] ) to obtain a wave equation for @xmath0 , as @xmath26 for @xmath27 , and the curl of @xmath20 is given by @xmath28 . in order to express the curl of the vector products we use the identity @xmath29 , where the doubly occurring index denotes a summation over the cartesian components . the operator @xmath30 represents differentiation with respect to the @xmath31th component and acts on the whole product which gives rise to terms containing the divergences of @xmath32 and @xmath15 . these terms are either zero , because @xmath33 and @xmath27 or they lead to terms which are of second order in @xmath22 and therefore negligible . the wave equation for @xmath0 is thus given by @xmath34 \left [ ( \dot{\mathbf{d } } \cdot \nabla ) \mathbf{v } - ( \mathbf{v } \cdot \nabla ) \dot{\mathbf{d } } \right ] \\ & + [ \epsilon(\omega ' ) - 1 ] \nabla \times \left [ ( \mathbf{h } \cdot \nabla ) \mathbf{v } - ( \mathbf{v } \cdot \nabla ) \mathbf{h } \right ] . \end{split}\ ] ] for a rotation @xmath5 we can specify terms of the form @xmath35 by expressing the components of the velocity @xmath22 using the levi - civitta symbol @xmath36 as @xmath37 . the components of @xmath38 are thus given by @xmath39_i = a_l \partial_l \varepsilon_{ijk } \omega_j r_k = a_l \varepsilon_{ijk } \omega_j \delta_{lk } = \left [ \mathbf{\omega } \times \mathbf{a } \right]_i.\]]if we use the results from ( [ eq : directderiv ] ) in ( [ eq : waveeq ] ) we find for @xmath40 : @xmath41 \left [ \mathbf{\omega } \times \dot{\mathbf{d } } - ( \mathbf{v } \cdot \nabla ) \dot{\mathbf{d } } \right ] \\ & + [ \epsilon(\omega ' ) - 1 ] \nabla \times \left [ \mathbf{\omega } \times \mathbf{h } - ( \mathbf{v } \cdot \nabla ) \mathbf{h } \right ] . \end{split}\ ] ] the curl of the last bracket requires some some additional calculations . the first term is given by : @xmath42 and the second term can be written as : @xmath43 where the last term originates from @xmath44 . the terms containing the divergence of @xmath15 cancel and the term @xmath45 can be added to the second term in ( [ eq : waveeqrot ] ) . the two remaining terms @xmath46 and @xmath47 together give @xmath48 : @xmath49 this concludes the derivation of the wave equation ( [ eq : finalwaveeq ] ) . it is possible to derive the same wave equation for @xmath3 using similar methods . for a rotation around the @xmath50 axis with constant angular velocity @xmath51 , the directional derivative @xmath52 is proportional to an azimuthal derivative , as @xmath53 . this allows us to identify the two terms @xmath48 and @xmath54 in the wave equation @xmath55 \left [ \mathbf{\omega } \times \dot{\mathbf{d } } - \omega \partial_\phi \dot{\mathbf{d } } \right]\ ] ] as the polarisation rotation and rotary fresnel drag terms , respectively . player s derivation does not contain the term proportional to @xmath56 because he treated only the case of a plane wave propagating in the @xmath50-direction and for such fields @xmath0 is independent of @xmath57 . on substituting a monochromatic ansatz of the form @xmath58 into ( [ eq : phiwaveeq ] ) , where @xmath59 is the optical angular frequency in the rest frame , we obtain : @xmath60 \omega \omega \left [ { \mathrm{i}}\mathbf{e}_z \times \mathbf{d}_0 - { \mathrm{i}}\partial_\phi \mathbf{d}_0 \right].\ ] ] if we make an ansatz for @xmath61 with a general polarisation given by the complex numbers @xmath62 and @xmath63 ( with @xmath64 ) in the form of @xmath65 , we find that the @xmath66 and @xmath67 components of the wave equation ( [ eq : monowaveeq ] ) decouple if @xmath68 corresponding to left- and right - circularly polarised light respectively . if we restrict the solutions to these two cases we can write the wave equation as : @xmath69 \omega \omega \left ( \pm 1 - { \mathrm{i}}\partial_\phi \right ) \mathcal{d},\ ] ] where the plus sign refers to left - circular polarisation and the minus sign to right - circular polarisation . we can then identify @xmath70 as the extreme values of the variable @xmath71 which corresponds to the circular polarisation or sam of the light beam . similarly we can identify @xmath72 as the oam operator , so that the wave equation contains a term which depends on the total angular momentum @xmath73 : @xmath74 \omega \omega \left ( \sigma + l_z \right ) \mathcal{d}.\ ] ] we shall see that it is the dependence on the optical angular momentum that is responsible for the rotation of both the polarisation and of a transmitted image . the rotation of the polarisation arises from the difference in the refractive indices for left- and right - circularly polarised light . the angle per unit length by which the polarisation is rotated is called the specific rotary power . for an optically active medium at rest the specific rotary power is characteristic for a given material , but from ( [ eq : transversewaveeq ] ) it can be seen that light propagates differently in a rotating medium , depending on whether the circular polarisation turns in the same rotation sense as the dielectric or in the opposite sense . this phenomenon is described by the effective specific rotary power ( jones 1976 ; player 1976 ) . the specific rotary power , defined as ( fowles 1975 ) : @xmath75 is the angle of rotation of the plane of polarisation in an optical active medium . here , the indices @xmath76 and @xmath77 refer to right- and left - circularly polarised light . it was convenient to set @xmath11 for our derivation in section [ sec : waveeq ] but we reintroduce it here to facilitate the calculation of measurable quantities . in order to illustrate the effect of the oam of light we choose a bessel beam as an ansatz for the electrical displacement in the @xmath78 plane : @xmath79 where @xmath80 and @xmath81 are the transverse and longitudinal components of the wavevector . bessel beams of this form carry oam of @xmath82 per photon ( allen _ et al . _ 1992 , 1999 , 2003 ) . substituting the bessel beam ansatz in the wave equation ( [ eq : transversewaveeq ] ) yields the following result for the overall wavenumber @xmath83 : @xmath84 \frac{\omega \omega}{c^2 } ( \sigma + m).\ ] ] the indices @xmath77 and @xmath76 denoting the circular polarisation correspond respectively to @xmath85 and @xmath86 . with the help of the relations @xmath87 and @xmath88 we can turn the equation for the wavenumbers into an equation for the effective refractive indices for left- and right - circularly polarised light : @xmath89 \frac{\omega}{\omega } ( \sigma + m).\ ] ] following player ( 1976 ) we assume that @xmath90 and we can therefore approximate the square root for the refractive indices @xmath91 by a small parameter expansion to the first order in @xmath92 : @xmath93 \frac{\omega}{\omega } \left ( \sigma + m \right).\ ] ] the frequency in the moving frame @xmath8 is different for left- and right - circularly polarised light ( garetz 1981 ) and , more generally , the azimuthal or rotational doppler shift is proportional to the total angular momentum @xmath94 ( allen _ et al . _ 1994 ; bialynicki - birula & bialynicka - birula 1997 ; courtial _ et al . _ 1998 ; allen _ et al . _ 2003 ) . for left - circularly polarised light with @xmath95 the frequency is thus @xmath96 , and for right - circularly polarised light with @xmath97 the frequency changes to @xmath98 . following player ( 1976 ) we expand the refractive index of the dielectric in a taylor series to calculate the difference @xmath99 : [ eq : refindices ] @xmath100 \frac{\omega}{\omega } \left(1 + m \right ) , \\ n_r(\omega ) & \simeq n(\omega ) - \frac{d n}{d \omega } \omega ( -1+m ) - \left [ n(\omega ) - \frac{1}{n(\omega ) } \right ] \frac{\omega}{\omega } \left(-1 + m \right).\end{aligned}\ ] ] higher order derivatives of @xmath101 become comparable in magnitude if @xmath102 . this will only be case for a strongly dispersive medium , such as atomic or molecular gases , near a resonance . for such gaseous media the dielectric response in a rotating medium has to examined more closely ( nienhuis _ et al . _ 1992 ) . for solid materials , such as a rotating glass rod , and for optical frequencies this condition is not fulfilled and we can neglect higher order derivatives in the expansion ( [ eq : refindices ] ) . within player s assumption that the refractive index is independent of the motion of the medium we find for the effective specific rotary power : @xmath103 on introducing the group refractive index @xmath104 and the phase refractive index @xmath105 , we can rewrite the rotary power as @xmath106 which is identical to player s ( 1976 ) expression . in this form the specific rotary power ( [ eq : playerrotpow ] ) can be used directly with experimental data in the si unit system . in the next section we look at image rotation caused by a difference in the effective refractive indices for different values of @xmath107 . the specific rotary power describes the rotation of the propagation , but we can define , analogously , a rotary power of image rotation . the image can simply be created by the superposition of two light beams carrying different values of oam which leads to an azimuthal variation of the intensity pattern . in particular we consider an incident superposition of two similarly circularly polarised bessel beams with opposite oam values of the form @xmath108 outside the medium the superposition can be written as one bessel beam with a trigonometric modulation @xmath109 but inside the medium the effective refractive index is different for the two components of the superposition ( allen & padgett 2007 ) . on propagation this leads to phase difference which causes a rotation of the image ( see figure [ fig : petalimage ] ) . we define @xmath110 which is the angle per unit length by which the image is rotated . the factor @xmath107 in the expression for @xmath111 appears because of the @xmath112 and @xmath113 phase structure of the interfering beams and the resulting @xmath114-fold symmetry of the created image ( pagdett _ et al . _ 2006 ) . image rotation , title="fig:",scaledwidth=49.0% ] image rotation , title="fig:",scaledwidth=49.0% ] the different effective refractive indices for the components of the superposition ( [ eq : superposition ] ) are given by : @xmath115 \frac{\omega}{\omega } ( \sigma \pm m).\ ] ] here , @xmath71 is fixed in contrast to ( [ eq : refindex ] ) . the roles of @xmath71 and @xmath107 are reversed for the image rotation and the refractive indices for positive and negative oam are given by : [ eq : rotdiffrefind ] @xmath116 \frac{\omega}{\omega } \left(\sigma + m \right ) , \\ n_-(\omega ) & \simeq n(\omega ) - \frac{d n}{d \omega } \omega ( \sigma - m ) - \left [ n(\omega ) - \frac{1}{n(\omega ) } \right ] \frac{\omega}{\omega } \left(\sigma - m \right).\end{aligned}\ ] ] on substituting ( [ eq : rotdiffrefind ] ) into ( [ eq : imgrot ] ) we find : @xmath117 which can be written in terms of the group and phase refractive indices as : @xmath118 this verifies the reasoning of padgett _ et al . _ ( 2006 ) that the polarisation and the image are turned by the same amount when passing through a rotating medium . it is the total angular momentum that determines the phase shifts and a linearly polarised image will undergo rotations of both the plane of polarisation and the intensity pattern or image . we have extended a theoretical study by player ( 1976 ) on the propagation of light through a rotating medium to include general electromagnetic fields . in the original analysis player ( 1976 ) showed that the rotation of the polarisation inside a rotating medium can be understood in terms of a difference in the propagation for left- and right - circularly polarised light . player s ( 1976 ) analysis was thus concerned solely with the spin angular momentum ( sam ) of light . our treatment has shown that the general wave equation has an additional term , which is of the same form as the fresnel drag term for a uniform motion . in the context of rotating motion , however , this term is connected to the orbital angular momentum ( oam ) of the light . by extending the theoretical analysis to include oam we have been able to attribute polarisation rotation and image rotation to sam and oam respectively we have shown that a superposition of bessel beams with the same oam but opposite sam states leads to the rotation of the polarisation , whereas a superposition of bessel beams with the same sam and opposite oam values gives rise to a rotation of the transmitted image . we have obtained quantitative expressions for the rotation of the polarisation and of the transmitted image and have verified that both are turned through the same angle , as recently suggested by padgett _ ( 2006 ) . player ( 1976 ) remarked that the derivation by fermi ( 1923 ) appears to be in error . the mistake in fermi s treatment seems to be in missing the transformation of the magnetic fields . whereas the change in the electric fields induced by the motion of the medium is explicitly given in terms of the electric polarisation @xmath119 , a similar transformation for the magnetic field is missing . in terms of our derivation this would mean that ( [ eq : bfirstorder ] ) changes to @xmath120 in the rest frame . this in turn causes that the term @xmath121 would be missing in ( [ eq : vsteady ] ) . this term and the term @xmath122 contribute equally to the wave equation ( [ eq : finalwaveeq ] ) , which explains why fermi s result for the specific rotary power is smaller than player s and ours by a factor of two . as pointed out by player ( 1976 ) this missing factor is cancelled by an additional factor of two in fermi s definition of the specific rotary power . the assumption that @xmath5 is steady is problematic for a rotating motion ; if we assume @xmath123 to be constant over time , then @xmath124 . in principle this would invalidate our initial considerations for the transformation of the electromagnetic fields ( [ eq : transforms ] ) which strictly hold only for uniform motion . including the time - derivative of @xmath22 would lead to additional terms in ( [ eq : vsteady ] ) of the form @xmath125 \dot{\mathbf{v } } \times \mathbf{e}$ ] . if we proceed in taking the curl of this vector product we produce four terms which either can be neglected because they are second order in @xmath9 , or they do not contain the time derivative of an optical field . the latter are smaller than terms that do contain a time derivative by @xmath126 . for our assumption @xmath90 all such terms are negligible . fermi , e. 1923 sul trascinamento del piano di polarizzazione da parte di un mezzo rotante . lincei _ * 32 * , 115118 . reprinted in : fermi , e. 1962 _ collected papers _ , vol . chicago : university of chicago press . padgett m. , whyte g. , girkin j. , wright a. , allen l. , hberg p. & barnett s. m. 2006 polarization and image rotation induced by a rotating dielectric rod : an optical angular momentum interpretation . _ optics lett . _ * 31 * ( 14 ) , 22052207 . | image rotation , polarisation , rotating dielectric , specific rotary power when light is passing through a rotating medium the optical polarisation is rotated .
recently it has been reasoned that this rotation applies also to the transmitted image ( padgett _ et al .
_ 2006 ) .
we examine these two phenomena by extending an analysis of player ( 1976 ) to general electromagnetic fields .
we find that in this more general case the wave equation inside the rotating medium has to be amended by a term which is connected to the orbital angular momentum of the light .
we show that optical spin and orbital angular momentum account respectively for the rotation of the polarisation and the rotation of the transmitted image .
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determining how well a particular graph is connected is a problem that arises often in various applications . for example , as defined in @xcite , _ resilience _ is the ability of a network to provide and maintain an acceptable level of service in the face of various faults and challenges to normal operation . examples of such networks include telecommunication networks , electrical grids , and commodity supply networks . one of the scientific disciplines that forms the basis of resilience is _ disruption tolerance _ the ability of a system to tolerate disruptions in connectivity among its components ; such disruptions could consist of environmental challenges such as weak and episodic channel connectivity , mobility , and unpredictably - long delay , as well as tolerance of energy challenges @xcite . the most common way to interpret such networks is as connected graphs , where the resilience of a network as a function of its disruption tolerance can be studied by analyzing the connectivity properties of the associated graph . graph connectivity has also been shown to play an important role in data security ; see @xcite and the references therein for more information . in 1973 , fiedler related the vertex - connectivity of a graph to the second - smallest eigenvalue of its laplacian matrix in the following way ( see section [ preliminaries ] for relevant definitions ) : [ thm1.1 ] @xcite if @xmath0 is a simple , non - complete graph , then @xmath1 . note that theorem [ thm1.1 ] also implies @xmath2 , since @xmath3 . this seminal result provided researchers with another parameter that quantitatively measures the connectivity of a graph ; hence , @xmath4 is known as the _ algebraic connectivity _ of @xmath0 . fiedler s discovery ignited interest in studying the connectivity of graphs by analyzing the spectral properties of their associated matrices . akin to other connectivity measures such as vertex - connectivity , edge - connectivity , and isoperimetric number , the algebraic connectivity of a graph has applications in the design of reliable communication networks @xcite and in analyzing the robustness of complex networks @xcite . recall that for a @xmath5-regular multigraph @xmath0 on @xmath6 vertices , @xmath7 for @xmath8 . thus , for regular multigraphs , spectral bounds related to connectivity are often expressed in terms of the second - largest eigenvalue , instead of the second - smallest laplacian eigenvalue . we now discuss several results relating the vertex- and edge - connectivity of graphs to their second - largest eigenvalues , which are similar in nature to the main results of the present paper . the following result of chandran @xcite relates the second - largest eigenvalue of an @xmath6-vertex @xmath5-regular graph to its edge - connectivity . @xcite let @xmath0 be an @xmath6-vertex @xmath5-regular simple graph with @xmath9 . then @xmath10 . in 2006 , krivelevich and sudakov improved the above result as follows . let @xmath0 be a @xmath5-regular simple graph with @xmath11 . then @xmath10 . the next result was shown in 2010 by ciob @xcite . [ cioba ] @xcite let @xmath12 be a nonnegative integer less than @xmath5 , and let @xmath0 be a @xmath5-regular , simple graph with @xmath13 . then @xmath14 . in the same paper , ciob also gave improvements of theorem [ cioba ] for the following two particular cases . [ cioba improvement for t=1 ] @xcite let @xmath15 be an odd integer and let @xmath16 denote the largest root of @xmath17 . if @xmath0 is a @xmath5-regular , simple graph such that @xmath18 , then @xmath19 . the value of @xmath16 above is approximately @xmath20 . for @xmath21 , it is still an open problem to find the best upper bound for @xmath22 in a @xmath5-regular simple graph @xmath0 to guarantee that @xmath23 . [ cioba improvement for t=2 ] @xcite let @xmath15 be any integer . let @xmath0 be a @xmath5-regular , simple graph with @xmath24 then @xmath25 . the value of @xmath26 above is approximately @xmath27 . in 2016 , o @xcite generalized fiedler s result to multigraphs , and established similar bounds as those above . [ o , bound 1 for multigraphs ] @xcite let @xmath0 be a connected , @xmath5-regular multigraph with @xmath28 then @xmath19 . [ o , bound 2 for multigraphs ] @xcite let @xmath29 and let @xmath0 be a connected , @xmath5-regular multigraph . if @xmath30 , then @xmath14 . furthermore , if @xmath12 is odd and @xmath31 , then @xmath14 . moreover , for every positive integer @xmath12 less than @xmath5 , o @xcite found the best upper bound for @xmath22 to guarantee that @xmath32 . note that the bounds for @xmath22 in theorems [ cioba improvement for t=1 ] and [ cioba improvement for t=2 ] are larger than those in theorems [ o , bound 1 for multigraphs ] and [ o , bound 2 for multigraphs ] ; this suggests that if multiple edges are not allowed , then a bound for @xmath22 that will guarantee a desired edge - connectivity may be larger . hence , we see that a @xmath5-regular multigraph @xmath0 may have high edge - connectivity while @xmath22 is small ; this supports the implicit meaning of cheeger s inequality ( see @xcite ) for a @xmath5-regular multigraph @xmath0 , which asserts that @xmath33 where @xmath34}{|s|}.\ ] ] the cheeger constant @xmath35 ( also known as the isoperimetric number ) can be considered as an approximation to the edge - connectivity of @xmath0 . the results above make assertions about the edge - connectivity of a graph based on its eigenvalues . in a more recent paper , o @xcite also established analogous results for vertex - connectivity . @xcite let @xmath0 be a @xmath5-regular multigraph that is not the @xmath36-vertex @xmath5-regular multigraph . if @xmath37 , then @xmath38 . algebraic connectivity has also been studied in the context of hypergraphs @xcite and directed graphs @xcite . see @xcite and the bibliographies therein for other recent results on algebraic connectivity , more general structural results on vertex- and edge - connectivity , and other algebraic measures of connectivity . the aim of the present paper is to investigate what upper bounds on the second - largest eigenvalues of regular simple graphs and multigraphs guarantee a desired vertex- or edge - connectivity . in other words , we address the following question : for a @xmath5-regular simple graph or multigraph @xmath0 and for @xmath39 , what is the best upper bound for @xmath22 which guarantees that @xmath14 or that @xmath40 ? the majority of the related results listed above were derived using a variety of combinatorial , linear algebraic , and analytic techniques ; moreover , they feature upper bounds for @xmath22 which do not depend on the order of the graph . in contrast , the results derived in the present paper feature bounds for @xmath22 which depend on both the degree and the order of the graphs , and as such are tight for infinite families of graphs . furthermore , the derivations of these results combine analytic techniques with computer - aided _ symbolic _ algebra ; this proves to be a powerful approach , easily establishing the desired results in all but finitely - many cases . the remaining cases are verified through a brute - force approach which relies on enumerating all multigraphs with certain properties . in order to avoid enumeration and post - hoc elimination of the exponential number of multigraphs without the desired properties , our approach required the development of novel combinatorial and graph theoretic techniques . while the problem of generating all non - isomorphic simple graphs having a certain degree sequence and other properties is well - studied ( cf . @xcite ) , there are not as many efficiently - implemented algorithms for constrained enumeration of multigraphs ( see @xcite for some results in this direction ) . thus , the developed enumeration procedure may also be of independent interest . the paper is organized as follows . in the next section , we recall some graph theoretic and linear algebraic notions , specifically those related to eigenvalue interlacing . in sections 3 and 4 , respectively , we give spectral bounds which guarantee a certain vertex- and edge - connectivity . we conclude with some final remarks and open questions in section 5 . the appendix includes further details and computer code for symbolic computations used in some of the proofs . in this paper , a _ multigraph _ refers to a graph with multiple edges but no loops ; a _ simple graph _ refers to a graph with no multiple edges or loops . the _ order _ and _ size _ of a multigraph @xmath0 are denoted by @xmath41 and @xmath42 , respectively . a _ double edge _ ( respectively _ triple edge _ ) in a multigraph is an edge of multiplicity two ( respectively three ) . the _ degree _ of a vertex @xmath43 of @xmath0 , denoted @xmath44 , is the number of edges incident to @xmath43 . the _ degree sequence _ of @xmath0 is a list @xmath45 of the vertex degrees of @xmath0 . we may abbreviate the degree sequence of @xmath0 by only writing distinct degrees , with the number of vertices realizing each degree in superscript . for example , if @xmath0 is the star graph on @xmath6 vertices , the degree sequence of @xmath0 may be written as @xmath46 . a _ vertex cut _ ( respectively _ edge cut _ ) of @xmath0 is a set of vertices ( respectively edges ) which , when removed , increases the number of connected components in @xmath0 . a multigraph @xmath0 with more than @xmath47 vertices is said to be _ @xmath47-vertex - connected _ if there is no vertex cut of size @xmath48 . the _ vertex - connectivity _ of @xmath0 , denoted @xmath49 , is the maximum @xmath47 such that @xmath0 is @xmath47-vertex - connected . similarly , @xmath0 is _ @xmath47-edge - connected _ if there is no edge cut of size @xmath48 ; the _ edge - connectivity _ of @xmath0 , denoted @xmath50 , is the maximum @xmath47 such that @xmath0 is @xmath47-edge - connected . cut - vertex _ ( respectively _ cut - edge _ ) is a vertex cut ( respectively edge cut ) of size one . given sets @xmath51 , @xmath52 $ ] denotes the number of edges with one endpoint in @xmath53 and the other in @xmath54 . the _ induced subgraph _ $ ] is the subgraph of @xmath0 whose vertex set is @xmath53 and whose edge set consists of all edges of @xmath0 which have both endpoints in @xmath53 . a _ matching _ is a set of edges of @xmath0 which have no common endpoints ; a @xmath47-matching is a matching containing @xmath47 edges . @xmath56 denotes the graph @xmath57 , and @xmath58 denotes the graph @xmath59 . the _ complete graph _ on @xmath6 vertices is denoted @xmath60 . an _ odd path _ ( respectively _ even path _ ) in a graph is a connected component which is a path with an odd ( respectively even ) number of vertices . for other graph theoretic terminology and definitions , we refer the reader to @xcite . the _ adjacency matrix _ of @xmath0 will be denoted by @xmath61 ; recall that in a multigraph , the entry @xmath62 is the number of edges between vertices @xmath63 and @xmath64 . the _ eigenvalues _ of @xmath0 are the eigenvalues of its adjacency matrix , and are denoted by @xmath65 . the _ laplacian matrix _ of @xmath0 is equal to @xmath66 , where @xmath67 is the diagonal matrix whose entry @xmath68 is the degree of vertex @xmath63 . the _ laplacian eigenvalues _ of @xmath0 are the eigenvalues of its laplacian matrix and are denoted by @xmath69 . the dependence of these parameters on @xmath0 may be omitted when it is clear from the context . a technical tool used in this paper is _ eigenvalue interlacing _ ( for more details see section 2.5 of @xcite ) . given two sequences of real numbers @xmath70 and @xmath71 with @xmath72 , we say that the second sequence _ interlaces _ the first sequence whenever @xmath73 for @xmath74 . if @xmath75 is a real symmetric @xmath76 matrix and @xmath77 is a principal submatrix of @xmath75 of order @xmath78 with @xmath72 , then for @xmath79 , @xmath80 , i.e. , the eigenvalues of @xmath77 interlace the eigenvalues of @xmath75 . let @xmath81 be a partition of the vertex set of a multigraph @xmath0 into @xmath82 non - empty subsets . the _ quotient matrix _ @xmath83 corresponding to @xmath84 is the @xmath85 matrix whose entry @xmath86 ( @xmath87 ) is the average number of incident edges in @xmath88 of the vertices in @xmath89 . more precisely , @xmath90}{|v_i|}$ ] if @xmath91 , and @xmath92)|}{|v_i|}$ ] . note that for a simple graph , @xmath86 is just the average number of neighbors between vertices in @xmath88 and vertices in @xmath89 . [ cor ] the eigenvalues of any quotient matrix @xmath83 interlace the eigenvalues of @xmath0 . in this section , we establish an upper bound for the second - largest eigenvalue of an @xmath6-vertex @xmath5-regular simple graph or multigraph which guarantees a certain vertex - connectivity . to our knowledge , this is the first spectral bound on the vertex - connectivity of a regular graph which depends on both the degree and the order of the graph . [ theorem1 ] let @xmath0 be an @xmath6-vertex @xmath5-regular simple graph or multigraph , which is not obtained by duplicating edges in a complete graph on at most @xmath94 vertices ; let @xmath95 where @xmath96 . if @xmath97 , then @xmath98 . assume to the contrary that @xmath99 . if @xmath0 is disconnected , then @xmath100 , a contradiction . now , assume that @xmath101 . hence , there exists a vertex cut @xmath102 of @xmath0 with @xmath103 . let @xmath104 be a union of some components of @xmath105 such that @xmath106=[c,\overline{s } ] \le \frac{cd}{2 } \le \frac{td}{2}$ ] , where @xmath107 and @xmath108 . see figure [ fig_ptd ] for an illustration of this partition . into @xmath109 and @xmath110 . ] let @xmath111=p$ ] , and @xmath112 ; then , we have @xmath113=d(s_1+c)-p$ ] , and @xmath114=d(n - s_1-c)-p$ ] , so the quotient matrix for the partition @xmath115 is @xmath116 and the characteristic polynomial of @xmath83 with respect to @xmath117 is @xmath118 . then by corollary [ cor ] , we have @xmath119 we now consider two cases based on whether @xmath0 is a simple graph or a multigraph . _ _ @xmath0 is a simple graph . if @xmath120 , since the degree of each vertex in @xmath104 is @xmath5 , it holds that @xmath121 . if @xmath122 , @xmath123 $ ] is a complete subgraph of @xmath0 , so the vertex in @xmath102 has degree greater than @xmath5 because @xmath124 ; this is a contradiction . thus @xmath125 , @xmath126 , and so @xmath127 , as desired . if @xmath128 , by the same argument as above , we must have @xmath129 , @xmath130 , and so @xmath131 , as desired . _ @xmath0 is a multigraph . if @xmath120 , then @xmath132 , @xmath133 , and so @xmath134 , as desired . if @xmath128 , then @xmath135 , @xmath136 , and so @xmath137 . consider the function @xmath138 ; then , @xmath139<0 $ ] . thus , @xmath140 is decreasing with respect to @xmath141 for @xmath142 , whence it follows that @xmath143 , as desired . we now improve the result of theorem [ theorem1 ] in the case when @xmath0 is a multigraph and @xmath120 . recall that in this case , theorem [ theorem1 ] states that if @xmath145 , then @xmath144 . [ theo23 ] let @xmath0 be an @xmath6-vertex @xmath5-regular multigraph with @xmath146 and @xmath147 . if @xmath148 , then @xmath144 . assume to the contrary that @xmath149 . if @xmath150 , then @xmath151 , a contradiction . thus , we can assume henceforth that @xmath152 . let @xmath43 be a cut - vertex of @xmath0 , and @xmath104 and @xmath153 be two components of @xmath154 with @xmath112 and @xmath155 . let @xmath156 $ ] and @xmath157 $ ] ; without loss of generality , we can assume that @xmath158 , and hence that @xmath159 ( otherwise the roles of @xmath104 and @xmath153 can be reversed ) ; note that since @xmath147 , we must have @xmath160 ; moreover , @xmath161 . see figure [ fig2 ] for an illustration of this partition in the case when @xmath162 . into @xmath104 , @xmath163 and @xmath153 , when @xmath164 . ] the quotient matrix for the partition @xmath165 is @xmath166 and its characteristic polynomial with respect to @xmath117 is @xmath167.\ ] ] then by corollary [ cor ] , we have @xmath168 , where @xmath169 is the second - largest root of the characteristic polynomial of @xmath83 ; it can be verified that @xmath169 can be expressed as follows : @xmath170.\ ] ] if we set the derivative of @xmath169 with respect to @xmath171 equal to zero and solve for @xmath171 , we obtain @xmath172 substituting @xmath173 for @xmath174 , and the right hand side of ( [ eq2 ] ) for @xmath171 in ( [ eq1 ] ) , and simplifying , we obtain @xmath175 finally , when we substitute @xmath176 for @xmath177 , the resulting expression has a minimum at @xmath162 , for @xmath146 , @xmath147 , and @xmath160 , with minimal value @xmath178 . this minimization and some of the algebraic manipulations described above were carried out using symbolic computation in mathematica ; for details , see the appendix . let @xmath0 be a multigraph with the following adjacency matrix : @xmath179 then @xmath180 . moreover , @xmath0 is a @xmath5-regular multigraph with 5 vertices , @xmath181 , @xmath182 , and @xmath152 . thus , the bound in theorem [ theo23 ] is the best possible for this infinite family of multigraphs . @xmath184 in this section , we first give an upper bound for @xmath22 in an @xmath6-vertex @xmath5-regular multigraph which guarantees that @xmath185 ; its proof is omitted , since it is similar to that of theorem [ theorem1 ] . theorem [ theorem4 ] extends a result of ciob @xcite to multigraphs . [ theorem4 ] let @xmath0 be an @xmath6-vertex @xmath5-regular multigraph , which is not obtained by duplicating edges in a complete graph on at most @xmath94 vertices . let @xmath186 where @xmath96 . if @xmath187 , then @xmath185 . now , we will improve the bound in theorem [ theorem4 ] for the case of @xmath120 ; see observation [ obs_improvement ] for an explanation of why theorem [ theo31 ] is an improvement . [ theo31 ] let @xmath0 be an @xmath6-vertex @xmath5-regular multigraph with @xmath188 , where @xmath189 is the second - largest eigenvalue of the following matrix : @xmath190 then @xmath191 . assume to the contrary that @xmath192 . if @xmath193 , then since the largest eigenvalue of @xmath83 equals @xmath5 , we have that @xmath194 , a contradiction . now , assume that @xmath195 . for any graph @xmath196 , define @xmath197 to be the number of vertices in the smallest connected component of @xmath196 . let @xmath198 be a cut - edge of @xmath0 such that @xmath199 . in other words , @xmath200 is a cut - edge such that one of the components of @xmath201 has minimum size among all subgraphs of @xmath0 which can be separated by removing a cut - edge of @xmath0 . let @xmath202 and @xmath203 be the two components of @xmath201 , where @xmath204 , @xmath205 , and @xmath206 . for @xmath207 , let @xmath208 and @xmath209 . by the degree - sum formula , @xmath210 , whence it follows that @xmath211 is odd . thus , both @xmath5 and @xmath212 are odd , and hence @xmath6 is even ; moreover , @xmath213 , and hence @xmath214 . see figure [ partition ] for an illustration . -regular multigraph with @xmath195 . ] we now consider three cases based on the cardinality of @xmath215 . case 1 : @xmath216 . : : in this case , the structure of the graph is determined uniquely , and the vertex partition @xmath217 corresponds to the quotient matrix @xmath83 defined in the statement of the theorem ; see figure [ case_n3 ] for an illustration . therefore , the inequality @xmath218 holds for all @xmath5 and @xmath6 . + [ case_n3 ] + -regular multigraph with @xmath195 and @xmath162 . ] case 2 : @xmath219 . : : consider the partition @xmath217 and the corresponding quotient matrix : + @xmath220 + let @xmath221 . by corollary [ cor ] , @xmath222 . note that @xmath5 is odd , and that due to the partition structure , @xmath223 . thus , to show that @xmath224 holds for all @xmath5 and @xmath6 , we will show that @xmath224 holds when @xmath225 and @xmath226 , and that + @xmath227 + holds for all other values of @xmath5 and @xmath6 . to verify that @xmath224 holds when @xmath225 and @xmath226 , we compute the second - largest eigenvalues of all possible multigraphs which have these parameters , and compare them to @xmath228 and @xmath229 , respectively ; the enumeration procedure is described in the appendix . for all other values of @xmath5 and @xmath6 , we verify ( [ ineq_rho_pp ] ) by separating it into the following cases and using symbolic computation in mathematica ; see the appendix for details . see also case 3 below for a more detailed explanation of why this computation is sufficient to establish the claim . + 1 . @xmath225 , @xmath230 . fix @xmath225 and @xmath231 . then , @xmath232 and @xmath233 hold for all @xmath230 . 2 . @xmath234 , @xmath226 . fix @xmath234 and @xmath235 . then , @xmath232 and @xmath233 hold for @xmath236 and @xmath237 . @xmath238 , @xmath236 . fix @xmath238 and @xmath239 . then , @xmath232 and @xmath233 hold for @xmath236 . @xmath234 , @xmath230 ; @xmath238 , @xmath240 ; @xmath241 , @xmath223 . fix @xmath242 . then , @xmath232 and @xmath233 hold for all values of @xmath5 and @xmath6 described in this case . case 3 : @xmath243 . : : in this case , we consider the vertex partition of @xmath0 with the sets @xmath244 and @xmath245 ; see figure [ case_n7 ] for an illustration . + into @xmath244 and @xmath245 . ] + the second - largest eigenvalue of the quotient matrix @xmath246 corresponding to this vertex partition is equal to @xmath247 . by corollary [ cor ] , @xmath248 , where the last inequality follows from the fact that @xmath249 . note that @xmath6 is even , @xmath5 is odd , @xmath147 , and due to the partition structure , @xmath230 . thus , to show that @xmath224 holds for all @xmath5 and @xmath6 , we will show that @xmath224 holds when @xmath225 and @xmath250 , and that + @xmath251 + holds for all other values of @xmath5 and @xmath6 . to verify that @xmath224 holds when @xmath225 and @xmath250 , we compute the second - largest eigenvalues of all possible multigraphs which have these parameters , and compare them to @xmath252 , @xmath253 , and @xmath253 , respectively ; the enumeration procedure is described in the appendix . for all other values of @xmath5 and @xmath6 , we verify ( [ ineq_rho_7 ] ) as follows . + note that @xmath254 is a monic polynomial of degree 4 , with roots @xmath255 , @xmath169 , @xmath256 , and @xmath257 ; all roots are real , since they interlace the eigenvalues of @xmath0 . moreover , @xmath258 and @xmath259 , which implies that @xmath260 . by theorem [ theorem4 ] , @xmath261 ; thus , @xmath262 for all @xmath147 and @xmath230 . since @xmath263 , it follows that @xmath264 . finally , note that @xmath265 thus , showing that ( [ ineq_rho_7 ] ) holds is equivalent to showing that + * @xmath266 for @xmath267 , and * @xmath233 for @xmath268 , + whence it follows that @xmath269 . using symbolic computation in mathematica , we can verify that a ) holds for all @xmath147 and @xmath230 , and b ) holds when @xmath225 and @xmath270 , and when @xmath271 and @xmath230 ; for details , see the appendix . since the case @xmath225 , @xmath272 was verified by enumeration , this completes the proof . [ obs_improvement ] when @xmath120 , theorem [ theorem4 ] states that if @xmath273 , then @xmath191 . case 3 of the proof of theorem [ theo31 ] guarantees that @xmath274 , which means that @xmath189 is a better bound than @xmath275 . let @xmath0 be the @xmath5-regular multigraph on 6 vertices with @xmath147 and @xmath195 . then @xmath276 , where @xmath189 is defined as in the statement of theorem [ theo31 ] . thus , the bound in theorem [ theo31 ] is the best possible for this infinite family of multigraphs . in this paper , we presented upper bounds for the second - largest eigenvalues of regular graphs and multigraphs , which guarantee a desired vertex- or edge - connectivity . the given bounds improve on several previous results , and hold with equality for infinite families of graphs . in deriving these bounds , we used computer - aided symbolic algebra , which synergizes well with the technique of eigenvalue interlacing ; this combination gives a viable approach to investigating spectral bounds guaranteeing graph theoretic properties , which differs from the typical analytic strategies used in similar derivations . as part of our proof of theorem [ theo31 ] , we developed an approach to enumerate all multigraphs with certain properties , by adding edges to certain simple graphs . since the theory and available software for enumerating simple graphs are generally better - developed than their analogues for multigraphs , it may be worth investigating this strategy further and adapting it to other applications . another problem of interest is to obtain bounds on the second - largest eigenvalues of a graph which guarantee a desired connectivity , and depend on other graph invariants such as girth or circuit rank . in particular , one could explore the utility of symbolic computer algebra used together with eigenvalue interlation in deriving such bounds . @xmath184 the authors gratefully acknowledge financial support for this research from the following grants and organizations : nsf - dms grants 1604458 , 1604773 , 1604697 and 1603823 ( all authors ) , the combinatorics foundation ( a. abiad , s. o ) , nsf 1450681 ( b. brimkov ) , institute of mathematics and its applications ( x. martnez - rivera ) , krf - msip grant 2016r1a5a1008055 ( s. o ) . xx n.m.m . de abreu , old and new results on algebraic connectivity of graphs . _ linear algebra and its applications _ , 423 : 5373 ( 2007 ) . brouwer and w.h . haemers , spectra of graphs , springer , new york , 2011 . chandran , minimum cuts , girth and spectral threshold . _ information processing letters _ , 89(3 ) : 105110 ( 2004 ) . j. chen and i. safro , algebraic distance on graphs . _ siam journal on scientific computing _ , 33(6 ) : 34683490 ( 2011 ) . cioaba , eigenvalues and edge - 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linear functions of gaussian random variables . _ stochastic processes and their applications _ , 13(2 ) : 121138 ( 1982 ) . west , _ introduction to graph theory _ , prentice hall , inc . , upper saddle river , nj , 2001 . wu , algebraic connectivity of directed graphs . _ linear and multilinear algebra _ , 53(3 ) : 203223 ( 2005 ) . below we provide the mathematica code used to calculate the minimum of the second root of the characteristic polynomial in the proof of theorem [ theo23 ] , and to check some cases in the proof of theorem [ theo31 ] . the version of mathematica used is 10.0.0.0 for 64-bit microsoft windows . we also include additional details about the procedure of enumerating certain multigraphs in the proof of theorem [ theo31 ] . case 2d : note that both inequalities hold for @xmath234 and @xmath230 , @xmath238 and @xmath240 , and @xmath241 and @xmath223 . + @xmath292;x = d-1/5 - 1/(n-5);}\\ \pmb{\text{reduce}[\text{polyq}<0\&\&n\geq 10\&\&d\geq 3,\text{integers}]}\\ \pmb{\text{reduce}[\text{polyqprim}>0\&\&n\geq 10\&\&d\geq 3,\text{integers}]}$ ] case 3 : note that both inequalities hold for @xmath225 and @xmath270 , and @xmath271 and @xmath230 . the case @xmath225 , @xmath272 is verified by enumeration in the next section . + @xmath295}\\ \pmb{\text{clear}[x];x = d-1/3 - 1/(n-3);}\\ \pmb{\text{reduce}[\text{polyq}>0\&\&n\geq 14\&\&d\geq 3,\text{integers}]}$ ] let @xmath298 and @xmath299 respectively be the sets of 3-regular multigraphs of order @xmath300 and @xmath301 with edge - connectivity 1 , such that the removal of any cut - edge of these graphs produces components of order at least 5 . let @xmath302 , @xmath303 , and @xmath304 respectively be the sets of 3-regular multigraphs of order @xmath305 , @xmath306 , and @xmath307 with edge - connectivity 1 , such that the removal of any cut - edge of these graphs produces components of order at least 7 . these constraints imply that a graph in @xmath298 or @xmath302 must have exactly one cut - edge , a graph in @xmath299 or @xmath303 can have one or two cut - edges , and a graph in @xmath304 can have one , two , or three cut - edges . for @xmath308 , let @xmath309 be the set of all connected multigraphs which have degree sequence @xmath310 and have no cut - edges . for any graph @xmath311 , @xmath308 , define @xmath312 to be the degree 2 vertex of @xmath196 . let @xmath313 be the graph consisting of two vertices joined by a double edge , let @xmath314 be the graph obtained by joining two copies of @xmath313 by one edge , and let @xmath315 be a complete graph on four vertices with one edge removed . for @xmath316 , define @xmath317 to be one of the degree 2 vertices of @xmath318 , and @xmath319 to be the other degree 2 vertex of @xmath318 . for any @xmath320 , define @xmath321 to be the set @xmath322 ( where @xmath323 denotes disjoint union ) . for any @xmath320 and @xmath316 , define @xmath324 to be the set @xmath325 . in other words , `` @xmath326 '' denotes the set obtained by joining all possible pairs of graphs from the indicated families by a cut - edge incident to their degree 2 vertices . with this in mind , it is easy to see that @xmath327 thus , to find the graphs in @xmath328 , @xmath329 , it suffices to find the graphs in @xmath330 , @xmath331 . since the graphs in @xmath330 are 3-regular and connected , they can not have triple edges ; moreover , they can have at most @xmath332 double edges . let @xmath333 be the set of multigraphs in @xmath330 which have @xmath334 double edges . then , @xmath335 . we will now describe a procedure for enumerating the graphs in @xmath333 . if the double edges of the graphs in @xmath333 are replaced by single edges , the resulting graphs will be simple , 2-vertex - connected , and have degree sequence @xmath336 . there are well - known algorithms for generating all nonisomorphic simple graphs with a given degree sequence ( cf . @xcite ) ; a practical algorithm is implemented in the software system sagemath . let @xmath337 be the set of nonisomorphic simple graphs with degree sequence @xmath336 . then , by adding double edges in all feasible ways to the simple graphs in @xmath337 , we can recover the multigraphs in @xmath333 . specifically , a double edge can be added to a graph in @xmath337 only where a single edge with two degree 2 endpoints already exists . moreover , not every graph in @xmath337 can have @xmath334 double edges added to it in a way that the resulting multigraph is in @xmath333 ; similarly , it may be possible to add @xmath334 double edges to a graph in @xmath337 in multiple ways so that the resulting multigraphs are in @xmath333 . let @xmath196 be a graph in @xmath337 and let @xmath338 be the subgraph induced by the degree 2 vertices of @xmath196 . since the maximum degree of @xmath338 is 2 , @xmath338 is the disjoint union of some paths and cycles . however , if @xmath338 contains a cycle with less than @xmath339 vertices , a multigraph in @xmath333 can not be obtained by doubling single edges of @xmath196 with two degree 2 endpoints ( since any resulting multigraph with degree sequence @xmath340 will be disconnected ) . similarly , if @xmath338 contains more than one odd path , a multigraph in @xmath333 can not be obtained by doubling single edges of @xmath196 with two degree 2 endpoints ( since any resulting multigraph with degree sequence @xmath340 will not have @xmath334 multiple edges ) . thus , let @xmath341 is either a cycle @xmath342 , or contains exactly one odd path@xmath343 . for any graph @xmath196 in @xmath344 , the different maximum matchings ( i.e. @xmath334-matchings ) of @xmath338 correspond to different ways to add double edges to @xmath196 . let @xmath345 be the set of multigraphs obtained by adding double edges to @xmath196 corresponding to the different @xmath334-matchings of @xmath338 . then , @xmath346 , @xmath347 , and @xmath328 can be obtained by joining pairs of graphs in @xmath330 as described earlier . note that the set of distinct maximum matchings of a graph whose components are paths ( one of which is odd ) can be found in linear time . see figure [ enumeration2 ] for an illustration of this enumeration for @xmath348 ; the other sets of multigraphs @xmath333 are handled analogously , and combined to obtain the graphs in @xmath328 . finally , for each multigraph in @xmath328 , we can easily compute and compare the second - largest eigenvalue to @xmath349 ; we have found that all of these eigenvalues are greater than or equal to @xmath349 , as desired . . _ top row _ : the graphs in @xmath350 ; the three graphs on the right are not 2-vertex - connected , so they are not considered further . _ second row _ : @xmath338 for the remaining graphs @xmath196 ; the graph on the left has multiple odd paths , so it is not considered further . _ third row _ : all possible 2-matchings of the remaining graphs in the second row . _ bottom row _ : adding double edges specified by the matchings to obtain the graphs in @xmath348 ; the two matchings of the graph on the right happen to result in isomorphic multigraphs . ] | the second - largest eigenvalue and second - smallest laplacian eigenvalue of a graph are measures of the graph s connectivity .
these parameters can be used to analyze the robustness , resilience , and synchronizability of networks , and are related to other connectivity attributes such as the vertex- and edge - connectivity , isoperimetric number , and characteristic path length . in this paper
, we give upper bounds for the second - largest eigenvalues of regular graphs and multigraphs which guarantee a desired vertex- or edge - connectivity .
the given bounds are in terms of the order and degree of the graphs , and hold with equality for infinite families of graphs .
second - largest eigenvalue ; vertex - connectivity ; edge - connectivity ; regular multigraph ; algebraic connectivity .
+ 05c50 , 05c40 . |
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theory of measurements has a special status in quantum mechanics . unlike classical mechanics , in quantum mechanics it can not be assumed that the effect of the measurement on the system can be made arbitrarily small . it is necessary to supplement quantum theory with additional postulates , describing the measurement . one of such additional postulate is von neumann s state reduction ( or projection ) postulate @xcite . the essential peculiarity of this postulate is its nonunitary character . however , this postulate refers only to an ideal measurement , which is instantaneous and arbitrarily accurate . real measurements are described by the projection postulate only roughly . the important consequence of von neumann s projection postulate is the quantum zeno effect . in quantum mechanics short - time behavior of nondecay probability of unstable particle is not exponential but quadratic @xcite . this deviation from the exponential decay has been observed by wilkinson _ _ @xcite . in 1977 , mishra and sudarshan @xcite showed that this behavior when combined with the quantum theory of measurement , based on the assumption of the collapse of the wave function , leaded to a very surprising conclusion : frequent observations slowed down the decay . an unstable particle would never decay when continuously observed . mishra and sudarshan have called this effect the quantum zeno paradox or effect . the effect is called so in allusion to the paradox stated by greek philosopher zeno ( or zenon ) of elea . the very first analysis does not take into account the actual mechanism of the measurement process involved , but it is based on an alternating sequence of unitary evolution and a collapse of the wave function . the zeno effect has been experimentally proved @xcite in a repeatedly measured two - level system undergoing rabi oscillations . the outcome of this experiment has also been explained without the collapse hypothesis @xcite . later it was realized that the repeated measurements could not only slow the quantum dynamics but the quantum process may be accelerated by frequent measurements as well @xcite . this effect was called a quantum anti - zeno effect by kaulakys and gontis @xcite , who argued that frequent interrogations may destroy quantum localization effect in chaotic systems . an effect , analogous to the quantum anti - zeno effect has been obtained in a computational study involving barrier penetration , too @xcite . recently , an analysis of the acceleration of a chemical reaction due to the quantum anti - zeno effect has been presented in ref . @xcite . although a great progress in the investigation of the quantum zeno effect has been made , this effect is not completely understood as yet . in the analysis of the quantum zeno effect the finite duration of the measurement becomes important , therefore , the projection postulate is not sufficient to solve this problem . the complete analysis of the zeno effect requires a more precise model of measurement than the projection postulate . the purpose of this article is to consider such a model of the measurement . the model describes a measurement of the finite duration and finite accuracy . although the used model does not describe the irreversible process , it leads , however , to the correct correlation between the states of the measured system and the measuring apparatus . due to the finite duration of the measurement it is impossible to consider infinitely frequent measurements , as in ref . the highest frequency of the measurements is achieved when the measurements are performed one after another , without the period of the measurement - free evolution between two successive measurements . in this paper we consider such a sequence of the measurements . our goal is to check whether this sequence of the measurements can change the evolution of the system and to verify the predictions of the quantum zeno effect . the work is organized as follows . in section [ sec : mod ] we present the model of the measurement . a simple case is considered in section [ sec : id ] in order to determine the requirements for the duration of the measurement . in section [ sec : meas ] we derived a general formula for the probability of the jump into another level during the measurement . the effect of repeated measurements on the system with a discrete spectrum is investigated in section [ sec : discr ] . the decaying system is considered in section [ sec : dec ] . section [ sec : concl ] summarizes our findings . we consider a system which consists of two parts . the first part of the system has the discrete energy spectrum . the hamiltonian of this part is @xmath0 . the other part of the system is represented by hamiltonian @xmath1 . hamiltonian @xmath1 commutes with @xmath0 . in a particular case the second part can be absent and @xmath1 can be zero . the operator @xmath2 causes the jumps between different energy levels of @xmath0 . therefore , the full hamiltonian of the system equals to @xmath3 . the example of such a system is an atom with the hamiltonian @xmath0 interacting with the electromagnetic field , represented by @xmath1 . we will measure in which eigenstate of the hamiltonian @xmath0 the system is . the measurement is performed by coupling the system with the detector . the full hamiltonian of the system and the detector equals to @xmath4 where @xmath5 is the hamiltonian of the detector and @xmath6 represents the interaction between the detector and the system . we choose the operator @xmath6 in the form @xmath7 where @xmath8 is the operator acting in the hilbert space of the detector and the parameter @xmath9 describes the strength of the interaction . this system detector interaction is that considered by von neumann @xcite and in refs . @xcite . in order to obtain a sensible measurement , the parameter @xmath9 must be large . we require a continuous spectrum of operator @xmath8 . for simplicity , we can consider the quantity @xmath10 as the coordinate of the detector . the measurement begins at time moment @xmath11 . at the beginning of the interaction with the detector , the detector is in the pure state @xmath12 . the full density matrix of the system and detector is @xmath13 where @xmath14 is the density matrix of the system . the duration of the measurement is @xmath15 . after the measurement the density matrix of the system is @xmath16 and the density matrix of the detector is @xmath17 where @xmath18 is the evolution operator of the system and detector , obeying the equation @xmath19 with the initial condition @xmath20 . since the initial density matrix is chosen in a factorizable form , the density matrix of the system after the interaction depends linearly on the density matrix of the system before the interaction . we can represent this fact by the equality @xmath21 where @xmath22 is the superoperator acting on the density matrices of the system . if the vectors @xmath23 form the complete basis in the hilbert space of the system we can rewrite eq . ( [ eq : supop ] ) in the form @xmath24 where the sum over the repeating indices is supposed . the matrix elements of the superoperator are @xmath25 due to the finite duration of the measurement it is impossible to realize the infinitely frequent measurements . the highest frequency of the measurements is achieved when the measurements are performed one after another without the period of the measurement - free evolution between two successive measurements . therefore , we model a continuous measurement by the subsequent measurements of the finite duration and finite accuracy . after @xmath26 measurements the density matrix of the system is @xmath27 further , for simplicity we will neglect the hamiltonian of the detector . after this assumption the evolution operator is equal to @xmath28 where the operator @xmath29 obeys the equation @xmath30 with the initial condition @xmath31 . then the superoperator @xmath22 is @xmath32 in order to estimate the necessary duration of the single measurement it is convenient to consider the case when the operator @xmath33 . in such a case the description of the evolution is simpler . the measurement of this kind occurs also when the influence of the perturbation operator @xmath34 is small in comparison with the interaction between the system and the detector and , therefore , the operator @xmath34 can be neglected . we can choose the basis @xmath35 common for the operators @xmath0 and @xmath1 , @xmath36 where @xmath37 numbers the eigenvalues of the hamiltonian @xmath0 and @xmath38 represents the remaining quantum numbers . since the hamiltonian of the system does not depend on @xmath39 we will omit the parameter @xmath11 in this section . from eq . ( [ eq : supmatr ] ) we obtain the superoperator @xmath40 in the basis @xmath35 @xmath41 where @xmath42 and @xmath43 represent the kronecker s delta in a discrete case and the dirac s delta in a continuous case . ( [ eq : s1 ] ) can be rewritten using the correlation function @xmath44 we can express this function as @xmath45 . since vector @xmath46 is normalized , the function @xmath47 tends to zero when @xmath48 increases . there exists a constant @xmath49 such that the correlation function @xmath50 is small if the variable @xmath51 . then the equation for the superoperator @xmath40 is @xmath52 using eqs . ( [ eq : rho1 ] ) and ( [ eq : s2 ] ) we find that after the measurement the non - diagonal elements of the density matrix of the system become small , since @xmath53 is small for @xmath54 when @xmath55 is large . the density matrix of the detector is @xmath56 from eqs . ( [ eq : evol1 ] ) and ( [ eq : det1 ] ) we obtain @xmath57 where @xmath58 the probability that the system is in the energy level @xmath37 may be expressed as @xmath59 introducing the state vectors of the detector @xmath60 we can express the density operator of the detector as @xmath61 the measurement is complete when the states @xmath62 are almost orthogonal . the different energies can be separated only when the overlap between the corresponding states @xmath62 is almost zero . the scalar product of the states @xmath62 with different energies @xmath63 and @xmath64 is @xmath65 the correlation function @xmath50 is small when @xmath51 . therefore , we have the estimation for the error of the energy measurement @xmath66 as @xmath67 and we obtain the expression for the necessary duration of the measurement @xmath68 where @xmath69 since in our model the measurements are performed immediately one after the other , from eq . ( [ eq : tt ] ) it follows that the rate of measurements is proportional to the strength of the interaction @xmath9 between the system and the measuring device . the operator @xmath2 represents the perturbation of the unperturbed hamiltonian @xmath70 . we will take into account the influence of the operator @xmath34 by the perturbation method , assuming that the strength of the interaction @xmath9 between the system and detector is large . the operator @xmath2 in the interaction picture is @xmath71 in the second order approximation the evolution operator equals to @xmath72 using eqs . ( [ eq : supmatr ] ) and ( [ eq : evol2 ] ) we can obtain the superoperator @xmath73 in the second order approximation , too . the expression for the matrix elements of the superoperator @xmath73 is given in the appendix ( eqs.([eq : ap1 ] ) , ( [ eq : ap2 ] ) and ( [ eq : ap3 ] ) ) . the probability of the jump from the level @xmath74 to the level @xmath75 during the measurement is @xmath76 . using eqs.([eq : ap1 ] ) , ( [ eq : ap2 ] ) and ( [ eq : ap3 ] ) we obtain @xmath77 the expression for the jump probability can be further simplified if the operator @xmath34 does not depend on @xmath39 . we introduce the function @xmath78 changing variables we can rewrite the jump probability as @xmath79 introducing the fourier transformation of @xmath80 @xmath81 and using eq . ( [ eq : w ] ) we obtain the equality @xmath82 where @xmath83 from eq . ( [ eq : pp ] ) , using the equality @xmath84 , we obtain @xmath85 the quantity @xmath86 equals to @xmath87 we see that the quantity @xmath88 characterizes the perturbation . let us consider the measurement effect on the system with the discrete spectrum . the hamiltonian @xmath0 of the system has a discrete spectrum , the operator @xmath89 , and the operator @xmath2 represents a perturbation resulting in the quantum jumps between the discrete states of the system @xmath0 . for the separation of the energy levels , the error in the measurement should be smaller than the distance between the nearest energy levels of the system . it follows from this requirement and eq . ( [ eq : tt ] ) that the measurement time @xmath90 , where @xmath91 is the smallest of the transition frequencies @xmath92 . when @xmath9 is large then @xmath93 is not very small only in the region @xmath94 . we can estimate the probability of the jump to the other energy level during the measurement , replacing @xmath47 by @xmath95 in eq . ( [ eq : w1 ] ) . then from eq . ( [ eq : w1 ] ) we obtain @xmath96 we see that the probability of the jump is proportional to @xmath97 . consequently , for large @xmath98 , i.e. for the strong interaction with the detector , the jump probability is small . this fact represents the quantum zeno effect . however , due to the finiteness of the interaction strength the jump probability is not zero . after sufficiently large number of measurements the jump occurs . we can estimate the number of measurements @xmath26 after which the system jumps into other energy levels from the equality @xmath99 where @xmath100 is the largest matrix element of the perturbation operator @xmath101 . this estimation allows us to introduce the characteristic time , during which the evolution of the system is inhibited @xmath102 we call this duration the inhibition time ( it is natural to call this duration the zeno time , but this term has already different meaning ) . the full probability of the jump from level @xmath74 to other levels is @xmath103 . from eq.([eq : jumpprob ] ) we obtain @xmath104 if the matrix elements of the perturbation @xmath101 between different levels are of the same size then the jump probability increases linearly with the number of the energy levels . this behavior has been observed in ref . @xcite . due to the unitarity of the operator @xmath29 it follows from eq.([eq : supmatr ] ) that the superoperator @xmath22 obeys the equalities @xmath105 if the system has a finite number of energy levels , the density matrix of the system is diagonal and all states are equally occupied ( i.e. , @xmath106 where @xmath107 is the number of the energy levels ) then from eq . ( [ eq : prop1 ] ) it follows that @xmath108 . such a density matrix is the stable point of the map @xmath109 . therefore , we can expect that after a large number of measurements the density matrix of the system tends to this density matrix . when @xmath98 is large and the duration of the measurement is small , we can neglect the non - diagonal elements in the density matrix of the system , since they always are of order @xmath97 . replacing @xmath47 by @xmath95 in eqs . ( [ eq : ap1 ] ) , ( [ eq : ap2 ] ) and ( [ eq : ap3 ] ) and neglecting the elements of the superoperator @xmath73 that cause the arising of the non - diagonal elements of the density matrix , we can write the equation for the superoperator @xmath73 as @xmath110 where @xmath111 then for the diagonal elements of the density matrix we have @xmath112 , or @xmath113 if the perturbation @xmath101 does not depend on @xmath39 then it follows from eq . ( [ eq : deriv ] ) that the diagonal elements of the density matrix evolve exponentially . as an example we will consider the evolution of the measured two - level system . the system is forced by the perturbation @xmath101 which induces the jumps from one state to another . the hamiltonian of this system is @xmath114 where @xmath115 here @xmath116 are pauli matrices and @xmath117 . the hamiltonian @xmath0 has two eigenfunctions @xmath118 and @xmath119 with the eigenvalues @xmath120 and @xmath121 respectively . the evolution operator of the unmeasured system is @xmath122 where @xmath123 if the initial density matrix is @xmath124 then the evolution of the diagonal elements of the unmeasured system s density matrix is given by the equations @xmath125 let us consider now the dynamics of the measured system . the equations for the diagonal elements of the density matrix ( eq . ( [ eq : deriv ] ) ) for the system under consideration are [ eq : eqs1 ] @xmath126 where the inhibition time , according to eq . ( [ eq : inht ] ) , is @xmath127 the solution of eqs . ( [ eq : eqs1 ] ) with the initial condition @xmath128 is @xmath129 from eq . ( [ eq : prop1 ] ) it follows that if the density matrix of the system is @xmath130 then @xmath131 . hence , when the number of the measurements tends to infinity , the density matrix of the system approaches @xmath132 . = .6 = .6 = .6 we have performed the numerical analysis of the dynamics of the measured two - level system ( [ eq : ham1])([eq : ham3 ] ) using eqs.([eq : rho1 ] ) , ( [ eq : supmatr ] ) and ( [ eq : evol4 ] ) with the gaussian correlation function ( [ f ] ) @xmath133 from the condition @xmath134 we have @xmath135 . the initial state of the system is @xmath119 . the matrix elements of the density matrix @xmath136 and @xmath137 are represented in fig . [ fig1 ] and fig . [ fig2 ] , respectively . in fig . [ fig1 ] the approximation ( [ eq : aprox1 ] ) is also shown . this approach is close to the exact evolution . the matrix element @xmath136 for two different values of @xmath9 is shown in fig . we see that for larger @xmath9 the evolution of the system is slower . the influence of the repeated non - ideal measurements on the two level system driven by the periodic perturbation has also been considered in refs . similar results have been found : the occupation of the energy levels changes exponentially with time , approaching the limit @xmath138 . we consider the system which consists of two parts . we can treat the first part as an atom , and the second part as the field ( reservoir ) . the energy spectrum of the atom is discrete and the spectrum of the field is continuous . the hamiltonians of these parts are @xmath0 and @xmath1 respectively and the eigenfunctions are @xmath23 and @xmath139 , @xmath140 there is the interaction between the atom and the field represented by the operator @xmath34 . so , the hamiltonian of the system is @xmath141 the basis for the full system is @xmath142 . when the measurement is not performed , such a system exhibits exponential decay , valid for the intermediate times . the decay rate is given according to the fermi s golden rule @xmath143 where @xmath144 and @xmath145 is the density of the reservoir s states . when the energy level of the atom is measured , we can use the perturbation theory , as it is in the discrete case . the initial state of the field is a vacuum state @xmath118 with energy @xmath146 . then the density matrix of the atom is @xmath147 or @xmath148 , where @xmath149 is an effective superoperator @xmath150 when the states of the atom are weakly coupled to a broad band of states ( continuum ) , the transitions back to the excited state of the atom can be neglected ( i.e. , we neglect the influence of emitted photons on the atom ) . therefore , we can use the superoperator @xmath149 for determination of the evolution of the atom . since the states in the reservoir are very dense , one can replace the sum over @xmath151 by an integral over @xmath152 @xmath153 where @xmath154 is the density of the states in the reservoir . the density matrix of the field is @xmath155 . the diagonal elements of the field s density matrix give the spectrum . if the initial state of the atom is @xmath156 then the distribution of the field s energy is @xmath157 . from eqs . ( [ eq : ap1]),([eq : ap2 ] ) and ( [ eq : ap3 ] ) we obtain @xmath158 where @xmath159 is given by the equation ( [ eq : pp ] ) . from eq . ( [ eq : spectr ] ) we see that @xmath160 is the measurement - modified shape of the spectral line . the integral in eq . ( [ eq : pp ] ) is small when the exponent oscillates more rapidly than the function @xmath161 . this condition is fulfilled when @xmath162 . consequently , the width of the spectral line is @xmath163 the width of the spectral line is proportional to the strength of the measurement ( this equation is obtained using the assumption that the strength of the interaction with the measuring device @xmath9 is large and , therefore , the natural width of the spectral line can be neglected ) . the broadening of the spectrum of the measured system is also reported in ref . @xcite for the case of an electron tunneling out of a quantum dot . the probability of the jump from the state @xmath164 to the state @xmath165 is @xmath166 . from eqs . ( [ eq : sef ] ) it follows @xmath167 using eq . ( [ eq : resw ] ) we obtain the equality @xmath168 where @xmath169 the expression for @xmath88 according to eq . ( [ eq : g ] ) is @xmath170 the quantity @xmath171 is the reservoir coupling spectrum . the measurement - modified decay rate is @xmath172 . from eq . ( [ eq : decprob2 ] ) we have @xmath173 the equation ( [ eq : result ] ) represents a universal result : the decay rate of the frequently measured decaying system is determined by the overlap of the reservoir coupling spectrum and the measurement - modified level width . this equation was derived by kofman and kurizki @xcite , assuming the ideal instantaneous projections . we show that eq . ( [ eq : result ] ) is valid for the more realistic model of the measurement , as well . an equation , similar to eq.([eq : result ] ) has been obtained in ref . @xcite , considering a destruction of the final decay state . depending on the reservoir spectrum @xmath88 and the strength of the measurement the inhibition or acceleration of the decay can be obtained . if the interaction with the measuring device is weak and , consequently , the width of the spectral line is much smaller than the width of the reservoir spectrum , the decay rate equals the decay rate of the unmeasured system , given by the fermi s golden rule ( [ eq : goldrule ] ) . in the intermediate region , when the width of the spectral line is rather small compared with the distance between @xmath174 and the nearest maximum in the reservoir spectrum , the decay rate grows with increase of @xmath98 . this results in the anti - zeno effect . if the width of the spectral line is much greater compared both with the width of the reservoir spectrum and the distance between @xmath174 and the centrum of the reservoir spectrum , the decay rate decreases when @xmath98 increases . this results in the quantum zeno effect . in such a case we can use the approximation @xmath175 where @xmath176 is defined by the equality @xmath177 and @xmath178 is the centrum of @xmath88 . then from eq . ( [ eq : result ] ) we obtain the decay rate @xmath179 . from eq . ( [ eq : pp ] ) , using the condition @xmath180 and the equality @xmath181 we obtain @xmath182 therefore , the decay rate is equal to @xmath183 the obtained decay rate is insensitive to the spectral shape of the reservoir and is inverse proportional to the measurement strength @xmath98 . in this work we investigate the quantum zeno effect using the definite model of the measurement . we take into account the finite duration and the finite accuracy of the measurement . the general equation for the probability of the jump during the measurement is derived ( [ eq : resw ] ) . the behavior of the system under the repeated measurements depends on the strength of measurement and on the properties of the system . when the the strength of the interaction with the measuring device is sufficiently large , the frequent measurements of the system with discrete spectrum slow down the evolution . however , the evolution can not be fully stopped . under the repeated measurements the occupation of the energy levels changes exponentially with time , approaching the limit of the equal occupation of the levels . the jump probability is inversely proportional to the strength of the interaction with the measuring device . in the case of a continuous spectrum the measurements can cause inhibition or acceleration of the evolution . our model of the continuous measurement gives the same result as the approach based on the projection postulate @xcite . the decay rate is equal to the convolution of the reservoir coupling spectrum with the measurement - modified shape of the spectral line . the width of the spectral line is proportional to the strength of the interaction with the measuring device . when this width is much greater than the width of the reservoir , the quantum zeno effect takes place . under these conditions the decay rate is inversely proportional to the strength of the interaction with the measuring device . in a number of decaying systems , however , the reservoir spectrum @xmath88 grows with frequency almost up to the relativistic cut - off and the strength of the interaction required for the appearance of the quantum zeno effect is so high that the initial system is significantly modified . when the spectral line is not very broad , the decay rate may be increased by the measurements more often than it may be decreased and the quantum anti - zeno effect can be obtained . we obtain the superoperator @xmath73 in the second order approximation substituting the approximate expression for the evolution operator ( [ eq : evol2 ] ) into eq . ( [ eq : supmatr ] ) . thus we have @xmath184 where @xmath185 is the superoperator of the unperturbed measurement given by eq . ( [ eq : s2 ] ) , @xmath186 is the first order correction , @xmath187 and @xmath188 is the second order correction , @xmath189 where @xmath190 j. von neumann , _ mathematisch grundlagen der quanten - mechanik _ ( springer , berlin , 1932 ) . english translation : _ mathematical foundations of quantum mechanics _ ( princeton university press , princeton , nj , 1955 ) . | in 1977 , mishra and sudarshan showed that an unstable particle would never be found decayed while it was continuously observed .
they called this effect the quantum zeno effect ( or paradox ) .
later it was realized that the frequent measurements could also accelerate the decay ( quantum anti - zeno effect ) . in this paper
we investigate the quantum zeno effect using the definite model of the measurement .
we take into account the finite duration and the finite accuracy of the measurement .
a general equation for the jump probability during the measurement is derived .
we find that the measurements can cause inhibition ( quantum zeno effect ) or acceleration ( quantum anti - zeno effect ) of the evolution , depending on the strength of the interaction with the measuring device and on the properties of the system .
however , the evolution can not be fully stopped . |
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quantum cryptography @xcite , being the first practical realization of quantum physics at the single quanta level , is the art of creating data channels physically secure against eavesdropping . most of its successful practical realizations are based on the use of single photon information coding though optical fiber links thus being based on a weak laser pulses . the security of the protocols is based on the state perturbation during the eavesdropping @xcite or the measurement correlations analysis with the bell inequalities check @xcite . since key bits are encoded in the single photon states the appearance of the additional photons may seriously undermine the protocol security by leading to the successful listening - in because additional photons states can be imperceptibly measured by an eavesdropper . thus the security reasons require the minimizing of the multi - photon pulses appearance probability making most of the pulses empty , which limits the key rate and results in an additional error rate caused by the single - photon counting detectors , which are inclined to `` dark counts '' , clicking when the photon is missing @xcite . this leads to some contradiction between the effectiveness and security of the existing quantum key distribution ( qkd ) schemes . the contradiction can be overcame by using multi - photon pulses and establishing the channel security on the realistic basis of multi - photon statistics rather then the fermion pairs statistics which the first qkd protocols where starting with . besides avoiding empty pulses by making carrier beams more intensive another point for increasing the effectiveness of quantum communications and qkd in particular is the alphabet size , which can be extended beyond the usual two - letter one , which corresponds to a classical bit . it was shown that the large - alphabet coding essentially increases the bit rate of a quantum channel with no loss in security even in the case of a single - photon polarization coding @xcite . in this work we prove the effectiveness of the alphabet extension for the new multiphoton qkd protocol which was proposed recently @xcite on the basis of the special beam states called two mode coherently correlated ( tmcc ) . these states have the strong correlation between the photon numbers in each of the two spatially parted modes . this correlation leads to the fact that laser shot noise shows itself equally in the both of the modes thus enabling the use of a tmcc - source as a generator of some random bit sequence which is then shared between two legitimate parties who perform independent photon - number measurements on each of the modes and extract the key bits from the measurement results comparing them to the constant average photon number . the quantum channel capacity for the 4- and 8-letter extended alphabets is estimated . the eavesdropping attacks on the extended - alphabet tmcc - channels are considered in terms of the introduced qber . it is shown that the alphabet extension leads to the increase of the effectiveness of the qkd protocol providing channel capacity of up to 3 bits per pulse . moreover the alphabet extension strengthen the protocol security against the intercept - resend attacks by making eavesdropper introduce larger qber on each successfully intercepted bit exceeding 70% qber per bit for the 8-letter alphabet . the tmcc - protocol is based on the use of a special states of light , the two mode coherently correlated ones . such states were mathematically studied by agarwal @xcite . they are defined as fully correlated and at once the eigenstates for the product of annihilation operators of the both modes . the latter condition makes them the special case of the wider class of the two - mode correlated states , also referred to as twin beams , which are broadly investigated in the past time @xcite and are usually obtained in the process of parametric down - conversion ( pdc ) in nonlinear crystals . the tmcc - states can be presented through series by fock states : @xmath0 here we use the designation @xmath1 , where @xmath2 and @xmath3stand for the states of the @xmath4 and @xmath5 mode accordingly , represented by their photon numbers . the main feature of the states ( [ eq : tmcc ] ) is that only the terms with equal photon numbers in the both of the modes are present in the expansion . this leads to the strong correlation between the observables concerned with each of the modes . at the same time the tmcc states differ from the usual two - mode correlated coherent states , because the average for any of the linear in field observable ( e.g. the vector - potential ) is equal to zero for each of the tmcc modes @xcite . thus each of the modes is not coherent to itself , but as the square in field observables ( e.g. correlation function , momentum , energy ) have non - zero average values , the tmcc states possess the second order coherence and so can be referred to as the coherently correlated states . the separate intensity measurements on each of the beams give the results which are proportional to the average of the @xmath6 , which corresponds to the number of photons in the mode . the probability distribution for different photon numbers detection is @xmath7 the photon statistics for each of the beams is sub - poisson as confirmed by the mandel parameter , which is negative even for the small intensities @xcite . this fact can be useful for the tmcc state identification and the quantum channel eavesdropping detection . the shot noise photon number fluctuations in the tmcc modes enable the use of the tmcc source as a generator of a random key encoded in the photon numbers value . the strong correlation between the independent measurements of two modes makes it possible to securely distribute such random key between two remote legitimate users , alice and bob . the security of the quantum channel is based on the fact that intermediate photon measurements will perturb the states which can be checked locally by measurement statistics @xcite . the simple protocol based on the two mode coherently - correlated states was described @xcite as follows : alice and bob simultaneously start the independent photon number measurements each on the corresponding tmcc mode . they compare the obtained photon number values for each next unit time to the average which is constant during the overall key transmission procedure . if the obtained number is larger than the average , the next bit is considered to be equal to `` 1 '' . if the photon number for the next time window is less than average , the corresponding bit is equal to `` 0 '' . thus , the protocol uses the two - bit photon number alphabet coding by the multiphoton two mode coherently correlated states . the state coherence is an important feature of the tmcc - protocol distinguishing it from the existing single - photon incoherent state protocols because coherence allows establishing the security for the multiphoton pulses transmission since the state perturbation leads to the decoherence . the loss of coherence can be revealed by checking the state statistics which can be done even locally by estimating differences between the obtained and expected state density matrices @xcite . the classical information channel is well known to be described by the shannon mutual information between the observables of some classical macroscopic systems . the measurements on such systems return the probability distributions for the sets of the observables discrete values . these value sets when used for encoding and decoding the information are called the alphabets which contain the discrete values of the observables as the letters . the input - output mutual information depends on the observables shannon entropies : @xmath8 where @xmath9 is the input observable entropy and the @xmath10 is the mutual entropy of the input relative to the output which describes the information loss in the channel : @xmath11 since @xmath10 depends on the probability distribution of the values set @xmath12 it depends on the alphabet size and increases with the increase of the number of its letters . this discourse clearly fits the quantum channels which differ from the classical ones by the fact that quantum observables , being the parameters of the quantum microscopic states , are used for information encoding and decoding . since quantum cryptography deals with the key sharing across the quantum channel it can also gain from the alphabet expansion . in 1999 bechmann - pasquinucci and tittel @xcite proposed the use of a larger alphabet for the bb84 single - qubit protocol @xcite , which is extended on the four - level quantum systems - the so - called quantum quarts ( qu - quarts ) . alice still selects randomly between the two possible bases , but she is now preparing one of the four states thus making eight possible choices for the qu - quart based protocol instead of the four choices for the qubit - based one . it was shown that such a development of the bb84 protocol increases the information flux and makes the qkd scheme more secure against realistic eavesdropping because eavesdropper introduces the much higher qber for the given amount of the acquired information . lately in 2003 sych , grishanin and zadkov @xcite made the further development of the single - qubit qkd scheme by proposing the use of the continuous alphabet for the key bits coding . the idea was to identify key bits from the unselected qubit states which was shown to result in the increase of the protocol effectiveness , security and reliability at noisy channels . thus it makes it clear that the protocol extension may be quite useful for the tmcc - based protocol especially since the multi - photon states in the fock presentation can be considered as the multi - dimensional systems . the maximum shannon entropy of a photon - counting beam measurement , i.e. the highest possible information capacity for a channel built on such beams can obviously be achieved if each of the different photon - number events are identified as the different measurement outcomes and correspond to the different alphabet letters . in other words , the m - letter alphabet for the m - photon state will give the maximum information which can be encoded into and transmitted by such a state . for the tmcc - beam this maximal information will be @xmath13 the dependence of this highest possible information gain of a state measurement on the average photon number of the state is given at ( [ maxent ] ) . one can easily see that depending on the beam intensity , the tmcc state can carry from 1 up to 4 bits of information . thus it is possible to build the effective tmcc - based qkd protocols utilizing 4-letter alphabets for beams carrying about 3 - 5 photons in average which will raise the measurement information gain to 2 bits and the 8-letter alphabets for more intensive beams which will result in the effectiveness of up to 4 bits per measurement . here we examine two possible alphabet extensions for the tmcc channels , containing 4 and 8 state `` letters '' . the quaternary channel with two bits per measurement capacity can be established on the basis of the tmcc beams which are distinguished by the photon numbers between 4 possible states ( see figure [ bits ] ) : @xmath14}-1 & 00 $ ( letter 0)$\\ \{n = { \left[\left\langle n \right\rangle\right ] } & 01 $ ( letter 1)$\\ \{n = { \left[\left\langle n \right\rangle\right]}+1 & 10 $ ( letter 2)$\\ \{n \geq { \left[\left\langle n \right\rangle\right]}+2 & 11 $ ( letter 3)$ \end{array}\right.}\ ] ] knowing the photon numbers registration probability distribution for a tmcc - beam ( [ eq : nphotprob ] ) one can easily estimate the probability for each of the 4 quart states ( [ eq : quart ] ) realization : @xmath15}-1 } { \frac{|\lambda|^{2n}}{n!^2 } } ; \ p_1(\lambda ) = \frac{1}{i_0 ( 2|\lambda| ) } \frac{|\lambda|^{2{\left[\left\langle n \right\rangle\right]}}}{{\left[\left\langle n \right\rangle\right]}!^2};\ ] ] @xmath16}+1)}}{({\left[\left\langle n \right\rangle\right]}+1)!^2};\ p_3(\lambda ) = \frac{1}{i_0 ( 2|\lambda| ) } \sum_{n={\left[\left\langle n \right\rangle\right]}+2}^\infty { \frac{|\lambda|^{2n}}{n!^2 } } \ ] ] these probability values can be then used for estimating the shannon entropy ( [ eq : shentropy ] ) of a photon - number measurement on a tmcc - beam with four possible interpretation outcomes , which is the information gain for such a measurement . the octuple tmcc - based channel with the capacity of three bits per measurement can be established in a similar way if the states are distinguished by 8 possible measurement outcomes which differ in photon numbers ( see figure [ bits ] ) : @xmath17}-3 & 000 $ ( letter 0)$ \\ \{n = { \left[\left\langle n \right\rangle\right]}-2 & 001 $ ( letter 1)$ \\ \{n = { \left[\left\langle n \right\rangle\right]}-1 & 010 $ ( letter 2)$ \\ \{n = { \left[\left\langle n \right\rangle\right ] } & 011 $ ( letter 3)$ \\ \{n = { \left[\left\langle n \right\rangle\right]}+1 & 100 $ ( letter 4)$ \\ \{n = { \left[\left\langle n \right\rangle\right]}+2 & 101 $ ( letter 5)$ \\ \{n = { \left[\left\langle n \right\rangle\right]}+3 & 110 $ ( letter 6)$ \\ \{n \geq { \left[\left\langle n \right\rangle\right]}+4 & 111 $ ( letter 7)$ \end{array}\right.}\ ] ] again , knowing the distribution ( [ eq : nphotprob ] ) we can obtain the probability for each octo - bit value occurrence and thus build the shannon information available from a measurement . the dependencies for the measurement information gain for 2- , 4- and 8-letter alphabets on the state average photon number are given on a graph at ( [ entropies ] ) . the security of a quantum channel i.e. the impossibility to carry out a successful eavesdropping is obviously the most important property for any qkd scheme . traditionally the security of a quantum channel is examined in the frames of the two approaches - the ideal and the realistic ones @xcite . in the ideal case the eavesdropper eve is supposed to have the unlimited technological power with possibilities restricted only by the laws of quantum mechanics @xcite . the realistic approach takes into account the technical possibilities of an eavesdropper compatible with today s and foreseeable technology at this the most successful realistic eavesdropping techniques on the secure quantum channels are the intercept - resend strategy also referred to as state cloning and the beam splitting @xcite . the most general security criterion known from the classical cryptography already states that bob has to possess more information on the transferred key than eve ( if the bob s mode was eavesdropped ) . in this case the privacy amplification post - transfer algorithms will be helpful in distilling the truly secure key , otherwise they will fail @xcite . the eavesdropping in quantum cryptography is usually detected by checking the qber ( quantum bit error rate ) which is the measure of the errors in the obtained key . since all of the errors are considered to be caused by an eavesdropper , knowing the qber one can estimate how much information on the key does the eavesdropper have and thus determine is the key distillation will be successful . the tmcc - based scheme was examined against the realistic eavesdropping and shown to be secure against splitting and cloning attacks . in the case of the beam splitting this is quite intuitive because the installation of a splitter at any of the modes removes the correlation between modes thus simply destroying the channel ( for more detailed examination of a beam splitting attack on a tmcc - channel see @xcite ) . in the case of a state cloning it was shown that eavesdropper significantly changes the statistics of the re - emitted mode which can be detected by the local calculations of the mandel parameter or the distances between the received and the expected states density matrices @xcite . in this work we estimate the security of the tmcc - based channel in the terms of the qber , which is introduced to the channel upon the eavesdropping . we use this measure to compare the security of the tmcc qkd scheme for different alphabet sizes . let s consider eve carrying out a cloning intercept - resend attack on a tmcc - based channel . in order to do so she installs a photon - counting detector which measures one of the modes ( suppose the one which goes to bob ) and a tmcc laser source assigned for re - creating the state ( fig [ cloning ] ) . for each incoming pulse ( i.e. in each time slot ) eve measures the photon number @xmath18 and tries to re - emit the same photon number in the bob s direction . we assume eve calculates the value of state parameter @xmath19 which corresponds to the photon number @xmath18 to be emitted and sets her source up to this value . though due to the quantum fluctuations she is unable to emit exactly the same number and thus bob obtains the state which density matrix @xmath20 , differs from the original @xmath21 measured by eve . this re - emitted state matrix is the mixture of the k - photon states @xmath22 with probabilities @xmath23 the probability for bob to incorrectly interpret the received state i.e. to obtain the wrong letter @xmath24 of the alphabet set @xmath25 of size @xmath26 ( so that @xmath27 ) is equal to @xmath28 where @xmath29 is the probability for bob to obtain the correct letter @xmath24 given alice obtained the same letter @xmath24 . the qber introduced by eve for each intercepted bit is then @xmath30 the sum in @xmath31 can be presented and calculated through the probability distributions as @xmath32 where @xmath33 is the photon number corresponding to the first alphabet letter @xmath34 and @xmath35 is the photon number corresponding to the last letter @xmath36 , @xmath37 is the probability distribution ( @xmath38 ) and @xmath39 hence the qber can be calculated numerically . the results are presented on the graphs at fig @xmath40 as the qber dependence on the average photon number ( i.e. on the original pulse intensity ) for different alphabet sizes . one can easily see that qber introduced for each intercepted bit during the state cloning attack on a 2-letter ( 1-bit ) alphabet channel is about 20% , growing to about 50% in the case of 4-letter ( 2-bit ) case and exceeding 70% for the 8-letter ( 3-bit ) alphabet . thus the tmcc - based qkd protocol turns out to be more secure at the larger alphabets . if eve uses a usual single mode laser source , producing a coherent beam with the poisson statistics for the tmcc - state cloning , the calculations give qber values of about 30% for 2-letter , up to 60% for 4-letter and over 80% for the 8-letter alphabet thus proving that tmcc - state cloning based on a usual laser source is even less effective . the quantum cryptography scheme with larger alphabets based on the use of the two - mode coherently correlated multiphoton beams is proposed . the alphabet extension is shown to result in the increase of the effectiveness of the qkd scheme . the protocol security against the realistic state cloning is examined in terms of the introduced qber . it is shown that the tmcc - based qkd scheme becomes more secure for the larger alphabet sets i.e. for the more intense laser pulses . usenko v. c. , usenko c. v. , proceedings of the international conference on quantum communication , measurement and computing ( glasgow , uk , 2004 ) , am . inst . of phys . cp734 , p. 319 / quant - ph/0407175 ( 2004 ) laurat j. , coudreau t. , treps n. , maitre a. and fabre c. , conditional preparation of a quantum state in the continuous variable regime : generation of a sub - poissonian state from twin beams , phys . 91 , 213601 / quant - ph/0304111 ( 2003 ) | the large - alphabet quantum cryptography protocol based on the two - mode coherently correlated multi - photon beams is proposed . the alphabet extension for the protocol
is shown to result in the increase of the qkd effectiveness and security .
pacs numbers : 03.67.-a,03.67.dd,03.67.hk |
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hh 1 and 2 were the first herbig - haro ( hh ) objects to be discovered ( herbig 1951 ; haro 1952 ) , and have played an important role in the field of hh objects ( see the historical review of raga et al . for example , hst images ( schwartz et al . 1993 ; hester et al . 1998 ) , proper motions ( ground based : herbig & jones 1981 ; hst : bally et al . 2002 ; ir : noriega - crespo et al . 1997 ; radio : rodrguez et al . 2000 ) , and detections in radio continuum ( pravdo et al . 1985 ) , uv ( ortolani & dodorico 1980 ) and x - rays ( pravdo et al . 2001 ) were first obtained for hh 1 and 2 . the hh 1/2 system has a central source detected in radio continuum ( see , e.g. , rodrguez et al . 2000 ) and a bipolar jet system , with a nw jet ( directed towards hh 1 ) which is visible optically , and a se jet ( directed towards hh 2 ) visible only in the ir ( see noriega - crespo & raga 2012 ) . hh 1 has a `` single bow shock '' morphology , and hh 2 is a collection of condensations , some of them also with bow - shaped morphologies ( see , e.g. , bally et al . the emission - line structure of these objects has been studied spectroscopically , with 1d ( solf , bhm & raga 1988 ) and 2d ( solf et al . 1991 ; bhm & solf 1992 ) coverage of the objects . it should be pointed out that the hh 1/2 outflow lies very close to the plane of the sky , so that projection effects do not have to be considered when interpreting the observations of these objects . the spatial structure of the optical line emission has been studied at higher angular resolution with hst images . schwartz et al . ( 1993 ) obtained h@xmath0 , [ s ii ] 6716 + 6730 and [ o i ] 6300 images . later images of hh 1 and 2 were all taken with filters isolating the h@xmath0 and the red [ s ii ] lines ( bally et al . 2002 ; hartigan et al . 2011 ) . in the present paper we describe a pair of new hst images of hh 1 and 2 obtained with filters isolating the h@xmath0 and h@xmath1 lines . these images were obtained in consecutive exposures , so that they are not affected by proper motions ( which become evident in hst observations of the hh 1/2 complex separated by more than a few weeks ) nor by differences in the pointing , and they therefore yield an accurate depiction of the spatial distribution of the h@xmath0/h@xmath1 ratio in these objects . these images show effects that have not been detected before in ground based studies of the emission line structure of hh 1 and 2 ( see , e.g. , solf et al . 1991 and bhm & solf 1992 ) nor in hst images of other hh objects ( since hst h@xmath1 images of hh objects have not been previously obtained ) . the paper is organized as follows . the new hst images are described in section 2 . the spatial distribution of the h@xmath0/h@xmath1 ratio , the line ratios as a function of h@xmath1 intensity and the distribution functions of the line ratios are presented in section 3 . finally , an interpretation of the results is presented in section 4 . the region around hh 1 and 2 was observed with the h@xmath0 ( f656n ) and h@xmath1 ( f487n ) filters on august 16 , 2014 with the wfc3 camera on the hst . the h@xmath0 image was obtained with a 2686 s exposure and the h@xmath1 image with a slightly longer , 2798 s exposure . the images were reduced with the standard pipeline , and a simple recognition / replacement algorithm was used to remove the cosmic rays . the final images have @xmath6 pixels , with a pixel size of @xmath7 . the images contain only two stars : the cohen - schwartz star ( cohen & schwartz 1979 ) and `` star no . 4 '' of strom et al . ( 1985 ) . these two stars have been used to determine astrometric positions in ccd images of the hh 1/2 region since the work of raga et al . ( 1990 ) , yielding better positions for hh 1 ( which is closer to the two stars ) than for hh 2 . we have carried out paraboloidal fits to the psfs of these two stars , and find no evidence for offsets and/or rotation , which shows the excellent tracking of the hst during the single pointing in which the two images were obtained . also , we have analyzed the h@xmath8h@xmath1 difference images of the two stars , and find no offsets between the two frames . the full h@xmath0 frame is shown in figure 1 , as well as blow - ups of regions around hh 1 and hh 2 in both h@xmath0 and h@xmath1 . as seen in the top frame , the h@xmath0 map shows the extended collection of hh 2 knots ( to the se ) and the more compact distribution of the hh 1 knots ( towards the nw ) . the central frames of figure 1 show the h@xmath0 emission of hh 2 ( left ) and hh 1 ( right ) at a larger scale . in the fainter h@xmath1 emission ( bottom frames of figure 1 ) only the brighter regions of hh 1 and 2 are detected . we have defined two boxes ( labeled a and b in the bottom frame of figure 1 ) enclosing the regions of the two objects which are detected in h@xmath1 . in the following section , the regions within these two boxes are used in order to study the spatial dependence of the h@xmath0/h@xmath1 ratio . frame of hh 1 and 2 obtained with the wfc3 camera of the hst ( the scale and orientation of the images is shown ) . the central and the bottom frames show the h@xmath0 and h@xmath1 images ( respectively ) of regions containing hh 2 ( left ) and hh 1 ( right ) . also , on the h@xmath1 frames we show boxes which include the brighter regions of hh 1 and hh 2 ( boxes b and a , respectively ) , which have been used for calculating the h@xmath0/h@xmath1 ratios shown in figures 2 to 5 . the images are displayed with a logarithmic greyscale.,width=302 ] as discussed in detail by odell et al . ( 2013 ) , the f656n filter is contaminated by emission from the [ n ii ] 6548 line , and both the f656n and f487n filters have contributions from the nebular continuum . using the fact that at all measured positions within hh 1 and 2 , the [ n ii ] 6548/h@xmath0 ratio does not exceed a value of @xmath9 ( see , e.g. , brugel , bhm & mannery 1981a and solf et al . 1988 ) and the transmission curve of the f656n filter ( see odell et al . 2013 and the wfc3 instrument handbook ) one then finds a peak contribution of @xmath10% to the measured flux . for estimating the effects of the continuum in the f656n and f487n images one can use the continuum and line fluxes obtained by brugel , bhm & mannery ( 1981a , b ) and the bandpasses of the filters to obtain estimates of @xmath11 and 5% ( for the f656n and f487n filters , respectively ) . therefore , when interpreting the h@xmath0/h@xmath1 ratios obtained from our hst images , it is necessary to keep in mind that there is an uncertainty of @xmath12% due to a possible spatial dependence in the h@xmath1 line to continuum ratio within the f487n filter . as this uncertainty is @xmath13 order of magnitude smaller than the h@xmath0/h@xmath1 ratio variations described below , we do not discuss it further . figure 2 shows the h@xmath0 map ( right ) and h@xmath0/h@xmath1 ratio map ( left ) for hh 2 . to avoid having extended regions dominated by noise , in order to calculate the line ratio map it is necessary to place a lower bound on the line fluxes . we have chosen to calculate the ratios only in regions in which the observed h@xmath1 flux is larger than @xmath14 erg s@xmath15pix@xmath15 . for calculating the intrinsic h@xmath0/h@xmath1 ratios we have applied the following reddening correction . we first calculate the observed ratios for all of the pixels with h@xmath1 intensities larger than @xmath16 ( see above ) for the a and b boxes ( shown in the bottom frames of figure 1 ) . for hh 2 we obtain a mean line ratio @xmath17 , and for hh 1 an almost identical @xmath18 value . considering an observed line ratio of 3.8 for both objects , comparing with the case b recombination cascade intrinsic h@xmath0/h@xmath1 ratio of 2.8 and using the average galactic extinction curve , we obtain an @xmath19 colour excess . this value is somewhat lower than the @xmath20 value deduced for hh 2 by brugel et al . ( 1981a ) , using the method of miller ( 1968 ) , based on the fixed ratios between the auroral and transauroral lines of [ s ii ] ( i.e. , not assuming a recombination cascade h@xmath0/h@xmath1 ratio ) . in order to calculate the dereddened h@xmath0/h@xmath1 ratios , we therefore multiply the observed ratios by a factor of 2.8/3.8 , basically assuming that the extinction towards hh 1 and 2 is position - independent . the dereddened h@xmath0/h@xmath1 ratios of hh 2 ( see figure 2 ) have values in the @xmath21 range , with the regions of higher values corresponding to filamentary structures in the leading edge of the emitting knots ( i.e. , in the edges directed away from the outflow source ) . in order to illustrate the positions of these `` high h@xmath0/h@xmath1 '' regions , we have superimposed an h@xmath0/h@xmath22 contour on the h@xmath0 emission map ( purple contour in the right frame of figure 2 ) . figure 3 shows the h@xmath0 map ( bottom ) and dereddened h@xmath0/h@xmath1 ratio map ( top ) for hh 1 . we have calculated the ratios only for pixels with an observed h@xmath1 flux larger than @xmath14 erg s@xmath15pix@xmath15 ( i.e. , the same cutoff used for hh 2 , see above ) . the region with h@xmath0/h@xmath23 is a thin filament on the e side of the leading edge of hh 1 ( see the purple contour on the h@xmath0 emission map in the bottom frame of figure 3 ) . it is clear that hh 1 shows a strong side - to - side asymmetry with respect to the outflow axis , as the sw region of the leading edge does not show high h@xmath0/h@xmath1 ratios ( see the top frame of figure 3 ) . the h@xmath0 emission also shows a strong side - to - side asymmetry . emission ( bottom , shown with a logarithmic colour scale ) and dereddened h@xmath0/h@xmath1 ratio ( top , shown with the linear colour scale given by the top bar ) for hh 1 . the outflow source lies towards the sse . on the h@xmath0 image ( bottom frame ) we have included a ( dereddened ) h@xmath0/h@xmath1=4 contour ( in purple ) . this contour shows that the higher values of the line ratio are distributed in a ridge along the e side of the leading edge of hh 1.,width=340 ] figure 4 shows the dereddened h@xmath0/h@xmath1 line ratio as a function of the ( observed ) h@xmath1 flux for all of the pixels with @xmath24 ( see above ) for hh 1 ( top frame ) and hh 2 ( bottom frame ) . it is clear that for low values of the h@xmath1 intensity in both hh 1 and 2 we have a relatively broad distribution of line ratios ( the width of this distribution representing the relatively large errors of the line ratio at low intensities ) centered on the h@xmath0/h@xmath25 recombination cascade value . for pixels with brighter intensities , we see a distribution of h@xmath0/h@xmath1 ratios extending from @xmath26 to larger values of @xmath12 ( for hh 1 ) or @xmath27 ( for hh 2 ) . /h@xmath1 ratio as a function of the observed h@xmath1 flux for the pixels of hh 1 ( top ) and hh 2 ( bottom ) . for the lower h@xmath1 intensities we see that the line ratios straddle a value of @xmath28 ( shown with a horizontal line in both frames ) , corresponding to the recombination cascade h@xmath0/h@xmath1 ratio . for larger h@xmath1 intensities we find line ratios extending from @xmath26 to @xmath29 ( for hh 1 ) or 6 ( for hh 2).,width=302 ] this result is seen more clearly in figure 5 , where we show the normalized distributions of the line ratios of pixels with @xmath30 erg s@xmath15 pix@xmath15 ( distribution @xmath31 , top frame ) , of pixels with @xmath32 erg s@xmath15 pix@xmath15 ( distribution @xmath33 , center ) , and of all pixels with @xmath34 ( distribution @xmath35 , bottom frame of figure 5 , with appropriate pixels found only in hh 2 ) . for both hh 2 ( left column ) and hh 1 ( right column of figure 5 ) , we see that the distribution @xmath31 of the lower intensity pixels is approximately symmetrical , centered at an h@xmath0/h@xmath36 line ratio . the distributions for higher intensity pixels ( @xmath33 and @xmath35 , see above and the central and bottom frames of figure 5 ) start at values of h@xmath0/h@xmath37 - 2.5 , have a peak at a line ratio of @xmath38 and have a wing extending to h@xmath0/h@xmath39 for hh 1 and @xmath27 for hh 2 . in the following section , we show that these high h@xmath0/h@xmath1 ratios coincide with the values expected for collisional excitation of the @xmath4 and 4 levels of h. /h@xmath1 ratio bins for hh 2 ( left ) and hh 1 ( right ) . we show three different distributions : @xmath31 ( top ) corresponding to pixels with @xmath14 erg s@xmath15 pix@xmath40 erg s@xmath15 pix@xmath15 , @xmath33 ( center ) of pixels with @xmath32 erg s@xmath15 pix@xmath15 and @xmath35 ( bottom ) of pixels with @xmath34 . the distribution function of the lower intensity pixels ( @xmath31 , top frames ) is approximately symmetrical , centered at a line ratio of @xmath28 , corresponding to the recombination cascade value ( the dashed , vertical line in all plots corresponds to h@xmath0/h@xmath25 ) . the distribution functions for higher intensity pixels ( @xmath33 and @xmath35 , central and bottom frames ) all show extended wings to higher values of h@xmath0/h@xmath1.,width=302 ] from our new h@xmath0 and h@xmath1 hst images we can compute dereddened h@xmath0/h@xmath1 maps for the brighter regions of hh 1 and 2 . for the reddening correction , we assume that the mean value of the h@xmath0/h@xmath1 ratio coincides with the recombination cascade value of 2.8 , as found previously by brugel et al . ( 1981a ) , who calculated the reddening correction with miller s method , based on the ratios of auroral to transauroral [ s ii ] lines . we find that in limited spatial regions the ( dereddened ) h@xmath0/h@xmath1 ratio has values of @xmath41 , which are inconsistent with the recombination cascade value . these high h@xmath0/h@xmath1 regions are filaments along the leading edges ( i.e. , the edges away from the outflow source ) of the brighter emitting regions of hh 1 and 2 ( see figures 2 and 3 ) . raga et al . ( 2014 ) show that the @xmath42h@xmath0/h@xmath1 ratio for a `` case b '' cascade fed by collisional excitations from the ground state of hydrogen has the approximate form : @xmath43 where @xmath44 is boltzmann s constant and @xmath45 is the energy difference between the @xmath46 and @xmath4 energy levels ( so that @xmath47 k ) . the first term of this functional form has a temperature dependence derived from the ratio of the @xmath48 and @xmath49 collisional excitation coefficients ( assuming temperature - independent collision strengths ) , and the second term is a correction necessary to match the results of a 5-level , collisionally fed cascade matrix description of the hydrogen atom in the @xmath50k temperature range ( see raga et al . 2014 ) . it is clear that the functional form of @xmath51 ( see equation [ r ] ) has high values for low temperatures , and has an asymptotic value of 4.35 for @xmath52 . from equation ( [ r ] ) , one obtains @xmath53 and @xmath54 . therefore , the wing of the line ratio distributions extending to h@xmath0/h@xmath55 ( see figure 5 ) can straightforwardly be explained as produced in regions with temperatures in the @xmath56 k range emitting collisionally excited balmer lines . this clear evidence that we are observing collisionally excited balmer lines together with the fact that the high h@xmath0/h@xmath1 regions are restricted to the leading edges of the outward moving condensations of hh 1 and 2 is quite conclusive evidence that we are observing the region of collisional excitation of h lines right after the shock waves driven into the surrounding medium by the condensations . most of the h@xmath0 emission , however , comes from a region further away from the shock , in which the balmer lines are produced through the standard recombination cascade ( as evidenced by the h@xmath0/h@xmath57 ratios , see figures 2 and 3 ) . the theoretical prediction of these two regions of balmer line emission ( a collisionally excited balmer line region immediately after the shock , and the recombination region with balmer lines dominated by the recombination cascade ) in hh shock wave models is already mentioned by raymond ( 1979 ) , and the h@xmath0 emission from the two regions was studied in more detail by raga & binette ( 1991 ) . these two regions are of course present in all shock models ( for example , in the plane - parallel , time - dependent shock models of teileanu et al . 2009 ) . in non - radiative shocks observed in some supernovae remnants or in pulsar cometary nebulae , the observed emission comes exclusively from the region of collisional excitation right behind the shocks ( see , e.g. , the review of heng 2010 ) . in hh objects , the only previous observational evidence of the emission from the immediate post - shock region ( as opposed to the emission from the recombination region ) were the h@xmath0 filaments seen in hst images of some bow shocks , notably in the hst images of hh 47 ( heathcote et al . 1996 ) , hh 111 ( reipurth et al . 1997 ) and hh 34 ( reipurth et al . 2002 ) . however , as only h@xmath0 was observed it was not possible to guarantee that these filaments did correspond to the region of collisionally excited balmer lines . our new h@xmath0 and h@xmath1 images for the first time show in a quite conclusive way that we have a detection of the immediate post - shock region of hh objects ( in which h is being collisionally ionized and the levels of h are being collisionally excited ) . the detection of this region provides a clear way forward for developing models of hh bow shocks , in which the position of the shock wave relative to the recombination region is directly constrained by the observations . we should note that throughout this paper we have assumed that the exctinction is uniform over the emission regions of hh 1 and 2 . in principle it could be possible that foreground structures in the vicinity of the objects might produce changes in the extinction on angular scales comparable to the size of the objects . however , estimates of the density of the pre - bow shock material of hh 1 and 2 ( based on observations of the post - shock density and on plane - parallel shock models , see , e.g. , hartigan et al . 1987 ) give values @xmath58 - 300 @xmath59 . clearly , such a low density environment will not produce appreciable extinction on spatial scales comparable to the size of the hh objects . because of this , if one wants to attribute the observed changes in the h@xmath0/h@xmath1 ratio to an angular dependence of the extinction , it is necessary to assume that still undetected , sharp - edged , high density regions are present in the immediate vicinity of hh 1 and 2 . support for this work was provided by nasa through grant hst - go-13484 from the space telescope science institute . ar and acr acknowledge support from the conacyt grants 101356 , 101975 and 167611 and the dgapa - unam grants in105312 and ig100214 . | we present new h@xmath0 and h@xmath1 images of the hh 1/2 system , and we find that the h@xmath0/h@xmath1 ratio has high values in ridges along the leading edges of the hh 1 bow shock and of the brighter condensations of hh 2 .
these ridges have h@xmath0/h@xmath2 , which is consistent with collisional excitation from the @xmath3 to the @xmath4 and 4 levels of hydrogen in a gas of temperatures @xmath5 k. this is therefore the first direct proof that the collisional excitation / ionization region of hydrogen right behind herbig - haro shock fronts is detected . |
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this paper is devoted to studying the stability of dynamical objects which are called by very different terms such as one - mode solutions ( omss ) @xcite , simple periodic orbits ( spos ) @xcite , low - dimensional solutions @xcite , one - dimensional bushes @xcite etc . below we refer to them as _ nonlinear normal modes _ ( nnms ) . let us comment on this terminology . the concept of similar nonlinear normal modes was developed by rosenberg many years ago @xcite . each nnm represents a periodic vibrational regime in the conservative @xmath1-particle mechanical system for which the displacement @xmath2 of every particle is proportional to the displacement of an arbitrary chosen particle , say , the first particle [ @xmath3 at any instant @xmath4 : @xmath5 where @xmath6 are constant coefficients . note that convenient _ linear normal modes _ ( lnms ) also satisfy eq . ( [ eqch1 ] ) since , for any such mode , one can write @xmath7 where @xmath8 are constant amplitudes of individual particles , while @xmath9 and @xmath10 are the frequency and initial phase of the considered mode . as a rule , nnms can exist in the mechanical systems with rather specific interparticle interactions , for example , in systems whose potential energy represents a _ homogeneous _ function with respect to all its arguments . however , in some cases , the existence of nnms is caused by certain symmetry - related reasons . we refer to such dynamical objects as _ symmetry - determined _ nnms . in @xcite , we have found all symmetry - determined nnms in all @xmath1-particle mechanical systems with any of 230 space groups . this proved to be possible due to the group - theoretical methods developed in @xcite for constructing _ bushes _ of vibrational modes . at this point , it is worth to comment on the concept of bushes of modes introduced in @xcite ( the theory of these dynamical objects can be found in @xcite ) . in rigorous mathematical sense , they represent symmetry - determined _ invariant manifolds _ decomposed into the basis vectors of _ irreducible representations _ of the symmetry group characterizing the considered mechanical system ( `` parent '' group ) . because of the specific subject of the present paper , it is sufficient to consider only bushes of vibrational modes in nonlinear monoatomic chains . such bushes have been discussed in @xcite . let us reproduce here some ideas and results from these papers . every bush b@xmath11 $ ] describes a certain vibrational regime , and some specific _ pattern _ of instantaneous displacements of all the particles of the mechanical system corresponds to it . in turn , this pattern is characterized by a symmetry group @xmath12 ( in particular , such group can be trivial ) which is a _ subgroup _ of the symmetry group @xmath13 of the mechanical system in its equilibrium state . for example , let us consider the _ two - dimensional _ bush b@xmath14 $ ] in the monoatomic chain with periodic boundary conditions whose displacement pattern @xmath15 can be written as follows @xmath16 this pattern is determined by two time - dependent functions @xmath17 , @xmath18 , and the corresponding _ vibrational state _ of the @xmath1-particle chain is fully described by displacements inside the _ primitive cell _ , which is four time larger than that of the equilibrium state . we will refer to the ratio of the primitive cell size of the vibrational state to that of the equilibrium state as _ multiplication number _ ( @xmath19 ) and , therefore , for the pattern ( [ eqch3 ] ) , one can write @xmath20 . the symmetry group @xmath21}$ ] of the bush b@xmath14 $ ] is determined by two _ generators _ : the translation ( @xmath22 ) by four lattice spacing ( @xmath23 ) and the inversion ( @xmath24 ) with respect to the center of the chain ( note that the condition @xmath25 must hold for existence of such bush ) . if we decompose the displacement pattern ( [ eqch3 ] ) into the linear normal coordinates @xmath26\right|n=1 .. n\right\}\nonumber\\ & ( j=0 .. n-1),\label{eqch10}\end{aligned}\ ] ] we get the following form of the bush b@xmath14 $ ] in the _ modal space _ : @xmath27 where @xmath28 @xmath29 while @xmath30 and @xmath31 are time - dependent coefficients in front of the normal coordinates @xmath32 and @xmath33 . thus , only @xmath34 normal coordinates from the full set ( [ eqch10 ] ) contribute to the `` configuration vector '' @xmath15 corresponding to the given bush and we will refer to @xmath35 as the _ bush dimension_. in @xcite , we developed a simple crystallographic method for obtaining the displacement pattern @xmath15 for any subgroup @xmath12 of the parent group @xmath13 . using this method one can obtain bushes of different dimensions for an arbitrary nonlinear chain . the _ one - dimensional bushes _ ( @xmath36 ) represent symmetry - determined nonlinear normal modes . the displacement pattern @xmath15 corresponding to a given nnm depends on only one ( time - periodic ) function @xmath31 : @xmath37 where @xmath38 is a constant vector , which is formed by the coefficients @xmath39 ( @xmath40 ) from eq . ( [ eqch1 ] ) , while the function @xmath31 satisfies a certain differential equation . this so - called `` governing '' equation can be obtained by substitution of the ansatz ( [ eqch20 ] ) into the dynamical equations of the considered chain . in some sense , the concept of bushes of vibrational modes can be considered as a certain _ generalization _ of the notion of nnms by rosenberg . indeed , if we substitute the ansatz ( [ eqch12 ] ) into dynamical equations of the chain , we obviously get two `` governing '' equations for the functions @xmath31 and @xmath30 , that determines the above - discussed two - dimensional bush ( note that , in contrast to a nnm , such dynamical object describes , in general , a _ motion ) . finally , one can conclude that @xmath35-dimensional bush is determined by @xmath35 time - dependent functions for which @xmath35 governing differential equations can be obtained from the dynamical equations of the considered mechanical system . let us emphasize that bushes of modes represent a new type of _ exact _ excitations in nonlinear systems with discrete symmetries and the excitation energy proves to be trapped in a given bush for the case of hamiltonian systems . it is very important to emphasize that there exist only a _ finite number _ of vibrational bushes of any fixed dimension in every @xmath1-particle mechanical system . as a consequence , there is sufficiently small number of nnms ( one - dimensional bushes ) in the fpu chains ( three nnms for the fpu-@xmath41 model and six for the fpu-@xmath0 model ) . all possible one - dimensional bushes are explicitly listed in our papers @xcite ( see also @xcite ) . the stability of some nnms in the fpu chains has been studied in @xcite by numerical and analytical methods . let us comment explicitly on the recent papers @xcite . in @xcite , t. bountis and coworkers have investigated the destabilization thresholds ( @xmath42 and @xmath43 ) of two nonlinear normal modes which they call spo-1 and spo-2 ( simple periodic orbits ) by numerical methods . the authors of the above papers try to reveal some relations between the destabilization thresholds @xmath42 , @xmath43 and the origin of the weak chaos in connection with the famous fermi - pasta - ulam problem of the energy equipartition between different modes . in particular , they conclude that the main role in the weak chaos appearance in the thermodynamic limit ( @xmath44 ) plays spo-2 , because @xmath45 , @xmath46 , and , therefore , @xmath47 . however , there are some other spos in the fpu-@xmath0 chain and one can be interested in their role in the origin of the weak chaos in the thermodynamic limit . some comments are appropriate at this point . according to lyapunov @xcite , some strictly periodic orbits for nonlinear systems can be obtained from the linear normal modes ( which are introduced in the harmonic approximation ) by continuation with respect to a parameter characterizing the strength of nonlinearity . from this point of view , there exist @xmath1 different spos for longitudinal vibrations of an @xmath1 particle monoatomic chain . however , only few of the modes , constructed in such a way , possess an _ identical _ time dependence of the displacements of all the particles . more exactly , only few of the lyapunov modes can be written in the form ( [ eqch20 ] ) implying a separation of time and space variables that is typical for the rosenberg nonlinear normal modes . indeed , in general case , @xmath48 where @xmath49 ( @xmath50 ) are _ different _ functions of time with identical periods . note that in the present paper we consider only _ extended _ spos , but the same problem there exists for _ localized _ periodic modes ( discrete breathers ) and we have discussed it in detail in @xcite . as far as we aware , all periodic solutions in monoatomic chains that have been studied up to now ( see the above cited papers @xcite and references therein ) belong namely to the class of the rosenberg nonlinear normal modes determined by eq . ( [ eqch20 ] ) . moreover , the spatial profiles @xmath51 of these modes possess certain symmetry properties . in particular , every such mode can be characterized by a multiplication number ( @xmath19 ) determining the enlargement of the primitive cell of the vibrational state in comparison with that of the equilibrium state . as was already noted , we refer to these modes as symmetry - determined nnms and there exist only finite number of such modes ( even for the case @xmath44 ! ) for each nonlinear chain @xcite . above considered spo-1 , spo-2 and the well - known @xmath52-mode ( zone boundary mode ) represent nnms with multiplication numbers 4 , 3 , and 2 , respectively . however , among _ six _ symmetry - determined nnms in the fpu-@xmath0 chain @xcite there exist another three nnms with @xmath53 , @xmath20 and @xmath54 . the stability properties of the second nnm with @xmath20 were studied by m. leo and r.a . leo in @xcite . the stability of this mode was investigated in the thermodynamic limit by both numerical and analytical methods . the stability diagrams for all the nonlinear normal modes in the fpu-@xmath0 chain , as well as for the fpu-@xmath41 chain , can be found in our paper @xcite . with the aid of these diagrams , one may reveal many stability properties of nnms for an arbitrary @xmath1 , in particular , for the thermodynamic limit ( @xmath44 ) . note that these diagrams were obtained numerically . in this paper , we present some _ analytical _ results for the stability properties of all nnms in the fpu-@xmath0 chain in the _ thermodynamic _ limit ( @xmath44 ) . we also compare our results with those by different authors when it is possible . in sec . 2 , we consider all the possible symmetry - determined nonlinear normal modes in the fpu-@xmath0 chain . in sec . 3 , the stability diagrams for these nnms are discussed . in sec . 4 , the analytical method for studying the stability of nnms in the thermodynamic limit is presented . in sec . 5 , we list our results on the stability properties for every nnm . as was already mentioned , there exists only _ finite _ number of symmetry - determined nnms in any monoatomic chain . every nnm corresponds to a certain _ subgroup _ of the symmetry group of the chain dynamical equations . the difference in the number of nonlinear normal modes for the fpu-@xmath41 chain ( three nnms ) and the fpu-@xmath0 chain ( six nnms ) is associated with the fact that the symmetry group of the fpu-@xmath0 chain dynamical equations is higher than that of the fpu-@xmath41 chain @xcite . in @xcite , we have investigated the stability of all nnms both in the fpu-@xmath41 and fpu-@xmath0 chains ( for the case @xmath55 ) by numerical methods . let us comment on the main idea of this investigation . following the standard method of the linear stability analysis , we linearize the fpu-@xmath0 dynamical equations near a given nnm and get the linearized system in the form @xmath56 , where @xmath57 represents the infinitesimal perturbation vector , while @xmath58 is the jacobian matrix of the original system of nonlinear differential equations . thus , we obtain @xmath1 linear differential equations with time - periodic coefficients depending on the considered nnm . then the floquet method can be applied for studying the stability of the zero solution of the system @xmath56 . however , such straightforward way for the stability analyzing becomes practically impossible for @xmath59 . in @xcite , the general group - theoretical method has been developed for splitting ( decomposition ) of the original system @xmath56 of @xmath1 linear differential equations into certain subsystems of sufficiently small dimensions @xmath60 . for the fpu-@xmath0 chain , these dimensions do not exceed three ( see below ) . then we have applied the floquet method for such subsystems of small dimensions . moreover , proceeding in this manner , one can reveal those subsets of the vibrational modes , which are responsible for the loss of stability of the considered nnm . as a consequence of this approach , it proves to be possible to construct very transparent diagrams , which demonstrate explicitly stability properties of each fpu nonlinear normal mode @xcite . the fpu-@xmath0 model represents a chain of unit masses coupled with each other by the appropriate nonlinear springs . the dynamical equations describing longitudinal vibrations of the fpu-@xmath0 chain can be written in the form @xmath61 where @xmath2 is the displacement of the @xmath24th particle from its equilibrium state at the instant @xmath4 , while the force @xmath62 depends on the spring deformation @xmath63 as @xmath64 the periodic boundary condition is assumed to hold : @xmath65 let us mention some results of the paper @xcite , which are necessary for our further discussions . every nnm in the fpu-@xmath0 chain can be written as follows [ see eq . ( [ eqch20 ] ) ] : @xmath66 where @xmath31 satisfies the duffing equation @xmath67 with different values @xmath9 and @xmath68 for different nnms . the function @xmath31 describes the time - evolution of a given nnm , while the @xmath1-dimensional vector @xmath38 determines the pattern of the displacements of all particles of the chain . below , we list all possible nnms in the fpu-@xmath0 chain . 1 . b@xmath69 $ ] : @xmath70 this is a boundary zone mode or @xmath52-mode . 2 . b@xmath71 $ ] : @xmath72 there exist three `` dynamical domains '' of this nnm ( see below ) . 3 . b@xmath73 $ ] : @xmath74 there exist two dynamical domains of this nnm . 4 . b@xmath75 $ ] : @xmath76 there exist three dynamical domains of this nnm . b@xmath77 $ ] : @xmath78 there exist two dynamical domains of this nnm b@xmath79 $ ] : @xmath80 there exist three dynamical domains of this nnm . let us comment on the above listed nnms in the fpu-@xmath0 chain . every nnm , denoted by the symbol b@xmath11 $ ] , is characterized by the corresponding symmetry group @xmath12 , that represents a certain subgroup of the symmetry group @xmath81}$ ] of the fpu-@xmath0 dynamical equations ( [ eqch35],[eqch36 ] ) . we determine every such group by the set of its generators using the following notations : @xmath23 : : the translation of the chain by one lattice spacing , @xmath24 : : the inversion with respect to the center of the chain , @xmath82 : : the operator , that changes signs of the displacements of all particles without any their transposition . the symmetry group @xmath81}$ ] of the fpu-@xmath0 dynamical equations is described by _ three _ generators ( @xmath23 , @xmath24 , and @xmath82 ) . the corresponding transformations @xmath23 , @xmath24 and @xmath82 of @xmath1-dimensional vectors @xmath83 _ do not _ change the dynamical equations ( [ eqch35],[eqch36 ] ) of the fpu-@xmath0 chain . all the above listed groups of nnms are fully described by only _ two _ generators , but these generators can be written as some _ products _ of the generators @xmath23 , @xmath24 , and @xmath82 of the group @xmath13 . for example , @xmath84 , @xmath85 , @xmath22 are translations of the chain by two , three and four lattice spacings , respectively . the transformation @xmath86 means that we must perform the inversion of the displacement pattern with respect to the chain center and then translate it by one lattice spacing . note that transformations @xmath23 and @xmath24 do not commute : @xmath87 [ the relation @xmath88 holds because of the periodic boundary condition ( [ eqch37 ] ) ] . on the other hand , the transformation @xmath82 does commute with both @xmath23 and @xmath24 transformations : @xmath89 the transformation @xmath90 means that we must change signs of all displacements and then translate the displacement pattern @xmath91 by two lattice spacings . some simple examples are worth mentioning at this point . for the chain with @xmath92 particles , we can write the following relations : @xmath93 the displacement pattern corresponding to a given nnm can be obtained as _ invariant vector _ of its symmetry group @xmath94 . for example , let us obtain the displacement pattern for the nnm with @xmath95 $ ] [ see eq . ( [ eqch_62 ] ) ] . for simplicity , we demonstrate the method for obtaining displacement patterns with the case @xmath96 . let @xmath97 where @xmath98 ( @xmath99 ) are arbitrary displacements of eight particles of the chain . the vector @xmath91 must be invariant with respect to the action of our two generators @xmath22 and @xmath86 of the symmetry group of the considered nnm : @xmath100 the former equation is reduced to the following form : @xmath101 from which we conclude that @xmath102 this displacement pattern is formed by two primitive cells whose size four times larger than that of the fpu-@xmath0 chain in its equilibrium state . the sets of the displacements in both cells are identical : @xmath103 now let us take into account the second equation ( [ eqch65 ] ) . acting on the vector ( [ eqch66 ] ) by @xmath86 , we obtain @xmath104 then using the equation @xmath105 , we get @xmath106 thus , the invariant ( under the action of the group @xmath107}$ ] ) vector @xmath91 depends on _ only one _ arbitrary parameter , which we denote by @xmath108 : @xmath109 ( note that this vector being invariant with respect to generators of the group @xmath107}$ ] will automatically be invariant relative to all its other elements ) . then the nnm corresponding to the invariant vector ( [ eqch60 ] ) can be written as follows @xmath110 to find all nnms , we can try _ all subgroups _ of the symmetry group @xmath81}$ ] to choose those displacement patterns , which depend on _ only one _ arbitrary parameter . the patterns depending on @xmath35 arbitrary parameters with @xmath111 form the @xmath35-dimensional bushes of vibrational modes . namely in this sense nonlinear normal modes may be called one - dimensional bushes . in @xcite , three different group - theoretical methods for constructing the bushes of vibrational modes in _ arbitrary _ @xmath1-particle nonlinear mechanical systems were developed . the most efficient of these methods uses the concept of irreducible representations of the symmetry groups . taking into account the above method that was used for constructing eq . ( [ eqch70 ] ) , we conclude that every nnm can be written in the form @xmath112 where @xmath38 is a certain time - independent vector . substituting ansatz ( [ eqch71 ] ) into the dynamical equations ( [ eqch35][eqch36 ] ) of the fpu-@xmath0 chain , with explicit forms of the vectors @xmath38 from eqs . ( [ eqch_60][eqch_65 ] ) , one can find that fpu-@xmath0 equations are reduced to only one differential equation ( governing equation of the corresponding nnm ) of the form : @xmath113 this is the duffing equation with different values @xmath9 and @xmath68 for different nnms which are listed in eqs . ( [ eqch_60][eqch_65 ] ) . above , we have mentioned the existence of so - called `` dynamical domains '' of all nonlinear normal modes presented in eqs . ( [ eqch_60][eqch_65 ] ) . let us comment on this notion borrowed from the theory of phase transitions . we have already emphasized that a certain symmetry group @xmath12 corresponds to every nnm . this group is a subgroup of the symmetry group of the considered mechanical system in its equilibrium state ( @xmath114 ) . if we act on the vector @xmath15 corresponding to a given nnm by operator @xmath115 , that _ does not _ belong to subgroup @xmath12 , we get the _ equivalent _ configuration vector @xmath116 . the equivalent vector @xmath117 corresponds to a new nnm , which is described by the _ same _ dynamical equations as that of the nnm associated with the vector @xmath15 . for example , three dynamical domains are associated with the nnm from eq . ( [ eqch_61 ] ) : @xmath118:&\vec c=\frac{3}{\sqrt{6n}}\{1,0,-1~|~1,0,-1~|~1,0,-1~|~\ldots\},\label{eqch90}\\ \text{b}[a^3,ai]:&\vec c=\frac{3}{\sqrt{6n}}\{0,1,-1~|~0,1,-1~|~0,1,-1~|~\ldots\},\label{eqch91}\\ \text{b}[a^3,a^2i]:&\vec c=\frac{3}{\sqrt{6n}}\{1,-1,0~|~1,-1,0~|~1,-1,0~|~\ldots\}.\label{eqch92}\end{aligned}\ ] ] all the displacement patterns ( [ eqch90][eqch92 ] ) differ from each other by a cyclic transposition of the displacements inside each primitive cell of the chain equilibrium state . let us note that the symmetry groups @xmath119 ( @xmath120 ) of nnms from eqs . ( [ eqch90][eqch92 ] ) prove to be _ conjugate _ subgroups in the parent group @xmath13 , for example , @xmath121 ( @xmath115 ) . since the above - discussed `` domains '' possess equivalent dynamical properties , we study below the stability of only one copy of the full set of dynamical domains for every nnm in the fpu-@xmath0 chain . all symmetry - determined nnms that can exist in the fpu-@xmath0 chain with an appropriate number of particles are listed in table [ table10 ] . [ cols="<,^,^,^ " , ] the time - depending function @xmath31 entering eq . ( [ eqch100 ] ) is a periodic solution to the governing eq . ( [ eqch38 ] ) . for every nnm in the fpu-@xmath0 chain , this governing equation represents duffing equation @xmath122 where @xmath23 is the squared frequency of the harmonic approximation , while @xmath123 is a nonlinearity coefficient . ( [ eqch39 ] ) is called the _ hard _ ( _ soft _ ) duffing equation if @xmath124 ( @xmath125 ) . for initial conditions @xmath126 , @xmath127 , the solution of the hard duffing equation can be written in the form @xmath128 here @xmath129 while modulus @xmath130 of the jacobi elliptic cosine is determined by the relation @xmath131 the solution ( [ eqch40a ] ) represents periodic function with the period @xmath132 where @xmath133 is the complete elliptic integral of the first kind . for the same initial conditions , the solution of the soft duffing equation ( @xmath125 ) can be written in the form @xmath134 with @xmath135 @xmath136 @xmath137 it is convenient to introduce the time scaling @xmath138 which transforms eq . ( [ eqch100 ] ) into the equation with @xmath52-periodic coefficients . as a result of this scaling the form of eq . ( [ eqch100 ] ) does not change , but the constant @xmath23 and @xmath139 , entering this equation , must be multiplied by @xmath140 . we do not change notations in eq . ( [ eqch100 ] ) , however , we imply below that the above transformations are already fulfilled . our further stability analysis of nnms reduces to investigating the stability of the zero solution of eq . ( [ eqch100 ] ) . the analysis consists of the following steps : * step 1 . * simplification of eq . ( [ eqch100 ] ) in the thermodynamic limit ( @xmath44 ) . in this case @xmath141 and we can decompose the coefficients of eq . ( [ eqch100 ] ) into power series with respect to the small dimensionless parameter @xmath142 * step 2 . * search for the general solution of the approximate equation that was obtained as a result of step 1 . * step 3 . * construction of the monodromy matrix with the aid of the above solution . * step 4 . * construction of the characteristic polynomial of the monodromy matrix . * step 5 . * analyzing discriminant of the characteristic polynomial in the limit @xmath143 . let us consider these steps in turn . the function @xmath31 in the form ( [ eqch40a ] ) for the case @xmath55 and in the form ( [ eqch50 ] ) for @xmath144 must be substituted into eq . ( [ eqch100 ] ) taking into account that modulus @xmath130 goes to zero when @xmath44 . to simplify @xmath31 , we use the following formulas from the theory of elliptic functions @xcite : @xmath145 @xmath146 where @xmath147 ^ 2\kappa^{2n}+\ldots\right\}\label{eqch122}\end{aligned}\ ] ] is the complete elliptic integral of the first kind , while @xmath148 ( @xmath149 is the complimentary modulus of the elliptic functions ) . note that the modulus @xmath130 depends on @xmath150 in a different manner for the cases @xmath55 and @xmath144 [ see eqs . ( [ eqch42 ] ) and ( [ eqch52 ] ) , respectively ] . now , we have to decompose the left - hand - side of eq . ( [ eqch100 ] ) into the power series with respect to the small parameter @xmath151 . this very cumbersome decomposition has been performed with the aid of the maple^^ mathematical package . the corresponding result can be written as follows : @xmath152{\varepsilon}+\right.\nonumber\\ & \quad\left[\frac{75}{128}\hat\omega^2+\left(-\frac{13}{32}-\frac{3}{8}\cos 2\tau+\frac{1}{32}\cos 4\tau\right)\hat m\right]{\varepsilon}^2+\label{eqmy40}\\ & \quad\left.\left[-\frac{243}{512}\hat\omega^2+\left(\frac{87}{256}+\frac{597}{2048}\cos 2\tau-\frac{3}{64}\cos 4\tau+\frac{3}{2048}\cos 6\tau\right)\hat m\right]{\varepsilon}^3\right\}\vec\mu+\nonumber\\ & o\left({\varepsilon}^4\right)=0,\nonumber\end{aligned}\ ] ] eq . ( [ eqmy40 ] ) represents a system of differential equations with time - periodic coefficients and to construct the corresponding monodromy matrix we must obtain its solution for @xmath153 . on the other hand , for small time intervals , the solution of eq . ( [ eqmy40 ] ) can be found by a simple perturbation theory . to that end we decompose @xmath154 into a formal series @xmath155 substitute it into eq . ( [ eqmy40 ] ) and equate to zero the terms with every fixed power of the small parameter @xmath156 . as a result , we get the following set of differential equations [ eqmy41 ] @xmath157\vec\mu_0,\label{eqmy41b}\\ \ddot{\vec\mu}_2+\hat\omega^2\vec\mu_2=&-\left[-\frac{3}{4}\hat\omega^2+\left(\frac{1}{2}+\frac{1}{2}\cos 2\tau\right)\hat m\right]\vec\mu_1\label{eqmy41c}\\ & -\left[\frac{75}{128}\hat\omega^2+\left(-\frac{13}{32}-\frac{3}{8}\cos 2\tau+\frac{1}{32}\cos 4\tau\right)\hat m\right]\vec\mu_0,\nonumber\\ \ldots\nonumber\end{aligned}\ ] ] because of the diagonal form of the matrix @xmath158 ( see table [ table20 ] ) , these equations determine certain sets of harmonic oscillators with different time - periodic external forces . each of these oscillators is described by equation @xmath159 the general solution to this equation , obtained by the method of variation of arbitrary constants , can be written in the form @xmath160dt,\ ] ] where @xmath161 , in our case , represents a _ sum _ of time - periodic functions whose frequencies are _ incommensurable_. indeed , for the most nnms from table [ table20 ] , @xmath158 are matrices with different diagonal elements and , therefore , the components of the vector @xmath162 from ( [ eqmy41a ] ) vibrate with different frequencies . substituting @xmath162 into ( [ eqmy41b ] ) leads to mixing its time - depended components because of multiplying by the matrix @xmath163 , and such a mixing produces more and more complicated terms in r.h.s . of eqs . ( [ eqmy41 ] ) when we take into account higher orders in the decomposition ( [ eqmy40 ] ) the usual way to study stability of a given periodic dynamical regime is the floquet method . in this method , we linearize nonlinear equations of motion in the vicinity of the periodic solution and calculate the _ monodromy matrix _ @xmath164 by integrating @xmath165 times the linearized equations with time - periodic coefficients over one period @xmath166 using specific initial conditions [ @xmath167 is the number of equations in ( [ eqmy40 ] ) ] . these conditions are determined by the successive columns of @xmath168 identity matrix . solving eqs . ( [ eqmy41 ] ) in step - by - step manner , we can construct the approximate analytical solution to eq . ( [ eqmy40 ] ) up to a fixed order of the small parameter @xmath156 . with the aid of this solution , we are able to obtain the monodromy matrix @xmath169 for eq . ( [ eqmy40 ] ) , where @xmath52 is the period of its coefficients . the stability of the considered periodic solution is determined by floquet multipliers representing eigenvalues of the monodromy matrix . if all these multipliers lie on the unit circle , the solution is _ linear stable_. in other case , the solution _ linear unstable_. we obtain eigenvalues of the monodromic matrix @xmath169 as the roots of its characteristic polynomial . let us remind that according to the newton formulas , the coefficients of the characteristic polynomial @xmath170 of any @xmath171 matrix @xmath172 can be expressed via the sums @xmath173 with the aid of the recurrence relation @xmath174 thus , we have @xmath175 on the other hand , all sums @xmath176 in eq . ( [ eqch500 ] ) can be found directly by means of traces of the matrix @xmath172 : @xmath177 it is well known , that in the case of any hamiltonian system with @xmath167 degree of freedom floquet multipliers @xmath178 form pairs @xmath179 ( @xmath50 ) and , as a consequence , the characteristic polynomial @xmath180 of the monodromic matrix @xmath169 proves to be _ palindromic _ : @xmath181 with the following coefficients @xmath8 : @xmath182,\\ & \cdots\end{aligned}\ ] ] now we have to obtain formulas for discriminants @xmath183 , @xmath184 and @xmath185 for the corresponding palindromic characteristic polynomials of @xmath186 degrees via traces of monodromy matrices . from the above newton formulas applied to the polynomial ( [ eqcha1 ] ) , we obtain the following formulas for its coefficients : @xmath187,\label{eqcha2}\\ a_3=&\frac{1}{6}\left[\operatorname{tr}^3\hat x(\pi)-3\operatorname{tr}\hat x(\pi)\operatorname{tr}\hat x^2(\pi)+2\operatorname{tr}\hat x^3(\pi)\right].\nonumber\end{aligned}\ ] ] on the other hand , one can express discriminant @xmath188 explicitly via these coefficients of the characteristic polynomial . with the aid of maple^^ , we finally find @xmath189 where @xmath190 , @xmath191 , @xmath192 are given by eqs . ( [ eqcha2 ] ) . now we present some illustrations of the above discussed technique . _ example 1 . _ nonlinear normal mode b@xmath69 $ ] ( @xmath52-mode ) : @xmath193 . firstly , let us discuss the case @xmath55 . one can see that there exist modes , corresponding to the left and right sides of the black region in fig . [ figstabdiag]a , which are not excited by parametric interactions with the @xmath52-mode . this fact was revealed analytically in @xcite with the aid of rotating wave approximation ( rwa ) . in @xcite , in the framework of the same approximation the following relation between the amplitude threshold value @xmath194 and the wavenumber @xmath195 ( i.e. the boundary curve of the black region in fig . [ figstabdiag]a ) was obtained : @xmath196 this analytical formula is in good agreement with the numerical results . let us now consider the stability threshold of the @xmath52-mode in the thermodynamic limit @xmath143 using the above discussed method . we have to consider the vicinity of the point @xmath197 on the @xmath198 plane ( @xmath199 ) . the one - dimensional constant matrices of the decoupled variational system can be found in table [ table20 ] : @xmath200 the monodromy matrix also depends on the wavenumber @xmath195 and we can decompose its trace @xmath201 into the taylor series in two small parameters @xmath202 and @xmath151 ( in our case , @xmath203 , @xmath204 and , therefore , @xmath205 ) . this decomposition read @xmath206 in the considered case , @xmath207 and the corresponding discriminant is @xmath208 where @xmath209 . the condition @xmath210 leads to the equations @xmath211 from the first of these equation , we find @xmath212 and then we obtain @xmath213 substitution of @xmath214 and @xmath215 leads us to the following result @xmath216 the corresponding energy per one particle is @xmath217 the second equation ( [ eqch261 ] ) leads to a contradiction with the condition of smallness of the parameter @xmath156 and , therefore , this equation does nt produce instability of the @xmath52-mode in the limit @xmath143 . note that the analytical dependence @xmath218 was revealed in @xcite , and later was recovered in @xcite . in the case @xmath144 , the stability properties of the @xmath52-mode are utterly different . indeed , one can see in fig . [ figstabdiag]a ( right column ) that this mode turns out to be stable up to the finite value of its amplitude @xmath194 . using a numerical method , we found that in the thermodynamic limit @xmath143 @xmath219 ( @xmath220 ) . _ example 2 . _ nonlinear normal mode b@xmath77 $ ] : @xmath221 . the variational system for this nnm is decoupled into @xmath222 subsystems with time - periodic coefficients . as a result , for studying the stability loss of the mode b@xmath77 $ ] we have to vanish the discriminant @xmath184 from ( [ eqch301 ] ) . the first factor of @xmath184 near the resonance wavenumber @xmath199 is equal to @xmath223 being positive , this factor can not lead to the condition @xmath224 . the second factor of @xmath184 from eq . ( [ eqch301 ] ) reads @xmath225 also does nt vanish in the thermodynamic limit @xmath143 . only the last factor of the discriminant @xmath184 @xmath226 can lead to fulfilment of the condition @xmath224 . this yields @xmath227 and , therefore , @xmath228 note that this nnm has been investigated quite enough in @xcite . above , we simply reproduced the main result of this paper by our method . _ example 3 . _ nonlinear normal modes b@xmath71 $ ] : @xmath229 and b@xmath75 $ ] : @xmath230 . the variational systems for these nnms can be decoupled into @xmath231 independent subsystems whose matrices are presented in table [ table20 ] . now one has to vanish the discriminant @xmath185 from eq . ( [ eqch302 ] ) . with the aid of our method , we get the following results . for both modes b@xmath71 $ ] and b@xmath75 $ ] , we have obtained identical scaling for the case @xmath143 : @xmath232 some numerical results on stability of the nnm b@xmath71 $ ] have been found in @xcite , but we do nt know any results on stability properties of the nnm b@xmath75 $ ] . in conclusion , let us consider another scenario of the stability loss of nnms . indeed , up to this point , we have discussed only the loss of stability associated with parametric interactions of a given nnm with other ( linear ) normal modes of the fpu-@xmath0 chain . some nnms in the fpu-@xmath0 chain , when @xmath233 , transform not into _ one _ linear normal mode ( lnm ) , but into a certain _ superposition _ of such modes . for example , nnm b@xmath73 $ ] @xmath234 transforms , in the case @xmath235 , into the linear combination @xmath236 of two linear normal modes @xmath237 where @xmath238 one can also say that nnm b@xmath73 $ ] is the result of the continuation of the superposition ( [ eqch701 ] ) with respect to the nonlinearity parameter @xmath156 of the considered fpu-@xmath0 chain . we have to emphasize that the continuation of an _ arbitrary _ linear combination of two above discussed lnms does nt represent an exact solution to nonlinear dynamical equations of the fpu-@xmath0 chain , while the superposition with @xmath239 produces an exact solution . then , the following question arises : if we will increase the parameter @xmath156 from zero , is it possible that the discussed nnm ( [ eqch700 ] ) loses stability because of transformation into a _ two - dimensional bush _ @xmath240 with two _ different _ functions of time @xmath31 , @xmath30 ? in general , such a bush describes not periodic , but a _ quasiperiodic _ dynamical regime in the fpu-@xmath0 chain . in contrast to the previous case , this stability loss scenario does nt imply any extension of the vibrational modes set , but it means breaking the correlation ( [ eqch701 ] ) between two lnms , @xmath241 and @xmath242 . we do nt discuss this scenario of the stability loss in the present paper . however , our analysis allows us to assert that such scenario is inessential for all six symmetry - determined nnms in the fpu-@xmath0 chain . with the aid of the above discussed method , we investigated scaling of stability thresholds in the thermodynamic limit @xmath143 for all possible in the fpu-@xmath0 chain nonlinear normal modes for both cases @xmath55 and @xmath144 . the summary results are presented in tables [ table3 ] , [ table4 ] . let us comment on these results . @xmath243 & \frac{\pi}{\sqrt{6\beta n } } & \frac{\pi^2}{3\beta n^2 } & \text{\cite{budinsky1983,bermankolovskij1984,sanduskypage1994,flach1996,dauxois1997,poggiruffo1997,bountis2006a , yoshimura2004,dauxois2005}}\\ b[a^3,i ] & \frac{2\pi}{3\sqrt{\beta n } } & \frac{2\pi^2}{3\beta n^2 } & \text{\cite{bountis2006}}\\ b[a^4,ai ] & \sqrt{\frac{2\pi}{3\beta } } & \frac{2\pi}{3\beta n } & \text{\cite{bountis2006,bountis2006a}}\\ b[a^3,iu ] & \frac{2\pi}{3\sqrt{\beta n } } & \frac{2\pi^2}{3\beta n^2 } & \\ b[a^4,iu ] & \frac{\sqrt{2}\pi}{\sqrt{3\beta n } } & \frac{2\pi^2}{3\beta n^2 } & \text{\cite{leoleo2007}}\\ b[a^6,ai , a^3u ] & \frac{2\pi}{\sqrt{3\beta n } } & \frac{2\pi^2}{3\beta n^2 } & \\ \hline \end{array}\ ] ] @xmath244 & \sqrt{\frac{2\pi}{3|\beta| } } & \frac{2\pi}{3|\beta|n } & \\ b[a^4,iu ] & \frac{\sqrt{2}\pi}{\sqrt{3|\beta|n } } & \frac{2\pi^2}{3|\beta|n^2}&\text{\cite{leoleo2007}}\\ b[a^6,ai , a^3u ] & 0 & 0 & \\ \hline \end{array}\ ] ] in the first column of tables [ table3 ] , [ table4 ] , the symbols of nnms are given . in two next columns , we present the scaling laws for @xmath143 of stability thresholds for the amplitude of each nnm , @xmath245 , and for its specific energy , @xmath246 ( the energy per one particle of the chain ) . in the last column of the above tables , we give references to the papers in which stability thresholds of the corresponding nnms are also discussed . firstly , let us pay attention to the following interesting fact : four of six nnms in the fpu-@xmath0 chain , namely , , obeys different duffing equations [ see eqs . ( [ eqch38])-([eqch_65 ] ) ] . ] @xmath247 & = & \nu(t)\cdot\frac{3}{\sqrt{6n}}\{1,0,-1\,|\,1,0,-1\,|\,\ldots\},\\ \text{b}[a^3,iu ] & = & \nu(t)\cdot\frac{1}{\sqrt{2n}}\{1,-2,1\,|\,1,-2,1\,|\,\ldots\},\\ \text{b}[a^4,iu ] & = & \nu(t)\cdot\frac{1}{\sqrt{n}}\{1,-1,-1,1\,|\,1,-1,-1,1\,|\,\ldots\},\\ \text{b}[a^3u , ai ] & = & \nu(t)\cdot\frac{3}{\sqrt{6n}}\{{\scriptstyle 0,1,1,0,-1,-1\,|\,0,1,1,0,-1,-1\,|\,\ldots}\},\\ \end{array}\ ] ] possess _ identical _ scaling of the stability threshold in the limit @xmath143 : @xmath248 the scaling of @xmath246 for the @xmath52-mode @xmath249 = \nu(t)\cdot\frac{1}{\sqrt{n}}\{1,-1\,|\,1,-1\,|\,\ldots\}\ ] ] is _ exactly _ twice less than that determined by eq . ( [ eqch_500 ] ) . only for the mode @xmath250 = \nu(t)\cdot\frac{2}{\sqrt{2n}}\{0,1,0,-1\,|\,0,1,0,-1\,|\,\ldots\}\\\ ] ] the scaling law of the stability threshold turns out to be cardinally different : @xmath251 a qualitative difference in scaling between this mode and all other nnms for the fpu-@xmath0 chain with @xmath55 can be seen in the left column of fig . [ figstabdiag ] . the stability properties of the same nnms in the fpu-@xmath0 chain with @xmath144 prove to be essentially different , as one can see in the right column of fig . [ figstabdiag ] . firstly , for three nnms , namely , b@xmath69 $ ] , b@xmath71 $ ] and b@xmath75 $ ] the stability thresholds @xmath246 do nt tend to zero when @xmath143 . these modes lose their stability for a certain _ finite _ value @xmath194 of the nnm s amplitude . for the above listed nonlinear normal modes , these values are equal respectively to @xmath252 , @xmath253 , and @xmath254 . the fundamental difference between scaling of @xmath246 for the modes b@xmath73 $ ] and b@xmath77 $ ] takes place in the case @xmath144 , as well as for the above discussed case @xmath55 : @xmath255 for b@xmath73 $ ] and @xmath256 for b@xmath77 $ ] ( see table [ table4 ] ) . studying of the stability threshold for the nnm b@xmath79 $ ] proves to be more difficult . in this case , the loss of stability is determined by the second scenario discussed at the beginning of the present section . normalizing the variational equations , described dynamics of the vibrational modes with basis vectors @xmath32 , @xmath257 , simultaneously with the duffing equation , corresponding to nnm b@xmath79 $ ] , we infer that for @xmath144 there exists an exponential _ detuning _ between the above modes for an arbitrary small amplitude of the investigated nnm . this means that @xmath246 turns out to be equal to zero as indicated in table [ table4 ] . in the present paper , a certain asymptotic technique for studying the stability loss of nonlinear normal modes in the fpu-@xmath0 chain in the thermodynamic limit @xmath143 is developed . using this technique we were able to obtain the scaling laws of the stability threshold @xmath246 for all six symmetry - determined nnms that are possible in the fpu-@xmath0 chain for both cases @xmath55 and @xmath144 . the general method @xcite for splitting the variational system for a given dynamical regime in a physical system with discrete symmetry into independent subsystems of small dimensions was applied for investigation of stability of nnms in the fpu-@xmath0 chain . the above dimensions for the considered case turn out to be equal to 1 , 2 and 3 . this splitting allows us to construct numerically the stability diagrams ( fig . [ figstabdiag ] ) that can help to reveal many interesting properties of nonlinear normal modes , such as qualitative behaviour of stability thresholds @xmath246 in the thermodynamic limit @xmath143 , the existence of the `` second stability threshold '' for some nnms , existence of finite limits of @xmath246 for certain modes , etc . chechin , d.s . ryabov , and v.p . sakhnenko , _ bushes of normal modes as exact excitations in nonlinear dynamical systems with discrete symmetry _ ( in : `` nonlinear phenomena research perspectives '' , pp . 225327 , ed . c. w. wang , nova science publishers , ny , 2007 ) . | all possible symmetry - determined nonlinear normal modes ( also called by simple periodic orbits , one - mode solutions etc . ) in both hard and soft fermi - pasta - ulam-@xmath0 chains are discussed . a general method for studying their stability in the thermodynamic limit , as well as its application for each of the above nonlinear normal modes
are presented . |
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systems with correlated electrons display a rich variety of physical phenomena and properties : different types of magnetic ordering , ( high - tc ) superconductivity , ferroelectricity , spin - charge separation , formation of spatial inhomogeneities @xcite ( phase separation , stripes , local gap and incoherent pairing , charge and spin pseudogaps ) . the realization of these properties in clusters and bulk depends on interaction strength @xmath1 , doping , temperature , the detailed type of crystal lattice and sign of coupling @xmath2 @xcite . studies of perplexing physics of electron behavior in non - bipartite lattices encounter enormous difficulties . exact solutions at finite temperatures exist only in a very few cases @xcite ; perturbation theory is usually inadequate while numerical methods have serious limitations , such as in the quantum monte - carlo method and its notorious sign problem . on the contrary , one can get important insights from the exact solutions for small clusters ( molecules " ) . for example , squares or cubes are the building blocks , or prototypes , of solids with bipartite lattices , whereas triangles , tetrahedrons , octahedrons without electron - hole symmetry may be regarded as primitive units of typical frustrated systems ( triangular , pyrochlore , perovskite ) . exact studies of various cluster topologies can thus be very useful for understanding nanoparticles and respective bulk systems . one can take a further step and consider an inhomogeneous bulk system as a collection of many such decoupled clusters , which do not interact directly , but form a system in thermodynamic equilibrium @xcite . thus we consider a collection of such molecules " , not at fixed average number of electrons per each cluster , but as a grand canonical ensemble , for fixed chemical potential @xmath3 . the electrons can be splintered apart by spin - charge separation due to level crossings driven by @xmath1 or temperature , so that the collective excitations of electron charge and spin of different symmetries can become quite independent and propagate incoherently . we have found that local charge and spin density of states or corresponding susceptibilities can have different pseudogaps which is a sign of spin - charge separation . for large @xmath1 near half filling , holes prefer to be localized on separate clusters having mott - hubbard ( mh ) like charge pseudogaps @xcite and nagaoka ferromagnetism ( fm ) @xcite ; otherwise , spin density waves or spin liquids may be formed . at moderate @xmath1 , this approach leads to reconciliation of charge and spin degrees with redistribution of charge carriers or holes between square clusters . the latter , if present , can signal a tendency toward phase separation , or , if clusters prefer " to have two holes , it can be taken as a signature of pairing @xcite . this , in turn , could imply imposed opposite spin pairing followed by condensation of charge and spin degrees into a bcs - like coherent state . although this approach for large systems is only approximate , it nonetheless gives very important clues for understanding large systems whenever correlations are local . we have developed this approach in refs . @xcite and successfully applied to typical unfrustrated ( linked squares ) clusters . our results are directly applicable to nanosystems which usually contain many clusters , rather well separated and isolated from each other but nevertheless being in thermodynamic equilibrium with the possibility of having inhomogeneities for different number of electrons per cluster . interestingly , an ensemble of square clusters displays checkerboard " patterns , nanophase inhomogeneity , incoherent pairing and nucleation of pseudogaps @xcite . the purpose of this work is to further conduct similar extensive investigations in frustrated systems @xcite , exemplified by 4-site tetrahedrons and 5-site square pyramids . as we shall see , certain features in various topologies are quite different and these predictions could be exploited in the nanoscience frontier by synthesizing clusters or nanomaterials with unique properties @xcite . and spin @xmath4 gaps versus @xmath1 in an ensemble of squares at @xmath5 and @xmath6 . phase a : charge and spin pairing gaps of equal amplitude at @xmath7 describe bose condensation of electrons similar to bcs - like coherent pairing with a single energy gap . phase b : mott - hubbard like insulator at @xmath8 leads to @xmath9 spin liquid behavior . phase c : parallel ( triplet ) spin pairing ( @xmath10 ) at @xmath11 displays @xmath12 saturated ferromagnetism ( see sec . [ squares ] ) . ] and @xmath13 versus @xmath3 in grand canonical ensemble of tetrahedrons and fcs at @xmath14 , @xmath15 and @xmath16 . mott - hubbard like ferromagnetism for @xmath17 at @xmath18 in tetrahedron occurs for @xmath19 , while absence of charge pseudogap near @xmath20 metallic state with spin rigidity manifests level crossing degeneracy related to pairing ( see inset ) . ] versus @xmath1 for one hole off half filling in tetrahedrons and fcs at @xmath6 . in tetrahedron , @xmath21 at @xmath22 implies phase separation and coherent pairing with @xmath23 , while @xmath24 for @xmath12 at @xmath19 leads to a ferromagnetic insulator ( @xmath10 ) for all @xmath1 . in fcs , @xmath24 at @xmath25 for @xmath26 describes mott - hubbard like antiferromagnetism ( @xmath27 ) . ] versus @xmath1 at @xmath18 and @xmath15 in tetrahedron ( @xmath22 ) and deformed tetrahedral clusters ( @xmath28 and @xmath29 ) . charge and spin pairing gaps of equal amplitude at @xmath22 imply coherent pairing , while @xmath30 and @xmath31 at @xmath32 correspond to an unsaturated ferromagnetic insulator for @xmath9 . coherent pairing is retained in a narrow range near @xmath33 . ] -@xmath3 phase diagram of tetrahedrons without electron - hole symmetry at optimally doped @xmath5 regime near @xmath34 at @xmath35 and @xmath22 illustrates the condensation of electron charge and onset of phase separation for charge degrees below @xmath36 . the incoherent phase of preformed pairs with unpaired opposite spins exists above @xmath37 . below @xmath38 , the paired spin and charge coexist in a coherent pairing phase . the charge and spin susceptibility peaks , denoted by @xmath39 and @xmath40 , define pseudogaps calculated in the grand canonical ensemble , while @xmath41 and @xmath42 are evaluated in canonical ensemble . charge and spin peaks reconcile at @xmath43 , while @xmath44 peak below @xmath38 signifies metallic ( charge ) liquid ( see inset for square cluster and ref . @xcite ) . ] .ground state ( gs ) in various cluster geometries for one hole off half filling at large @xmath45 and @xmath26 having saturated ferromagnetism ( sf ) , unsaturated ferromagnetism ( uf ) , antiferromagnetism ( af ) or coherent pairing ( cp ) . [ cols="^,^,^,^,^,^,^,^",options="header " , ] exact diagonalization of the hubbard model ( hm ) @xmath46 and quantum statistical calculations of charge @xmath47 and spin @xmath48 susceptibilities , i.e. , _ fluctuations _ , in a grand canonical ensemble and pseudogaps @xmath49 and @xmath50 for canonical energies @xmath51 and @xmath52 ( @xmath53 and @xmath54 being the total number of electrons and spin respectively ) yield valuable insights into quantum critical points and various phase transitions as shown in ref . @xcite . by monitoring the peaks in @xmath44 and @xmath55 one can identify charge / spin pseudogaps and relevant crossover temperatures ; nodes , sign and amplitude of pseudogaps determine energy level crossings , phase separation , electron pairing ranges , spin - charge separation and reconciliation regions . [ squares ] for completeness and to facilitate the comparison with frustrated clusters , we first summarize the main results obtained earlier for small 2@xmath562 and 2@xmath564-sites bipartite clusters in refs . the energies are measured in units @xmath57 in all results that follow . [ fig : gap-4-site ] illustrates @xmath58 and @xmath59 in ensemble of 2@xmath562 square clusters at @xmath60 and @xmath61 . vanishing of gaps indicates energy ( multiple ) level crossings and corresponding quantum critical points , @xmath62 and @xmath63 . the negative gaps show phase separations for charge below @xmath64 and spin degrees above @xmath65 @xcite . phase a : negative charge gap below @xmath62 displays electron pairing @xmath66 and charge phase separation into hole - rich ( charged ) metal and hole - poor ( neutral ) cluster configurations . in a grand canonical approach @xmath67 at @xmath7 corresponds to electron charge redistribution with opposite spin ( singlet ) pairing . this picture for electron charge and spin gaps of equal amplitude @xmath68 of purely electronic nature at @xmath60 is similar to the bcs - like coherency in the attractive hm and will be called coherent pairing ( cp ) . in equilibrium , the spin singlet background ( @xmath69 ) stabilizes phase separation of paired electron charge in a quantum cp phase . the unique gap @xmath70 at @xmath6 in fig . [ fig : gap-4-site ] is consistent with the existence of a single quasiparticle energy gap in the bcs theory for @xmath71 @xcite . positive spin gap in fig . [ fig : gap-4-site ] at @xmath72 provides pairs rigidity in response to a magnetic field and temperature ( see sec . [ diagram ] ) . however , unlike in the bcs theory , the charge gap differs from spin gap as temperature increases . this shows that coherent thermal excitations in the exact solution are not quasiparticle - like renormalized electrons , as in the bcs theory , but collective paired charge and coupled opposite spins . the spin gap @xmath59 for excited @xmath12 configuration in fig . [ fig : gap-4-site ] above @xmath62 is shown for canonical energies in a stable mh - like state , @xmath30 . phase b : unsaturated ferromagnetism ( uf ) for unpaired @xmath9 with zero field @xmath55 peak for gapless @xmath73 projections and gapped @xmath27 for @xmath12 excitations at @xmath74 will be called a spin liquid . phase c : negative @xmath10 at @xmath11 defines @xmath12 saturated ferromagnetism ( sf ) . localized holes at @xmath30 rule out possible nagaoka fm in a metallic phase @xcite . field fluctuations lift @xmath75-degeneracy and lead to segregation of clusters into magnetic domains . it appears that the ensemble of square clusters share common and important features with larger bipartite clusters in the ground state and at finite temperatures @xcite ( see sec . [ diagram ] ) . for example , in 2x4 ladders ( fig . 5 of ref . @xcite ) , we have identified the existence of ( negative and positive ) oscillatory behavior in ( @xmath6 ) charge gaps as a function of @xmath1 . similar to what was seen in square clusters at low temperatures , we observe level crossing degeneracies in charge and spin sectors in bipartite 2@xmath564 clusters at relatively small and large @xmath1 values , respectively . thus the use of chemical potential and departure from zero temperature singularities in the canonical and grand canonical ensembles appear to be essential for understanding important physics related to the pseudogaps , phase separation , pairings and corresponding crossover temperatures . a full picture of coherent and incoherent pairing , electronic inhomogeneities and magnetism emerges only at finite , but rather low temperatures . ( if we set @xmath22 ev , the most of the interesting physics is seen to occur below a few hundred degrees k. ) the topology of the tetrahedron is equivalent to that of a square with next nearest neighbor coupling ( @xmath76 ) while the square pyramid of the octahedral structure in the htscs is related to face centered squares ( fcs ) . the average electron number @xmath77 and magnetization @xmath13 versus @xmath3 in fig . [ fig : num_mag ] for @xmath15 shows contrasting behavior in pairing and magnetism at @xmath22 and @xmath19 for the tetrahedron at @xmath78 and fcs at @xmath79 . different signs of @xmath2 in these topologies for one hole off half filling lead to dramatic changes in the electronic structure . [ fig : gap ] illustrates the charge gaps at small and moderate @xmath1 . tetrahedron at @xmath19 : sf with a negative spin gap in a mh - like phase exists for all @xmath1 . tetrahedron at @xmath22 : metallic cp phase with charge and spin gaps of equal amplitudes similar to the bcs - like pairing , discussed for squares in sec . [ squares ] , forms at all @xmath1 . fcs at @xmath19 : mh - like insulator displays two consecutive crossovers at @xmath80 from ( @xmath81 ) antiferromagnetism ( af ) into ( @xmath82 ) uf and into ( @xmath83 ) sf above @xmath84 . fcs at @xmath22 : mh - like insulator shows crossover at @xmath85 from ( @xmath81 ) af into ( @xmath82 ) uf . in triangles , sf and af are found to be stable for all @xmath86 at @xmath19 and @xmath22 respectively . finally table [ nagaoka ] illustrates magnetic phases at large @xmath1 and @xmath6 . for example , the squares and all frustrated clusters at @xmath19 exhibit stable sf ; tetrahedron and triangle at @xmath22 retain cp and af respectively ; uf for the @xmath82 state , separated by @xmath87 from @xmath81 , exists in fcs at @xmath22 . [ fig : pyramid ] shows charge gap at two coupling values @xmath88 between the vertex and base atoms in the deformed tetrahedron . vanishing of gap , driven by @xmath88 , manifests level crossings for @xmath28 , while @xmath32 and @xmath22 cases describe a single phase with avoided crossings . [ diagram ] fig . [ fig : ph4_6 ] illustrates a number of nanophases , defined in refs . @xcite , for the tetrahedron similar to bipartite clusters . the curve @xmath41 below @xmath36 signifies the onset of charge paired condensation . as temperature is lowered below @xmath39 , a spin pseudogap is opened up first , as seen in nmr experiments @xcite , followed by the gradual disappearance of the spin excitations , consistent with the suppression of low - energy excitations in the htscs probed by stm @xcite . the opposite spin cp phase with fully gapped collective excitations begins to form at @xmath89 . the charge inhomogeneities @xcite of hole - rich and charge neutral _ spinodal _ regions between @xmath90 and @xmath91 are similar to those found in the squares and resemble important features seen in the htscs . [ fig : ph4_6 ] shows the presence of bosonic modes below @xmath41 and @xmath38 for paired electron charge and opposite spin respectively . this picture suggests condensation of electron charge and spin at various crossover temperatures while condensation in the bcs theory occurs at a single t@xmath0 value . the temperature driven spin - charge separation above @xmath92 resembles an incoherent pairing ( ip ) phase seen in the htscs @xcite . the charged pairs without spin rigidity above @xmath92 , instead of becoming superconducting , coexist in a nonuniform , charge degenerate ip state similar to a ferroelectric phase @xcite . the unpaired weak moment , induced by a field above @xmath38 , agrees with the observation of competing dormant magnetic states in the htscs @xcite . the coinciding @xmath55 and @xmath44 peaks at @xmath93 show full reconciliation of charge and spin degrees seen in the htscs above t@xmath0 . in the absence of electron - hole symmetry , the tetrahedral clusters near optimal doping @xmath94 undergo a transition with temperature from a cp phase into a mh - like phase . it is clear that our exact results , discussed above , provide novel insight into level crossings , spin - charge separation , reconciliation and full bose condensation @xcite . separate condensation of electron charge and spin degrees offers a new route for superconductivity , different from the bcs scenario . the electronic instabilities found for various geometries , in a wide range of @xmath1 and temperatures , will be useful for the prediction of electron pairing , ferroelectricity @xcite and possible superconductivity in nanoparticles , doped cuprates , etc . in contrast to the squares , exact solution for the tetrahedron exhibits coherent and incoherent pairings for all @xmath1 . our findings at small , moderate and large @xmath1 carry a wealth of new information at finite temperatures in bipartite and frustrated nanostructures regarding phase separation , ferromagnetism and nagaoka instabilities in manganites / cmr materials . these exact calculations illustrate important clues and exciting opportunities that could be utilized when synthesizing potentially high - tc superconducting and magnetic nanoclusters assembled in two and three dimensional geometries @xcite . ultra - cold fermionic atoms in an optical lattice @xcite may also offer unprecedented opportunities to test these predictions . we thank daniil khomskii , valery pokrovsky for helpful discussions and tun wang for valuable contributions . this research was supported in part by u.s . department of energy under contract no . de - ac02 - 98ch10886 . 0 y. kohsaka _ et al_. , science * 315 * , 1380 ( 2007 ) . t. valla _ et al_. , science * 314 * , 1914 ( 2006 ) . a. c. b@xmath95ia , r. laihob and e. l@xmath96hderantab , physica c*411 * , 107 ( 2004 ) . h. e. mohottala _ et al_. , nature materials * 5 * , 377 ( 2006 ) . k. k. gomes _ et al_. , nature * 447 * , 569 ( 2007 ) . r. moro , s. yin , x. xu , and w. a. de heer , phys . lett . * 93 * , 086803 ( 2004 ) ; 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these exact calculations of charge and spin collective excitations and pseudogaps yield intriguing insights into level crossing degeneracies , phase separation and condensation .
criteria for spin - charge separation and reconciliation driven by interaction strength , next nearest coupling and temperature are found .
phase diagrams resemble a number of inhomogeneous , coherent and incoherent nanoscale phases seen recently in high t@xmath0 cuprates , manganites and cmr nanomaterials .
= 1.5 cm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
alfvn waves play important roles in strongly magnetized media . they propagate along magnetic field lines with the alfvn speed @xmath2 , where @xmath3 is the strength of the mean magnetic field and @xmath4 is density . alfvn waves moving in opposite directions can interact and result in alfvnic magnetohydrodynamic ( mhd ) turbulence . alfvnic mhd turbulence in the non - relativistic limit has been studied for many decades and the best available mhd turbulence model is , in spite of all existing controversies ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the one by goldreich & srdihar ( 1995 ; henceforth gs95 ) which was first numerically tested by cho & vishniac ( 2000 ) . the gs95 model predicts a kolmogorov spectrum ( @xmath5 ) and scale - dependent anisotropy ( @xmath6 ) , where @xmath7 and @xmath8 are wave - numbers along and perpendicular to the local mean magnetic field directions , respectively , and @xmath9 . when @xmath3 goes to infinity and/or @xmath4 goes to zero , alfvn speed approaches the speed of light and a new regime of turbulence emerges . more precisely , when the magnetic energy density is so large that the inertia of the charge carriers can be neglected , the medium can be described by relativistic force - free mhd equations @xcite . cho ( 2005 ) numerically studied three - dimensional mhd turbulence in this extreme relativistic limit and found the following results . first , the energy spectrum is consistent with a kolmogorov spectrum : @xmath5 . second , turbulence shows the goldreich - sridhar type anisotropy : @xmath6 . these scaling relations are in agreement with earlier theoretical predictions by thompson & blaes ( 1998 ) . the similarity between non - relativistic alfvnic mhd turbulence and relativistic force - free mhd turbulence leads us to the question : to what extent are relativistic and non - relativistic alfvnic turbulence similar ? in this paper , we try to answer this question . strong imbalanced alfvnic turbulence is an ideal problem for that purpose because interactions between eddies are very complicated in strong imbalanced alfvnic turbulence . in imbalanced alfvnic turbulence , the waves traveling in one direction ( dominant waves ) have higher amplitudes than the opposite - traveling waves ( sub - dominant waves ) . by ` strong ' imbalanced turbulence , we mean the dominant waves satisfy the condition of critical balance , @xmath10 , at the energy injection scale , where @xmath11 is the strength of the fluctuating magnetic field . many studies exist for strong imbalanced alfvnic turbulence in the non - relativistic limit @xcite , but no study is available yet for its relativistic counterpart . in this paper , we compare our relativistic simulations with non - relativistic ones . our study can have many astrophysical implications . so far , we do not fully understand turbulence processes in extremely relativistic environments , such as black hole / pulsar magnetospheres , or gamma - ray bursts . if we can verify close similarities between extremely relativistic and newtonian alfvnic turbulence , we can better understand physical processes , e.g. reconnection , particle acceleration , etc . , in such media . we describe the numerical methods in section 2 and we present our results in section 3 . we give discussions and summary in section 4 . we solve the following system of equations in a periodic box of size @xmath12 : @xmath13 where @xmath14 here @xmath15 is the electric field , @xmath16 the poynting flux vector , and we use units such that the speed of light and @xmath17 do not appear in the equations ( see @xcite for details ) . one can derive this system of equations from @xmath18 where @xmath19 is the dual tensor of the electromagnetic field , @xmath20 the fluid four velocity , and @xmath21 the stress - energy tensor of the electromagnetic field @xmath22 where @xmath23 is the metric tensor and @xmath24 is the electromagnetic field tensor . we ignore the stress - energy tensor of matter . we use flat geometry and greek indices run from 1 to 4 . one can obtain the force - free condition from maxwell s equations and the energy - momentum equation @xmath25 . from equation ( [ eq_cond ] ) , one can derive @xmath26 in our simulations , the mhd condition @xmath27 is enforced all the time . we solve equations ( 1)-(6 ) using a monotone upstream - centered schemes for conservation laws ( muscl ) type scheme with hll fluxes ( harten , lax , van leer 1983 ; in fact , in force - free mhd these fluxes reduce to lax - friedrichs fluxes ) and monotonized central limiter ( see kurganov et al . the overall scheme is second - order accurate . after updating the system of equations along the @xmath28 direction , we repeat similar procedures for the @xmath29 and @xmath30 directions with appropriate rotation of indexes . gammie , mckinney , & tth ( 2003 ) used a similar scheme for general relativistic mhd and del zanna , bucciantini , & londrillo ( 2003 ) used a similar scheme to construct a higher - order scheme for special relativistic mhd . while the magnetic field consists of the uniform background field and a fluctuating field , @xmath31 , the electric field has only a fluctuating one . the strength of the uniform background field , @xmath3 , is set to 1 . at @xmath32 , no fluctuating fields are present . we isotropically drive turbulence in the wave - number range @xmath33 . we adjust the amplitude of forcing to maintain @xmath34 after saturation , where the subscript ` + ' denotes dominant waves . therefore , we have @xmath35 after saturation . since the energy injection rates for the sub - dominant waves ( @xmath36 ) are equal to or less than those of dominant waves ( @xmath37 ) , where @xmath38 s are forcing vectors , we have @xmath39 and @xmath40 . simulation parameters are listed in table 1 . to check the stability of our code , we perform a simulation of relativistic alfvn waves moving in the same direction . since alfvn waves moving in one direction do not interact each other , their energy spectrum should not change in time . indeed figure [ fig:1](a ) confirms this : the initial energy spectrum ( the thick solid line ) does not show much change even after t@xmath4163 , which corresponds to @xmath4110 wave crossing times over the box size . lcl 256-bal & 256@xmath42 & 1 + 256-r0.75 & 256@xmath42 & 0.75 + 256-r0.5 & 256@xmath42 & 0.5 + 256-r0.33 & 256@xmath42 & 0.33 + 512-r0.33 & 512@xmath42 & 0.33 figure [ fig:1](b ) shows time evolution of the energy densities of the dominant waves . we have @xmath43 at @xmath32 and we drive the medium for @xmath44 . the energy densities of the dominant waves initially rise quickly and reach saturation states . the values of @xmath45 during saturation in those runs are between 0.5 and 1.0 . since we drive turbulence isotropically , critical balance is roughly satisfied . in general , the larger the imbalance , the slower the approach to the saturation state . the largest imbalanced run ( run 256-r0.33 ) shows very slow approach to the saturation state . figure [ fig:1](c ) shows time evolution of energy densities of the sub - dominant waves . from top to bottom , the degree of imbalance increases . the top curve corresponds to the balanced turbulence ( run 256-bal ) and the bottom curve to the largest imbalance ( run 256-r0.33 ) . note that , in run 256-r0.33 , @xmath46 goes up very quickly for @xmath47 and then gradually goes down , which may be due to the increase of @xmath48 . figures [ fig:1](d ) and ( e ) show time evolution of the ratio @xmath49 and @xmath50 , respectively . figure [ fig:1](d ) clearly shows that the value of @xmath49 increases substantially as the degree of imbalance increases . for @xmath50 , we actually plot @xmath51 , where @xmath52 . since different theories on imbalanced non - relativistic alfvnic turbulence predict different relations between @xmath49 and @xmath50 , it will be useful to plot the relation for our simulations . figure [ fig:1](f ) shows the relation between the two ratios . roughly speaking , the ratio @xmath49 exhibits a power - law dependence on the ratio @xmath53 : @xmath0 with @xmath1 . in figure [ fig:1 ] , all simulations are performed on a grid of @xmath54 points . the top panel of figure [ fig:2 ] , which compares results of runs 512-r0.33 and 512-r0.33 , implies that numerical resolution of @xmath54 would be enough for our current study . note that two runs have identical numerical set - ups except the numerical resolution ( @xmath54 versus @xmath55 ) . the values of @xmath45 ( upper curves ) almost coincide , but the value of @xmath46 for @xmath55 is slightly higher than that for @xmath54 ( see lower curves ) . the bottom panel of figure [ fig:2 ] shows energy spectra for run 512-r0.33 . although we have only about 1 decade of inertial range , we can clearly observe that the spectral slopes for dominant and sub - dominant waves are different . the spectrum of the dominant waves ( upper curve ) is slightly steeper than @xmath56 , while that of the sub - dominant ones ( lower curve ) is a bit shallower than @xmath56 . in the presence of a strong mean magnetic field , structure of turbulence tends to elongate along the direction of the mean field . therefore elongation of structures , or anisotropy , is an important aspect of mhd turbulence . both relativistic force - free and non - relativistic balanced alfvnic turbulence . imbalanced non - relativistic alfvnic turbulence is also anisotropic ( e.g. , @xcite ) . since interactions between eddies are very complicated in imbalanced alfvnic turbulence , it will be interesting to study anisotropy of imbalanced relativistic force - free mhd turbulence . figure [ fig:3 ] shows the shapes of eddies . in the figure , we plot a contour diagram of the second - order structure function for the magnetic field in a local frame , which is aligned with the local mean magnetic field @xmath57 : @xmath58 where @xmath59 and @xmath60 and @xmath61 are unit vectors parallel and perpendicular to the local mean field @xmath57 , respectively ; see cho & vishniac ( 2000 ) and cho et al . ( 2002 ) for the detailed discussion of the local frame . the left and middle panels of figure [ fig:3 ] show shapes of dominant and sub - dominant eddies , respectively . we can clearly see that the dominant eddies ( left panel ) are less anisotropic than the sub - dominant ones ( middle panel ) . if we plot the relation between perpendicular sizes of eddies ( or , y intercepts of the contours ; @xmath62 ) and the parallel ones ( or , x intercepts ; @xmath63 ) , than we can see that the dominant eddies show anisotropy weaker than @xmath64 and the sub - dominant ones show anisotropy stronger than @xmath64 . our simulations are consistent with the theory and simulations of the imbalanced non - relativistic mhd turbulence ( beresnyak & lazarian 2008 , 2009 ) . indeed , the latter results are consistent with our finding of the relation between the ratio of the energy densities of the sub - dominant and dominant waves , their spectral slopes and their anisotropy . this is suggestive of a close relation between the non - relativistic and relativistic turbulence and implies that the existing theories of non - relativistic turbulence , e.g. theories for magnetic reconnection , particle acceleration , etc . , can be generalized for the relativistic limit . this has not yet been done and , naturally , more theoretical / numerical research , especially with high numerical resolutions , for the relativistic case is necessary . imbalanced turbulence is a generic incarnation of turbulence in the presence of sources and sinks of turbulent energy . we know from the studies of non - relativistic imbalanced turbulence that its slower decay compared to the balanced one allows the energy transfer over larger distances and its transfer to the balanced one due to parametric instabilities or the reflection of waves from density inhomogeneities can result in local deposition of energy and momentum which provide many astrophysically important consequences . the properties of imbalanced relativistic turbulence are important for many astrophysical settings including the magnetosphere of pulsars , environments of gamma ray bursts and relativistic jets . 1 . the magnetic spectrum of dominant waves is steeper than that of sub - dominant waves . 2 . the dominant waves show anisotropy weaker than and the sub - dominant waves show anisotropy stronger than @xmath64 . the energy density ratio @xmath49 is roughly proportional to @xmath65 , where @xmath66 s are energy injection rates and @xmath1 . all these results are consistent with the theory and simulations in beresnyak & lazarian @xcite . therefore we can conclude that relativistic force - free mhd turbulence is indeed very similar to its non - relativistic counterpart . our results imply that many results in non - relativistic alfvnic turbulence can be carried over to relativistic force - free mhd turbulence . for example , theories on magnetic reconnection ( e.g. , @xcite ) , particle acceleration ( e.g. , @xcite ) and thermal diffusion ( e.g. , cho et al . 2003 ) obtained in non - relativistic alfvnic turbulence can also be applicable to relativistic force - free mhd turbulence . the close similarity between the properties of non - relativistic and relativistic imbalanced turbulence found in this paper elucidates the nature of magnetic turbulence that preserves its properties in both regimes irrespectively of whether turbulence is balanced or imbalanced . from the practical point of numerical studies , this allows us to test or double - check theories on non - relativistic alfvnic turbulence using a completely different numerical scheme . j.c.s work is supported by the national r & d program through the national research foundation of korea ( nrf ) , funded by the ministry of education ( no . 2011 - 0012081 ) . a.l . is supported by nsf grant ast 1212096 , the center for magnetic self - organization and the vilas associate award . we thank the international institute of physics ( natal ) for their hospitality . 99 beresnyak , a. 2011 , phys . lett . , 106 , 075001 beresnyak , a. , & lazarian , a. 2006 , apjl , 640 , l175 beresnyak , a. , & lazarian , a. 2008 , apj , 682 , 1070 beresnyak , a. , & lazarian , a. 2009 , apj , 702 , 1190 | when magnetic energy density is much larger than that of matter , as in pulsar / black hole magnetospheres , the medium becomes force - free and we need relativity to describe it . as in non -
relativistic magnetohydrodynamics ( mhd ) , alfvnic mhd turbulence in the relativistic limit can be described by interactions of counter - traveling wave packets . in this paper
we numerically study strong imbalanced mhd turbulence in such environments . here
, imbalanced turbulence means the waves traveling in one direction ( dominant waves ) have higher amplitudes than the opposite - traveling waves ( sub - dominant waves ) .
we find that ( 1 ) spectrum of the dominant waves is steeper than that of sub - dominant waves , ( 2 ) the anisotropy of the dominant waves is weaker than that of sub - dominant waves , and ( 3 ) the dependence of the ratio of magnetic energy densities of dominant and sub - dominant waves on the ratio of energy injection rates is steeper than quadratic ( i.e. , @xmath0 with @xmath1 ) .
these results are consistent with those obtained for imbalanced non - relativistic alfvnic turbulence .
this corresponds well to the earlier reported similarity of the relativistic and non - relativistic balanced magnetic turbulence . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
flows of most liquid substances are usually studied by modeling the liquid as a continuum , but there are some substances that allow the study of flows at the kinetic level , i.e. , at the level of the individual constituent particles . as examples , we can mention chute flows in granular materials @xcite and capillary flows in colloids @xcite . the solid particles in these soft materials are large enough that their motion can be tracked by video microscopy , allowing experimenters to record their positions and velocities . like granular materials and colloids , dusty plasmas also allow direct observation of individual particle motion . dusty plasma @xcite is a four - component mixture consisting of micron - size particles of solid matter , neutral gas atoms , free electrons , and free positive ions . these particles of solid matter , which are referred to as `` dust particles , '' gain a large negative charge @xmath2 , which is about @xmath3 elementary charges under typical laboratory conditions . the motion of the dust particles is dominated by electric forces , corresponding to the local electric field @xmath4 , where @xmath5 is due to confining potentials and @xmath6 is due to coulomb collisions with other dust particles . due to their high charges , coulomb collisions amongst dust particles have a dominant effect . the interaction force @xmath7 amongst dust particles is so strong that the dust particles do not move easily past one another , but instead self - organize and form a structure that is analogous to that of atoms in a solid or liquid @xcite . in other words , the collection of dust particles is said to be a strongly - coupled plasma @xcite . in a strongly - coupled plasma , the pressure @xmath8 is due mainly to interparticle electric forces , with only a small contribution from thermal motion @xcite . even when it has a solid structure , a collection of dust particles is still very soft , as characterized by a sound speed on the order of 1 cm / s @xcite . as a result , a dusty plasma in a solid phase is very easily deformed by small disturbances , and it can be made to flow . flows can be generated , for example , by applying shear using a laser beam that exerts a spatially - localized radiation force @xcite . in such an experiment , the reynolds number is usually very low , typically @xmath9 , indicating that the flow is laminar @xcite . this paper provides further analysis and details of the experiment that was reported in @xcite . we now list some of the major points of these two papers , to indicate how they are related and how they differ . in this paper , we present : ( 1 ) a detailed treatment of the continuity equations for both momentum and energy , ( 2 ) our method of simultaneously determining two transport coefficients ( viscosity and thermal conductivity ) , ( 3 ) values of these two coefficients , and ( 4 ) spatially - resolved profiles of the terms of the energy equation , including the terms for viscous heating and thermal conduction , as determined by experimental measurements . in @xcite , we reported : ( 1 ) a discovery of peaks in a spatially - resolved measurement of kinetic temperature , ( 2 ) a demonstration that these peaks are due to viscous heating in a region of a highly sheared flow velocity , and ( 3 ) a quantification of the role of viscous heating , in competition with thermal conduction , by reporting a dimensionless number of fluid mechanics called the brinkman number @xcite which we found to have an unusually large value due to the extreme properties of dusty plasma as compared to other substances . the values of viscosity and thermal conduction found in this paper are used as inputs for the calculations of dimensionless numbers in @xcite . the identification of viscous heating as the cause of the temperature peaks reported in @xcite is supported by the spatially - resolved measurements reported here . in the experiment , the dust particles are electrically levitated and confined by the electric field in a sheath above a horizontal lower electrode in a radio - frequency ( rf ) plasma , forming a single layer of dust particles , fig . the dust particles can move easily within their layer , but very little in the perpendicular direction , so that their motion is mainly two dimensional ( 2d ) . they interact with each other through a shielded coulomb ( yukawa ) potential , due to the screening provided by the surrounding free electrons and ions @xcite . as the dust particles move , they also experience frictional drag since individual dust particles in general move at velocities different from those of the molecules of the neutral gas . this friction can be modeled as epstein drag @xcite , and characterized by the gas damping rate @xmath10 . using experimental measurements of their positions and velocities , the dust particles can of course be described in a _ particle _ paradigm . they can also be described by a _ continuum _ paradigm by averaging the particle data on a spatial grid . in transport theory , momentum and energy transport equations are expressed in a continuum paradigm , while transport coefficients such as viscosity and thermal conductivity are derived using the particle paradigm because these transport coefficients are due to collisions amongst individual particles . in our experiment , we average the data for particles , such as velocities , to obtain the spatial profiles for the continuum quantities , such as flow velocity . in the continuum paradigm , a substance obeys continuity equations that express the conservation of mass , momentum , and energy . these continuity equations , which are also known as navier - stokes equations , characterize the transport of mass , momentum , and energy . in a multi - phase or multi - component substance , these equations can be written separately for each component . the component of interest in this paper is dust . in sec . ii , we review the continuity equations for dusty plasmas . in secs . iii and iv , we provide details of our experiment and data analysis method . we designed our experiment to have significant flow velocity and significant gradients in the flow velocity , i.e. , velocity shear . in sec . v , we simplify the continuity equations using the spatial symmetries and steady conditions of the experiment , and including the effects of external forces . we will use our experimental data as inputs in these simplified continuity equations in secs . vi and vii . we now review the continuity equations for mass , momentum , and energy for dusty plasmas . we will then discuss the significance of some of the terms for our experiments . the equation of mass continuity , i.e. , conservation of mass , is @xmath11 in this paper , the mass density @xmath12 and the fluid velocity @xmath13 describe the dust continuum . the momentum equation @xcite is @xmath14\nabla(\nabla \cdot { \bf v } ) & + & { \bf f}_{\rm ext},\end{aligned}\ ] ] here , @xmath15 , @xmath0 , and @xmath16 are the charge density , shear viscosity , and bulk viscosity , respectively ; and @xmath17 is called the kinematic viscosity . in this paper , these parameters describe the dust continuum . equation ( [ momentum ] ) describes the force per unit mass , i.e. , acceleration , for the continuum . the confining field @xmath18 and pressure @xmath8 have been discussed in sec . i. the last term in eq . ( [ momentum ] ) is due to the momentum contribution from forces such as gas friction , laser manipulation , ion drag , and any other forces that are external to the layer of dust particles , as discussed in sec . the other terms on the right - hand - side of eq . ( [ momentum ] ) correspond to viscous dissipation , which arises from coulomb collisions amongst the charged dust particles . the viscous term @xmath19 was studied in a previous dusty plasma experiment @xcite . for our experiment , we can consider eq . ( [ momentum ] ) for two conditions , with and without the application of the external force @xmath20 . without the external force , the fluid velocity @xmath13 is zero so that eq . ( [ momentum ] ) is reduced to @xmath21 , meaning that the confining electric force is in balance with the pressure ( which mainly arises from interparticle electric forces ) . with the external force , the external confining force remains as it was without @xmath20 ; moreover , the pressure @xmath8 will be affected only weakly because the density is unchanged ( as we will show later ) so that the interparticle electric forces that dominate the pressure will also be unchanged . therefore , in using eq . ( [ momentum ] ) , we will assume that the first two terms on the right hand side always cancel everywhere . the internal energy equation , as it is expressed commonly in fluid dynamics @xcite , is @xmath22 here , @xmath23 is the entropy per unit mass , @xmath1 is the thermal conducitivity , and @xmath24 is the thermodynamic temperature of the dust continuum . the last term @xmath25 is due to the energy contribution from any external forces @xmath20 . we assume that the continuity equations , eqs . ( [ mass]-[energy ] ) , are valid for the dust particles separately from other components of dusty plasmas that occupy the same volume , such as neutral gas atoms . the coupling between the dust particles with other components is indicated in @xmath20 and @xmath25 , so that the momentum and energy of the dust particle motion is treated for the dust particles separately from other components . other external forces , such as those due to laser manipulation , are also indicated in @xmath20 and @xmath25 . the first term on the right - hand - side of eq . ( [ energy ] ) is due to viscous heating . the viscous heating term @xmath26 depends on the square of the shear , i.e. , the square of the gradient of flow velocity . a general expression for @xmath26 has many terms ( cf . ( 3.4.5 ) of ref . @xcite or eq . ( 49.5 ) of ref . @xcite ) , but it can be simplified for our experiment by taking advantage of symmetries , as explained in sec . iv . the second term on the right - hand - side of eq . ( [ energy ] ) is due to thermal conduction . it arises from a temperature gradient . previous experiments with 2d dusty plasmas include a study of the thermal conduction term in eq . ( [ energy ] ) @xcite . in this paper , most of our attention will be devoted to the first two terms on the right - hand - side of eq . ( [ energy ] ) . using our experimental data , we will compare the magnitudes of these terms . in @xcite , we demonstrated that viscous heating is measurable and significant , when evaluated using only _ global _ measures like the brinkman number . here we further evaluate viscous heating by characterizing it _ locally _ using spatially - resolved profiles for the terms in eq . ( [ energy ] ) . we also develop a method to simultaneously obtain values of two transport coefficients of @xmath0 and @xmath1 . here we provide a more detailed explanation of the experiment than in @xcite . an argon plasma was generated in a vacuum chamber at @xmath27 ( or 2.07 pa ) , powered by rf voltages at @xmath28 and @xmath29 peak - to - peak . we used the same chamber and electrodes as in @xcite . the dust particles were @xmath30 diameter melamine - formaldehyde microspheres of mass @xmath31 . the dust particles settled in a single layer above the powered lower electrode . the layer of dust particles had a circular boundary with a diameter of @xmath32 and contained @xmath33 dust particles . as individual dust particles moved about within their plane , they experienced a frictional damping @xcite with a rate @xmath34 due to the surrounding argon gas . the particles were illuminated by a @xmath35-nm argon laser beam that was dispersed to provide a thin horizontal sheet of light , fig . 1(a ) . using a cooled 14-bit digital camera ( pco model 1600 ) viewing from above , we recorded the motion of individual dust particles . this top - view camera imaged a central portion of the dust layer , as sketched in fig . the movie is available for viewing in the supplemental material of @xcite . the portion of the camera s field of view that we will analyze was @xmath36 , and it contained @xmath37 particles . we recorded @xmath38 frames at a rate of 55 frame / s , with a resolution of @xmath39 . our choice of 55 frame / s was sufficient for accurate measurement of various dynamical quantities , including the kinetic temperature , although a slightly higher frame rate would have been optimal @xcite . in addition to the top - view camera , we also operated a side - view camera to verify that there was no significant out - of - plane motion ; this was due to a strong vertical confining electric field . thus , we will analyze the particle motion data taking into consideration only the motion within a horizontal plane . at first , the dust particles self - organized in their plane to form a 2d crystalline lattice . the particle spacing , as characterized by a wigner - seitz radius @xcite , was @xmath40 , corresponding to a lattice constant @xmath41 , an areal number density @xmath42 , and a mass density @xmath43 . using the wave - spectra method for thermal motion of particles in the undisturbed lattice , we found the following parameters for the dust layer : @xmath44 , @xmath45 , and @xmath46 , where @xmath47 is the nominal 2d dusty plasma frequency @xcite , @xmath2 is the particle charge , @xmath48 is the elementary charge , and @xmath49 is the screening length of the yukawa potential . we used laser manipulation @xcite to generate stable flows in the same 2d dusty plasma layer . a pair of continuous - wave @xmath50-nm laser beams struck the layer at a @xmath51 downward angle . in this manner , we applied radiation forces that pushed particles in the @xmath52 directions , as shown in fig . 1 . the power of each beam was @xmath53 as measured inside the vacuum chamber . to generate a wider flow than in @xcite , the laser beams were rastered in the both @xmath54 and @xmath55 directions in a lissajous pattern as in @xcite , with frequencies of @xmath56 and @xmath57 . we chose these frequencies to be @xmath58 to avoid exciting coherent waves . the rastered laser beams filled a rectangular region , which crossed the entire dust layer and beyond , as sketched in fig . this laser manipulation scheme resulted in a circulating flow pattern with three stable vortices , as sketched in fig . we designed the experiment so that in the analyzed region the flow is straight in the @xmath52 directions , without any significant curvature . curvature in the flow , which is necessary for the flow pattern to close on itself as sketched in fig . 1(b ) , was limited in our experiment to the extremities of the dust layer , where it would not affect our observations . ( color online ) . ( a ) side - view sketch of the apparatus , not to scale . a single layer of dust particles of charge @xmath2 and mass @xmath59 are levitated against gravity by a vertical dc electric field . there is also a weaker radial dc electric field @xmath60 which prevents the dust particles from escaping in the horizontal direction . further details of the chamber are shown in @xcite . the two laser beams are rastered in the @xmath55 direction so that they have a finite expanse , and they are offset in the @xmath55 direction as shown in ( b ) . ( b ) top - view sketch of laser - driven flows in the 2d dusty plasma . in the region of interest , the flow is straight , with curvature of the flow limited to the extremities of the dust layer . a video image of the dust particles within the region of interest is also shown . the region of interest is divided into 89 bins of width @xmath61 so that particle data can be converted to continuum data . ] we start our data analysis by analyzing data for individual particles , i.e. , by working in the _ particle _ paradigm . using image analysis software @xcite , with a method optimized as in @xcite to minimize measurement error , we identify individual particles in each video image and calculate their @xmath62 coordinates . we then track a dust particle between two consecutive frames and calculate its velocity @xmath63 as the difference in its position divided by the time interval between frames @xcite . now having the position and velocity of all the particles in the analyzed region , we can study motion at the particle level . for example , in sec . vi we will use data for the individual particles to calculate the rate of their energy dissipation due to their frictional drag with the neutral gas , @xmath25 . we next convert our data for individual particles to continuum data , i.e. , we change from the _ particle _ paradigm to the _ continuum _ paradigm . this is done by averaging particle data within spatial regions of finite area , which we call bins . there are 89 bins , which are all long narrow rectangles aligned in the @xmath55 direction , as shown in fig . each bin contains @xmath64 dust particles . we choose the shape of these bins to exploit the symmetry of the experiment , which in the analyzed region has an ignorable coordinate , @xmath54 . the width of each bin is the same as the wigner - seitz radius , @xmath65 . to reduce the effect of particle discreteness as a particle crosses the boundaries between bins , we use the cloud - in - cell weighting method @xcite which has the effect of smoothing data so that a particle contributes its mass , momentum and energy mostly to the bin where it is currently located and to a lesser extent to the next nearest bin . the data are binned in this way regardless of their @xmath54 positions , since @xmath54 is treated as an ignorable coordinate . we also time - average these binned data , exploiting the steady conditions of the experiment . this procedure ( binning , cloud - in - cell - weighting , and time averaging ) yields our continuum quantities , such as the flow velocity @xmath66 . it also yields a kinetic temperature @xmath67 which is calculated from the individual particle velocities ; this kinetic temperature is not necessarily identical to the thermodynamic temperature @xmath24 . here , @xmath68 is the boltzmann constant . we assume that it is valid to use a continuum model when gradients are concentrated in a region as small as a few particle spacings . in fact , it has been shown experimentally that the momentum equation @xcite and the energy equation @xcite remain useful in 2d dusty plasma experiments with gradients that are as strong as in our experiment . the notation we use in this paper distinguishes velocities and other quantities according to whether they correspond to individual particles or continuum quantities . parameters for individual dust particles are denoted by a subscript @xmath69 , for example @xmath63 for the velocity of an individual dust particle . continuum quantities in a theoretical expression are indicated without any special notation , for example @xmath13 for the hydrodynamic velocities in eqs . ( [ mass ] ) and ( [ momentum ] ) . finally , continuum quantities that we compute with an input of experimental data , as described above , are indicated by a bar over the symbol , for example @xmath70 . here , we present our simplification of the continuity equations , eqs . ( [ mass]-[energy ] ) , to describe our 2d dust layer . these simplifications involve three approximations suitable for the conditions in our 2d layer , and a treatment of two external forces , laser manipulation and gas friction , that are responsible for @xmath20 and @xmath25 . we describe these simplifications and the resulting continuity equations , next . the first of our three approximations is @xmath71 . this approximation is suitable for the steady overall conditions of our experiment . aside from the particle - level fluctuations that one desires to average away , when adopting a continuum model , the only time - dependent processes in the experiment were the rastering of the laser beam at @xmath72 and the @xmath28 rf electric fields that powered the plasma . these frequencies are too high for the dust particles to respond , and the rastering of the beams is not a factor anyway because we will only use the continuum equations outside the laser beams . the second approximation is that @xmath73 is negligibly small . due to the symmetry in our experiment design , as mentioned in sec . iii , @xmath54 is treated as an ignorable coordinate , i.e. , @xmath74 . as a verification of this assumption , we observe that the ratio @xmath75 is of order @xmath76 , as a measure of the slightly imperfect symmetry of our experiment . thus , for our experiment , we consider @xmath77 to be of zeroth order and @xmath73 to be two orders of magnitude smaller , when we approximate eqs . ( [ mass]-[energy ] ) . the third approximation is that @xmath78 is negligibly small , based on our observation of the flow velocity of our 2d layer . our results for the calculated flow velocity of the dust layer are shown in fig . 2 . from the velocity @xmath79 in fig 2(a ) , the flow can be easily identified from the two peaks with broad edges . however , the flow velocity in the @xmath55 direction is two orders of magnitude smaller than in the @xmath54 direction , as shown in fig . 2(b ) . ( color online ) . profiles of continuum parameters during laser manipulation , including ( a - b ) flow velocity , ( c ) areal number density , and ( d ) a measure @xmath80 of local structural order that would be a value of 1 for a perfect crystal @xcite . disorder , as indicated by a small value in ( d ) , is found to be greatest where the shear is largest , not where the flow is fastest . ] using the second and third approximations , and omitting terms that are small by at least two orders of magnitude , we can easily see that we can approximate @xmath81 and @xmath82 . the latter indicates that the dust layer can be treated as an incompressible fluid in our experiment . using these three approximations in eq . ( [ mass ] ) , we find that @xmath83 . in other words , for our approximations , the density @xmath12 is uniform , which is confirmed by our experimental observation of fig . because the density is so uniform , we can also assume that plasma parameters , such as the dust charge @xmath2 , are also spatially uniform within the analyzed region . in addition to these approximations , we also assume that @xmath0 and @xmath1 are valid transport coefficients for our system . there are theoretical reasons to question whether 2d systems ever have valid transport coefficients , and this is typically tested _ theoretically _ using long - time tails in correlation functions @xcite . it could be tested _ experimentally _ by repeating the determination of @xmath0 and @xmath1 for vastly different length scales for the gradients of velocity and temperature and verifying that they do not depend on the length scale . however , such a test is not practical for experiments like ours , which tend to have a limited range of diameters of dust layers that can be prepared . now we consider external forces that contribute momentum and energy to the two equations , eqs . ( [ momentum ] ) and ( [ energy ] ) . for our experiment , we can mention six _ external _ forces acting on the dust layer : gas friction , laser manipulation , electric confining force , gravity , electric levitating force , and ion drag . since we only study the 2d motion of dust particles within their plane , the last three forces are of no interest because they are in the perpendicular direction , and will not affect the horizontal motion of particles that is of interest here . the electric confining force is balanced by the pressure inside the 2d dusty plasma lattice , @xmath84 , as described in sec . thus , only two of the six forces need to be considered : gas friction and laser manipulation . gas friction is the main dissipation mechanism in our experiment . we can consider the effect of this friction first at the level of a single dust particle , and then at the level of a continuum . for the _ momentum _ equation , we note first that at the particle level , a single dust particle moving at a speed of @xmath63 experiences a drag acceleration of @xmath85 . at the continuum level , the contribution of this drag to eq . ( [ momentum ] ) is simply the average acceleration experienced by all dust particles in a given spatial region , @xmath86 . for the _ energy _ equation , at the particle level the rate of energy dissipation for one dust particle is @xmath87 , which is the product of a drag force and velocity , where @xmath88 is the kinetic energy of one dust particle . at the continuum level , averaging over all the dust particles in the given spatial region , the rate of energy loss per unit mass in eq . ( [ energy ] ) is @xmath89 @xcite . using the three approximations listed above and taking into account gas friction and laser manipulation forces , the mass , momentum , and energy continuity equations become @xmath90 @xmath91 @xmath92 where @xmath93 is the viscous heating term , after simplifications based on the assumptions of @xmath74 , @xmath94 , and @xmath82 . in this paper , we will restrict our analysis to a spatial region where the laser force and power are zero , i.e. , @xmath95 and @xmath96 . in this region , the momentum and energy continuity equations will be further simplified : @xmath97 @xmath98 to simplify the problem , we will assume that @xmath0 and @xmath1 are independent of temperature , as discussed in @xcite . we can comment on the meaning of these two equations . equation ( [ momentum_dpg ] ) indicates a balance of the sideways transfer of momentum due to two mechanisms : viscosity arising from interparticle electric forces and frictional loss of momentum due to collisions with gas atoms . equation ( [ energy_dpg ] ) describes the energy transferred from the viscous heating @xmath26 and thermal conduction ( the second term ) as being balanced by the energy dissipated due to friction as expressed in the last term of eq . ( [ energy_dpg ] ) . we will use eqs . ( [ momentum_dpg]-[energy_dpg ] ) only in spatial regions where @xmath95 and @xmath96 , i.e. , outside the laser beam . in the next section , we will present our calculation of terms appearing in eqs . ( [ momentum_dp]-[energy_dpg ] ) using the dust particles position and velocity data from our experiment . the terms of interest in these equations are @xmath99 , @xmath100 , @xmath101 , and @xmath102 . ( color online ) . profiles of the first ( a ) and second ( b ) derivatives of flow velocity . to make a comparison , the flow velocity profile @xmath103 is also provided in ( c ) . ] our results for the first and second derivatives of the flow velocity are presented in fig . these results are calculated using the flow velocity profile in fig . 2(a ) , which we reproduce in fig . 3(c ) . the first and second derivatives in fig . 3(a - b ) will be used in eqs . ( [ viscous_heat ] ) and ( [ momentum_dpg ] ) , respectively . from fig . 3(a ) , we can identify four points of maximum shear , i.e. , maximum @xmath99 ; these are at @xmath104 , @xmath105 , @xmath106 , and @xmath107 . these points of maximum shear coincide with other features of interest : the minimum in the structural order , fig . 2(d ) , and peaks in the mean - square velocity fluctuation profile , which we will present below . we find that disorder , as indicated by a small value in fig . 2(d ) , is greatest where the shear is largest , not where the flow is fastest . comparing panels ( b ) and ( c ) of fig . 3 , we find that the profiles for the flow velocity @xmath108 and its second derivative are similar , in regions without laser manipulation , for example in the central region @xmath109 . this similarity is expected from the momentum equation , eq . ( [ momentum_dpg ] ) , provided that the viscosity @xmath0 is spatially uniform . we do not expect that @xmath0 would be spatially uniform since viscosity in general depends on temperature and the temperature is highly non - uniform , as we will show below . nevertheless , we find that the two curves are nearly similar , with a small discrepancy that we will quantify below when we present data for the residual of eq . ( [ momentum_dpg ] ) . ( color online ) . profiles of the mean squared particle velocity shown separately for particle motion in the @xmath54 ( a ) and @xmath55 ( b ) directions . these quantities correspond to the mean particle kinetic energy @xmath101 , i.e. , @xmath110 . the thin curve in each panel is the flow velocity profile @xmath103 ; its scale is shown in fig . 3(c ) . ] profiles of the mean squared particle velocity , corresponding to the kinetic energy in the energy equation , eq . ( [ energy_dpg ] ) , are shown in fig . 4 . this kinetic energy includes energy associated with both the macroscopic flow and the fluctuations at the particle level . we will use these profiles in determining @xmath101 in the next section , where we will find the residual of eq . ( [ energy_dpg ] ) . ( color online ) . profiles of the mean - square particle velocity fluctuation shown separately for particle motion in the @xmath54 and @xmath55 directions , ( a ) and ( b ) , respectively . these quantities combined correspond to the kinetic temperature @xmath111 , eq . ( [ kt ] ) . the profile of @xmath111 were reported in @xcite , where we discovered peaks in the kinetic temperature where the shear is largest . the second derivative of the averaged mean - square particle velocity fluctuation for the motion in the @xmath54 and @xmath55 directions ( c ) will be used to determine the second derivative of the thermodynamic temperature in eq . ( [ energy_dpg ] ) . the thin curve in each panel is the flow velocity profile @xmath103 ; its scale is shown in fig . ] in fig . 5 , we present our results for the mean - square velocity fluctuation , which corresponds to the kinetic temperature as in eq . ( [ kt ] ) . we will use this kinetic temperature in place of the thermodynamic temperature @xmath24 in the energy equation , eq . ( [ energy_dpg ] ) . unlike the kinetic energy @xmath101 , the kinetic temperature only includes the energy associated with the fluctuations of particle velocity about the flow velocity @xmath112 @xcite . as reported in @xcite , there are peaks in the kinetic temperature profile that coincide with the position of maximum shear . these peaks can also be seen in fig . 5(a - b ) . in @xcite , we attributed these peaks to viscous heating . as an intuitive explanation of viscous heating , consider that higher shear conditions lead to collisions of particles flowing at different speeds , causing scattering of momentum and energy that leads to higher random velocity fluctuations , and therefore higher kinetic temperature . in the next section , we provide further verification that the temperature peaks are due to viscous heating ; we do this by confirming that three terms in the energy equation , including viscous heating , are in balance as indicated by their summing to zero . we now examine the momentum and energy equations , eqs . ( [ momentum_dpg ] ) and ( [ energy_dpg ] ) , which are written so that the right - hand - side is zero . when we use these equations with an input of experimental data , however , the terms will not sum exactly to zero , but will instead sum to a finite residual . we calculate these residuals , and we vary two free parameters , the viscosity @xmath0 and the thermal conductivity @xmath1 , to minimize the residuals . ( specifically , we minimize the square residual summed over all bins in the central region of @xmath113 . ) this minimization procedure yields the best estimation for the values for @xmath0 and @xmath1 , which will be our first chief result . we will then , as our second chief result , be able to make a spatially - resolved comparison of the magnitude of different terms in the energy equation , eq . ( [ energy_dpg ] ) . ( color online ) . ( a ) a profile of the residual of the momentum equation , eq . ( [ momentum_dpg ] ) , assuming a kinematic viscosity of @xmath114 . for the central region , magnified in ( b ) , the summation of the squared residual reaches its minimum when @xmath114 ; this minimization process is how we determine @xmath0 , which is one of our main results . the thin curve in ( a ) is the flow velocity profile @xmath103 ; its scale is shown in fig . 3(c ) . ] figure 6 shows the residual of the momentum equation , eq . ( [ momentum_dpg ] ) . this result is shown for @xmath115 , which is the best estimation of the kinematic viscosity . the data are shown as a spatial profile because we calculated eq . ( [ momentum_dpg ] ) separately for each bin , i.e. , each value of @xmath55 . two peaks in fig . 6(a ) , located within the laser manipulation region , are due to the momentum contribution from the laser . we do not use eq . ( [ momentum_dpg ] ) with these peaks because of a finite laser force @xmath116 there . instead we will use the flatter region between these peaks , where @xmath95 , as magnified in fig . the small residuals in this flatter region indicate that the momentum equation , eq . ( [ momentum_dpg ] ) , is able to accurately account for the momentum of our 2d dust layer , and that the minimization process in this region yields a value for the viscosity . ( color online ) . ( a ) a profile of the residual of the energy equation , eq . ( [ energy_dpg ] ) , assuming a thermal diffusivity of @xmath117 . for the central region , magnified in ( b ) , the summation of the squared residual reaches its minimum when @xmath118 . this minimization process is how we determine @xmath1 , which is another of our main results . the thin curve in ( a ) is the flow velocity profile @xmath103 ; its scale is shown in fig . 3(c ) . ] figure 7 shows the residual of the energy equation , eq . ( [ energy_dpg ] ) , for @xmath119 , which is the best estimation of the thermal diffusivity . in this calculation , we used the value of @xmath120 from the momentum equation above , and we varied the value of @xmath1 to minimize the residuals as described above . two large negative peaks in fig . 7(a ) are due to the energy contribution from the laser manipulation @xmath121 , which is not included in eq . ( [ energy_dpg ] ) . the small values of residuals in the flatter region between these peaks , as magnified in fig . 7(b ) , show that the energy equation , eq . ( [ energy_dpg ] ) , accurately describe energy transport in our 2d dust layer , and that the minimization process in this region yields a value for the thermal diffusivity . by achieving a small value of the residual , we have verified that the three terms in the energy equation , eq . ( [ energy_dpg ] ) , are in balance . since one of these terms is viscous heating and another is computed from the temperature profile which has peaks , the balance we observe here is consistent with the conclusion of @xcite that the temperature peaks are due to viscous heating . as the first chief result of this paper , we obtain the kinematic viscosity value of @xmath122 and thermal diffusivity value of @xmath119 . these values are obtained simultaneously in a single experiment . a source of uncertainty in these values is systematic error in @xmath10 , for example due to particle size dispersion or uncertainty in the epstein drag coefficients @xcite . this is so because our method actually yields results for @xmath123 and @xmath124 . another source of uncertainty is our simplification that we neglect heating sources other than laser manipulation @xcite ; in a test we determined that @xmath125 is in a range from 7.5 to @xmath126 , depending on whether these small heating effects are accounted for . we note that our results of @xmath17 and @xmath127 agree with the results of previous experiments @xcite using the same size of dust particles and a similar value of @xmath128 . ( color online ) . ( a ) profiles of three terms in eq . ( [ energy_dpg ] ) , assuming the transport coefficients , @xmath0 and @xmath1 , obtained above . for the central region without laser manipulation ( b ) , thermal conduction is one order magnitude larger than the viscous heating . this is our second chief result : a spatially - resolved comparison of the different mechanisms for energy transfer in our 2d dust layer . the thin curve in ( a ) is the flow velocity profile @xmath103 ; its scale is shown in fig . 3(c ) . ] our experiment allows us to obtain spatially - resolved quantitative measurements of three heat transfer effects : viscous heating , thermal conduction , and dissipation due to gas friction ( i.e. , cooling ) . these three effects appear as the three terms on the left - hand - side of eq . ( [ energy_dpg ] ) . as the second chief result of this paper , we plot these three terms , presented as spatial profiles , in fig . 8 . examining the spatial profiles for these terms , in fig . 8 , we see the most prominent features are two large peaks for the gas dissipation term where the flow velocity @xmath103 is fastest ; in this region the energy dissipation due to gas friction reaches its maximum . despite the prominence of these features , however , they are not what interest us here . instead , we are more interested in the regions of high shear , near the edge of the laser manipulation . recall that in these high shear regions , the flow velocity gradient @xmath99 is largest , and there are peaks in the profiles of the kinetic temperature and the second derivative of the flow velocity , figs . 3 and 5 . in fig . 8 , our spatially - resolved profiles reveal that viscous heating and thermal conduction terms are peaked in regions of high shear . the viscous heating term , eq . ( [ viscous_heat ] ) , is always positive , meaning that viscous dissipation is always a source of heat wherever it occurs . the thermal conduction term partial @xmath129 , on the other hand , can be either positive or negative , indicating that heat is conducted toward or away from the point of interest , respectively . in locations where the shear is strongest , for example at @xmath104 , the thermal conduction term is negative indicating that heat is conducted away from that point . although viscous heating has great importance in all kinds of fluids , and it has been understood theoretically for a very long time @xcite , a spatially - resolved measurement of it is uncommon . in most physical systems viscous heating is usually hard to measure either because the temperature increase is overwhelmingly suppressed by rapid thermal conduction , as we discussed in @xcite , or because the thinness of the shear layer does not allow convenient _ in - situ _ temperature measurements . most experimental observations of temperature increases due to viscous heating are either external or global measurements , and not spatially - resolved measurements like those that we report here . indeed , in our literature search , we found no previous spatially - resolved experimental measurements of the viscous heating term , not only for dusty plasma , but also for any other physical system . as we explained in @xcite , our ability to detect strong effects of viscous heating is due to the extreme properties of dusty plasma , as compared to other substances . our ability to make spatially - resolved measurements is due to our use of video imaging of particle motion . ( color online ) . profiles of ( a ) the ratio of the viscous heating and thermal conduction terms in eq . ( [ energy_dpg ] ) , and ( b ) the absolute value of this ratio . the sign of @xmath129 determines the sign of the ratio in ( a ) . a positive ratio indicates that heat is conducted toward the position of interest . the large values of the ratio in ( b ) at high shear regions indicate significant viscous heating at those locations . for comparison , in @xcite we found a brinkman number br = 0.5 ( indicated by the arrow ) , which is a global measure of the flow that provides less detailed information than the spatially resolved ratio shown here . the thin curve in ( b ) is the flow velocity profile @xmath103 ; its scale is shown in fig . 3(c ) . ] to further analyze the second chief result of this paper , the spatial profiles of the viscous heating and thermal conduction terms in fig . 8(a ) , we plot the ratio of these two terms in fig . 9(a ) . this ratio has its largest positive and negative values in the regions of high shear . in fig . 9(a ) , negative values of this ratio are observed to occur in the high shear regions , which indicates that heat is conducted away from these regions . this result is consistent with observation of kinetic temperature peaks here . to characterize the magnitude of these two terms , we plot in fig . 9(b ) the absolute value of this ratio . we can see that , within regions of high shear , this ratio can be as large as unity , or even larger . a typical value of this ratio in the shear region is of order 0.5 for our experiment . this matches the value of the brinkman number , br = 0.5 that we found in @xcite for the same experiment . the brinkman number is a global measure of the viscous heating , in competition with thermal conduction . in summary , we reported further details of the laser - driven flow experiment in a dusty plasma that was first reported in @xcite . we simplified the momentum and energy continuity equations , exploiting the symmetry and steady conditions of the experiment . we developed a method to obtain transport coefficients by minimizing the residuals of continuity equations using the input of experimental data . as our first chief result , we use this method to simultaneously determine , from the same experiment , two transport coefficients : kinematic viscosity and thermal diffusivity , which are based on viscosity and thermal conductivity , respectively . as our second chief result , we obtained spatially - resolved measurements of various terms in the energy equation . we found that , in a laser - driven dusty plasma flow , viscous heating is significant in regions with high shear , which is consistent with the interpretation of @xcite that the peaks in the temperature profile are due to viscous heating . a. melzer and j. goree , in _ low temperature plasmas : fundamentals , technologies and techniques _ , 2nd ed . , edited by r. hippler , h. kersten , m. schmidt , and k. h. schoenbach ( wiley - vch , weinheim , 2008 ) , p. 129 . the external forces can add or remove energy from the collection dust particles , as expressed by @xmath25 , which is the final term of the energy equation , eq . ( [ energy ] ) . besides heating from laser manipulation and cooling from the gas friction as discussed in the text , we can mention two other contributions to @xmath25 for our experiment : heating from random kicks from collisions with the gas atoms and heating by fluctuating electric fields in the plasma . in the absence of any laser manipulation , the other three terms have their background levels which are in balance , so that their total @xmath25 is zero . in the presence of laser manipulation , even outside the region where the laser beams strike the dust particles , there are flows that result in a cooling by gas friction that is enhanced above its background level . when using the energy equation with data from our experiment with laser manipulation , we ignore two heating effects , gas atom collisions and fluctuating electric fields . using data from an experimental run without laser manipulation , we estimated those effects , so that we could report the range of values for @xmath125 in the text . our laser manipulation method introduces anisotropy in the particle velocities . since our laser manipulation drives a flow only in the @xmath52 directions , @xmath130 . similarly , the kinetic energy is mostly due to the flow motion in the @xmath54 direction , i.e. , @xmath131 , as shown in fig . 4(a - b ) . however , the kinetic temperature due to the velocity fluctuations in the @xmath54 direction is almost the same as for fluctuations in the @xmath55 direction . | a shear flow of particles in a laser - driven two - dimensional ( 2d ) dusty plasma are observed in a further study of viscous heating and thermal conduction .
video imaging and particle tracking yields particle velocity data , which we convert into continuum data , presented as three spatial profiles : mean particle velocity ( i.e. , flow velocity ) , mean - square particle velocity , and mean - square fluctuations of particle velocity . these profiles and
their derivatives allow a spatially - resolved determination of each term in the energy and momentum continuity equations , which we use for two purposes .
first , by balancing these terms so that their sum ( i.e. , residual ) is minimized while varying viscosity @xmath0 and thermal conductivity @xmath1 as free parameters , we simultaneously obtain values for @xmath0 and @xmath1 in the same experiment .
second , by comparing the viscous heating and thermal conduction terms , we obtain a spatially - resolved characterization of the viscous heating .
= 1 |
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x - ray imaging observations have shown that the x - ray emitting , hot gas in a large fraction of all galaxy clusters reaches high enough densities in the cluster centers that the cooling time of the gas falls below the hubble time , and gas may cool and condense in the absence of a suitable fine - tuned heating source ( e.g. silk 1976 , fabian & nulsen 1977 ) . from the detailed analysis of surface brightness profiles of x - ray images of clusters obtained with the _ einstein _ , _ exosat _ , and _ rosat _ observatories , the detailed , self - consistent scenario of inhomogeneous , comoving cooling flows emerged ( e.g. fabian et al . 1984 , nulsen 1986 , thomas , fabian , & nulsen 1987 , fabian 1994 ) . the main assumptions on which the cooling flow model is based and some important implications are : ( i ) each radial zone in the cooling flow region comprises different plasma phases covering a wide range of temperatures . the consequence of this temperature distribution is that gas will cool to low temperature and condense over a wide range of radii . ( ii ) the gas features an inflow in which all phases with different temperature move with the same flow speed . ( iii ) there is no energy exchange between the different phases , between material at different radii , and no heating . now the first analysis of high resolution x - ray spectra and imaging spectroscopy obtained with _ xmm - newton _ has shown to our surprise that the spectra show no signatures of cooler phases of the cooling flow gas below an intermediate temperature which constitutes a problem for the interpretation of the results in the conventional cooling flow picture ( e.g. peterson et al . 2001 , tamura et al . another result is that the spectroscopic data are better explained with local isothermality in the cooling flow region ( e.g. bhringer et al . 2001a , matsushita et al . 2001 , molendi & pizzolato 2001 ) also in conflict with the inhomogeneous cooling flow model . here , we discuss these new spectroscopic results and their implications and point out the way to a new possible model for this phenomenon . the results are mostly based on the detailed observations of the m87 x - ray halo . a detailed description of this study is provided by bhringer et al . ( 2001b ) . xmm reflection grating spectrometer ( rgs ) observations of several cooling core regions show signatures of different temperature phases ranging approximately from the hot virial temperature of the cluster to a lower limiting temperature , @xmath0 . clearly observable spectroscopic features of even lower temperature gas expected for a cooling flow model are not observed . a1835 with a bulk temperature of about 8.3 kev has @xmath0 around 2.7 kev ( peterson et al . 2001 ) and similar results have been derived for a1795 ( tamura et al . these results are very well confirmed by _ xmm _ observations with the energy sensitive imaging devices , epn and emos , providing spectral information across the entire cooling core region , yielding the result that ( for m87 , a1795 , and a1835 ) single temperature models provide a better representation of the data than cooling flow models ( bhringer et al . 2001a , molendi & pizzolato 2001 ) also implying the lack of low temperature components . the very detailed analysis of m87 by matsushita et al . ( 2001 and the contribution to this workshop ) has shown that the temperature structure is well described locally by a single temperature over most of the cooling core region , except for the regions of the radio lobes and the very center ( @xmath1 arcmin , @xmath2 kpc ) . among the spectroscopic signatures which are sensitive to the plasma temperature in the relevant temperature range , the complex of iron l - shell lines is most important . 1 shows simulated x - ray spectra as predicted for the xmm epn instrument in the spectral region around the fe l - shell lines for a single - temperature plasma at various temperatures from 0.4 to 2.0 kev and 0.7 solar metallicity . there is a very obvious shift in the location of the peak making this feature an excellent thermometer . for a cooling flow with a broad range of temperatures one expects a composite of several of the relatively narrow line blend features , resulting in a quite broad peak . 2a shows for example the deprojected spectrum of the m87 halo plasma for the radial range 1 - 2 arcmin ( outside the inner radio lobes ) and a fit of a cooling flow model with a mass deposition rate slightly less than 1 m@xmath3 yr@xmath4 as expected for this radial range from the analysis of the surface brightness profile ( e.g. stewart et al . 1984 , matsushita et al . it is evident that the peak in the cooling flow model is much broader than the observed spectral feature . for comparison fig . 2b shows the same spectrum fitted by a cooling flow model where a temperature of 2 kev was chosen for the maximum temperature and a suitable lower temperature cut - off ( 1.44 kev ) was determined by the fit . the very narrow temperature interval ( almost isothermality ) is well consistent with the narrow peak . a similar result is obtained for other clusters , e.g. a1795 as shown in fig . since this diagnostics of the temperature structure is essentially based on the observation of metal lines , an inhomogeneous distribution of the metal abundances in the cluster icm and a resulting suppression of line emission at low temperatures was suggested as a possible way to reconcile the above findings with the standard cooling flow model by fabian et al . ( 2001a and contribution in these proceedings ) . as shown by bhringer et al . ( 2001b ) such a scenario will still result in a relatively broad fe l - line feature and does not solve the problem in this case of m87 . another possible attempt to obtain consistency is to allow the absorption parameter in the fit to adjust freely . this is demonstrated in fig . 2c with the same observed spectrum where the best fitting absorption column density is selected in such a way by the fit that the absorption edge limits the extent of the fe - l line feature towards lower energies . this is actually the general finding with asca observations which has shown two possible options for the interpretation of the spectra of cluster core regions : ( 1 ) an interpretation of the results in form of an inhomogeneous cooling flow model which than necessarily includes an internal absorption component ( e.g. allen 2000 , allen et al . 2001 ) , or ( 2 ) an explanation of the spectra in terms of a two - temperature component model ( e.g. ikebe et al . 1997 , 1999 , makishima 2001 ) where the hot component is roughly equivalent to the hot bulk temperature of the clusters and the cool component corresponds approximately to @xmath0 . thus for the cooling flow interpretation to work and to produce a sharp fe - l line feature as observed , the absorption edge has to appear at the right energy and therefore values for the absorption column of typically around @xmath5 @xmath6 are needed ( e.g. allen 2000 and allen et al . 2001 who find values in the range @xmath7 @xmath6 ) . it is therefore important to perform an independent test on the presence of absorbing material in the cluster cores . thanks to _ chandra _ and _ xmm - newton _ we can now use central cluster agn as independent light sources for probing . using the nucleus and jet of m87 ( with _ xmm _ , see fig . 3 ) and the nucleus of ngc1275 ( with _ chandra _ ) we find no signature of internal absorption . thus at least for these two cases it is difficult to argue for internal absorption to obtain consistency of the observations with the cooling flow model . in view of these difficulties of interpreting the observations with the standard cooling flow model , we may consider the possibility that the cooling and mass deposition rates are much smaller than previously thought , that is reduced by at least one order of magnitude . to decrease the mass condensation under energy conservation some form of heating is clearly necessary . three forms of heat input into the cooling flow region have been discussed : ( i ) heating by the energy output of the central agn ( e.g. pedlar et al . 1990 , binney & tabor 1993 , mcnamara et al . 2000 , \(ii ) heating by heat conduction from the hotter gas outside the cooling flow ( e.g. tucker & rosner 1983 , bertschinger & meiksin 1986 ) , and ( iii ) heating by magnetic fields , basically through some form of reconnection ( e.g. soker & sarazin 1990 , makishima et al . the latter two processes depend on poorly known plasma physical conditions and are thus more speculative . the energy output of the central agn , however , can be determined as shown below . a heating scenario can only successfully explain the observations if among others the two most important requirements are met : ( i ) the energy input has to provide sufficient heating to balance the cooling flow losses , that is about @xmath8 to @xmath9 erg in 10 gyr or on average about @xmath10 erg s@xmath4 , and ( ii ) the energy input has to be fine - tuned . too much heating would result in an outflow from the central region and the central regions would be less dense than observed . too little heat will not reduce the cooling flow by a large factor . therefore the heating process has to be self - regulated : mass deposition triggers the heating process and the heating process reduces the mass deposition . @xmath11 [ tab1 ] further constraints are discussed by bhringer et al . ( 2001b ) . the total energy input into the icm by the relativistic jets of the central agn can be estimated by the interaction effect of the jets with the icm by means of the scenario described in churazov et al . it relies on a comparison of the inflation and buoyant rise time of the bubbles of relativistic plasma which are observed e.g. in the case of ngc 1275 ( bhringer et al . 1993 , fabian et al . 2001b ) and requires as observational input parameters the bubble size , @xmath12 , the ambient pressure , @xmath13 , and the keplerian velocity at the bubble radius in the cluster , @xmath14 . the parameters and the estimated total energy output is given in table 1 for three examples , m87 , perseus , and hydra a. these values for the energy input have to be compared with the energy loss in the cooling flow , which is of the order of @xmath15 erg s@xmath4 for m87 and about @xmath16 erg s@xmath4 for perseus . thus in these cases the energy input is larger than the radiation losses in the cooling flow for at least about the last @xmath17 yr . we have , however , evidence that this energy input continued for a longer time with evidence given by the outer radio halo around m87 with an outer radius of 35 - 40 kpc ( e.g. kassim et al . 1993 , rottmann et al . 1996 ) . owen et al . ( 2000 ) give a detailed physical account of the halo and model the energy input into it . they estimate the total current energy content in the halo in form of relativistic plasma to @xmath18 erg and the power input for a lifetime of about @xmath17 years , which is also close to the lifetime of the synchrotron emitting electrons , to the order of @xmath16 erg s@xmath4 , consistent with our estimate . the very characteristic sharp outer boundary of the outer radio halo of m87 , noted by owen et al . ( 2000 ) , has the important implications , that this could not have been produced by magnetic field advection in a cooling flow . thus , we find a radio structure providing evidence for a power input from the central agn into the halo region of the order of about ten times the radiative energy loss rate over at least about @xmath17 years ( for this representative example of m87 ) . the energy input could therefore balance the heating for at least about @xmath19 years . the observation of active agn in the centers of cooling flows is a very common phenomenon . e.g. ball et al . ( 1993 ) find in a systematic vla study of the radio properties of cd galaxies in cluster centers , that 71% of the cooling flow clusters have radio loud cds compared to 23% of the non - cooling flow cluster cds . therefore we can safely assume that the current episode of activity was not the only one in the life of m87 and its cooling flow . the mechanism for a fine - tuned heating of the cooling flow region should most probably be searched for in a feeding mechanism of the agn by the cooling flow gas . the most simple physical situation would be given if simple bondi type of accretion from the inner cooling core region would roughly provide the order of magnitude of the power output that is observed and required . using the classical formula for spherical accretion from a hot gas by bondi ( 1952 ) we can obtain a very rough estimate for this number . for the proton density near the m87 nucleus ( @xmath20 arcsec ) of about 0.1 @xmath21 , a temperature of about @xmath22 k ( e.g. matsushita et al . 2001 ) , and a black hole mass of @xmath23 m@xmath3 ( e.g. ford et al . 1994 ) we find a mass accretion rate of about 0.01 m@xmath3 yr@xmath4 and an energy output of about @xmath24 erg s@xmath4 , where we have assumed the canonical value of 0.1 for the ratio of the rest mass accretion rate to the energy output . the corresponding accretion radius is about 50 pc ( @xmath25 arcsec ) . this accretion rate is more than a factor of 1000 below the eddington value and thus no reduction effects of the spherical accretion rate by radiation pressure has to be expected . small changes in the temperature and density structure in the inner cooling core region will directly have an effect on the accretion rate . therefore we have all the best prospects for building a successful self - regulated agn - feeding and cooling flow - heating model . several observational constraints have let us to the conclusion that the mass deposition rates in galaxy cluster cooling cores are not as high as previously predicted . the new x - ray spectroscopic observations with a lack of spectral signatures for the coolest gas phases expected for cooling flows and the lower mass deposition rates indicated at other wavelength bands than x - rays are more consistent with mass deposition rates reduced by one or two orders of magnitude below the previously derived values . this can , however , only be achieved if the gas in the cooling flow region is heated . the most promising heating model is a self - regulated heating model powered by the large energy output of the central agn in most cooling flows . most of the guidance and the support of the heating model proposed here ( based on concepts developed in churazov et al . 2000 , 2001 ) is taken from the detailed observations of a cooling core region in the halo of m87 and to a smaller part from the observations in the perseus cluster . these observations show that the central agn produces sufficient heat for the energy balance of the cooling flow , that the most fundamental and classical accretion process originally proposed by bondi ( 1952 ) provides an elegant way of devising a self - regulated model of agn heating of the cooling flow , and that most of the further requirements that have to be met by a heating model to be consistent with the observations can most probably be fulfilled . since these ideas are mostly developed to match the conditions in m87 , it is important to extent such detailed studies to most other nearby cooling flow clusters . in this new perspective the cooling cores of galaxy clusters become the sites where most of the energy output of the central cluster agn is finally dissipated . strong cooling flows should therefore be the locations of agn with the largest mass accretion rates . while in the case of m87 with a possible current mass accretion rate of about 0.01 m@xmath3 y@xmath4 the mass addition to the black hole ( with an estimated mass of about @xmath26 m@xmath3 ) is a smaller fraction of the total mass , the mass build - up may become very important for the formation of massive black holes in the most massive cooling flows , where mass accretion rates above 0.1 m@xmath3 y@xmath4 become important over cosmological times . | new x - ray observations with xmm - newton show a lack of spectral evidence for large amounts of cooling and condensing gas in the centers of galaxy clusters believed to harbour strong cooling flows . here , we explore these diagnostics of the temperature structure of cooling cores with xmm - spectroscopy .
we further find no evidence of intrinsic absorption in the center of the cooling flows of m87 and the perseus cluster .
to explain these findings we consider the heating of the core regions of clusters by jets from a central agn .
we find that the power of the agn jets as estimated by their interaction effects with the intracluster medium in several examples is more then sufficient to heat the cooling flows .
we explore which requirements such a heating model has to fulfill and find a very promising scenario of self - regulated bondi accretion of the central black hole . in summary
it is argued that most observational evidence points towards much lower mass deposition rates than previously inferred for cooling flow clusters .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
You are an expert at summarizing long articles. Proceed to summarize the following text:
equal - sized hard spheres constitute probably the simplest example of a purely entropic material . in a hard - sphere system there is no contribution to the internal energy @xmath1 from interparticle forces so that @xmath1 is a constant , at a fixed temperature . minimising the free energy , @xmath2 , is thus simply equivalent to maximising the entropy @xmath3 . consequently , the structure and phase behaviour of hard spheres is determined solely by entropy . although the hard - sphere model was originally introduced as a mathematically simple model of atomic liquids@xcite recent work has demonstrated it usefulness as a basic model for complex fluids@xcite . suspensions of submicron poly(methyl methacrylate ) or silica colloids , coated with a thin polymeric layer so that strong repulsions dominate the attractive dispersion forces between the colloidal cores , behave in many ways as hard spheres . in the last decade , a great deal of effort has been devoted to systematic studies of such colloidal ` hard - sphere ' systems@xcite . for sufficiently monodisperse colloids , crystallisation is observed at densities similar to those predicted by computer simulation for hard spheres@xcite . measurements of the osmotic pressure and compressibility show a similar dramatic agreement with predicted hard sphere properties@xcite . there is however one important and unavoidable difference between colloids and the classical hard sphere model which is frequently overlooked . whereas the spheres in the classical model are identically - sized ( _ i.e. _ monodisperse ) colloidal particles have an inevitable spread of sizes which is most conveniently characterised by the polydispersity , @xmath4 , defined as @xmath5 where @xmath6 with @xmath7 the density distribution and @xmath8 . recent work has revealed that as soon as a hard sphere suspension is allowed to enjoy a significant degree of polydispersity , several interesting new phenomena arise . experiments find that the crystallisation transition is suppressed altogether at @xmath9 while samples with @xmath10 crystallise only slowly in the coexistence region and not above the melting transition@xcite . other examples of the effects of polydispersity are the appearance of a demixing transition in a polydisperse fluid of hard spheres@xcite and the observation of a liquid - vapour transition in polydisperse adhesive hard spheres@xcite . the effect of polydispersity on the crystallisation of hard sphere colloids has been investigated by computer simulation@xcite , density functional@xcite , and analytical theories@xcite . the picture that emerges is remarkably consistent . all calculations find that the fluid - solid phase transition vanishes for polydispersities above a certain critical level @xmath11 . the phase diagram for small polydispersities ( @xmath12 ) has been rationalised @xcite in terms of the appearance of an additional high density crystal - to - fluid transition , in a polydisperse system . while , at low polydispersities hard spheres display , with increasing density , the conventional fluid - to - crystal transition , at higher polydispersities re - entrant behaviour is predicted . the two freezing transitions converge to a single point in the @xmath13 plane which is a polydisperse analogue of the point of equal concentration found in molecular mixtures@xcite . at this singularity , the free energies of the polydisperse fluid and crystal phases are equal . the purpose of this note is to examine the fate of a _ highly _ polydisperse ( @xmath14 ) hard sphere fluid . previous theoretical research has not been able to identify a fluid - solid transition for @xmath14 so it is generally believed that the equilibrium phase is disordered at all densities up to close packing . several years ago , pusey@xcite suggested that a highly polydisperse suspension might crystallise by splitting the broad overall distribution into a number of narrower distributions of polydispersity @xmath15 , each of which could be accommodated within a single crystalline phase . each crystal would then have a correspondingly different mean size with the number of crystalline phases increasing with the overall polydispersity . for fractionated crystallisation to occur the total free energy of the set of multiple crystals must be lower than that of the equivalent polydisperse fluid . this can only happen if the reduction in free energy as particles are removed from a fluid and placed in a crystal is sufficiently large to exceed the loss of entropy of mixing as the distribution is partitioned . this is a delicate balance and it is far from obvious where the result lies . pusey , for instance , originally suggested that fractionation would generate crystals with a polydispersity of @xmath11 so as to minimise the number of crystal phases required and the subsequent loss of entropy of mixing . however , as noted above , at @xmath11 the free energy of the polydisperse fluid and crystal phases are equal@xcite and so there is no driving force for fractionation . in earlier work@xcite the possibility of fractionated crystallisation for @xmath14 was considered but no conditions where two crystal phases could coexist could be found . here we re - examine the stability of a polydisperse hard - sphere fluid using a much simpler approach . rather than solving the equations of phase equilibria in a polydisperse system we restrict ourselves to the easier task of comparing the free energies , at the same density and temperature , of crystal and fluid phases . we find , in agreement with pusey@xcite , that fractionation occurs in polydisperse hard sphere mixtures but that the polydispersity of the resulting crystals is substantially less than @xmath11 . the rest of the paper is organised as follows : in section [ model ] we present our model for the free energies of the polydisperse fluid and crystal phases . the stability diagram is present in section [ results ] . finally , in section [ discussion ] we summarise and conclude . our model consists of @xmath16 hard sphere particles in a volume @xmath17 , at an overall density of @xmath18 . each particle has a diameter @xmath19 drawn from some overall distribution @xmath7 so that @xmath8 . previous work has suggested@xcite that the thermodynamic properties of a polydisperse system are relatively insensitive to the detailed _ shape _ assumed for the diameter distribution , at least when the polydispersity is small ( @xmath20 ) . indeed the phase behaviour has been found@xcite , to a rather good approximation , to be a function only of the first three moments of the diameter distribution , @xmath21 , @xmath22 , and the normalised width @xmath4 . for a broad diameter distribution , higher moments will presumably also need to be considered . consequently , the phase behaviour will become increasingly sensitive to the shape of the distribution assumed . however we expect that the width of the distribution will still play an important if not the dominant role in determining the generic features of the phase behaviour of a polydisperse system . since our concern here is with establishing these general features of the polydisperse phase diagram we have taken @xmath7 as a simple rectangular distribution , @xmath23 where @xmath24 is the mean diameter and the normalised width @xmath25 is related directly to the conventional polydispersity @xmath4 by @xmath26 . it will prove convenient to choose as variables the mean diameter @xmath24 , the polydispersity @xmath4 and the volume fraction @xmath27 . our first task is to calculate the thermodynamics of the fluid state . the free energy density ( @xmath28 ) may be conveniently split into ideal and excess portions @xmath29 the ideal part of the free energy density is given by a straightforward generalisation of the free energy of an ideal gas as @xmath30 with @xmath31 . for the case of the rectangular distribution , the ideal free energy density is simply @xmath32 we calculate the excess free energy density , @xmath33 , of the polydisperse fluid from the general equation of state given by salacuse and stell@xcite . this equation of state is a straightforward generalisation of the highly successful eos for an arbitrary hard sphere mixture@xcite first considered by mansoori _ the mansoori eos has been checked against simulation data by a number of authors and agreement is generally excellent . the free energy per particle ( @xmath34 ) is then @xmath35 . now we consider the free energy of the fractionated solid phase . since as mentioned above , we expect the number of crystals to depend upon the overall polydispersity @xmath4 of the parent distribution , we consider the general case of @xmath36 coexisting crystals . in order to establish some relation between the parent distribution @xmath7 and the sub - distribution , @xmath37 , found in the @xmath38 crystal we make two assumptions . first we choose the distribution in each crystal to be rectangular and second , to keep the theory computationally manageable , we force the width ( or polydispersity ) to be equal in each of the @xmath36 coexisting phases . with these two restrictions the housekeeping of the fractionation process is straightforward . the @xmath38 crystal , for instance , contains all @xmath39 particles whose diameters lie between @xmath40 $ ] and @xmath41 $ ] with @xmath42 , the width of the crystal distribution , given by @xmath43 . the mean diameter of particles in the @xmath38 crystal is then simply given as @xmath44 \label{meandiameter}\ ] ] and the polydispersity of each crystal is @xmath45 . the diameter distribution in the @xmath38 crystal is then @xmath46 at an overall density of @xmath47 with @xmath48 the volume of the @xmath38 phase . although the number of particles in each crystal phase is the same for all crystals and equal to @xmath49 ( since the polydispersity of each crystal is assumed equal ) the volume @xmath48 of each phase is so far undetermined . to fix the volume , or equivalently the number density @xmath50 , of each crystal we use the fact that the crystals are in equilibrium with each other this constraint is straightforward to apply since , in our model of the partition , no particle may exist in anymore than one crystal . consequently we do not need to consider exchange of particles . equilibrium simply requires that the osmotic pressure of each crystal , @xmath51 , be fixed and equal to an external pressure of , say , @xmath52 @xmath53 with the density of each crystal phase determined , the total number density of the set of @xmath36 coexisting crystals is then @xmath54 and the corresponding composite packing fraction is @xmath55 . the mean free energy per particle is @xmath56 where @xmath57 is the free energy density of the @xmath38 crystal . to complete our model for the fractionated phases we need an expression for the excess free energy of a polydisperse crystal . for that purpose we use a scaled particle theory . since the details have been described elsewhere@xcite we just outline the approach here . the model exploits the picture of particles in a solid as being confined in cells or cages formed by the neighbours from which they can not escape . while , in a monodisperse crystal the cells are all identical , in a polydisperse crystal the size and shape of the cell varies . the idea is to recognise that one can allow for the variability in the cell shape , to a first approximation , by considering just a finite number of different - sized neighbours . the number of different - sized particles is determined by the accuracy with which the properties of the finite mixture approximate those of the polydisperse system . if the excess free energy of the polydisperse system is a function of the first @xmath58 moments of the diameter distribution @xmath59 $ ] different - sized spheres are needed to match the diameter moments of the continuous distribution ( @xmath60 $ ] is the smallest integer not less than @xmath61 ) . for a polydisperse system of hard spheres , highly successful mean - field theories such as the percus - yevick approximation suggest that only four diameter moments are significant@xcite . in this case , the properties of a polydisperse system should be well approximated by a binary mixture of spheres , chosen so that the first four moments of the polydisperse and binary distributions are equal . the two diameter distributions are ` equivalent ' and all excess properties are , to a first approximation , equal . for the rectangular distribution the equivalent binary distribution has an equal number of large and small spheres with diameters @xmath62 and @xmath63 respectively . specifically then , we equate the excess free energy of the polydisperse crystal with that of an equivalent binary substitutional crystal . for the evaluation of the latter we take advantage of the analytical expressions quoted by kranendonk and frenkel@xcite as fits to monte carlo simulation results . combining the excess free energy with the ideal free energy density , @xmath64 yields the free energy density @xmath57 of the @xmath38 crystal . finally , we note that our model for the polydisperse crystal implicitly assumes that all different - sized spheres are placed randomly on the sites of a common fcc lattice . at high polydispersities , the smallest particles in the distribution could also be accommodated within the interstices of a crystal of larger particles . the current calculations ignore this possibility and so are valid only for polydispersities of @xmath65 . putting everything together , we now calculate the difference free energy per particle , @xmath66 , between the polydisperse fluid and the fractionated set of @xmath36 crystals as follows . the overall particle density distribution of either the set of crystals or the polydisperse fluid is characterised by identical values for the overall volume fraction @xmath67 , the polydispersity @xmath4 and the mean diameter @xmath24 . for fixed @xmath68 , @xmath4 and @xmath24 , the free energy of the fluid @xmath69 is calculated from eqs . [ fideal ] and the mansoori eos . combining eqs . [ constantp ] and [ effectiverho ] we see that the overall volume fraction @xmath67 of the solid phases ( or equivalently the mean density @xmath70 ) may be considered as a function of the osmotic pressure @xmath52 . inverting this relation yields the osmotic pressure @xmath52 and so , from eq . [ constantp ] , the densities @xmath50 of each of the coexisting crystals which taken together have an overall volume fraction of @xmath67 . knowing the density , polydispersity @xmath71 and mean diameter @xmath72 ( from eq . [ meandiameter ] ) of each crystal it is straightforward to calculate the corresponding free energy density @xmath57 and the mean free energy @xmath73 of the multiple crystals from eq . [ freesolid ] . we now present in detail results for the stability of a system of polydisperse hard spheres obtained from the theory described above . in fig . [ fig1 ] , we show the free energy difference , @xmath74 , between the fractionated crystals and the fluid phase as a function of polydispersity , for the representative density of @xmath75 . the first question to be addressed is the relative stability of the various fractionated crystals and the fluid phase . referring to fig . [ fig1 ] , we see that for low levels of polydispersity , at this density , the crystal has a significantly lower free energy than the fluid phase . however , with increasing polydispersity , the stability of the polydisperse crystal reduces rapidly until for @xmath76 the unfractionated polydisperse crystal is unstable relative to the fluid phase . the origin of this behaviour lies in the different ways polydispersity affects the maximum packing fraction , or equivalently the free volume , of ordered and disordered structures . in a fluid , smaller particles are free to pack in the cavities between larger particles and so the free volume and thus the entropy increases with polydispersity . conversely , the periodicity of a crystal causes the free volume and entropy of a polydisperse crystal to decrease with increasing polydispersity . consequently the free energy of a fixed - density polydisperse crystal will diverge at a critical level of polydispersity at which the fluid free energy will remain finite . second , we note from fig . [ fig1 ] that at low polydispersities , fractionating the distribution into two or more crystals always raises the free energy of the solid state . this is because of the loss of entropy of mixing as the diameter distribution is split up . the contribution to the free energy difference , @xmath74 , due to the loss of entropy of mixing is simply : @xmath77 for fractionation into @xmath36 crystal phases which is very close to the differences seen in fig . [ fig1 ] as @xmath78 . however at finite polydispersities , eq . [ mixing ] , does not give a reasonable estimate of the free energy differences between the various sets of crystals . as is evident , the free energy @xmath74 reduces with increasing polydispersity . the reduction being smaller as the number of crystals considered increases this is because the polydispersity of each fractionated crystal drops as the total number of crystals , between which the overall distribution is split , increases . for instance , fractionating a distribution with a polydispersity of 6% between two phases would result in crystals with 3% polydispersity , three phases would lead to 2% polydispersity and so on . as the polydispersity of each of the individual crystals decreases the influence of the divergence of the free energy at the close - packing limit is reduced and so the free energy is less affected by the overall polydispersity . as a consequence the fractionated two - crystal phase takes the place of the polydisperse single - phase crystal , as the state of lowest free energy , for @xmath79 and remains the most stable phase until @xmath80 , at which point the fluid appears . by a procedure similar to that described above , we have mapped out the range of stability of the polydisperse hard sphere system for @xmath81 . our results are shown in fig . the regions are labelled by the number of coexisting crystals which minimise the total free energy of the polydisperse system while the boundaries indicate the specific densities and polydispersities at which competing phases have equal free energies . as is evident , there are appreciable areas of density and polydispersity where fractionation into multiple crystals is predicted . at low volume fractions , @xmath82 , the polydisperse fluid is the most stable phase at all polydispersities . however with an increase in packing fraction a crystalline phase takes the place of the polydisperse fluid as the most stable phase . the nature of the crystalline phase depends on the magnitude of the polydispersity @xmath4 . so while for @xmath83 we have crystallisation into the usual single polydisperse crystal , with increasing polydispersity the fluid fractionates upon solidification into a rapidly increasing number of crystals . the diagram gets more and more complicated as the polydispersity grows since , by necessity , the number of crystalline phases required to accommodate the increasing width of the diameter distribution increase . so while for polydispersities in the range @xmath84 two fractionated crystals are first formed , three crystals are required for @xmath85 , four for @xmath86 , five for @xmath87 while at @xmath88 complete crystallisation requires six different crystal phases . furthermore , we note that as the polydispersity increases the packing fraction at which the fluid becomes unstable increases from @xmath89 at @xmath90 to @xmath91 at @xmath88 . the degree of fractionation predicted varies with the overall level of polydispersity . referring to fig . [ fig3 ] , we see that the polydispersity of the crystal(s ) at the fluid - solid boundary reaches a maximum of 0.085 before on average falling with increasing fluid polydispersity , although in a far from continuous fashion . where fractionation occurs , fig . [ fig3 ] reveals that the polydispersity of each crystal formed is considerably lower than the critical polydispersity , estimated here as @xmath92 . in addition to the fluid - multiple crystal transitions discussed above , fig . [ fig2 ] reveals that there is a further sequence of demixing transitions in the solid phase . compressing a polydisperse crystal causes a solid - state phase separation in which the diameter distribution is fractionated . for instance a polydisperse crystal with @xmath93 is stable up to a density of @xmath94 . at this point the system separates into two crystals , each of polydispersity @xmath95 , which are stable until a total density of @xmath96 is reached . further compression results in three crystals , followed at still high densities by a cascade of demixing transitions . the explanation for this behaviour lies in the effect of polydispersity on the limiting packing fraction of a crystal . as mentioned previously , the density at which the free energy of a polydisperse crystal starts to diverge reduces with increasing polydispersity . as a polydisperse crystal is compressed there comes a point at which the reduction in the excess free energy which occurs when the polydispersity is reduced is sufficient to exceed the increase in the ideal free energy of mixing after fractionation . at this point the crystal phase separates . finally , we emphasise that we have calculated only the stability boundaries , not the full equilibrium phase diagram of a system of polydisperse spheres . the reason for this is that a full calculation of the phase equilibria , particularly in view of the large number of competing phases found in this work , would be a large and complicated problem . however we expect our stability diagram ( fig . [ fig2 ] ) to be a useful guide to the form of the phase diagram . confirmation of this point of view is provided by the comparison , seen in fig . [ fig4 ] , between the current model and the fluid - single crystal coexistence region established in earlier work@xcite . the dashed curves indicate the positions of the polydisperse cloud - point boundaries and were calculated by a novel moment projection method using , as input , polydisperse free energies similar to those described above . referring to fig . [ fig4 ] , we see that the stability boundary lies approximately midway between the two coexistence densities . inspection of fig . [ fig4 ] reveals , rather intriguingly , that the polydisperse point of equal concentration , marked by the large circle in fig . [ fig4 ] , is practically coincident with the lowest density at which the binary system of fractionated crystals is stable . this seems to be a chance occurrence . however , the proximity of the two points suggests that , at equilibrium , the re - entrant melting predicted in ref . @xcite will in fact be pre - empted by a solid state phase separation . we have analysed the stability of a polydisperse fluid of hard spheres with respect to a process of simultaneous fractionation and crystallisation . although the model is quite simple we find several interesting features . in particular , we predict that for a sufficiently polydisperse system of hard spheres fractionation occurs upon solidification . the fluid diameter distribution separates into a number of fractions of narrower polydispersity which then crystallise . furthermore , we find that compressing a polydisperse crystal induces a sequence of demixing transitions in the crystal . to keep the theory presented here manageable we have been forced to make a number of assumptions . we have , for instance , imposed a somewhat arbitrary model for the fractionation process . in particular we have assumed that a rectangular diameter distribution fractionates into a number of equal - width daughter distributions , each of which is also rectangular in form . in principle , we could relax the constraint of equal width . however , since the densities of the coexisting crystals are similar it seems likely that this modification would have little effect . the more critical restriction is probably the assumption made for the form of the daughter distribution . however , in view of the relative insensitivity of polydisperse systems to the detailed form chosen for the diameter distribution , we expect that the current predictions will be in at least qualitative if not quantitative agreement with more rigorous calculations , when available . we now speculate briefly on the feasibility of observing solid - state fractionation in experiments on hard - sphere colloids . the first point to note is that fractionation requires colloid diffusion over distances comparable to the size of the growing crystallite . the rate of such large scale diffusive motion reduces with increasing colloid density , as a result of caging effects , and essentially vanishes at the glass transition density , @xmath97@xcite . for uniformly - sized hard spheres simulations@xcite find a long - lived metastable glassy state around @xmath98 which persists up to the random close packing limit at @xmath99 . the glass transition in a system of polydisperse hard spheres has not been studied to the best of our knowledge although schaertl and silescu@xcite have estimated @xmath100 as a function of polydispersity from simulation . assuming a simple linear relationship between the two densities gives a crude estimate of the glass transition density as @xmath101 . referring to fig . [ fig4 ] , we see that if this estimate for the glass transition is used , then fractionated crystallisation might be experimentally observable for polydispersities in the narrow range @xmath102 where diffusive motion is still possible although slow . if conversely , the dynamics of the fractionation process turn out to be appreciably slower than the experimental timeframe then experiments should follow our earlier predictions@xcite and exhibit a polydisperse point of equal concentration . a clear resolution of this ambiguity must await experiment . finally , our results suggest a possible explanation for the apparently anomalous simulations reported by bolhuis and kofke@xcite . in contrast with other theoretical work , this study found that : ( i ) the particle size distributions in the fluid and crystal phases were significantly different and ( ii ) that the coexistence region although initially narrowing with increasing polydispersity finally widened . the phase diagram was traced out by integrating along the fluid - crystal coexistence line . implicit in this approach is the assumption that only one crystalline phase exists at the fluid - solid boundary . referring to fig . [ fig2 ] , we envisage that at the point where two polydisperse crystals become stable the simulation will follow one of the two fluid - solid branches until it halts at the discontinuity marking the start of the three - crystal region . with this picture in mind , we may readily account for the observation that at the point , where the simulation could no longer follow the fluid - solid boundary , the crystal polydispersity was almost exactly half that of the fluid value . at the same time , we note that their limiting polydispersity ( @xmath103 ) lies close to our estimate for the highest polydispersity ( 0.117 ) at which two polydisperse crystals remain the equilibrium phase . | we consider the nature of the fluid - solid phase transition in a polydisperse mixture of hard spheres . for a sufficiently polydisperse mixture
( @xmath0 ) crystallisation occurs with simultaneous fractionation . at the fluid - solid boundary ,
a broad fluid diameter distribution is split into a number of narrower fractions , each of which then crystallise .
the number of crystalline phases increases with the overall level of polydispersity . at high densities , freezing is followed by a sequence of demixing transitions in the polydisperse crystal . |
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the problem of the transcritical flow of a bec through the penetrable barriers has been under recent active investigations @xcite . the damping processes for the superfluid flow moving through the barrier are of a fundamental interest . in multidimensional case above some critical velocity of the obstacle motion the damping accompanied by the radiation emission @xcite is observed . thus in the region when the motion is still superfluid , the velocity is bounded above . the damping is associated with the landau type damping and related to the emission of the elementary excitations . landau damping can be described in the framework of the mean field theory and is not associated with thermalization processes @xcite . the critical velocity value at which the damping is observed , differs essentially from the values predicted by the landau theory . as it was shown firstly by feynman @xcite , the reason is in the nonlinearity of the system . in the case of a quasi 1d bose - einstein condensate flow , when passing through a penetrable barrier , some interval of velocities @xmath2 exists , where trains of dark solitons are generated , that leads to deviation from predictions based on the matching with the spectrum of elementary linear excitations @xcite.in addition in this range of velocities , generation of dispersive shock waves occurs . experimental proof of the existence of the velocities interval was given in the work @xcite . hakim @xcite has indicated that for supersonic velocities ( including ones above supercritical velocity @xmath3 ) some radiation is still nonzero and its amplitude rapidly decreases at the ratio of the potential variation length to the gpe coherence length . the amplitude of the wake can be characterized by the fourier transform of the obstacle potential @xcite . thus , wide and smooth potentials can be considered as radiationless at velocities above _ supercritical_. seemingly in one dimensional case only stable dark solitons can exist . peculiarity of the one dimension is in the fact that generation of the solitons is possible till some _ supercritical _ velocity , @xmath3 . above this velocity the emission is strongly damped and the quasi - superfuidity is restored . the radiation exists , but exponentially small - decay rate is proportional to @xmath4 , where @xmath5 is the healing length of the order of the dark soliton width . in this work we consider the phenomena occurring in the flow of a quasi 1d bec past _ a nonlinear _ barrier which is a localized space inhomogeneity of the the nonlinearity coefficient in the gross - pitaevskii equation . such a type of barriers can be formed by some area of bec where the effective value of the atomic scattering length is varied in _ the space_. it can be achieved both by the feshbach resonance techniques @xcite , and by the local variation of the transverse frequency of the trap potential . in the former case , varying external magnetic field in space near the resonance , one can vary the value of the atomic scattering length @xmath6 . another way is to use optically induced feshbach resonances @xcite . in this case the variation can be achieved by local change in the intensity of a laser field . variation of @xmath6 in a half space recently has been suggested to generate vortices in bec as a nonlinear piston method @xcite . the present paper is motivated by the works @xcite where flow of a bec past an obstacle in one dimension was investigated . we consider two cases , wide obstacle potential and short range one . let us consider a nonlinear penetrable barrier moving through the elongated bec . a quasi one dimensional bec can be described by the gross - pitaevsky ( gp ) equation with standard dimensionless variables @xmath7 where @xmath8 @xmath6 is the atomic scattering length , @xmath9 is the transverse frequency of the trap , @xmath10 , @xmath11 is the background value of the scattering length @xmath6 . for the further study of the flow problem it is useful to pass to the reference frame moving with the barrier @xmath12 . so we come to the equation @xmath13 the scattering length can be manipulated with a laser field tuned near a photo association transition , e.g. , close to the resonance of one of the bound @xmath14 levels of the excited molecules . virtual radiative transitions of a pair of interacting atoms to this level can change the value and even reverse the sign of the scattering length @xcite . recently spatial modulations of the atomic scattering length by the optical feshbach resonance method was realized experimentally in bec @xcite . such approach implies some spontaneous emission loss which is inherent in the optical feshbach resonance technique . here we assume that such dissipative effects can be ignored , since they become possible if one uses laser fields of sufficiently high intensity detuned from the resonance . thus the repulsive nonlinear barrier can be formed by an focused external laser beam with the parameters lying near the optically induced feshbach resonance . we analyze this case following the method developed in @xcite for the linear barrier case . let us pass to the hydrodynamical form for the gp equation ( [ eq1 ] ) . it can be obtained by the following transformation @xmath15 substituting it into ( [ eq1 ] ) and introducing @xmath16 we obtain the system @xmath17 for a wide smooth obstacle potential we can neglect the terms in the bracket in the second equation that corresponds to the hydrodynamical approximation . omitting also primes , for stationary solutions we can put @xmath18 and @xmath19 , and obtain the following system of equations @xmath20 @xmath21 with the boundary conditions @xmath22 integrating over @xmath23 we find @xmath24 @xmath25 eliminating the function @xmath26 from these equations , we get @xmath27\equiv f(u ) . \label{fu}\ ] ] since we consider repulsive obstacle potential @xmath28 we have the condition @xmath29 . maximum of @xmath30 is realized at @xmath31 . thus the maximum of the function @xmath30 is @xmath32 = \mu(v ) = \frac{1}{v}\sqrt{\left(\frac{v^2 + 2}{3}\right)^{3 } } - 1 . \label{eqmaxfu}\ ] ] stationary solution @xmath33 is obtained by solving the equation ( [ fu ] ) with respect to @xmath34 . this equation has a real solution defined for all @xmath23 provided that @xmath35 \leq \max[f(u ) ] , \label{maxvmaxfu}\ ] ] i.e. the range of values of @xmath36 , which is @xmath37 $ ] , lies within the range of values of the function @xmath30 @xcite . ( see eq.([eqmaxfu ] ) ) versus @xmath23 . for given obstacle potential maximum @xmath38 , critical values of the velocity @xmath39 , @xmath40,width=302 ] . [ maxfu ] maximum of the function @xmath30 versus the obstacle velocity @xmath41 of bec is presented in fig . [ maxfu ] . as seen for any value of @xmath42 two critical values of the velocity exist , @xmath43 , determined by equation @xmath44 . in transcritical regime , in the interval @xmath45 , the condition of the stationary flow ( [ maxvmaxfu ] ) does not hold . out of this region , in subcritical ( @xmath46 ) and supercritical ( @xmath47 ) regimes the radiation phenomena are negligible and the motion of the system can be considered as superfluid . analyzing expression ( [ eqmaxfu ] ) and [ maxfu ] it should be noted that unlike the case of a wide linear barrier , considered in @xcite , the velocity @xmath48 is not vanish and there always exists an interval @xmath49 where the flow is superfluid . eq.([fu ] ) can be rewritten as @xmath50 which is a cubic equation with respect to @xmath33 . solving it we obtain the following solutions for @xmath33 satisfying the boundary conditions @xmath51 where @xmath52 . the barrier velocities are equal to @xmath53 and @xmath54 for lower and upper lines respectively.,width=302 ] spatial profiles of the local velocity @xmath34 for subcritical @xmath55 ( @xmath46 ) and supercritical @xmath56 ( @xmath47 ) regimes are depicted in fig . [ ux ] . the nl obstacle potential is taken in the form @xmath57 with its maximum value @xmath38 . and ( b ) supercritical regime , @xmath56 through a nonlinear repulsive potential barrier @xmath58 with @xmath59 . the initial wave packet and distribution of the bec local velocities @xmath33 are taken in the form determined by formulas ( [ sol_u- ] ) , ( [ sol_u+ ] ) and eq . ( [ pu1]).,width=302 ] and ( b ) supercritical regime , @xmath56 through a nonlinear repulsive potential barrier @xmath58 with @xmath59 . the initial wave packet and distribution of the bec local velocities @xmath33 are taken in the form determined by formulas ( [ sol_u- ] ) , ( [ sol_u+ ] ) and eq . ( [ pu1]).,width=302 ] fig . [ repdef ] depicts time evolution of a bec flow through a repulsive non - linear potential @xmath60 with @xmath59 in ( a ) subcritical @xmath61 and ( b ) supercritical @xmath62 regimes , respectively . initial form of the condensate density @xmath63 is determined by eq.([pu1 ] ) as @xmath64 , where initial distribution of local velocities @xmath33 is given by eqs . ( [ sol_u- ] ) , ( [ sol_u+ ] ) . one can see that in these regimes the flow through the barrier is steady . existence of small amplitude waves , spreading from the hump in the beginning is a result of neglecting small terms in the course of derivation of eqs.([difpu1 ] ) and ( [ difpu2 ] ) . in fig . [ repdef]b one can see that in supercritical regime the solution at the center has the hump form . the numerical simulations show stability of this kind of steady flows . ( @xmath2 ) . the nl barrier is taken in the form of @xmath58 with @xmath59 . during the time period from @xmath65 to @xmath66 ( that is not presented in the figure ) the value of @xmath67 is adiabatically being increased from @xmath68 to @xmath69 . further evolution is given at @xmath59.,width=302 ] in order to carry out numerical simulations of the behavior of a bec at transcritical velocities ( @xmath70 ) , we can not use eqs . ( [ sol_u- ] ) , ( [ sol_u+ ] ) as an initial wave packets , because they have been derived for a steady flow . in numerical simulations it is more convenient to increase adiabatically the strength of nl potential @xmath67 . in fig . [ movrepdef ] we show time evolution of bec flow through a nl potential barrier in the transcritical regime with @xmath71 @xmath72 . the nl potential is taken in the form @xmath60 . @xmath67 is increasing from @xmath68 to @xmath69 in the time interval @xmath73 and then is kept constant . one can see that in the transcritical regime the flow becomes unsteady and a train of dark solitons emerges from the nl barrier at the barrier potential strength @xmath59 . in this section we follow the approach used in the work @xcite . let us suppose the condensate to have a chemical potential @xmath74 . then in the frame of the moving obstacle with the velocity @xmath41 equation ( [ eq1 ] ) takes the form @xmath75 with uniform boundary conditions @xmath76 at @xmath77 . looking for time independent solution in the form @xmath78 we get equations for amplitude @xmath79 and phase @xmath80 @xmath81 in the case of the @xmath1 function barrier potential ( a sharp jump in the nonlinearity ) @xmath82 the solution @xmath79 has the form @xmath83 } \ \mbox{at } \ x \lessgtr 0 , \label{r2}\end{aligned}\ ] ] substituting obtained r(x ) into eq . ( [ eqphi ] ) and solving it we obtain phase @xmath80 as @xmath84 where unknown parameter @xmath85 depending on the potential strength @xmath86 is determined from the relation @xmath87 obtained from matching condition for derivatives @xmath88 at @xmath89 @xmath90 fig . [ gamma ] depicts a typical relation between the potential strength @xmath86 and parameter @xmath85 at @xmath91 . as seen for given strength @xmath86 there are two values of the parameter @xmath85 ( or not a single ) corresponding to a pair of steady solutions . one of the solutions ( @xmath92 ) is unstable and another ( @xmath93 ) is stable @xcite . time evolution of stable and unstable steady solutions corresponding to @xmath94 and @xmath95 are shown in fig . [ twosol ] . and @xmath96 corresponding relatively to ( a ) stable and ( b ) unstable bec flows past a nonlinear repulsive delta potential barrier in subcritical regime . the other parameters @xmath91 and @xmath97 . for this case @xmath98.,width=302 ] and @xmath96 corresponding relatively to ( a ) stable and ( b ) unstable bec flows past a nonlinear repulsive delta potential barrier in subcritical regime . the other parameters @xmath91 and @xmath97 . for this case @xmath98.,width=302 ] as seen the unstable solution decays into a gray soliton moving upstream with the velocity less than @xmath41 and a stable solution localized at the barrier position . the decay is accompanied by the radiation emitted downstream in front of the barrier . unlike the case of a wide barrier , in the case of the @xmath1 function nonlinear barrier potential , localized steady states exist only at @xmath99 where @xmath100 is the sound velocity . in our case @xmath101 . critical velocity @xmath102 is determined by the potential strength @xmath86 @xmath103 where @xmath104 . in order to cover a wide range of velocities we have carried out numerical simulations of the flow of a bec through the delta potential nonlinear barrier moving with small acceleration beginning from zero velocity . [ movenl ] depicts the time evolution of a bec flow when the acceleration @xmath105 . the barrier potential strength @xmath97 . the initial wave packet is taken in the form of eq . ( [ r2 ] ) . . the barrier potential strength @xmath97 , initial velocity of the flow @xmath106 . the initial wave packet is taken in the form eq . ( [ r2]).,width=302 ] time interval @xmath107 ( @xmath108 ) corresponds to a superfluid flow . at tmes @xmath109 ( @xmath110 ) one can observe generation of grey solitons chain . in time interval @xmath111 ( @xmath112 ) corresponding to transcritical flow of a bec at supersonic velocity one can observe qualitatively the same wave pattern obtained in the work @xcite where a dispersive shock propagates upstream with generation of soliton - like waves propagating downstream . in conclusion , we studied steady flow in a defocusing quasi 1d bec moving through a nonlinear _ repulsive _ barrier . such a kind of barriers can be formed by variation of the atomic scattering length of bec in _ space_. for the case of a wide nonlinear barrier we have found critical velocities of steady flows . within the interval of velocities @xmath2 , in the transcritical regime we observed generation of a slow moving train of dark solitons . at velocities above supercritical the train disappears . at the same time in this regime one can observe formation of a hump localized at the place of _ the barrier_. for the case of a @xmath1 function nonlinear barrier potential the dependence of the steady solution parameters and a critical velocity on the potential strength @xmath86 was found in analytical form . as numerical simulations show , in subcritical regime @xmath113 an unstable solution decays into a gray soliton moving upstream and a stable solution localized at the barrier position . the decay is accompanied by a dispersive shock wavepropagating downstream in front of the barrier . the dynamics of flows past through a linear and nonlinear barriers are qualitatively similar except the following . in the case of a wide _ linear _ barrier , the superfluidity is broken at any small velocities if the barrier potential strength greater than some threshold value ( see fig . 2 in @xcite ) . for a wide _ nonlinear _ barrier an interval of velocities @xmath49 _ always _ exists , where the flow is superfluid regardless of the barrier potential strength . when using the optically induced feshbach resonance technique to generate a repulsive nonlinear barrier by focused laser beam , one should in general take into account the losses , induced by spontaneous emission of atoms . phenomenologically it can be described by adding a nonlinear loss term @xmath114 in the gp equation.atom feeding can be described by linear gain term @xmath115 . this case requires a separate investigation . it should be noted that this problem relates to one considered in the recent work @xcite , where the flow of polariton condensate @xcite past a linear barrier was studied taking into account linear amplification and nonlinear damping . | the problem of a quasi 1d _ repulsive _ bec flow past through a nonlinear barrier is investigated .
two types of nonlinear barriers are considered , wide and short range ones .
steady state solutions for the bec moving through a wide repulsive barrier and critical velocities have been found using hydrodynamical approach to the 1d gross - pitaevskii equation .
it is shown that in contrast to the linear barrier case , for a wide _ nonlinear _ barrier an interval of velocities @xmath0 _ always _ exists , where the flow is superfluid regardless of the barrier potential strength .
for the case of the @xmath1 function - like barrier , below a critical velocity two steady solutions exist , stable and unstable one .
an unstable solution is shown to decay into a gray soliton moving upstream and a stable solution .
the decay is accompanied by a dispersive shock wave propagating downstream in front of the barrier . |
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the supernova remnant ( snr ) b [email protected] lies to the north and east of the region lh@xmath1 120-n206 in the large magellanic cloud ( lmc ) . the snr was first identified in radio by @xcite , who referred to it simply as n206 . although this is more properly the name of the larger complex , for simplicity we shall use the same notation in the following discussion . h@xmath1 images of n206 obtained from the magellanic cloud emission - line survey @xcite show largely shell - like emission , with complex filamentary structure along the limb @xcite noted that these features were in sharp contrast to the x - ray emission as observed by _ rosat _ , which shows the x - ray emission brightening toward the center of the remnant . the striking difference between x - ray and optical morphologies led the authors to categorize n206 as a centrally brightened " snr , and to suggest that it might undergo the same physical processes as mixed morphology " snrs @xcite , which have centrally brightened x - ray emission and a radio shell . @xcite studied n206 at 5 and 14.7 ghz , and found a spectral index @xmath1 ( as in @xmath2 ) over the remnant of about @xmath00.33 . this value is fairly flat for a snr ; it is on the borderline between the ranges for a pulsar wind nebula ( pwn ) , for which spectral indices tend to be flatter than @xmath00.4 , and for a typical shell - type remnant , for which spectral indices tend to be between @xmath00.8 and @xmath00.3 @xcite . low resolution ( several - arcminute scales ) and interference from the nearby region made features in the snr interior difficult to examine . of particular interest , therefore , are the observations by @xcite of n206 at 3 and 6 cm wavelengths ( 8.0 and 4.8 ghz ) , using the australia telescope compact array ( atca ) with resolutions of 18 and 11 ( hpbw ) , respectively . they saw emission over the face of the entire snr , somewhat brightened toward the limb . the spectral index calculated from their findings and those of other radio observations was @[email protected] , in the normal range for filled - center snrs . @xcite also discerned a peculiar linear feature " within the snr : a narrow wedge of radio emission aligned in projection with the remnant s center . a spectral index map of the snr showed this feature to have a similar index to the rest of the remnant , @xmath40.2 . in the absence of high - resolution x - ray data , the authors could not isolate a point source , and therefore could only speculate that the feature was produced by a low - mass star or compact object . " using the opening angle " of the linear feature to calculate the mach number of the presumed object s travel , and the length of the linear feature , the authors obtained an estimate for the snr age of 23,000 yr . x - ray observations of n206 were made with both the _ chandra _ and _ xmm - newton _ observatories . _ chandra _ observations used the advanced ccd imaging spectrometer ( acis ) , primarily the s3 back - illuminated chip . datasets were sequence number 500327 , observations 3848 ( 33.1 ks ) and 4421 ( 32.5 ks ) . data were reduced following procedures recommended by the chandra x - ray center ( cxc ) and analyzed using the cxc s ciao program and xspec . the two datasets were filtered for high - background times and poor event grades , resulting in a total good time " interval of 65.6 ks . for each dataset , the 05 pixel randomizations were removed and the subpixel resolution " technique10% without decreasing the overall count rates . documentation at http://cxc.harvard.edu/cont-soft/software/subpixel_resolution.1.4.html ] was applied , allowing us to more closely examine the structure on small spatial scales . the filtered datasets were then merged , and spectral results extracted from the merged file . as spectral analysis required us to choose fairly wide regions for reasonable counting statistics , the subpixel resolution " files were not used for this purpose . a background region was taken from an annulus surrounding the snr , and the spectrum of this background region was scaled and subtracted from the source spectra . individual spectra for regions of interest selected from radio and x - ray images , and the corresponding primary and auxiliary response files , were extracted with the ciao psextract script and analyzed in xspec . spectra were rebinned by spectral energy to achieve a signal - to - noise ratio of 4 in each bin . we received the pipeline - processed _ xmm - newton _ data from the science operations centre ( soc ) . observations were made simultaneously with multiple _ xmm - newton_instruments ; in this paper we will concentrate on the european photon imaging camera ( epic ) mos and pn detectors . the epic - mos data were taken over two intervals , in 2001 november ( observation i d 089210901 , 41.4 ks ) and 2001 december ( 0089210101 , 14.5 ks ) . the latter observation also included an epic - pn exposure ( 12.0 ks ) . initial reduction and analysis were carried out using the science analysis software ( sas ) package provided by the soc , with subsequent spectral analysis in xspec . the data were filtered to remove high background times or poor event grades , reducing the total good " dataset time to 24.2 ks for the first observation ( epic - mos only ) and 12.0 ks for the second observation ( epic - mos and epic - pn ) . images and spectra were then extracted from the filtered event files . background regions immediately surrounding the snr but free of point sources were used to produce background spectra , which were then scaled and subtracted from the source spectra . joint spectral fits were performed with the data from the two epic - mos observations . images of n206 were obtained with the _ hubble space telescope _ using the wide field planetary camera 2 ( wfpc2 ) with the f656n ( h@xmath1 ) , f673n ( [ ] @xmath5 6347 , 6371 ) and f502n ( [ ] @xmath5 5007 ) filters . the southwestern side of the snr was observed in h@xmath1 ( 3@xmath6800 s ) , [ ] ( 3@xmath6800 s ) and [ ] ( 6@xmath6800 s ) , all at the same pointing and position angle , but the northeastern side of the snr was observed only in h@xmath1 ( 3@xmath6800 s ) . the data were reduced using the iraf program and stsdas package . multiple exposures were combined to remove cosmic - ray events , and the resulting files were bias - subtracted . the images were divided by exposure times to produce count - rate maps , which were then multiplied by the photflam parameter ( provided in the image header ) to convert these to flux - density maps . we used the synphot task to determine widths for each filter , and multiplied the flux - density maps by the filter widths to produce flux maps . the files for the wfc and pc were then mosaicked together for the final images . images for [ ] and h@xmath1 were clipped at 3@xmath7 of sky background to reduce noise , and then divided to produce a calibrated [ ] /h@xmath1 ratio map . high - dispersion spectra of n206 were obtained with the echelle spectrograph on the 4 m telescope at cerro tololo inter - american observatory ( ctio ) from two observing runs , 2000 december 6 and 2004 january 14 . both runs used the 79 line mm@xmath8 echelle grating in the single - order , long - slit observing configuration , in which a flat mirror replaced a cross disperser and a post - slit h@xmath1 filter ( @xmath9 = 6563 , @xmath10 = 75 ) was inserted to isolate a single order . the red long focus camera and the 2000@xmath62000 site2k#6 ccd were used to record data . the 24 @xmath11 m pixel size corresponds to roughly 0.082 along the dispersion and 026 perpendicular to the dispersion . limited by the optics of the camera , the useful spatial coverage is @xmath123@xmath13 . the spectral coverage is wide enough to include both the h@xmath1 line and the flanking [ ] @xmath56548 , 6583 lines . a slit width of 250 @xmath11 m ( 164 ) was used and the resultant fwhm of the instrumental profile was @xmath14 km s@xmath8 . the spectral dispersion was calibrated by a th - ar lamp exposure taken in the beginning of the night , but the absolute wavelength was calibrated against the geocoronal h@xmath1line present in the nebular observations . the journal of echelle observations is given in table [ tab : echobs ] . we undertook new observations using the parkes 64-m radio telescope at the australia telescope national facility . we observed n206 for 12 hours on 2003 september 29 using the central beam of the 20-cm multibeam receiver at 1374 mhz on the parkes telescope with the aim of detecting pulsed emission from a small x - ray source . following the recent successful surveys for pulsars in radio nebulae at parkes @xcite , we recorded total - power signals from 96 frequency channels which provide a bandwidth of 288mhz for two polarizations every 1ms . the sensitivity of the system was @xmath15jy which is a factor of @xmath122.5 more sensitive than an earlier , large - scale survey of the large magellanic cloud @xcite . off - line processing was carried out to search for any dispersed , periodic signal using standard procedures with the seek software . no signal was detected . the morphology of the snr in the _ xmm - newton _ and _ chandra _ observations is complex , with particular structures revealing themselves variously under _ xmm - newton _ s high sensitivity and _ chandra _ s high resolution . in both cases ( fig . [ fig : xmm_acis ] ) diffuse emission can be seen over the entire face of the remnant , all the way out to the rim ; and a significant increase in x - ray emission appears toward the center . in the _ xmm - newton _ epic - pn and epic - mos observations there is an extension of this central emission that corresponds well to the position of the radio linear feature " . this apparent correlation is confirmed by the _ chandra _ observations , in which x - ray emission is clearly seen along the length of the linear feature , broadening in the same region of the wedge " as in radio ( fig . [ fig : acis_atca ] ) . an x - ray point source appears in the _ chandra _ observation at 05@xmath1632@xmath1703@xmath18 , @xmath0710051 . no counterpart to this source is found in the optical or radio data . a search of the simbad database shows these coordinates to be within the error circle for _ einstein _ source 2e 0532.6@xmath07102 @xcite . in the absence of features indicating an association between this source and the snr , we presume it to be a background source ; possibly , given its hard x - ray emission , an active galactic nucleus . we will not include this source in the subsequent discussion . the _ hubble space telescope _ wfpc2 h@xmath1 mosaic of n206 in figure [ fig : hsthafull ] shows the circular , limb - brightened structure of this snr . an intricate array of loop - like filaments extends from the outermost limb well within the remnant . toward the southwest of the snr the h@xmath1 emission becomes more prominent ; this is unsurprising , as in this region the n206 snr may overlap slightly with the larger n206 region . the [ ] emission , as seen in figure [ fig : hst3band]c , tends to the outer edges of the filaments . [ ] is a useful tracer " of shock fronts , due to the relatively narrow range of temperatures and densities in which this emission dominates . thus the tendency of the [ ] to lie in the limbward direction of the filaments implies a fairly regular expansion . there do not appear to be any counterparts in h@xmath1 or [ ] to the linear feature seen in radio by @xcite . the h@xmath1 and [ ] morphologies , shown in figure [ fig : hst3band]a and b , are quite similar , including the filamentary structure , as is typical for cooled ( @xmath1210@xmath19 k ) post - shock gas . spectral fits to the x - ray data from the various instruments ( _ chandra _ acis , _ xmm - newton _ epic - mos and epic - pn ) show that the emission from the snr is dominated by thermal emission ; a number of thermal emission lines are visible ( fig . [ fig : xray_spec ] ) . we therefore utilized thermal plasma models to determine the plasma parameters . as it is uncertain whether the plasma has reached ionization equilibrium , we fit both a non - equilibrium ionization model ( nei ; pshock " in xspec ) and a collisional - ionization equilibrium model ( cie ; mekal " in xspec ) ] . we selected the following regions for further analysis : ( region 1 ) one comprising the entire snr for comparison with other x - ray studies ; ( 2 ) a region enclosing much of the snr limb , which is expected to be dominated by emission from recently shocked gas ; ( 3 ) a region enclosing the central x - ray brightening but excluding emission thought to be associated with the linear feature " ; ( 4 ) a region covering an apparent wedge - shaped x - ray brightening north and south of the linear feature " , excluding a possible point source ; ( 5 ) a region corresponding to the linear feature " seen in radio ; ( 6 - 8 ) three equal - area regions along this feature to examine possible changes over its length ; ( 9 ) a small region surrounding a possible point source at one end of the linear feature " ; ( 10 ) a region covering emission immediately behind the possible point source , excluding a narrow area corresponding to the brightest radio emission from the linear feature " ; and ( 11 ) a region covering the aforementioned radio - bright linear feature . a list of the regions used for x - ray analysis is given in table [ tab : reglist ] and shown in figure [ fig : xray_regions ] . we chose two of these regions thought to include only thermal x - ray emission : outer limb " ( region 2 , fig . [ fig : xray_regions]b ) and central " ( region 3 , fig . [ fig : xray_regions]c ) . we compared the fits of simple thermal models to the data from each of the x - ray instruments used ( table [ tab : snrspecfit ] ) . best fits are obtained with sub - solar metal abundances , consistent with the ambient metal abundances in the lmc of about 30% solar @xcite . fits to the absorption column density are reasonably consistent for regions with good statistics , giving @xmath20 = 3@xmath31 @xmath6 10@xmath21 @xmath22 . fits to the temperature differ by a factor of two depending on whether cie or nei is assumed ; cie fits have @xmath23 = [email protected] kev and nei fits have @xmath23 = [email protected] kev ( table [ tab : snrspecfit ] ) . the fits to the ionization parameter @xmath24 ( @xmath25 , where @xmath26 is the hot gas density and @xmath27 is the time since ionization ) indicate that the hot gas deviates considerably from ionization equilibrium , with @xmath24 = 3@xmath28 @xmath29 s. in general , the nei model also provides a better statistical fit to the data . we will therefore refer primarily to the results of the nei fits in the subsequent discussion . in order to facilitate comparison of our x - ray results with other x - ray studies at lower resolution , we include the whole snr " region ( region 1 in table [ tab : reglist ] , fig . [ fig : xray_regions]a ) , which includes all x - ray emission from the remnant , including that from the area of the linear feature " . these fits ( table [ tab : wholesnrspec ] ) largely fall within the range described above for the outer limb " and central " regions , indicating that the snr s emission is dominated by other sources of emission than that of the linear feature . " while an additional power - law component with a normalization of one - third that of the thermal plasma component slightly improves the fit , this improvement is not significant . the nei fit yields an absorbed flux of 7@xmath30 erg @xmath22 s@xmath8 , an unabsorbed flux of 4@xmath31 erg @xmath22 s@xmath8 , and a luminosity of 8@xmath32 erg s@xmath8 , all over the [email protected] kev range . the normalization of the nei model fits to x - ray data can be used to calculate the rms electron density ( @xmath33 ) of the hot gas within the snr . we assume the remnant to be roughly spherical , with a radius of 21 pc , at a distance of 50 kpc . we further assume that the gas is composed of fully ionized hydrogen and helium ( @xmath34 ) . then , using the normalization from nei fits to the chandra acis data for the whole snr " region ( table [ tab : wholesnrspec ] ) we find a density in the hot gas of @xmath35 @xmath36 @xmath29 . using the outer " and central " regions ( table [ tab : outerlimbspec]-[tab : centerspec ] ) , which do not include emission from the area around the linear feature , " for this calculation gives a very similar density of @xmath37 @xmath36 @xmath29 . the parameter @xmath38 is a volume filling factor for the hot gas . if we assume @xmath38=1 , to reflect the centrally filled x - ray morphology of the remnant , the hot gas density is @xmath39 @xmath29 ; if the gas only partially fills this volume , the density will rise in inverse proportion to the square root of this filling factor . from this derived density of the x - ray emitting material , we obtain a mass of 5.3@xmath40 @xmath41 g , or 270@xmath310 @xmath41 m@xmath42 . presuming that the hot gas fills the entire remnant ( @xmath38=1 ) , this mass is greater than that expected from sn ejecta alone , emphasizing that n206 is an older snr whose emission is dominated by swept - up material from the surroundings ; and supports the lmc - like abundances found from the spectral fit . using the temperature and density from spectral fits to the x - ray data , we find a thermal pressure , @xmath43 , of 3.3@xmath44 @xmath36 dyne @xmath22 in the hot gas . as the analysis in 3.2 will show , this is significantly greater than the thermal pressure we find for the cool shell . we can also calculate the thermal energy in this gas , @xmath45 . this gives a thermal energy of 6@xmath46 @xmath41 erg for the hot gas . finally , we can estimate the shock velocity under the ( somewhat dubious ) assumption that the bulk of the x - ray emission is being generated at the current shock front , using @xmath47 , with the reduced mass @xmath48 ( for he : h = 1:10 ) . if we presume the newly shocked gas to be at 0.45 kev , we find that such a temperature would result from a shock speed of 620 km s@xmath8 , implying an overall expansion velocity of 470 km s@xmath8 . for simplicity , we have assumed in the above calculations that the ions have equilibrated with the electrons ; although this assumption may well not be valid , it provides us with at least initial estimates . the derived energy and pressure are similar to those for other snrs in the adiabatic stage of evolution . n206 is clearly more complex than the simple shell approximated above . it features an increase in surface brightness toward the remnant center , a wedge " of emission appearing to surround the linear feature " , and the feature itself , terminating in a possible compact source . the properties of these features will be discussed in more detail in 4 . the central brightening on the snr is quite puzzling . the spectrum of this region ( table [ tab : centerspec ] ) appears similar to that for the limb of the remnant , with slightly higher abundances , particularly oxygen . there is no evidence for a strong power - law component in this region ; fits with such a component require it to be normalized to a small fraction of the thermal component s emissivity . toward the eastern side of the remnant , surrounding the linear feature , " the x - rays show a slightly brighter broad wedge " of emission ( region 4 in table [ tab : reglist ] , fig . [ fig : xray_regions]c ) . fits to the spectrum of this region ( table [ tab : wedgespec ] ; this spectrum does not include contributions near the compact source " ) suggest a markedly higher plasma temperature than elsewhere in the snr . while a nonthermal contribution might artificially skew a thermal model fit toward higher temperatures , fits which included nonthermal emission over a tenth of the thermal emissivity could be statistically ruled out at the 90% level . we also examined a region covering the area of the linear feature " , as seen from radio images ( region 5 in table [ tab : reglist ] , fig . [ fig : xray_regions]d ) . we defined a hardness ratio ( h@xmath0s / h+s ) such that s=0.3 - 1.0 kev and h=1.0 - 8.0 kev ( table [ tab : reglist ] ) . using these hardness ratios to compare this region to others throughout the snr , we find that , while the contribution of soft - x - rays still dominates , the proportion of hard x - rays in this region is greater than elsewhere in the snr . this increase in the hardness ratio is consistent with the combination of thermal emission from the snr and emission from a harder source , as for example the small - diameter bright source at the tip of this region . while a localized increase in temperature could also explain this hardening , it would be quite coincidental that this increase appears only in this portion of the snr that also shows the small - diameter x - ray source and elongated radio feature ( fig . [ fig : xmm_acis]c - e ) . we performed spectral fits to data from region 5 , though we note that these fits are somewhat limited by the relatively low number of counts . we find that either a thermal plasma model or a power - law model can provide a statistically adequate fit to this source ( table [ tab : linearspec ] ) . best " fit arises from a combination of nonequilibrium thermal plasma and power - law models , providing roughly equal contributions to the spectrum . although the improvement in the fit is not statistically significant , such a combination is consistent with the scenario inferred from the hardness ratios . to examine the detailed properties of the linear feature " , we used the _ chandra _ acis data to analyze emission from smaller segments along that feature ( regions 6 - 8 in table [ tab : reglist ] , fig . [ fig : xray_regions]e ) . these fits , summarized in table [ tab : smregspec ] , should be considered as preliminary estimates only . figure [ fig : hst3band]d shows an [ ] /h@xmath1 ratio map of the southwestern side of n206 . [ ] /h@xmath1 ratios range from 0.7 to 1.2 across the snr , typical for shocked gas , with an average value of 0.9 . this is initially somewhat surprising , as it indicates that there has not been much ionization of material from the nearby ob association . however , maps of the vicinity show an shell , with the n206 region largely situated within the central cavity , so that the shell walls probably block most of the ionizing radiation before it can reach the snr @xcite . we measured the average h@xmath1 surface brightness of filaments in the flux - calibrated wfpc2 image of n206 to be ( 0.8 - 1.8)@xmath610@xmath49 erg @xmath22 s@xmath8 pix@xmath8 , or ( 0.8 - 1.8)@xmath50 erg s@xmath8 @xmath22 arcsec@xmath51 . assuming a cool shell temperature of 10@xmath19 k , this surface brightness implies an optical emission measure of 680@xmath3 260 pc @xmath52 . we presumed the average filament thickness along the line of sight ( 20 @xmath0 30 ) to be equal to its width perpendicular to the line of sight , and used this as a representative number for the path length @xmath53 through the warm ionized gas . using this @xmath53 , we calculate an rms electron density in the shell of about 10@xmath34 @xmath29 . if we presume the snr to be in the adiabatic point - blast " stage of expansion @xcite , and the cool shell to be representative of the ambient ism , we would expect the ambient ism density to be roughly one - quarter of that in the shell , or about 3@xmath31 @xmath29 . the optical echelle data ( fig . [ fig : echelle ] ) show several lines in the h@xmath1 spectral region , including the narrow geocoronal h@xmath1 ( 6562.85 ) and telluric oh 6 - 1 p2(3.5 ) 6568.779 and 6 - 1 p1(4.5)e / f 6577.183/.386 lines @xcite , as well as broader lines corresponding to doppler - shifted nebular h@xmath1 emission toward the snr . the latter include both a velocity component constant along the slit , showing the systemic velocity of the snr , and the characteristic bow - shaped pattern deviating from this systemic velocity , showing motions within the expanding gas . in order to measure the systemic velocity , we extracted a velocity profile from a region 13 wide outside of the emission of the snr expansion pattern . the profile showed two components , one of which was identified as the telluric oh line . the other component is the h@xmath1 line with a measured doppler shift of 241@xmath34 km s@xmath8 ; as the snr expansion pattern appears to converge to this component , we take this as the systemic velocity for the n206 snr . for comparison , we note that the nearest position to n206 in the maps of @xcite shows velocity components at 240@xmath37 and 262@xmath326 km s@xmath8 . to characterize the expansion of the n206 snr , we measure the velocity offsets ( @xmath54 ) from the systemic velocity in both the blue and red directions . as the material at the forefront of the snr expansion may be reasonably expected to show the highest velocity , we take the greatest of these velocity offsets to represent the overall expansion velocity of the remnant . the greatest offset in the blue direction ( @xmath55 ) is @xmath0202@xmath35 km s@xmath8 , while the largest @xmath56= @xmath57193@xmath35 km s@xmath8 . we therefore estimate the expansion velocity @xmath58 = 202@xmath35 km s@xmath8 . using this value for expansion , and assuming that the snr is in the sedov phase , we would expect the shock velocity @xmath59 = @xmath60 = 270@xmath35 km s@xmath8 . it should be noted that the errors given here are only the random errors of the measurements . it is possible that some of the optically emitting material is too faint for detection in these observations . if the highest - velocity material is undetected , the actual expansion velocity may be higher than our estimate . for example , @xcite found a higher expansion rate of 250 km s@xmath8 ; the discrepancy may be due to the difficulty of discerning motions in the faint outer material . to an extent , this may also apply to the large discrepancy between the shock velocity calculated from the temperature of the hot gas in 3.1.1 , and that measured from the warm ionized gas . more likely , however , this discrepancy is due to two factors . ( 1 ) the x - ray temperature may not be a reliable representation of the newly shocked gas , particularly given its large deviation from a limb - brightened shell morphology . ( 2 ) we expect the x - ray emission to arise from areas where the shock front is moving through diffuse gas , while the optical emission is expected to be generated where the shock is moving through higher density gas , as seen in the relative densities of the hot gas to the warm ionized gas . thus , we expect that the shock within the optically - emitting clumps has been somewhat slowed by its progress through this denser material . the overall expansion velocity from the blast wave is probably intermediate between the measured optical expansion of 202 km s@xmath8 and the calculated expansion from x - ray temperatures of 470 km s@xmath8 . using the shell electron density of n=10 @xmath29 calculated above , and presuming @xmath61 and singly ionized he ( @xmath62 ) , we can infer the mass of gas in the shell according to @xmath63 , where @xmath64 is the shell volume . from the filamentary structure at the edge of the shell , we measure an average shell thickness of about 11@xmath305 ( [email protected] pc at a distance of @xmath1250 kpc to the lmc ) . if we assume a simple spherical shell of this thickness , we calculate a shell mass of 9@xmath65 g , or 460@xmath3300 m@xmath42 . this is almost certainly an underestimate , as the filamentary structure shows a much greater extent of cool gas than such a simplistic scenario . if instead we presume the cool gas occupies a volume filling factor of as much as 0.1 , we obtain a mass estimate of 2.4@xmath66 g , or 1200@xmath3800 m@xmath42 . using these as low- and high - end estimates of the mass range , and the expansion velocity above , we calculate the kinetic energy in the cool shell , @xmath67 , to find values of 2@xmath68 erg and 5@xmath69 erg , respectively . one can also use the density calculated above to calculate the thermal pressure in the shell , @xmath70 , where @xmath71 is the boltzmann constant and @xmath72 is the temperature , presumed here to be @xmath73 k. this equation gives a pressure of 3@xmath74 dyne @xmath22 . again , these values for kinetic energy and shell pressure are typical for middle - aged snrs . the physical characteristics of the n206 snr are summarized in table [ tab : parameters ] . the high spatial resolution of _ chandra _ s acis allows us to examine specific regions in more detail . we have isolated three features that appear to be of particular interest : x - ray emission associated with the radio - identified linear feature ; " the x - ray knot and surrounding emission which coincides with the very tip of that linear feature ; and the region of enhanced x - ray surface brightness toward the center of the snr . below , we describe each of these features individually and then discuss the possible origins for these features . the interior x - ray emission is brightest from an irregular region extending n@xmath0s for half the diameter of the remnant . this central emission is clearly dominated by thermal x - rays , with prominent line features including the blends of he - like lines of mg and si . the emission is largely soft ( @xmath75 2 kev ) and thermal plasma fits to data from this region give parameters for @xmath20 and @xmath23 similar to those found for the snr as a whole . however , there does appear to be some difference in abundance distributions between the central area and the outer regions of the snr ( excluding emission associated with the linear feature " ) . the oxygen lines in the central region are more prominent , with respect to the iron blends ( fe l ) , than in other regions of the snr ( table [ tab : centerspec ] ) . these differences should be treated with caution , as there are large uncertainties in the abundance determinations ; however , the difference in oxygen abundance between the two regions within the snr is significant when compared to the 90% error ranges for this quantity . if the central emission is due to fossil radiation , " i.e. gas that was shock - heated during earlier phases of the snr expansion , we would expect that this gas would have a higher proportion of ejecta to swept - up matter than the more recently - shocked gas , and therefore that the higher oxygen abundance reflects the presence of oxygen - rich ejecta , as seen in the much younger mixed - morphology snr 0103 - 72 in the smc @xcite . the presence of o - rich material can not be confirmed from _ hubble space telescope _ this probably reflects the fact that the x - ray emission which shows the oxygen excess is primarily from the hot central cavity , where the temperatures are too high and the densities too low for [ ] to show up . it is worth noting that in addition to the similar oxygen - rich and mixed - morphology snr 0103 - 72 @xcite , the oxygen - rich snr n132d shows , likewise , a higher ratio of oxygen ( o@xmath76 ) to other elements in the remnant interior than at the bright rim @xcite . nucleosynthesis models indicate that a high oxygen abundance within a snr s ejecta is the result of a type ii sn ( e.g. , * ? ? ? the examples of mixed - morphology " snrs which show signs of enhanced oxygen abundances , therefore , may imply that the mixed - morphology " snrs are likely to have originated from type ii sne . as with other mixed - morphology snrs @xcite , the expanding shock of n206 has slowed to the point where bright x - rays from the limb do not dominate the overall emission from the remnant . thus , it is reasonable to think that n206 follows the typical pattern for mixed - morphology snrs , wherein the central emission is dominated by fossil radiation from earlier large - scale shock heating during the snr s evolution . an elliptical region ( 65 e@xmath0w @xmath6 21 n@xmath0s ) surrounding the linear feature " in radio was selected from the 6 cm radio image @xcite , and was used to extract spectra from the corresponding x - ray data ( region 5 ) . comparison of the hardness ratio for this region to those elsewhere in the snr , as discussed in 3.1.2 , suggests the presence of a hard component in addition to the soft emission seen throughout n206 . a combined nei and power - law model with @xmath77 and @xmath78 provided the best fit . fits to these spectra are summarized in table [ tab : linearspec ] . ( note that the value of @xmath24 obtained for this fit would indicate a plasma in collisional ionization equilibrium ; we retain the nei model , however , for consistency in intercomparison with the other fits . ) using these fitted values , we obtained an absorbed x - ray flux of 4@xmath79 10@xmath80 erg @xmath22 s@xmath8 over the 0.3 - 8.0 kev range , which corresponds to an unabsorbed flux of 1@xmath6 10@xmath81 erg @xmath22 s@xmath8 . at the lmc distance of 50 kpc , this gives a luminosity of 3@xmath6 10@xmath82 erg s@xmath8 over this energy range . to further examine the nature of this linear feature , a series of three equal - area regions ( 20@xmath6 15 ) e@xmath0w along the feature were identified in _ chandra _ images for further spectral examination ( regions 6 - 8 in table [ tab : reglist ] ) . comparison of the hardness ratios defined above for regions 6 - 8 shows a pronounced shift from regions dominated by high - energy photons to those dominated by low - energy photons . we also performed spectral fits for these regions ( table [ tab : smregspec ] ) . it should be stressed that , due to the low numbers of counts , these spectral fits are quite uncertain . cited values of @xmath83 are low due to the large error bars for the spectral bins ; however , further binning was deemed undesirable due to the loss of remaining spectral information . however , as with the hardness ratios discussed above , the fits illustrate the shift from harder to softer emission along the feature . in order to see whether the relative fluxes of thermal to nonthermal emission would change substantially as one moved from the remnant center toward the x - ray bright knot at the east end of the linear feature , we adopted a dual approach to these spectral fits . initially , we fit a two - component ( power - law and nei ) model jointly to all three regions , and determined the flux in each component for each region . as an alternate approach , we fit the power - law component to the high - energy end of the spectrum only ( @xmath842 kev ) and fixed those parameters , then fit the joint model to the low - energy end of the spectrum . again , the flux in each component was determined . the results , summarized in table [ tab : linearflux ] , indicate that the x - ray emission along the linear feature is more strongly dominated by nonthermal emission the further one moves eastward ( closer to the knot ) along the feature , and dies off on the west side . while this result is largely qualitative , given the errors , we argue that the increase in hard " emission toward the knot is most likely to result from a nonthermal source . the nonthermal spectra of the x - ray emission and radio emission in the linear feature " imply the presence of a pulsar - wind nebula ( pwn ) . the most likely source for nonthermal x - rays within a snr is synchrotron radiation . while nonthermal x - ray emission may occasionally be generated by a fast snr shock at the outer limb of a snr , both the relatively slow expansion of this snr and the shape of the emitting region of the nonthermal emission , i.e. over a small spatial segment perpendicular to the shell , argue against this scenario . likewise , it seems unlikely that the motion of a compact source would be sufficient to produce a bow shock capable of generating significant synchrotron x - rays . the most plausible explanation , then , is that the acceleration of particles by a hidden " pulsar is the source of the nonthermal emission ( in both radio and x - rays ) seen in the linear feature . the westward decrease in the ratio of nonthermal to thermal x - ray flux along the linear feature suggests that it is a pwn generated by an eastward moving pulsar . the lifetime of the high - energy particles generated in a pwn is comparatively short ; the radio - emitting particles persist and so trail farther behind the x - ray extent . the radio pulsar observations were unable to find any periodicity greater than 2 ms to the radio signal from the area of the linear feature " . it is by no means uncommon for a pwn to be detected in radio and x - rays without a radio point - source counterpart ; for example , n157b , also in the lmc , was deduced to have a pwn long before the x - ray pulsar was discovered , and a radio counterpart for that pulsar has yet to be found @xcite . another possibility that must be considered , of course , is that these features are associated with a background source rather than the snr . @xcite discuss this scenario , but conclude that this is unlikely . the presence of extended nonthermal x - ray emission provides an additional reason to believe the feature is not a background source , as such a feature would be highly unusual in a background galaxy . if we accept the premise that the linear feature " is a pwn , we may anticipate that in part , the morphology of the pwn is created by a bow - shock structure . the dynamics of a rapidly moving pulsar and evolving snr lead to the prediction of the formation of a bow shock when the snr expansion speed has decelerated sufficiently for the pulsar s motion relative to the shocked snr material to become supersonic ( e.g. * ? ? ? at this point the pwn is deformed , leading to a substantial offset between the pulsar and the center of the pwn ; this is certainly consistent with the observed bright knot at the outer tip of the linear feature " presumptive pwn . numerical simulations predict that for a remnant in the sedov stage , this should occur when the distance traveled by the pulsar from the center of the snr , @xmath85 , approaches the snr radius @xmath86 according to @xmath85/@xmath86 @xmath87 0.677 @xcite . presuming that the geometric center of the snr shell accurately represents the site of the sn , the current position of the small - diameter x - ray source is such that the transverse component of @xmath85 is 0.85 @xmath86 , well over the point for bow shock formation . if the direction of pulsar motion is not perpendicular to the line of sight , @xmath85 increases accordingly . these findings are in general accord with those of @xcite , and their assumptions in their estimation of the pulsar velocity and age . the fact that emission from the linear feature " in x - ray and radio does not dominate the overall emission from the remnant presents an additional question . its spectral index would place it among the most flat radio snr shells , although overlapping ( within the error bars ) with remnants such as ic443 @xcite . it is difficult to see why the snr as a whole should show a radio index typical of filled - center remnants @xcite if the putative pwn is largely confined to this feature . however , the radio emission from n206 is faint compared to that from other snrs , and therefore the uncertainties in the spectral index determination are considerable . in addition , residual emission from the n206 region may contaminate the radio emission , adding additional uncertainty . in the _ chandra _ images , a distinct knot of x - ray emission appears to the far eastward end of the linear feature , near the snr limb . to investigate whether this knot is consistent with an unresolved source , we generated a point source of comparable x - ray flux using the marx simulator , and compared the profile of this simulated source to that of the knot . we conclude that the knot is a small diffuse region of dimension 2 . this can also by seen by comparison with the appearance of the moderately bright point source @xmath1250 sw of the knot ( identified above with 2e 0532.6 - 7102 ) . if a point source is embedded within the knot , only @xmath8850% of the counts from the knot could be due to that point source . we extracted counts within a 2 radius of the knot from the merged _ chandra _ acis events file ( region 9 in table [ tab : reglist ] , fig . [ fig : xray_regions]e ) , obtaining a count rate after background subtraction of 7.9 @xmath6 10@xmath89 ct s@xmath8 , or 52 counts over the 65.6 ks exposure time , an insufficient quantity to obtain a meaningful spectrum . fig . [ fig : xmm_acis ] , however , shows that the knot shows noticeably harder emission than that from the surrounding snr . once again using the hardness ratio defined above , ( table [ tab : reglist ] ) , we find a hardness ratio of 0.5 for the knot , in contrast to hardness ratios of -0.45 and -0.58 for the outer " ( region 2 ; snr limb excluding the putative pwn ) and central " ( region 3 ) areas . to obtain a first - order estimate of spectral properties for this knot , we fixed @xmath20 ( and abundances , for the thermal case ) to the values determined by fits to the x - ray emission from the rest of the snr . the data are consistent with a thermal plasma model with temperatures @xmath84 1 kev , or a power - law model with a spectral index @xmath90 between 2 and 3 . using a power - law model with @xmath91 , we obtain an estimate for the absorbed flux of 6@xmath6 10@xmath92 erg @xmath22 s@xmath8 , an unabsorbed flux of 1@xmath6 10@xmath80 erg @xmath22 s@xmath8 , and a luminosity of 3@xmath6 10@xmath93 erg s@xmath8 at the lmc distance , all over the 0.3 - 8.0 kev energy range . the maximum luminosity for an embedded point source is therefore @xmath121.5@xmath6 10@xmath93 erg s@xmath8 over this range . an elliptical region ( 5@xmath68 ) immediately west of the knot , along the linear feature , yielded a count rate of 1.4 @xmath6 10@xmath94 ct s@xmath8 . this region has a very similar spectrum to that from the knot , suggesting that the spectrum determined from the knot is dominated by emission from its immediate surroundings . combining the spectra from these two regions allowed us to increase the signal - to - noise ratio for a slightly better spectral fit . while these fits would still not allow us to rule out a thermal plasma interpretation at 90% confidence , they confirm that a relatively high temperature @xmath95 kev , is required . more plausible is a power - law fit , with @xmath96 , roughly consistent with that measured from the linear feature " as a whole . while fits to smaller regions along the linear feature suggest a possible spatial variation in the power - law spectral index , the sensitivity of the x - ray observations is insufficient to make this determination at a statistically significant level . a brief analysis of the x - ray power spectrum shows no evidence for periodic emission from an embedded pulsar . however , the timing resolution of these observations , 3.2 seconds , would not be sufficient to detect the pulsations from a typical pulsar . in addition , the estimated maximum of 25 counts from a point source is insufficient to determine a period for its emission , even if the timing resolution were available . proceeding westward from the knot , x - ray emission associated with the linear feature " broadens into a wedge of brighter x - ray emission which then appears to merge with the central emission . faint emission connects this patch to the compact knot close to the eastern rim . we interpret this connection as a bow shock , " clear in the south and barely discernable in the north . this bow shock merges with the radio linear feature " close to the knot . a bow shock in front of a hypothetical moving pulsar might re - heat the material through which it moves , leaving a trail of fossil radiation " back to the remnant center and merging there with fossil radiation from much earlier fast - moving shocks . one approach to these features would be to consider much of the x - ray emission between the bright linear feature and the central region as the actual boundary of the bow - shocked gas . in this case we would have a much larger opening angle for this structure than that found by @xcite . using the x - ray opening angle of about 57@xmath3 5(1.0 @xmath97 rad ) we obtain a mach number @xmath98 = 2.1 @xmath99 . this is considerably lower than than the @xmath100 value of @xcite , and closer to the theoretical value for a pulsar moving through a snr interior ( @xmath101 , * ? ? ? we can calculate the sound speed of the hot gas within the snr according to @xmath102 where @xmath103 is the adiabatic index , here taken as 5/3 ; @xmath71 is boltzmann s constant ; @xmath104 is the electron temperature ; @xmath11 is the reduced mass ; and @xmath105 is the hydrogen particle mass . presuming the reduced mass @xmath106 , and using the temperature found above , we obtain a sound speed in the hot gas of 230 @xmath3 110 km s@xmath8 . according to @xmath107 , where @xmath108 is the motion of the pulsar relative to the snr material , this gives a mean relative motion of @xmath108=480 @xmath3 240 km s@xmath8 . if the pulsar is , as it appears , moving radially outward from the snr center , we can presume its motion to be parallel to that of the expanding snr material around the pulsar . in order to obtain a mean relative velocity of @xmath108=480 km s@xmath8 with material moving at the expansion velocity @xmath58 = 202 km s@xmath8 ( slow - expansion case ) , we would require the pulsar to be moving at a speed of roughly 680 km s@xmath8 with respect to the snr center . if instead we use the x - ray derived expansion speed of @xmath58 = 470 km s@xmath8 ( fast - expansion case ) , the pulsar must be moving with a speed of @xmath12950 km s@xmath8 with respect to the snr center . if the motion is entirely perpendicular to the line of sight , and the pulsar began at the geometric center of the snr , it would have taken the pulsar about 28,000 yr to reach its current transverse distance of 19.4 pc in the slow - expansion case , and 20,000 yr in the fast - expansion case . these can , of course , be affected by the viewing angle ; but if we assume the pulsar to be interacting with the snr interior , its maximum distance of travel is the snr radius of 21 pc ( at a 67 angle to the line of sight ) , which puts its time of travel at 30,000 yr in the slow - expansion case and 22,000 yr in the fast - expansion case . in our suggested picture , the bow - shocked gas is primarily thermal , representing material within the snr that is shocked to higher temperatures by the encounter with the bow shock . presumably , the nonthermal x - ray emission is associated with the elongated pwn , as would be the radio emission . thus we would expect a broad outer cone " of thermal ( shocked snr material ) emission , with an interior cone " of nonthermal ( shocked pwn ) emission . such a scenario is in accord with hydrodynamic models of bow - shock pwn , in which the mach cone should manifest itself only in the outer bow shock " ( * ? ? ? * and references therein ) . the shocked pwn material forms a trail of synchrotron emitting particles opposite the direction of pulsar motion , creating a cometary " or linear " morphology . to examine this picture for consistency with the data , we fit a simple thermal plasma plus power - law model combination to data from two regions within the linear feature : the thin trail " region , corresponding to the narrow location of the brightest radio feature ( region 11 in table [ tab : reglist ] , fig . [ fig : xray_regions]f ) , and the bow " region , comprised of emission around the linear feature excluding the aforementioned trail " ( region 10 , fig . [ fig : xray_regions]f ) . both regions excluded emission within 6 of the presumed pulsar . the best - fit model combination gave thermal and power - law parameters similar to those given in previous sections ( @xmath109 @xmath22 , @xmath110 kev , @xmath1112.3 ) . calculating the fraction of the total flux for each model component , we find that the trail " region shows 96% nonthermal flux and only 4% due to thermal plasma , while the bow " region shows 49% thermal flux and 51% nonthermal flux . errors on these flux ratios , based on the differences in flux due to uncertainties in the fitted parameters at the 90% confidence level , are of order 10% . alternately , it is possible that some of the x - ray brightening we consider to be associated with the radio linear feature " ( excluding the x - ray knot ) is in fact only a surface brightness fluctuation , and does not accurately represent the mach cone " of the presumptive moving pulsar . if the mach cone " is more tightly confined than our x - ray based estimate , the resulting mach number may be somewhat higher , up to a value of about @xmath112 when one considers only the angular extent of the bright x - ray emission immediately surrounding the x - ray knot . while it is also possible that the emission from these features is due to the pwn alone , with no substantial bow - shock emission , the highly elongated morphology and off - center pulsar position argue strongly for the presence of a significant bow - shock contribution . the ages of supernova remnants are notoriously ill - characterized . we consider a number of different approaches to this age , in order to find the range of reasonable age estimates . from the fitted x - ray parameters , several estimates are possible . combining the derived hot gas density with the fitted ionization parameter @xmath24 gives an upper limit age estimate of @xmath1240,000 years for this remnant . however , an age this large would be difficult to reconcile with the fact that the hot gas in the snr has apparently not yet reached ionization equilibrium . lowering the filling factor shortens this age estimate significantly . a filling factor of 0.25 , for instance , gives an age estimate of 23,000 yr , similar to the estimate based on the radio linear feature " by @xcite . we can also make another estimate from the sedov relations , in which the relative fractions of thermal and kinetic energies to total energy are constant . taking the theoretical relation of @xmath113 , our estimate for thermal energy ( for @xmath114 ) would yield an initial explosion energy @xmath115 of @xmath116 erg . assuming the density behind the shock is a factor of 4 greater than that of the unshocked ism , we obtain an ism mass density @xmath117 of 1.2 @xmath118 g @xmath29 . then from the sedov relation @xmath119 , we have an age of only 9,000 yr . if the x - ray filling factor is lower than 1 , the age estimate will rise accordingly ; @xmath120 , for instance , gives an age of 17,000 yr . if we instead presume the overall ambient density to be represented by the current filaments of warm ionized gas , we obtain an ism mass density of 5.3 @xmath121 g @xmath29 . the sedov relation then gives an age as high as 57,000 yr , demonstrating the critical role played by the ambient density assumption . using the the simple expansion relation @xmath122 , and assuming sedov expansion ( @xmath123=0.4 ) , we can calculate the age of the remnant based on the expansion velocity . the expansion velocity of 470 km s@xmath8 derived from the x - ray temperature gives a remnant age of 17,000 yr . the expansion velocity obtained from optical echelle spectroscopy of 202 km s@xmath8 , however , leads to a calculated age of 41,000 yr . again , however , this must be considered an upper limit , as this value of @xmath58 is , as stated above , a lower limit to the blast - wave expansion . we can try to narrow down these estimates by considering the consistency of the entire picture . ages above @xmath1240,000 yr are clearly ruled out by the ionization timescale of the x - ray gas and by the observed optical expansion . the value of 9000 yr from energetics arguments is highly dependent on the assumption of ambient density ; a better lower limit might be that of 17,000 yr from the expansion velocity derived from the x - ray temperature . when we consider the estimates of pulsar motion ( 4.4 ) we find further limitations : the minimum time for the pulsar to reach its current location is @xmath1220,000 yr , while the maximum is @xmath1230,000 yr . if we choose a velocity intermediate between the x - ray and optically derived values ( e.g. , @xmath12300 km s@xmath8 ) and a hot gas filling factor of 0.25 , we can reconcile most of these numbers . the ionization timescale for the hot gas then gives an approximate age of 23,000 yr . pulsar travel times give ages ( depending on viewing angle ) between 24,000 26,000 yr . simple sedov expansion at this velocity gives an age of 27,000 yr . while arguments from sedov energetics would , for this same filling factor , give us a somewhat lower age of 17,000 yr , this estimate is highly dependent on the assumption used for ambient density , a very uncertain parameter . we therefore suggest the snr s age most likely falls in the range between 23,000 27,000 yr , with hard " limits of 17,000 yr 40,000 yr . we have observed the snr in n206 with high - resolution optical and x - ray instruments , and analyzed the results in concert with additional radio data and optical echelle spectra . we find it highly probable that n206 is the result of a type ii sn , due to its proximity to other massive - star phenomena , enhancement of oxygen abundances , and the presence of a probable compact object . we use these data to calculate overall properties for the physical components of the snr , and find these typical of a middle - aged snr in the adiabatic stage . the remnant is over - pressured , and the bulk of the energy budget still resides in thermal energy from the hot interior . evidently , therefore , the hot gas within the snr still plays a significant role in the remnant s development . this remnant is particularly unusual for the very different characteristics it displays in different wavelength regimes . the optical morphology is limb - brightened and highly filamentary ; the radio morphology is center - filled , with diffuse emission over the remnant s face , but some snr shell structure ; and the x - ray morphology appears somewhat centrally brightened . the best categorization for this remnant appears to be that of mixed - morphology " @xcite , but the picture is complicated by the presence of radio and x - ray emission near the linear feature " . we analyze the x - ray data for the area surrounding the linear feature " seen in radio . we find a small , hard x - ray source located at the tip of the radio feature , with a surface brightness profile consistent with the presence of an embedded compact source . emission from this source and its surroundings is nonthermal , with a power - law index similar to that seen in crab - type objects . the ratio of nonthermal to thermal x - ray flux decreases with increasing distance from this source . we conclude that the most probable scenario for this feature is a pulsar moving at moderate velocity through the surrounding snr . this creates a bow - shock structure in the direction of motion , deforms the surrounding pwn , and leaves behind a trail of synchrotron emission along the line of travel . however , it would be difficult to attribute most , or even the majority , of the central x - ray emission to that linear feature . the x - ray emission from the central region is clearly dominated by thermal line emission , rather than nonthermal continuum . the lifetime of relativistic particles energetic enough to generate x - ray emission is not sufficient for those particles to be responsible for the central x - rays , as indicated by the fact that the radio emission , generated by longer - lived particles , shows no particular increase in this central region . this indicates , somewhat surprisingly , that n206 is both a composite " snr , containing a pwn and shell structure together , and a mixed morphology " snr , with centrally concentrated x - ray emission that can not be simply associated with the pwn . indeed , n206 is not the only snr to show this combination ; for instance , the galactic remnant w28 combines centrally concentrated x - ray emission with a probable off - center pwn @xcite . other examples include w44 @xcite and ic443 @xcite . we shall call this category mixed - composite " snrs , reflecting the combined action of separate physical processes influencing their emission . the authors thank the anonymous referee for a very detailed critique of this work , which has much improved it , and also thank brian d. fields for valuable discussions . rmw and yhc acknowledge support from stsci grant sti go-08110 . rmw , yhc , and jrd acknowledge support from nasa grant nag 5 - 11159 . rmw , yhc , fds and jrd acknowledge support from sao grant g03 - 4096 . mag acknowledges support from the grant aya 2002 - 00376 of the spanish mcyt ( cofunded by feder funds ) . behar , e. , rasmussen , a. p. , griffiths , r. g. , dennerl , k. , audard , m. , aschenbach , b. , & brinkman , a. c. 2001 , a&al , 365 , 242 bocchino , f. & bykov , a. m. 2001 , a&a , 376 , 248 camilo , f. et al . 2002 , apjl , 567 , 71 camilo , f. et al . 2002 , apjl , 595 , 25 chevalier , r. a. , & liang , e. p. 1989 , apj , 344 , 332 chu , y .- h . , & kennicutt , r. c. 1988 , aj , 95 , 1111 cox , d. , shelton , s. , maciejewski , w. , smith , r. c. , plewa , t. , pawl , a. , & ryczka , m. 1999 , apj , 524 , 179 dunne , b. , chu , y .- h . , & stavely - smith , l. 2005 , in preparation fan , g. l. 2002 , phd thesis , the university of hong kong gaensler , b. , van der swaluw , e. , camilo , f. , kaspi , v. m. , baganoff , 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& dickel , j. r. 2001 , apj , 559 , 275 williams , r. m. , chu , y .- h . , dickel , j. r. , petre , r. , smith , r. c. , & tavarez , m. 1999 , apjs , 123 , 467 lccccc 2000 dec 6 & h@xmath1 + & 1200 s & e@xmath0w through linear feature " + 2000 dec 6 & h@xmath1 + & 1200 s & n@xmath0s through snr center + 2004 jan 14 & h@xmath1 + & 1200 s & e@xmath0w through bright filament + [ tab : echobs ] lccccc 1 & whole snr & 19,950 & 10,920 & @xmath00.49 + 2 & outer limb & 7,450 & 4,130 & @xmath00.45 + 3 & central & 6,350 & 3,320 & @xmath00.58 + 4 & wedge & 2,375 & 1,400 & @xmath00.46 + 5 & linear feature & 1350 & 430 & @xmath00.24 + 6 & knot & 456 & & 0.26 + 7 & 22 w of knot & 301 & & @xmath00.35 + 8 & 44 w of knot & 264 & & @xmath00.43 + 9 & point src & 94 & & 0.50 + 10 & bow shock & 754 & & @xmath00.45 + 11 & trail & 196 & & 0.076 + [ tab : reglist ] lccccccccccccccccc [ tab : snrspecfit ] pshock & 2.7@xmath124 & 0.47@xmath125 & 0.25@xmath126 & 1.4@xmath127 & 7@xmath128 & 1.38 & 154 + mekal & 3.6@xmath129 & 0.22@xmath130 & 0.14@xmath131 & & 6@xmath132 & 1.93 & 155 + pshock & 7@xmath133 & 0.50@xmath134 & 0.20@xmath135 & 3.4@xmath136 & 8@xmath137 & 1.17 & 83 + mekal & 2.3@xmath138 & 0.24@xmath139 & 0.08@xmath140 & & 0.1@xmath141 & 1.48 & 84 + pshock & 2.2@xmath142 & 0.44@xmath143 & 0.22@xmath144 & 2.1@xmath145 & 7@xmath146 & 1.54 & 241 + & & & & & 1.5@xmath147 + mekal & 3.1@xmath148 & 0.23@xmath149 & 0.12@xmath150 & & 6@xmath137 & 1.96 & 242 + & & & & & 0.12 @xmath3 0.03 + pshock & 3.5 @xmath151 & 0.34 @xmath3 0.08 & 0.25 @xmath3 0.05 & 4 @xmath152 & 1.4 @xmath153 & 3.09 & 91 + mekal & 3.9 @xmath154 & 0.22 @xmath3 0.08 & 0.4 @xmath3 0.2 & & 3 @xmath137 & 3.79 & 92 + pshock & 2.0 @xmath151 & 0.38 @xmath3 0.04 & 0.14 @xmath3 0.03 & 2.7 @xmath155 & 2 @xmath156 & 2.10 & 64 + mekal & 2.7 @xmath154 & 0.22 @xmath3 0.01 & 0.11 @xmath3 0.02 & & 1.0 @xmath157 & 2.98 & 65 + pshock & 2.9@xmath154 & 0.37@xmath3 0.02 & 0.25@xmath3 0.03 & 3.0@xmath158 & 9@xmath159 & 3.21 & 159 + & & & & & 2.1@xmath153 + mekal & 3.6@xmath160 & 0.22@xmath3 0.06 & 0.4@xmath3 0.1 & & 3@xmath161 & 3.97 & 160 + & & & & & 6@xmath137 + [ tab : snrspecfit ] cccccccccccccccccc [ tab : snrspecfit3 ] component & vmekal & vpshock & vpshock & powerlaw + @xmath20 ( @xmath22 ) & 2.8@xmath162 & 2.2@xmath162 & + @xmath23 ( kev ) & 0.264@xmath163 & 0.453@xmath164 & 0.43@xmath165 & + o / o@xmath166 & 0.36@xmath167 & 0.32@xmath168 & 0.30@xmath165 & + ne / ne@xmath166 & 0.24@xmath169 & 0.22@xmath168 & 0.24@xmath165 & + mg / mg@xmath166 & 0.66@xmath170 & 0.39@xmath171 & 0.41@xmath171 & + si / si@xmath166 & 0.9@xmath172 & 0.28@xmath173 & 0.3@xmath174 & + fe / fe@xmath166 & 0.18@xmath165 & 0.20@xmath175 & 0.21@xmath167 & + @xmath24 ( @xmath29 s ) & & 3.5@xmath176 & 3.3@xmath176 & + @xmath90 & & & & 2.2@xmath177 & + acis norm ( @xmath178 ) & 5.25@xmath179 & 1.75@xmath180 & 1.76@xmath180 & 2.3@xmath181 + mos norm ( @xmath178 ) & 1.05@xmath182 & 3.47@xmath183 & 3.59@xmath184 & 7@xmath185 + @xmath83 & 2.73 & 2.23 & + dof & 294 & 294 & + [ tab : wholesnrspec ] cccccccccccccccccc [ tab : snrspecfit3 ] component & vmekal & vpshock & vpshock & powerlaw + @xmath20 ( @xmath22 ) & 2.9@xmath186 & 1.7@xmath162 & + @xmath23 ( kev ) & 0.237@xmath169 & 0.46@xmath168 & 0.46@xmath165 & + o / o@xmath166 & 0.25@xmath187 & 0.22@xmath175 & 0.22@xmath188 & + ne / ne@xmath166 & 0.20@xmath169 & 0.25@xmath189 & 0.25@xmath190 & + mg / mg@xmath166 & 0.7@xmath191 & 0.43@xmath192 & 0.43@xmath193 & + si / si@xmath166 & 0.8@xmath194 & 0.2@xmath195 & 0.2@xmath195 & + fe / fe@xmath166 & 0.21@xmath196 & 0.24@xmath167 & 0.24@xmath167 & + @xmath24 ( @xmath29 s ) & & 2.2@xmath176 & 2.2@xmath176 & + @xmath90 & & & & 2@xmath197 & + acis norm ( @xmath178 ) & 2.69@xmath198 & 5.1@xmath199 & 5.0@xmath200 & 2@xmath201 + mos norm ( @xmath178 ) & 5.8@xmath202 & 1.11@xmath203 & 1.11@xmath204 & 0@xmath205 + @xmath83 & 1.71 & 1.47 & + dof & 203 & 203 & + [ tab : outerlimbspec ] cccccccccccccccccc [ tab : snrspecfit3 ] component & vmekal & vpshock & vpshock & powerlaw + @xmath20 ( @xmath22 ) & 2.8@xmath206 & 3.0@xmath162 & + @xmath23 ( kev ) & 0.278@xmath207 & 0.335@xmath208 & 0.37@xmath168 & + o / o@xmath166 & 0.56@xmath209 & 0.46@xmath210 & 0.46@xmath187 & + ne / ne@xmath166 & 0.16@xmath211 & 0.24@xmath169 & 0.27@xmath187 & + mg / mg@xmath166 & 0.9@xmath191 & 0.5@xmath195 & 0.6@xmath191 & + si / si@xmath166 & 0.7@xmath196 & 0.3@xmath172 & 0.3@xmath172 & + fe / fe@xmath166 & 0.22@xmath188 & 0.21@xmath190 & 0.23@xmath167 & + @xmath24 ( @xmath29 s ) & & 9@xmath212 & 5.5@xmath212 & + @xmath90 & & & & 3@xmath213 + acis norm ( @xmath178 ) & 1.22@xmath214 & 1.13@xmath215 & 7.7@xmath216 & 2@xmath217 + mos norm ( @xmath178 ) & 2.72@xmath218 & 2.5@xmath219 & 1.7@xmath219 & 0@xmath220 + @xmath83 & 2.50 & 2.22 & + dof & 150 & 150 & + [ tab : centerspec ] cccccccccccccccccc [ tab : snrspecfit3 ] component & vmekal & vpshock & vpshock & powerlaw + @xmath20 ( @xmath22 ) & 2.3@xmath221 & 2.2@xmath162 & + @xmath23 ( kev ) & 0.32@xmath165 & 0.90@xmath222 & 0.66@xmath169 & + o / o@xmath166 & 0.31@xmath223 & 0.30@xmath189 & 0.39@xmath171 & + ne / ne@xmath166 & 0.09@xmath190 & 0.12@xmath223 & 0.14@xmath224 & + mg / mg@xmath166 & 0.36@xmath225 & 0.2@xmath174 & 0.4@xmath191 & + si / si@xmath166 & 0.8@xmath196 & 0.2@xmath195 & 0.1@xmath195 & + fe / fe@xmath166 & 0.06@xmath175 & 0.19@xmath190 & 0.19@xmath210 & + @xmath24 ( @xmath29 s ) & & 7@xmath226 & 2.1@xmath227 & + @xmath90 & & & & 1.6@xmath228 & + acis norm ( @xmath178 ) & 5.9@xmath229 & 1.05@xmath230 & 1.14@xmath231 & 2@xmath232 + mos norm ( @xmath178 ) & 4.3@xmath233 & 7.5@xmath234 & 7.9@xmath235 & 2.3@xmath236 + @xmath83 & 1.49 & 1.07 & + dof & 102 & 102 & + [ tab : wedgespec ] cccccccccccccccccc [ tab : snrspecfit3 ] component & vmekal & vpshock & vpshock & powerlaw & powerlaw + @xmath20 ( @xmath22 ) & 1.0@xmath237 & 3.5@xmath238 & & 2.6@xmath239 + @xmath23 ( kev ) & 2.0@xmath240 & 2.8@xmath241 & 0.4@xmath195 & & + o / o@xmath166 & 2@xmath197 & 0.15@xmath242 & 0.2@xmath243 & & + ne / ne@xmath166 & 2@xmath244 & 0.18@xmath245 & 0.2@xmath246 & & + mg / mg@xmath166 & 1@xmath247 & 0.3@xmath195 & 0.7@xmath248 & & + si / si@xmath166 & 0.1@xmath249 & 0.2@xmath250 & 0.4@xmath251 & & + fe / fe@xmath166 & 0.1@xmath252 & 0.11@xmath253 & 0.1@xmath191 & & + @xmath24 ( @xmath29 s ) & & 4@xmath254 & 5@xmath255 & & + @xmath90 & & & & 2.2@xmath256 & 2.8@xmath196 + acis norm & 4.5@xmath257 & 4.2@xmath258 & 3@xmath259 & 1.5@xmath260 & 2.4@xmath261 + mos norm & 1.1@xmath262 & 1.0@xmath200 & 2.0@xmath263 & 2.7@xmath264 & 6@xmath265 + @xmath83 & 1.13 & 1.04 & & 1.08 + dof & 92 & 92 & & 92 + [ tab : linearspec ] lcccccccc knot & 0.9@xmath266 & 4@xmath267 & 1.7@xmath172 & 2.6@xmath268 & 7@xmath269 & 0.89/18 + 22 w of knot & 2.0@xmath270 & 3@xmath226 & 2@xmath271 & 7@xmath269 & 1.2@xmath272 & 0.67/15 + 44 w of knot & 0.36@xmath209 & 8@xmath273 & 3@xmath271 & 2.4@xmath274 & 1.8@xmath275 & 0.78/15 + point source & & & 2.1@xmath276 & & 2.4@xmath277 & 1.4/6 + [ tab : smregspec ] lccccc cool shell & density & 10@xmath34 @xmath29 + cool shell & mass & 160 - 2000 @xmath278 + cool shell & expansion velocity & 202 @xmath35 km s@xmath94 + cool shell & kinetic energy & 1 - 8 @xmath279 erg + cool shell & thermal pressure & 3@xmath280 dyne @xmath22 + hot gas & temperature & [email protected] kev + hot gas & density & 0.24 @xmath281 @xmath29 + hot gas & mass & 270 @xmath310 @xmath282 @xmath278 + hot gas & thermal energy & 6@xmath68 @xmath282 erg + hot gas & thermal pressure & 3.3 @xmath44 @xmath282 dyne @xmath22 + [ tab : parameters ] lccccccccc knot & 2.2@xmath283 & 2@xmath284 & 0.9/0.1 + 22 w of knot & 7@xmath285 & 7.5@xmath286 & 0.5/0.5 + 44 w of knot & 2.7@xmath287 & 7.9@xmath288 & 0.3/0.7 + knot & 2.3@xmath289 & 4.3@xmath290 & 0.8/0.2 + 22 w of knot & 8.3@xmath291 & 7.3@xmath292 & 0.5/0.5 + 44 w of knot & 8@xmath293 & 7.3@xmath291 & 0.1/0.9 + [ tab : linearflux ] | the n206 supernova remnant ( snr ) in the large magellanic cloud ( lmc ) has long been considered a prototypical mixed morphology " snr .
recent observations , however , have added a new twist to this familiar plot : an elongated , radially - oriented radio feature seen in projection against the snr face . utilizing the high resolution and sensitivity available with the _ hubble space telescope _ , _ chandra _ , and _ xmm - newton _ , we have obtained optical emission - line images and spatially resolved x - ray spectral maps for this intriguing snr .
our findings present the snr itself as a remnant in the mid to late stages of its evolution .
x - ray emission associated with the radio linear feature " strongly suggests it to be a pulsar - wind nebula ( pwn ) .
a small x - ray knot is discovered at the outer tip of this feature .
the feature s elongated morphology and the surrounding wedge - shaped x - ray enhancement strongly suggest a bow - shock pwn structure . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the ability of light to influence the kinetic motion of microscopic and atomic matter has had a profound impact in the last three decades . the optical manipulation of matter was first seriously studied by ashkin and co - workers in the 1970s @xcite , and led ultimately to the demonstration of the single beam gradient force trap @xcite , referred to as optical tweezers , where the gradient of an optical field can induce dielectric particles of higher refractive index than their surrounding medium to be trapped in three dimensions in the light field maxima @xcite . much of ashkin s early work centered not on gradient forces , but on the use of radiation pressure to trap particles @xcite , and a dual beam radiation pressure trap was demonstrated in a which single particle was confined . this work ultimately contributed to the development of the magneto - optical trap for neutral atoms @xcite . recently we observed one - dimensional _ arrays _ of silica spheres trapped in a dual beam radiation pressure trap @xcite . these arrays had an unusual property in that the particles that formed the array were regularly spaced from each other . the particles were redistributing the incident light field , which in turn redistributed the particle spacings , allowing them to reside in equilibrium positions . this effect , known as optically bound matter " was first realised in a slightly different context via a different mechanism to ours some years ago @xcite using a single laser beam and was explained as the interaction of the coherently induced dipole moments of microscopic spheres in an optical field creating bound matter . in the context of our study optically bound matter is of interest as it relates to the way in which particles interact with the light field in extended optical lattices , which may prove useful for the understanding of three - dimensional trapping of colloidal particles @xcite . indeed optically bound matter may provide an attractive method for the creation of such lattices that are not possible using interference patterns . bound matter may also serve as a test bed for studies of atomic or ionic analogues to our microscopic system @xcite . subsequent to our report a similar observation was made in an experiment making use of a dual beam fiber trap @xcite . in this latter paper a theory was developed that examined particles of approximately the same size as the laser wavelength involved . in this paper we develop a numerical model that allows us to simulate the equilibrium positions of two and three particles in a counter - propagating beam geometry , where the particle sizes are larger than the laser wavelength , and fall outside the upper bound of the limits discussed in @xcite . the model can readily be extended to look at larger arrays of systems . we discuss the role of the scattering and refraction of light in the creation of arrays . in the next section we describe the numerical model we use for our studies and derive predictions for the separation of two and three spheres of various sizes . we then compare this with both previous and current experiments . = 1000 our model comprises two monochromatic laser fields of frequency @xmath0 counter - propagating along the z - axis which interact with a system of @xmath1 transparent dielectric spheres of mass @xmath2 , refractive - index @xmath3 , and radius @xmath4 , with centers at positions @xmath5 , and which are immersed in a host medium of refractive - index @xmath6 . the electric field is written @xmath7 , \label{efield}\ ] ] where @xmath8 is the unit polarization vector of the field , @xmath9 are the slowly varying electric field amplitudes of the right or forward propagating @xmath10 and left or backward propagating @xmath11 fields , and @xmath12 is the wavevector of the field in the host medium . the incident fields are assumed to be collimated gaussians at longitudinal coordinates @xmath13 for the forward field and @xmath14 for the backward field @xmath15 where @xmath16 , @xmath17 is the initial gaussian spot size , and @xmath18 is the input power in each beam . it is assumed that all the spheres are contained between the beam waists within the length @xmath19 . consider first that the dielectric spheres are in a fixed configuration at time @xmath20 specified by the centers @xmath21 . then the dielectric spheres provide a spatially inhomogeneous refractive index distribution which can be written in the form @xmath22 where @xmath23 is the heaviside step function which is unity within the sphere of radius @xmath4 centered on @xmath24 , and zero outside , and @xmath3 is the refractive - index of the spheres . then , following standard approaches @xcite , the counter - propagating fields evolve according to the paraxial wave equations @xmath25 along with the boundary conditions in eq . ( [ bcond ] ) , where @xmath26 and @xmath27 is the transverse laplacian describing beam diffraction . thus , a given configuration of the dielectric spheres modifies the fields @xmath9 in a way that can be calculated from the above field equations . we remark that even though the spheres move , and hence so does the refractive - index distribution , the fields will always adiabatically slave to the instantaneous sphere configuration . to proceed we need equations of motion for how the sphere centers @xmath21 move in reaction to the fields . the time - averaged dipole interaction energy @xcite , relative to that for a homogeneous dielectric medium of refractive - index @xmath6 , between the counter - propagating fields and the system of spheres is given by @xmath28<\vec{e}^2 > \nonumber \\ & = & -\frac{\epsilon_0}{4}(n_s^2-n_h^2)\sum_{j=1}^{n}\int _ { } ^ { } dv\theta(r-|\vec{r}-\vec{r}_j(t)|)\left[|{\cal e}_+(\vec{r})|^2+|{\cal e}_-(\vec{r})|^2 \right ] , \label{u}\end{aligned}\ ] ] where the angled brackets signify a time - average which kills fast - varying components at @xmath29 . the most important concept is that the dipole interaction potential depends on the spatial configuration of the spheres @xmath30 since the counter - propagating fields themselves depends on the sphere distribution via the paraxial wave equations ( [ parax ] ) . this form of the dipole interaction potential ( [ u ] ) shows explicitly that we pick up a contribution from each sphere labelled @xmath31 via its interaction with the local intensity . assuming over - damped motion of the spheres in the host medium with viscous damping coefficient @xmath32 , the equation of motion for the sphere centers become @xmath33 where @xmath34 signifies a gradient with respect to @xmath35 , and @xmath36 are the gradient and the scattering forces experienced by the j@xmath37 sphere , the latter of which we shall give an expression for below . carrying through simulations for a 3d system with modelling of the electromagnetic propagation in the presence of many spheres poses a formidable challenge , so here we take advantage of the symmetry of the system to reduce the calculation involved . first , for the cylindrically symmetric gaussian input beams used here we assume that the combination of the dipole interaction potential , and associated gradient force , and the scattering force supplies a strong enough transverse confining potential that the sphere motion remains directed along the z - axis . this means that the positions of the sphere centers are located along the z - axis @xmath38 , and that the gradient and scattering forces are also directed along the z - axis @xmath39 . second , we assume that the sphere distribution along the z - axis is symmetric around @xmath40 , the beam foci being at @xmath41 . this means , for example , that for one sphere the center is located at @xmath40 , for two spheres the centers are located at @xmath42 , @xmath43 being the sphere separation distance , and for three spheres the centers are at @xmath44 . for three or less spheres the symmetric configuration of spheres is captured by the sphere spacing @xmath43 , and we shall consider this case here . for more than three spheres the situation becomes more complicated and we confine our discussion to the simplest cases of two and three spheres . with the above approximations in mind the equations of motion for the sphere centers become @xmath45 at this point it is advantageous to consider the case of two spheres , @xmath46 , to illustrate how calculations are performed . for a given distance @xmath43 between the spheres we calculate the counter - propagating fields between @xmath47 $ ] using the beam propagation method . from the fields we can numerically calculate the dipole interaction energy @xmath48 for a given sphere separation , and the resulting axial ( z - directed ) gradient force is then @xmath49 . thus , by calculating the counter - propagating fields for a variety of sphere separations we can numerically calculate the gradient force which acts on the relative coordinate of the two spheres . for our system we approximate the scattering force @xcite along the positive z - axis for the j@xmath37 sphere as @xmath50 , \ ] ] with @xmath51 the scattering cross - section . this formula is motivated by the generic relation @xmath52 for unidirectional propagation , with the scattered power @xmath53 , and @xmath54 the incident intensity . the integral yields the difference in power between the two counter - propagating beams integrated over the sphere cross - section , and when this is divided by the sphere cross - sectional area @xmath55 we get the averaged intensity difference over the spheres . for the case of two spheres we calculate the scattering force @xmath56 , evaluated at the position of the sphere at @xmath57 , and for a variety of sphere spacings @xmath43 . a similar procedure can readily be applied to the case of three spheres . the theory described above has some limitations that we now discuss . first , we assume that the spheres are trapped on - axis by a combination of the scattering and/or dipole forces acting transverse to the propagation axis . for this to be possible we require that the sphere diameter be less than the laser beam diameter @xmath58 . furthermore , we have assumed paraxial propagation that neglects any large angle or back - scattering of the laser fields . however , when light is incident on a sphere of diameter @xmath43 there is an associated wavevector uncertainty @xmath59 , and when @xmath60 back - scattering can occur , as it is within the uncertainty that an incident wave of wavevector @xmath61 along a given direction is converted into @xmath62 . this yields the condition @xmath63 , with @xmath64 the free - space wavelength , to avoid back - scattering and so that our paraxial assumptions are obeyed . our goal is to compare the axial gradient and scattering forces for an array of two and three spheres and compare with the experimental results . however , the scattering cross - section for our spheres , which incorporates all sources of scattering in a phenomenological manner , can not be calculated with any certainty . our approach , therefore , will be to calculate the equilibrium sphere separation @xmath65 for the gradient and scattering forces separately , which does not depend on the value of the cross - section , and compare the calculated sphere separations with the experimental values . by comparing the theoretical predictions with the experiment for @xmath66 we can determine the dominant source of the axial force acting on the spheres . to compare our theory with experiment we use data from our previous work @xcite and also recreate that experiment , but using a different laser wavelength and particle sphere size . the previously reported experiment @xcite makes use of a continuous - wave 780 nm ti : sapphire laser , which is split into two beams with approximately equal power ( 25mw ) in each arms . each of the beams is focussed down to a spot with a 3.5 @xmath67 beam waist and then passed , counterpropagating , through a cuvette with dimensions of 5 mm x 5 mm x 20 mm . the beam waists were separated by a finite amount , which is discussed further below . uniform silica spheres with a 3@xmath67 diameter ( bangs laboratories , inc ) in a water solution were placed in the cuvette , and the interaction of the beams with the sample caused one - dimensional arrays of particle to be formed . the refractive index of the spheres is approximately 1.43 . we also carried out a similar experiment using a 1064@xmath68 nd : yag laser where the beam waists were 4.3@xmath67 and we used 2.3@xmath67 diameter spheres . the particles were viewed by looking at the scattered light orthogonal to the laser beam propagation direction viewed on a ccd camera with an attached microscope objective ( x20 , na=0.4 , newport ) . to compare our theory with experimental results we need to concentrate on a small number of parameters , the sphere size , the beam waist , the refractive index of the spheres and the beam waist separation . we know the particle sizes and can make a good estimate as to their refractive index , further we can measure the beam waist to a high degree of accuracy . the only problematic factor is the beam waist separation . due to experimental constraints , this is quite difficult to measure . we estimate the waist separation by filling the cuvette with a high density particle solution and looking at the scattered light from the sample . the high density of particle allows us to map out the intensity pattern of the two beams and hence make an estimate as to the waist separation . this is , however , an inaccurate method and leaves us with an error of more than 100% . we therefore use our model to help us fix the beam waist separation on a single result and then examine the behavior of the model when varying other parameters . the error in the beam waist separation is not as extreme as first it sounds however . modelling the system for a range of beam waist separations from 80@xmath69 m to 200@xmath69 m results in a predicted range of sphere separations as shown in figure 1 , 2.3@xmath69 m diameter spheres . we see that although initially the beam waist separation difference makes a reasonable difference to the predicted sphere separation the region that we believe we are working in , @xmath70 m waist separation , is relatively flat . therefore even if we do have a large error in this value , the predicted result does not vary significantly . this increases our confidence that we have the correct beam waist separation with a higher uncertainty that our experimental measurements of this parameter suggests . we begin by examining the case of the 2.3 @xmath67 diameter spheres . the rate of change of sphere separation is seen to drop off as the waist separation increases . the fit to a parabola is to aid the eye , rather than to suggest a quantitative relationship.,width=302 ] we consider the case for chains of both two and three spheres . for two spheres we measure a sphere separation of 34@xmath67 , for a beam waist , @xmath71 at a laser wavelength , @xmath72 . using a beam waist separation of @xmath73 our model predicts an equilibrium in the scattering force of 34@xmath74 , as is shown in figure 2 . the intensity in the x - z plane for this configuration is shown in figure 3 . we see no such equilibrium in the gradient force , shown in figure 4 and conclude that the scattering force is the dominant factor in this instance . using the same parameters for the three sphere case give us a sphere separation prediction of 62@xmath67 , as shown in figure 5 . again this dominates over the gradient force , this assumption being valid , as the theory gives a good prediction of our experimental observations . our experimental result is 57@xmath67 , but we estimate our model value falls within the standard deviation error we observe on our experimental measurements . diameter silica spheres with the beam waists 180 @xmath67 apart . @xmath75 and @xmath72.,width=302 ] diameter silica spheres with the beam waists 180 @xmath67 apart . @xmath75 and @xmath72.,width=302 ] diameter silica spheres with the beam waists 180 @xmath67 apart . @xmath75 and @xmath72.,width=302 ] diameter silica spheres with the beam waists 180 @xmath67 apart . @xmath76 and @xmath72 . the plot shows the separation between two of the three spheres , and the scattering forces are symmetric about the center sphere.,width=302 ] the data for 3 micron spheres carried out at a different wavelength than the 2.3 @xmath67 data ( @xmath77 ) also fits well with our theory . for two spheres , with the beam waists @xmath78 apart , we predict a sphere separation of 47@xmath67 ( figure 6 ) while our experiment predicts a distance of 45@xmath67 . using the same parameters for the three sphere case we predict a sphere separation of 27@xmath67 ( figure 7 ) , while our experiment shows a separation of 35@xmath67 . again , as we predict equilibrium positions with the scattering force component , but not with the gradient force component , we conclude that the scattering force is the dominant factor in determining the final sphere separations . diameter silica spheres with the beam waists 150 @xmath67 apart . @xmath71 and @xmath77.,width=302 ] diameter silica spheres with the beam waists 150 @xmath67 apart . @xmath79 and @xmath77 . the plot shows the separation between two of the three spheres , and the scattering forces are symmetric about the center sphere.,width=302 ] our model accurately predicts separations for the case of two and three spheres , at certain sizes . however we also performed experiments using @xmath80 diameter spheres and could not find any agreement between experiment and theory . since our model uses a paraxial approximation , the assumption is that in these smaller size regimes the model breaks down . this in contrast to the work detailed in @xcite which works in size regimes closer to the laser wavelength , @xmath64 , and begins to break down in the larger size regimes ( @xmath81 ) , where @xmath82 is the sphere diameter . we also note that the beam separation distance becomes less critical as it becomes larger . for small beam waist separation distances ( @xmath83 , say ) , any change in this parameter leads to a sharp change in the sphere separation distance , whereas at the waist separation distances we work at the change in sphere separation distance is far more gentle , and hence gives less rise to uncertainty over exact fits with theory and experiment . the other main parameter is sphere size , which has an appreciable effect on the predicted sphere separation . the incident power on the spheres does not make much of a difference and is more of a scaling factor in the forces involved rather than a direct modifier in the model . predicted sphere separation is also sensitive to the refractive index difference between the spheres and the surrounding medium , so it is important that the spheres refractive index is well known . it should also be possible to create two - dimensional and possibly three dimensional arrays from optically bound matter . the extension to two dimensions is relatively simple to envisage with the use of multiple pairs of counterpropagating laser beams . in three dimensions the formation of such optically bound arrays may circumvent some of the problems associated with loading of three - dimensional optical lattices @xcite . it is often assumed that the creation of an optical lattice ( via multibeam interference , say ) will allow the simple , unambiguous trapping of particles in all the lattice sites , thereby making an extended three - dimensional array of particles . such arrays may be useful for crystal template formation @xcite and in studies of crystallization processes @xcite . however crystal formation in this manner is not particularly robust in that as the array is filled the particles perturb the propagating light field such that they prevent the trap sites below them being efficiently filled . arrays of optically bound matter do not suffer from such problems , as they are organized as a result of the perturbation of the propagating fields . further the fact that the particles are bound together provides more realistic opportunities for studying crystal and colloidal behaviour than that in unbound optically generated arrays , such as those produced holographically @xcite . we have developed a model by which the propagation of counter - propagating lasers beams moving past an array of silica spheres may be examined . analysis of the resulting forces on the spheres allows us to predict the separation of the spheres which constitute the array . we have compared this model with experimental results for different beam parameters ( wavelength , waist separation , waist diameter ) and found the results to be in good agreement with our observations . the model , does not however , work with sphere sizes much less than approximately twice the laser wavelength . our model is readily extendable to larger number of spheres , and will be of great use in the study of such one- and higher - dimensional arrays of optically bound matter . dm is a royal society university research fellow . this work is supported by the royal society and the uk s epsrc . | counter - propagating light fields have the ability to create self - organized one - dimensional optically bound arrays of microscopic particles , where the light fields adapt to the particle locations and vice versa .
we develop a theoretical model to describe this situation and show good agreement with recent experimental data ( phys .
rev .
lett .
89 , 128301 ( 2002 ) ) for two and three particles , if the scattering force is assumed to dominate the axial trapping of the particles .
the extension of these ideas to two and three dimensional optically bound states is also discussed . |
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the basic motivation behind a quantum cosmological model is two - fold . one is the fact that when the linear dimension of the universe reaches the planck scale ( @xmath0 cm ) , the universe is indeed governed by a quantum picture . the second is the hope that a quantum description might be able to produce a singularity free birth of our universe . the quantum state of the universe is described by a wave function @xmath1 which is the solution of the wheeler dewitt equation on a minisuperspace . for a comprehensive review , we refer to @xcite . + one critical problem in quantum cosmology is certainly that of a suitable choice of time against which the evolution of the universe is investigated . this is because the notion of time has different implications in general relativity and quantum mechanics . systematic attempts towards resolving this problem started in the 90s@xcite . the strategies also include a scenario where the notion of time plays no role whatsoever@xcite . for a recent review of different strategies , see @xcite . + if matter is taken as a perfect fluid , the strategy adopted by schutz@xcite becomes extremely useful as a set of canonical transformations leads to one conjugate momentum associated with the fluid giving a linear contribution to the hamiltonian . the corresponding fluid variable thus qualifies to play the role of time in the relevant schr@xmath2dinger equation . in an ever expanding model , the fluid density has a monotonic temporal behaviour , the time orientability is thus ensured . + schutz s formalism has been extensively used by alvarenga _ et al_@xcite for isotropic cosmological models . the investigation involves spatially flat , open and closed models . matter content is taken as a perfect fluid with an equation of state @xmath3 . they found finite norm wave packet solutions of the wheeler - dewitt equations . one important finding is that some singularity free models could be constructed even without violating the energy conditions . a later work involves anisotropic bianchi i cosmology@xcite , which is the anisotropic generalization of a spatially flat isotropic model . anisotropic models have problems , particularly that of non - unitarity . the hamiltonian , although hermitian , is not self - adjoint . this is discussed in some detail in ref@xcite . + it is well known that a standard copenhagen interpretation of a quantum cosmological model is not tenable . a `` bohm - de broglie '' interpretation normally does better . for a comprehensive review , we refer to @xcite . in the anisotropic case , the non unitarity leads to futher problems in the interpretation . as the norm of the wave function becomes time dependent , bohmian trajectories are not conserved and the bohm - de broglie interpretation also becomes vulnerable@xcite . it appears that the philosophy of interpretation might require a dramatic extension so as to incorporate anisotropic cosmologies . in the present work schutz s formalism is utilized to quantize bianchi type v and type ix perfect fluid cosmological models which are the anisotropic generalzations of open and closed isotropic models respectively . in section 2 we discuss the formalism and use that to quantize bianchi type v models . in section 3 , a type ix model is quantized . some concluding remarks are made in section 4 . the relevant action for gravity with a perfect fluid can be written as @xmath4 where @xmath5 is the induced metric over three dimensional spatial hypersurface which is the boundary @xmath6 of the four dimensional manifold @xmath7 and @xmath8 is the extrinsic curvature . here units are so chosen that @xmath9 is equal to one . @xmath10 is the pressure due to the perfect fluid . the perfect fluid satisfies an equation of state @xmath11 where @xmath12 . this restriction stems from the consideration that sound waves can not propagate faster than light . in schutz s formalism @xcite the fluid s four velocity can be expressed in terms of six potentials . however , two of them are connected with rotation . in bianchi v or ix models permit timelike geodesics which are hypersurface orthogonal , the rotation tensor @xmath13 vanishes , and one can write the four velocity in terms of only four independent potentials as @xmath14 here @xmath15 , @xmath16 , @xmath17 and @xmath18 are the velocity potentials having their own evolution equations , where the potentials connected with vorticity are dropped . the four velocity is normalized as @xmath19 although the physical identification of velocity potentials are irrelevant for the formulation , @xmath15 and @xmath16 can be identified with the specific enthalpy and specific entropy respectively . this identification facilitates the representation of fluid parameters in terms of thermodynamic quantities . the metric for the bianchi v anisotropic model is written as @xmath20\ ] ] where @xmath21 is called the lapse function . while @xmath22 are functions of the cosmic time @xmath23 , @xmath24 is a constant . bianchi type i model is recovered when @xmath25 . eliminating the surface terms , the first and second terms of the action ( 1 ) give @xmath26 \ , , \ ] ] where an overhead dot indicates a differentiation with respect to time @xmath23 and @xmath27 is contribution of geometry to the action . so the gravitational lagrangian density can be easily identified as @xmath28 if we now choose the three metric coefficients as @xmath29 then we get @xmath30 and @xmath31 - 6e^{\beta_0}nm^2 \ , .\ ] ] now @xmath32 , @xmath33 and @xmath34 will be treated as the relevant variables in place of @xmath35 , @xmath36 and @xmath37 . the corresponding conjugate momenta are then @xmath38 the hamiltonian of the gravity sector is now given by @xmath39 where momentum @xmath40 is replaced in terms of @xmath41 using equation ( [ mom ] ) . for the fluid part , the action can be written using thermodynamic relations for @xmath15 and @xmath16 @xcite . the relevant equations are @xmath42 where @xmath43 , @xmath44 , @xmath45 and @xmath46 are temperature , total mass energy density , rest mass density and specific internal energy respectively . rewriting the third equation of ( [ thermo ] ) we get @xmath47 \quad .\ ] ] it then follows that , @xmath48 and @xmath49 . we can show that the equation of state takes the form @xmath50 in a comoving system @xmath51 , and equation ( [ nor ] ) yields @xmath52 \ , , \ ] ] @xmath53 being the fluid part of the action . as @xmath54 so @xmath55 . if we try the canonical methods used , for example , in @xcite the hamiltonian for this action can be written in a very simple form with the canonical transformations @xmath56 along with @xmath57 . here @xmath58 , @xmath59 and @xmath60 , the lagrangian density of the fluid , is the expression inside the square bracket of equation ( [ af ] ) . the hamiltonian for this perfect fluid can now be written as @xmath61 the advantage of using this method , i.e. , using canonical transformations , is that we could find a set of variables where the system of equations is more tractable , while the hamiltonian structure of the system remains intact . it also deserves mention that amongst the four velocty potentials mentioned , actually two are used , namely @xmath17 and @xmath16 . this is because @xmath17 and @xmath15 are related by @xmath62 ( see ref@xcite ) and one other , namely @xmath18 can be settled using the normalization ( [ nor ] ) . the super hamiltonian for the minisuperspace of this anisotropic quantum model is @xmath63 \ , .\end{aligned}\ ] ] here @xmath64 acts as a lagrange multiplier taking care of the classical constraint equation @xmath65 . using the usual quantization procedure @xcite , we write the schr@xmath2dinger - wheeler - dewitt equation for our super hamiltonian with the ansatz that the super hamiltonian operator annihilates the wave function , @xmath66 there are attempts@xcite to show that the classical hamiltonian for a cosmological spacetime is zero , if one takes into account both the matter sector and the geometry sector together , like the present situation . but these attempts involves pseudotensorial calculations and the result depends on the minisuperspace chosen@xcite . the problem perhaps stems from the fact that localization of energy in general relativity is not uniquely defined . whatever be the status of the constraint ( [ wdwe1 ] ) in the most general case , we shall be using this following the standard practice . + with @xmath67 , @xmath68 , @xmath69 equation ( [ wdwe1 ] ) can now be written as @xmath70\ , \psi(\beta_0,\beta_+,t ) = 0 \ , \ , .\ ] ] in this equation @xmath71 is the cosmic time co - ordinate if we choose the gauge @xmath72 and this follows from hamilton s classical equations as @xmath73 . now it must be mentioned that while constructing the schr@xmath2dinger - wheeler - dewitt equation ( [ wdwe ] ) we have considered a particular choice of factor ordering for @xmath74 and @xmath75 . we will discuss other choices of factor orderings and its consequences on our prime results in the subsequent section . we require that the super hamiltonian must be hermitian , so the wave function @xmath1 must satisfy these conditions @xcite ; @xmath76 in order to solve for the wave function @xmath1 from equation ( [ wdwe ] ) we employ a separation of variables as , @xmath77 we get @xmath78 where @xmath79 is the separation constant . further if we write @xmath80 we get @xmath81 the solution for @xmath82 is @xmath83 where @xmath84 is the integration constant and @xmath85 is again a constant of separation which has to be real so that the wave function is normalizable . the equation for @xmath86 is @xmath87 for @xmath25 , this equation reduces to the corresponding equation for the bianchi i model @xcite . it is very difficult to solve this equation analytically for all allowed values of @xmath88 . here we will study the solution for some particular values of @xmath88 . with @xmath90 equation ( [ vs1 ] ) becomes @xmath91 the solution of equation ( [ phi1 ] ) is known in terms of bessel function and now we can write the wave function of equation ( [ wdwe ] ) as @xcite @xmath92 ~,\ ] ] where @xmath93 , @xmath94 are the integration constants . now we can construct a regular wave packet superposing these solutions . in principle this can be easily done by considering the arbitrary integration constants to be suitable gaussian functions of the parameters @xmath85 and @xmath79 ( see ref @xcite ) . defining @xmath95 we can write the form of the wave packet as @xmath96 where @xmath97 and @xmath98 and @xmath99 are arbitrary positive constants . the above integrals can be explicitly evaluated and the wave packet becomes @xmath100 where @xmath101 . the norm of the wave packet is @xmath102 where @xmath103 . clearly we can see that the norm is time dependent and hence the model is not unitary . this is not surprising as we know that the anisotropic quantum cosmological models are non - unitary @xcite . the expectation value of any variable @xmath104 can now be calculated as @xmath105 for @xmath106 we find @xmath107 where @xmath108 and @xmath109 is a numerical factor @xmath110 . now @xmath111 this describes the cosmological evolution of the spatial volume and this model predicts a bounce from a minimum volume universe with no singularity . of course this result is facilitated by the choice of the gauge @xmath112 as mentioned after equation(21 ) . it is easy to see that the minimum volume is obtained at @xmath113 and the corresponding length scale is @xmath114 . as already mentioned , @xmath115 . so the linear size of the universe can be set in the model by choosing the vaule of @xmath98 . for @xmath116 , @xmath117 it deserves mention that @xmath33 is in fact a constant in time . the plot of the norm of the wave packet against @xmath32 and @xmath33 shows a well behaved pattern , and even at @xmath113 , does not show any sign of blowing up ( see figure [ [ fig3 ] ] ) . this is consistent with the fact that there exists a minimum of the proper volume and thus the pathology of a singular state of the universe can be avoided in this model . furthermore , the plots are given for two different time , namely @xmath113 and @xmath118 in some units , and the qualitative nature of the plots remain similar . now let us have a closer look at equation ( [ wnorm ] ) . as @xmath119 so we infer that the norm of the wave packet is time - dependent hence the model is non - unitary . we now investigate the time dependence of the norm ( @xmath120 ) as a function of time ( @xmath121 ) in figure [ [ norm1 ] ] . it is observed that with the increase of @xmath121 , the norm flattens , i.e. , almost becomes a constant . this indicates that the problem of non - unitarity is somewhat diluted at least for a large time . it deserves mention that this feature may be facilitated by our choice of gauge @xmath122 which makes @xmath123 a constant . [ cols="^ " , ] with @xmath125 equation ( [ vs1 ] ) becomes @xmath126 the solution is obtained in terms of bessel function and we can write @xmath127 where @xmath128 and @xmath129 are the integration constants . the wave function can now be written as @xmath130~.\ ] ] with @xmath132 equation ( [ vs1 ] ) becomes @xmath133 the solution can be written in terms of confluent hypergeometric function ( @xmath134 ) and associated laguerre function ( @xmath135 ) , @xmath136 ~,\end{aligned}\ ] ] where @xmath137 , @xmath138 are the integration constants . the expression for the wave function can be written as @xmath139 ~.\end{aligned}\ ] ] the bianchi ix metric is written as @xmath140 d\phi^2 + 2 a^2 \cos \theta ~ dr d\phi ~.\ ] ] the gravitational lagrangian density is written as @xmath141 where @xmath142 . the hamiltonian for this gravitational lagrangian density becomes @xmath143 where @xmath144 and @xmath145 are the canonical conjugate momenta for the variables @xmath35 and @xmath146 . applying schutz s mechanism @xcite and the canonical methods described in @xcite we can evaluate the hamiltonian for the fluid as @xmath147 the super hamiltonian for the minisuperspace of this model is @xmath148 we shall calculate the wave function of the bianchi ix universe with @xmath149 . using the quantization procedure as described in @xcite we write the schr@xmath2dinger - wheeler - dewitt equation as @xmath150 now we have to apply the method of separation of variables for the solution of this equation . we write the wave function as @xmath151 which leads to @xmath152 with the separation @xmath153 we get @xmath154 and @xmath155 here @xmath85 is the separation parameter . the solutions of ( [ pv ] ) and ( [ pu ] ) are known in terms of modified bessel functions of first ( @xmath156 ) and second ( @xmath157 ) kind and can be written as @xmath158\ ] ] and @xmath159 ~,\ ] ] where @xmath160 and @xmath161 . @xmath162 s are the arbitrary integration constants . so the final expression of the wave function is @xmath163~[~ c_9 ~{\cal i}_{\nu'}~ ( ia^2)~ + ~ c_{10 } ~{\cal k}_{\nu ' } ~(ia^2)~]\end{aligned}\ ] ] bianchi type v and type ix perfect fluid models are quantized following schutz s formalism . as the wheeler - dewitt equation for a general equation of state @xmath3 is too involved , some specific examples are taken up . in bianchi type v models , the solution for the wave function for the universe comes out as a combination of bessel functions for the case @xmath164 . but it deserves mention that the problem of non - unitarity exists in this model , the norm of the wave packet is indeed time dependent . a clever operator ordering can only alleviate the problem in the sense that the norm becomes a constant for a large time . but the problem is only alleviated and not eradicated ! in addition to the operator ordering , the choice of a gauge as @xmath122 also may have its say . the proper volume shows a bounce from a finite minimum and thus the singularity of a zero proper volume can be avoided . by a choice of parameter , the minimum length scale obtained can be tuned to a desired value , say the planck length . however , this is obtained as a bonus , as the chief motivation of the work had been to check the merits and problems of the particular method of quantization for the anisotropic cosmological models . only one case , namely @xmath164 has been solved here as an example . this is simply because this case could be integrated anayltically . for some other cases also the wave functions are obtained analytically . but the complete analysis could not be done as the wave packet could not be integrated . so the work is nowhere near a general investigation of the problem . however , so long there is not a complete resolution of the quantum cosmology , one has to look for various examples . so although potentially schutz formalism gives us a beautiful way of handling the problem of the choice of time , it also leads to other problems , namely that of non unitatiry . this could be a generic feature of anisotropic models as indicated in @xcite . the philosophy of interpretation also seems to be awaiting a new direction . 100 s.w . hawking . quantum cosmology from _ relativity groups and topology _ , eds . b.dewitt and r. stora , north holland ( 1984 ) . k.v.kuchar in _ conceptual problems of quantum gravity _ , eds a.ashtekar and j.stachel ( birkhauser , boston , 1991 ) + k.v.kuchar in _ arguments of time _ , ed j.butterfield ( oxford university press , oxford , 1999 ) + c.j.isham in _ integrable systems , quantum groups and quantum field theory _ , eds l.a.ibort and m.a.rodriguez , ( kluwer , dordrecht , 1993 ) . c. rovelli ; arxiv : gr - qc/0903.3832 . e. anderson ; arxive : gr - qc/1009.2157 . b.f . schutz , phys . rev . * d2 * , 2762 ( 1970 ) . b.f . schutz , phys . * d4 * , 3559 ( 1971 ) . f.g.alvarenga , j.c.fabris , n.a.lemos and g.a.monerat ; gen . relativ . gravit . * 34 * , 651 ( 2002 ) . n. pinto - neto , quantum cosmology , _ notas de fisica _ , cbpf - nf-006/97 ; cnpq , brazil . alvarenga , a.b . batista , j.c . fabris and s.v.b . goncalves , gen . . grav . * 35 * , 1659 ( 2003 ) . v.g . lapchinskii and v.a . rubakov , theor . phys . * 33 * , 1076 ( 1977 ) . charles w. misner , phys . rev . * 186 * , 1319 ( 1969 ) . bryce s. dewitt , phys . rev . * 160 * , 1113 ( 1967 ) . cooperstock , gen . * 26 * , 323 ( 1994 ) ; + n. rosen , gen . * 26 * , 319 ( 1995 ) . n. banerjee and s. sen , pramana , * 49 * , 609 ( 1997 ) + j.m.nester , l.l.so and t.vargas , phys . d , * 78 * , 044035 ( 2008 ) . lemos , j. math . phys . * 37 * , 1449 ( 1996 ) . batista , j.c . fabris , s.v.b . gonalves and j. tossa , phys . rev . * d65 * , 063519 ( 2002 ) . w.w . bell , _ secial functions for scientists and engineers _ ; van nostrand reinhold co. , london ( 1968 ) . | the present paper deals with quantization of perfect fluid anisotropic cosmological models .
bianchi type v and ix models are discussed following schutz s method of expressing fluid velocities in terms of six potentials .
the wave functions are found for several examples of equations of state . in one case
a complete wave packet could be formed analytically .
the initial singularity of a zero proper volume can be avoided in this case , but it is plagued by the usual problem of non - unitarity of anisotropic quantum cosmological models .
it is seen that a particular operator ordering alleviates this problem .
* perfect fluid quantum anisotropic universe : merits and challenges * + barun majumder and narayan banerjee + department of physical sciences , + indian institute of science education and research - kolkata , + mohanpur campus , p.o .
bckv main office , district nadia , + west bengal 741252 , india .
_ pacs numbers : 04.20.cv . , 04.20.me_ |
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systems with asymmetric transport characteristics are also known as ratchets and brownian motors since they can be used to generate dc currents from an ac or noisy voltage . this mechanism has important applications in physics and biology @xcite . as specific examples , we mention diodes and photovoltaic current rectifiers . here we address the possibility of realizing such effects in luttinger liquids , which is of interest in connection with nanostructured devices . motivated by recent experiments on quantum dots @xcite , carbon nanotubes @xcite , and quantum hall systems @xcite , we examine ratchet effects caused by two unequal constrictions as point scatterers . we consider a one - dimensional spinless electron liquid at zero temperature subject to an impurity potential @xmath0 and a pair interaction @xmath1 with the hamiltonian @xmath2 the impurity potential @xmath0 is chosen to describe two unequal point scatterers at a distance @xmath3 . in the absence of interactions , the application of a voltage @xmath4 leads to a current @xmath5 with a finite conductance @xmath6 . the conductance is proportional to @xmath7 with the transmission probability @xmath8 for electrons with incident energy @xmath9 in the vicinity of the fermi energy @xmath10 . as a consequence of time reversal symmetry , @xmath8 does not depend on the direction of the incoming momentum and therefore noninteracting electrons have a symmetric transport characteristic with an odd function @xmath11 . therefore , the inclusion of interactions is mandatory for the analysis of ratchet effects . for the inclusion of interactions , the bosonized representation of the model is particularly convenient . the bosonization technique @xcite maps the quantum dynamics of the electron liquid onto a path integral for a bosonic field @xmath12 which essentially describes collective displacements of the electron liquid . in terms of this field , the particle density of the electrons reads @xmath13 } + \rm{h.c . } \right ) \ . \label{rho}\end{aligned}\ ] ] here , @xmath14 is the fermi wave vector and @xmath15 is a microscopic cutoff length scale of the order of @xmath16 . the particle current is given by @xmath17 , and the charge current by @xmath18 . in terms of the field @xmath12 , the relevant contributions to the action corresponding to the hamiltonian ( [ h ] ) read @xmath19 } + { \rm h.c . } ] \nonumber \\ & & - \frac{w_0}{\pi^{3/2 } \alpha } \partial_x \vartheta ( x ) [ e^{i [ 2 k_f x + 2 \sqrt \pi \vartheta(x ) ] } + { \rm h.c . } ] \bigg\ } \ . \label{s}\end{aligned}\ ] ] forward scattering by the interaction is included in the `` free '' bosonic theory via the luttinger parameter @xmath20^{-1/2}$ ] which is less than unity for repulsive interaction . assuming that @xmath1 is short ranged due to screening effects , only its weight @xmath21 is effective . the second and third lines of eq . ( [ s ] ) describe backscattering off the impurity potential and by the interaction , respectively . for a single point - like scatterer , @xmath22 , it was shown @xcite that repulsive electron interactions ( @xmath23 ) lead to a vanishing linear conductance . a weak scatterer was found to suppress the conductance according to @xmath24 where @xmath25 is conductance change due to the presence of the scatterer . on the other hand , for a strong scatterer , the conductance vanishes like @xmath26 with the amplitude @xmath27 for hopping across the scatterer . we subsequently focus on the weak - barrier case , deferring the strong currugation limit to ref . @xcite . the suppression of the conductance in the presence of interactions can be attributed to the backscattering of electrons from a hartree - type potential caused by friedel oscillation in the vicinity of the impurity . in the framework of the bosonic action ( [ s ] ) , this backscattering arises from the second - line contribution , and the third - line contribution is irrelevant for the asymptotics ( [ dg ] ) . to illustrate the mechanism at work , it is instructive to briefly recall the scattering features of single electrons by a double barrier @xmath28 . we assume the amplitudes @xmath29 and @xmath30 to be positive . @xmath3 is the distance between the two barriers . comparing the probability density @xmath31 for a particle incident from @xmath32 with wave vector @xmath33 to the density @xmath34 for a particle incident from @xmath35 with wave vector @xmath36 , one observes that the densities are not only different but also _ not _ related by reflection symmetry , @xmath37 ) . in this term , the hartree approximation amounts to replacing @xmath38 which is proportional to the slowly varying part of the density , cf . ( [ rho ] ) by its average value @xmath39 . this average is readily calculated from the action perturbatively to second order in @xmath40 . then , the third - line contribution can be absorbed into the second - line contribution by replacing the bare potential @xmath0 with the effective potential @xmath41 including the correction @xmath42 . in analogy to the analysis @xcite yielding eq . ( [ dg ] ) for a single scatterer , one then can calculate the corrections to conductance to second order in this effective potential . to estimate the strength of this effect , it is important to notice that , to leading order , this correction is proportional to the backscattering current off the impurities , i.e. , one expects @xmath43 corresponding to eq . ( [ dg ] ) . the reinsertion of this correction into eq . ( [ dg ] ) suggests subleading ratchet corrections to the conductance of order @xmath44 . performing the systematic calculation outline above , we obtain an asymmetric ratchet contribution to the conductance @xmath45 \label{res}\end{aligned}\ ] ] with the function @xmath46 invoking the bessel function @xmath47 . equation ( [ res ] ) was obtained within the hartree approximation . it becomes exact in a model of many bands @xmath48 interacting through the coupling @xmath49 . however , the order - of - magnitude estimate of the current is valid in a more general case including our one - band model ( [ s ] ) . this can be verified by analyzing the expression for the current in the forth order of the perturbation theory . equation ( [ res ] ) is our main result . its proportionality to @xmath50 reflects the fact that the ratchet effect vanishes in the absence of interactions . it is valid provided the backscattering current can be obtained within perturbation theory . this is the case for weak scattering , more specifically for @xmath51 within this limit , one can distinguish two regmies . * at lower voltages @xmath52 , @xmath53 and the effect is in agreement with the above estimate . additional oscillating factors reflect resonances due to quantum interferences @xcite . the absolute value of the ratchet current grows with decreasing voltage for @xmath54 . * in the high - voltage regime @xmath55 , @xmath56 . then @xmath57 . in this regime , the ratchet current grows with decreasing votalge for @xmath58 . this result shows that the ratchet effect can be increasing with decreasing voltage . at low voltages beyond the point where the condition ( [ valid ] ) breaks down and the total current vanishes with decreasing voltage , also the ratchet current has to vanish . nevertheless , it can give a sizeable contribution to the total current . this explains the pronounced asymmetry observed experimentally in corrugated nanotubes @xcite . although the hamiltonian ( [ h ] ) does not describe quantum hall systems , the bosonized form ( [ s ] ) captures tunneling of electron or quasiparticles between edges @xcite . in a fractional quantum hall state with filling factor @xmath59 , the tunneling of quasiparticles corresponds to the case @xmath60 . thus , the interesting regime with small @xmath61 is physically accessible . | we investigate a one - dimensional electron liquid with two point scatterers of different strength . in the presence of electron interactions ,
the nonlinear conductance is shown to depend on the current direction .
the resulting asymmetry of the transport characteristic gives rise to a ratchet effect , i.e. , the rectification of a dc current for an applied ac voltage . in the case of strong repulsive interactions , the ratchet current grows in a wide voltage interval with decreasing ac voltage . in the regime of weak
interaction the current - voltage curve exhibits oscillatory behavior .
our results apply to single - band quantum wires and to tunneling between quantum hall edges
. ratchets , luttinger liquids , conductance , current rectification , bosonization , impurity scattering , hartree approximation |
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@xmath0ne has been known as a typical example of a nucleus which has @xmath1 cluster structure . there have been numerous works based on the cluster model , which explain the observed doublet rotational band structure . in addition to the ground @xmath7 band , the negative parity band ( @xmath8 ) starting with the @xmath9 state at @xmath10 mev has been observed , and existence of this `` low - lying '' negative parity band is the strong evidence that simple spherical mean field is broken . these bands are well explained by the picture that @xmath1 cluster is located at some distance from the @xmath2o core @xcite . recently `` container picture '' has been proposed to describe the non - localization of the @xmath1 cluster around @xmath2o @xcite . however , according to the shell model , four nucleons perform independent particle motions around the @xmath2o core , which has doubly closed shell of the @xmath11 shell , and the spin - orbit interaction acts attractively to them . if we apply simple @xmath1 cluster models , we can not take into account this spin - orbit effect . in traditional @xmath1 cluster models , @xmath1 cluster is defined as @xmath12 configuration centered at some localized point , and the contributions of non - central interactions vanish . if we correctly take into account the spin - orbit effect , @xmath1 cluster structure competes with the @xmath3-coupling shell model structure . previously we have investigated this competition in @xmath0ne based on the antisymmetrized quasi - cluster model ( aqcm ) @xcite . aqcm is a method that enables us to describe a transition from the @xmath1 cluster wave function to the @xmath3-coupling shell model wave function @xcite . in this model , the cluster - shell transition is characterized by only two parameters ; @xmath4 representing the distance between @xmath1 cluster and core nucleus and @xmath5 describing the breaking of the @xmath1 cluster . by introducing @xmath5 , we transform @xmath1 cluster to quasi cluster , and the contribution of the spin - orbit interaction , very important in the @xmath3-coupling shell model , can be taken into account . it was found that the level structure of the yrast states of @xmath0ne strongly depends on the strength of the spin - orbit interaction in the hamiltonian . in this article we apply aqcm again to @xmath0ne and introduce @xmath2o plus one quasi cluster model . particularly we focus on the effect of cluster - shell competition on the @xmath6 transition . the @xmath6 transition operator has the form of monopole operator , @xmath13 , and this operator changes the nuclear sizes . however , changing nuclear density uniformly requires quite high excitation energy . on the other hand , clusters structures are characterized as weakly interacting states of strongly bound subsystems . thus it is rather easy for the cluster states to change the sizes without giving high excitation energies ; this is achieved just by changing the relative distances between clusters . therefore , @xmath6 transitions in low - energy regions are expected to be signatures of the cluster structures , and many works along this line are going on @xcite . in our preceding work for @xmath2o @xcite , we found that the ground state has a compact four @xmath1 structure and is almost independent of the strength of the spin - orbit interaction ; however the second @xmath14 state , which has been known as a @xmath15c+@xmath1 cluster state , is very much affected by the change of the strength . with increasing the strength , the level repulsion and crossing occur , and the @xmath16 cluster part changes from three @xmath1 configuration to the @xmath17 subclosure of the @xmath3-coupling shell model . the @xmath6 transition matrix elements are strongly dependent on this level repulsion and crossing , and they are sensitive to the persistence of @xmath18 correlation in the excited states . here , `` larger cluster '' part of binary cluster system ( @xmath15c part of @xmath15c+@xmath1 ) has been changed into quasi cluster . the present study on @xmath0ne is different from the preceding work on @xmath2o in the following two points . one is that we focus on the change of `` smaller cluster '' part of the binary cluster system , and in this case , we change @xmath1 cluster around the @xmath2o core to quasi cluster . another difference is that this change influences very much the ground state ( in the case of @xmath2o , the second @xmath14 state with the @xmath15c+@xmath1 configuration is affected by the spin - orbit interaction ) . since other higher nodal states are determined by the orthogonal condition to the ground state , this change also has influences on the wave functions of the excited states . naturally @xmath6 transition matrix elements are also affected by this change . the paper is organized as follows . the formulation is given in sect . [ model ] . in sect . [ results ] , the results for @xmath0ne are shown . finally , in sect . [ summary ] we summarize the results and give the main conclusion . the wave function of the total system @xmath19 is antisymmetrized product of these single particle wave functions ; @xmath20 the projection onto parity and angular momentum eigen states can be numerically performed . the number of mesh points for the integral over euler angles is @xmath21 . for the single particle orbits of the @xmath2o part , we introduce conventional @xmath1 cluster model . the single particle wave function has a gaussian shape @xcite ; @xmath22 \eta_{i } , \label{brink - wf}\ ] ] where @xmath23 represents the spin - isospin part of the wave function , and @xmath24 is a real parameter representing the center of a gaussian wave function for the @xmath25th particle . for the width parameter , we use the value of @xmath26 fm , @xmath27 . in this brink - bloch wave function , four nucleons in one @xmath1 cluster share the common @xmath24 value . hence , the contribution of the spin - orbit interaction vanishes . we introduce four different kinds of @xmath24 values , and four @xmath1 clusters are forming tetrahedron configuration . when we take the limit of the relative distances between @xmath1 cluster to zero , the wave function coincide with the closed @xmath11 shell configuration of the shell model @xcite , and this limit is called elliot su(3 ) limit @xcite . in our model , the relative distance is taken to be a small value , 0.1 fm . we add one quasi cluster around the @xmath2o core based on aqcm . in the aqcm , @xmath1 clusters are changed into quasi clusters . for nucleons in the quasi cluster , the single particle wave function is described by a gaussian wave packet , and the center of this packet @xmath28 is a complex parameter ; @xmath29 \chi_{i } \tau_{i } , \label{aqcm_sp}\ ] ] @xmath30 where @xmath31 and @xmath32 in eq . represent the intrinsic spin and isospin part of the @xmath25th single particle wave function , respectively . in eq , @xmath33 is a unit vector for the orientation of the intrinsic spin @xmath31 . here , @xmath5 is a real control parameter describing the dissolution of the @xmath1 cluster . the width parameter is the same as nucleons in the @xmath2o cluster ( @xmath26 fm , @xmath27 ) . as one can see immediately , the @xmath34 aqcm wave function , which has no imaginary part , is the same as the conventional brink - bloch wave function . the aqcm wave function corresponds to the @xmath3-coupling shell model wave function when @xmath35 and @xmath36 . the mathematical explanation is summarized in ref . @xcite . gaussian center parameters for the four nucleons in the quasi cluster ( @xmath37 ) are given in the following way . we place quasi cluster on the @xmath38 axis , and the real part of the gaussian center parameters ( @xmath39 ) are given as @xmath40 here @xmath4 is a parameter , which describes the distance between quasi cluster and the @xmath2o cluster , and @xmath41 is the unit vector in the @xmath38 direction . next , we give the imaginary parts . here we quantize the spin of the nucleons along the @xmath42 axis , and in order to satisfy the condition of eq . , we must give the imaginary parts in the @xmath43 direction as , @xmath44 @xmath45 @xmath46 @xmath47 where @xmath48 and @xmath49 are unit vectors in the @xmath38 and @xmath50 direction , respectively . the gaussian center parameter @xmath51 is for a proton with spin up ( @xmath52 direction ) , @xmath53 is for a proton with spin down ( @xmath54 direction ) , @xmath55 is for a neutron with spin up ( @xmath52 direction ) , and @xmath56 is for a neutron with spin down ( @xmath54 direction ) . when @xmath5 is set to zero , the wave function consisting the quasi clusters agrees with that of an @xmath1 cluster . if we take the limit of @xmath57 and @xmath58 , four nucleons in the quasi cluster occupy @xmath59 orbits of the @xmath3-coupling shell model . for the hamiltonian , we use volkov no.2 @xcite as an effective interaction for the central part with the majorana exchange parameter of @xmath60 . for the spin - orbit part , g3rs @xcite , which is a realistic interaction originally determined to reproduce the nucleon - nucleon scattering phase shift , is adopted ; @xmath61 where @xmath62 @xmath63 @xmath64 , and @xmath65 is a projection operator onto a triplet odd state . the operator @xmath66 stands for the relative angular momentum and @xmath67 is the spin ( @xmath68 ) . in the present work , the strength of the spin - orbit interaction , @xmath69 , is a parameter as in ref . @xcite and we compare the results by changing the value . in this section , we apply our aqcm wave function introduced in the previous section to @xmath0ne and discuss the @xmath69 ( strength of the spin - orbit interaction ) dependence of energy levels and @xmath6 transition probabilities . the @xmath69 value is changed from 0 mev to 3000 mev , and reasonable value of around @xmath70 mev has been suggested in our preceding work @xcite . we prepare aqcm wave functions with different @xmath4 and @xmath5 values as basis states of generator coordinate method ( gcm ) . the adopted values are @xmath71 fm and @xmath72 . the @xmath14 energies of these basis states are presented in table [ basis ] , and here , we show the values for two extreme cases for the strengths of the spin - orbit interaction ; ( a ) @xmath73 mev and ( b ) @xmath74 mev . the @xmath14 energies of the gcm basis states corresponding to other @xmath69 values can be estimated just by interpolating these values linearly . in table [ basis ] ( a ) , we find that @xmath34 basis states give lower energies than @xmath5 finite basis states . this is because of the absence of the spin - orbit interaction ; introducing imaginary part for the gaussian center parameters does not work for the spin - orbit interaction and that simply increases the kinetic energy of four nucleons in the quasi cluster . here the basis state with @xmath75 fm ( @xmath34 ) gives the lowest energy of @xmath76 mev . on the contrary , table [ basis ] ( b ) is the case of @xmath74 mev , and basis states with finite @xmath5 values get much lower , since the contribution of the spin - orbit interaction can be taken into account by transforming the @xmath1 cluster to quasi cluster . the basis state which gives the lowest energy has the values of @xmath77 and @xmath78 fm ( @xmath79 mev ) . this result suggests that when the spin - orbit interaction is switched on , the @xmath4 value of the optimal basis state becomes smaller and the @xmath5 value increases . this means that not only the @xmath1 cluster dissolutes into quasi cluster , the relative distance between the cluster and the @xmath2o core decreases . ( a ) .the @xmath14 energies of gcm basis states for the cases of different strengths of the spin - orbit interaction ; ( a ) @xmath73 mev and ( b ) @xmath74 mev . the @xmath14 energies of the gcm basis states with other @xmath69 values can be estimated by linearly interpolating these two . [ cols="^,^,^,^,^",options="header " , ] [ sqol-1- ] we have applied aqcm , which is a method to describe a transition from the @xmath1-cluster wave function to the @xmath3-coupling shell model wave function , to @xmath0ne . @xmath0ne has been known as a nucleus which has @xmath2o+@xmath1 structure , and we investigated how the @xmath1 cluster structure competes with independent particle motions of these four nucleons by changing the strength of the spin - orbit interaction ( @xmath69 ) . we focused on the @xmath6 transition matrix element , which was found to be sensitive to @xmath69 . based on aqcm , @xmath0ne is characterized by only two parameters ; @xmath4 representing the relative distance between @xmath2o and @xmath1 and @xmath5 describing the breaking of @xmath1 cluster . when the spin - orbit interaction is switched off ( @xmath73 mev ) , the ground @xmath14 state has the squared overlap of 0.92 with the gcm basis state which has @xmath75 fm and @xmath34 . the second @xmath14 state is a higher nodal state and it has 0.74 with @xmath80 fm and @xmath81 . the third @xmath14 state also has large @xmath2o-@xmath1 distance , and the fourth @xmath14 state has overlaps with basis states with finite @xmath5 values . when the spin - orbit interaction is switched on , we found that the decrease of the energy for the fourth @xmath14 state at @xmath73 is much steeper than other states . eventually the wave function of the fourth @xmath14 state at @xmath73 mev strongly mixes in the ground and second @xmath14 states at @xmath74 mev . on the other hand , the wave function of the second @xmath14 state at @xmath73 mev almost stays at this energy . similar thing can be found for the third @xmath14 state at @xmath73 mev . these two states correspond to the third and fourth @xmath14 states at @xmath74 mev , and @xmath1 cluster structure becomes important again there . the @xmath6 transition matrix elements from the ground state to the second @xmath14 state is calculated as 10.0 @xmath82 @xmath83 at @xmath73 mev , which is slightly larger than the experimental value ( 6.914 @xmath82 @xmath83 ) . with increasing @xmath69 value , the mixing of basis states with finite @xmath5 values becomes important in both the ground and second @xmath14 states , and the @xmath6 transition matrix decreases . the value agrees with the experimental one around @xmath84 mev . this deduced strength is consistent with our preceding work on the level structure of this nucleus . at @xmath84 mev , which is the spin - orbit strength deduced from the present analysis on the @xmath6 transition matrix element , the ground state has the squared overlap of 0.78 with the basis state which has @xmath75 fm and @xmath81 , and the character at @xmath73 mev still remains . however the largest squared overlap of 0.82 is with the basis state which has @xmath85 fm and @xmath86 . therefore , the @xmath1 breaking effect due to the spin - orbit interaction is also important in the ground state . the second @xmath14 states has squared overlap of 0.40 with the basis state which has @xmath87 fm and @xmath34 ; however , it has also components of the basis states with finite @xmath5 values . the third and fourth @xmath14 states are @xmath1 cluster states and contain the components of basis states with large @xmath4 values . the presence of low - lying negative parity band starting with the first @xmath9 has been the key evidence for the @xmath1 cluster structure . we also investigated the @xmath9 states and found that the energy of the first @xmath9 state is almost constant even if the spin - orbit interaction is switched on and @xmath1 breaking basis states are introduced . the @xmath1 cluster structure is really important in this state . there have been discussions that the @xmath15c+@xmath1+@xmath1 cluster states appear in this energy region of the third @xmath14 state , and inclusion of this configuration can be done by applying aqcm to the three @xmath1 clusters in the @xmath2o core . also , here we transformed an @xmath1 cluster to four independent nucleons , in which the spin - orbit interaction acts attractively . however , in principle it is possible to introduce other shell model configurations , for instance configurations where the spin - orbit interaction acts repulsively , or one of the nucleon is excited from @xmath88-upper orbit to @xmath88-lower orbit . the analysis aiming at the unified view is going on . 100 hisashi horiuchi and kiyomi ikeda , prog . ( 1968 ) * 40 * 277 . bo zhou , zhongzhou ren , chang xu , y. funaki , t. yamada , a. tohsaki , h. horiuchi , p. schuck , and g. r pke , phys . c * 86 * , 014301 ( 2012 ) . n. itagaki , j. cseh , and m. poszajczak , phys . c * 83 * , 014302 ( 2011 ) . n. itagaki , h. masui , m. ito , and s. aoyama , phys . rev . c * 71 * 064307 ( 2005 ) . h. masui and n. itagaki , phys . c * 75 * 054309 ( 2007 ) . t. yoshida , n. itagaki , and t. otsuka , phys . c * 79 * 034308 ( 2009 ) . t. suhara , n. itagaki , j. cseh , and m. poszajczak , phys . c * 87 * , 054334 ( 2013 ) . n. itagaki , h. matsuno , and t. suhara , arxiv : 1507.02400 . t. kawabata _ et al_. , phys . b * 646 * , 6 ( 2007 ) . y. sasamoto _ et al_. , mod . a * 21 * , 2393 ( 2006 ) . t. yoshida , n. itagaki , and t. otsuka , phys . c * 79 * , 034308 ( 2009 ) . taiichi yamada and yasuro funaki , phys . c * 92 * , 034326 ( 2015 ) . t. ichikawa , n. itagaki , t. kawabata , tz . kokalova and w. von oertzen , phys . c * 83 * , 061301(r ) ( 2011 ) . t. ichikawa , n. itagaki , y. kanada - enyo , tz . kokalova , and w. von oertzen , phys . c * 86 * , 031303(r ) ( 2012 ) . h. matsuno and n. itagaki , arxiv : 1601.05892 . d. m. brink , in _ proceedings of the international school of physics `` enrico fermi '' course xxxvi _ , edited by c. bloch ( academic , new york , 1966 ) , p. 247 . j. p. elliot , proc . a * 245 * 128 , 562 ( 1958 ) a. b. volkov , nucl . phys . * 74 * , 33 ( 1965 ) . r. tamagaki , prog . theor . phys . * 39 * , 91 ( 1968 ) . | @xmath0ne has been known as a typical example of a nucleus with @xmath1 cluster structure ( @xmath2o+@xmath1 structure ) . however according to the spherical shell model , the spin - orbit interaction acts attractively for four nucleons outside of the @xmath2o core , and this spin - orbit effect can not be taken into account in the simple @xmath1 cluster models .
we investigate how the @xmath1 cluster structure competes with independent particle motions of these four nucleons . the antisymmetrized quasi - cluster model ( aqcm )
is a method to describe a transition from the @xmath1 cluster wave function to the @xmath3-coupling shell model wave function . in this model ,
the cluster - shell transition is characterized by only two parameters ; @xmath4 representing the distance between clusters and @xmath5 describing the breaking of @xmath1 clusters , and the contribution of the spin - orbit interaction , very important in the @xmath3-coupling shell model , can be taken into account by changing @xmath1 clusters to quasi clusters . in this article ,
based on aqcm , we apply @xmath2o plus one quasi cluster model for @xmath0ne . here
we focus on the @xmath6 transition matrix element , which has been known as the quantity characterizing the cluster structure .
the @xmath6 transition matrix elements are sensitive to the change of the wave functions from @xmath1 cluster to @xmath3-coupling shell model . |
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we will treat a class of models of ` dense ' random graphs , that is , simple graphs on @xmath0 vertices in which the average number of edges is of order @xmath1 . more specifically we will consider exponential random graphs in which dependence between the random edges is defined through some finite graph , in imitation of the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics . exponential random graphs have been widely studied ( see @xcite for a range of recent work ) since the pioneering work on the independent case by erds and rnyi @xcite . we will concentrate on the phenomenon of phase transitions which can emerge for dependent variables . following analyses using mean - field and other uncontrolled approximations ( see @xcite ) there has recently been important progress by chatterjee and diaconis @xcite , including the first rigorous proof of singular dependence on parameters . we will extend their result both in the class of models and parameter values under control and provide an appropriate formalism of phase structure for such models . we consider the class of models in which the probability of the simple graph @xmath2 on @xmath0 vertices is given by : @xmath3},\ ] ] where : @xmath4 is an edge , @xmath5 is any finite simple graph with @xmath6 edges , @xmath7 is the normalization constant , @xmath8 is the density of graph homomorphisms @xmath9 : @xmath10 and @xmath11 denotes the vertex set . expectation of a real function of a random graph is denoted @xmath12 . our main results are the following . .1truein [ one ] restrict to @xmath13 . then the pointwise limit @xmath14 exists and is analytic in @xmath15 and @xmath16 off a certain curve @xmath17 which includes the end point @xmath18 the derivatives @xmath19 and @xmath20 are both discontinuous across the curve , except at @xmath21 where , however , all the second derivatives @xmath22 , @xmath23 , and @xmath24 diverge . .1truein [ two ] if in theorem [ one ] the graph @xmath5 is a @xmath25-star , @xmath26 , the analogous result also holds for @xmath27 . a @xmath25-star has @xmath25 edges meeting at a vertex . .1truein [ corr ] the parameter space @xmath28 of each of the models consists of a single phase with a first order phase transition across the indicated curve and a second order phase transition at the critical point @xmath21 . to explain the language of phase transitions in corollary [ corr ] we first give a superficial introduction to the formalism of classical statistical mechanics within @xmath29-dimensional lattice gas models ; for more details see for instance @xcite . assume each point in a @xmath29-dimensional cube @xmath30 is randomly occupied ( by one particle ) or not occupied , and assume there is a ( many - body ) potential energy of value @xmath31 associated with every occupied subset of @xmath32 congruent to a certain @xmath33 . the interaction is attractive if @xmath34 and repulsive if @xmath35 . we define the probability that the occupied sites in @xmath32 are precisely @xmath36 by @xmath37 } } { { \mathbb z}_n},\ ] ] where the parameter @xmath38 is called the inverse temperature , the parameter @xmath39 is called the chemical potential , the normalization constant @xmath40 is called the partition function , @xmath4 and @xmath5 are subsets of @xmath32 with @xmath4 a singleton and the cardinality @xmath41 , and @xmath42 is the number of copies of @xmath43 in @xmath36 . ( we are using ` free ' boundary conditions . ) one of the basic features of the formalism is that the free energy density , @xmath44/n^d$ ] , contains all ways to interact with or influence the system , so that ` all ' physically significant quantities can be obtained by differentiating it with respect to @xmath45 and @xmath46 . for instance @xmath47 the ( average ) particle density . to model materials in thermal equilibrium , calculations in this formalism normally require that the system size be sufficiently large , and in practice one often resorts to using @xmath48 . with this as motivation we tentatively define a ` phase ' as a set of states ( i.e. probability distributions ) corresponding to a connected region of the @xmath49 parameter space , which is maximal for the condition that @xmath50 are analytic in @xmath45 and @xmath46 for all @xmath51 . intuitively one associates a ` phase transition ' with singularities which develop in some of these quantities as the system size diverges . an important simplification was proven by yang and lee @xcite who showed that the limiting free energy density @xmath52 always exists and that certain limits commute : @xmath53 this implies that phases and phase transitions can be determined from the limiting free energy density , and so a phase is commonly defined ( see for instance @xcite ) as a connected region of the @xmath54 parameter space maximal for the condition that @xmath55 is analytic . using the obvious analogues for random graphs , with @xmath15 playing the role of @xmath56 and @xmath16 the role of @xmath57 ( and therefore positive if and only if the model is ` attractive ' ) , @xmath7 plays the key role of free energy density . we will show in theorems [ a ] and [ b ] below that the limiting free energy density @xmath58 exists , and the proof of theorem 2 by yang and lee @xcite , on the commutation of limits , then goes through without any difficulty in this setting , so we can again define phases and phase transitions through the limiting free energy density , as follows . a phase is a connected region , of the parameter space @xmath28 , maximal for the condition that the limiting free energy density , @xmath58 , is analytic . there is a @xmath59-order transition at a boundary point of a phase if at least one @xmath59-order partial derivative of @xmath60 is discontinuous there , while all lower order derivatives are continuous . theorems [ one ] and [ two ] thereby justify our interpretation in corollary [ corr ] that each of our models consists of a single phase with a first order phase transition across the indicated curve except at the end ( or ` critical ' ) point @xmath61 , where the transition is second order , superficially similar to the transition between liquid and gas in equilibrium materials . chatterjee and diaconis have proven that the main object of interest , @xmath62 , exists for all @xmath15 and nonnegative @xmath16 and is the solution of a certain optimization problem : [ cd ] fix one of our models and assume @xmath5 has @xmath26 edges . then for any @xmath63 and @xmath64 , the pointwise limit @xmath65 exists : @xmath66 } \big(\beta_1u+\beta_2u^p-\frac{1}{2}u\log u-\frac{1}{2}(1-u)\log(1-u)\big).\ ] ] the following detailed analysis of this maximization problem is therefore fundamental to understanding the phase structure of our models , and we now address it . [ max ] fix an integer @xmath67 . consider the maximization problem for @xmath68 on the interval @xmath69 $ ] , where @xmath70 and @xmath71 are parameters . then there is a v - shaped region in the @xmath72 plane with corner point @xmath21 such that outside this region , @xmath73 has a unique local maximizer ( hence global maximizer ) @xmath74 ; whereas inside this region , @xmath73 always has exactly two local maximizers @xmath75 and @xmath76 . moreover , for every @xmath15 inside this v - shaped region ( @xmath77 ) , there is a unique @xmath17 such that the two local maximizers of @xmath78 are both global maximizers . furthermore @xmath79 is a continuous and decreasing function of @xmath15 . plane . graph drawn for @xmath80.,width=384 ] by the lebesgue differentiation theorem , @xmath79 being monotone guarantees that it is differentiable almost everywhere . the location of maximizers of @xmath73 on the interval @xmath69 $ ] are closely related to properties of its derivatives @xmath81 and @xmath82 : @xmath83 @xmath84 we first analyze properties of @xmath82 on the interval @xmath69 $ ] . consider instead the function @xmath85 simple optimization shows @xmath86 and the equality holds if and only if @xmath87 . thus @xmath88 the graph of @xmath89 is concave up with two ends both growing unbounded , and the global minimum is achieved at @xmath87 . this implies that for @xmath90 , @xmath91 on the whole interval @xmath69 $ ] ; whereas for @xmath92 , @xmath82 will take on both positive and negative values , and we denote the transition points by @xmath93 and @xmath94 ( @xmath95 ) . based on properties of @xmath82 , we next analyze properties of @xmath81 on the interval @xmath69 $ ] . for @xmath90 , @xmath81 is monotonically decreasing . for @xmath92 , @xmath81 is decreasing from @xmath96 to @xmath93 , increasing from @xmath93 to @xmath94 , then decreasing again from @xmath94 to @xmath97 . based on properties of @xmath81 and @xmath82 , we analyze properties of @xmath73 on the interval @xmath69 $ ] . independent of the choice of parameters @xmath15 and @xmath16 , @xmath73 is a bounded continuous function , @xmath98 , and @xmath99 , so @xmath73 can not be maximized at @xmath96 or @xmath97 . for @xmath90 , @xmath81 crosses the @xmath100-axis only once , going from positive to negative . thus @xmath73 has a unique local maximizer ( ( hence global maximizer ) @xmath74 . for @xmath101 , the situation is more complicated and deserves a careful analysis . if @xmath102 ( resp . @xmath103 ) , @xmath73 has a unique local maximizer ( hence global maximizer ) at a point @xmath104 ( resp . @xmath105 ) . if @xmath106 , then @xmath73 has two local maximizers @xmath75 and @xmath76 , with @xmath107 . notice that @xmath93 and @xmath94 are solely determined by the choice of parameter @xmath92 , and vice versa . by ( [ beta2 ] ) , @xmath108 @xmath109 consider the function @xmath110 it is not hard to see that @xmath111 , @xmath112 , @xmath113 is decreasing from @xmath96 to @xmath114 , then increasing from @xmath114 to @xmath97 , and the global minimum value is @xmath115 this implies in particular that @xmath102 for @xmath116 . the only possible region in the @xmath72 plane where @xmath106 is thus bounded by @xmath117 and @xmath92 . we now analyze the behavior of @xmath118 and @xmath119 more closely when @xmath15 and @xmath16 are chosen from this region . recall that by construction , @xmath95 . by monotonicity of @xmath113 on the intervals @xmath120 and @xmath121 , there exist continuous functions @xmath122 and @xmath123 of @xmath15 , such that @xmath124 for @xmath125 and @xmath126 for @xmath127 . @xmath122 is an increasing function of @xmath15 , whereas @xmath123 is a decreasing function , and they satisfy @xmath128 also , as @xmath129 , @xmath130 and @xmath131 . by ( [ beta2 ] ) , the restrictions on @xmath93 and @xmath94 yield restrictions on @xmath16 . we have @xmath124 for @xmath132 and @xmath126 for @xmath133 . notice that @xmath134 and @xmath135 are both decreasing functions of @xmath15 , and as @xmath129 , they both grow unbounded . by construction , for every parameter value @xmath72 , @xmath136 . also , for fixed @xmath15 , @xmath134 is the value of @xmath16 for which @xmath137 , and @xmath135 is the value for which @xmath138 . thus the curve @xmath135 must lie below the curve @xmath134 . and together they generate the bounding curves of the v - shaped region in the @xmath72 plane where two local maximizers exist for @xmath73 . it is not hard to see that the corner point is given by @xmath139 . ( see figures 16 . ) fixing an arbitrary @xmath77 , we examine the effect of varying @xmath16 on the graph of @xmath81 . it is clear from ( [ l ] ) that @xmath81 shifts upward as @xmath16 increases . as a result , as @xmath16 gets large , the positive area bounded by the curve @xmath81 increases , whereas the negative area decreases . by the fundamental theorem of calculus , the difference between the positive and negative areas is the difference between @xmath140 and @xmath141 , which goes from negative ( @xmath138 , @xmath75 is the global maximizer ) to positive ( @xmath137 , @xmath76 is the global maximizer ) as @xmath16 goes from @xmath135 to @xmath134 . thus there must be a unique @xmath142 such that @xmath75 and @xmath76 are both global maximizers . we denote this @xmath16 by @xmath143 ; see figures [ v - shape ] and [ qcurve ] . by analyzing the graph of @xmath81 , we see that the parameter values of @xmath144 are exactly the ones for which positive and negative areas bounded by @xmath81 equal each other . an increase in @xmath15 will induce an upward shift of @xmath81 , which must be balanced by a decrease in @xmath17 . similarly , a decrease in @xmath15 will induce a downward shift of @xmath81 , which must be balanced by an increase in @xmath17 . this justifies that @xmath79 is monotonically decreasing in @xmath15 . furthermore , the continuity of @xmath81 as a function of @xmath15 and @xmath16 implies the continuity of @xmath79 as a function of @xmath15 . the transition curve @xmath17 displays a universal asymptotic behavior as @xmath145 : @xmath146 by proposition [ max ] , it suffices to show that as @xmath147 , @xmath148 has two global maximizers @xmath75 and @xmath76 . this is easy when we realize that as @xmath149 , @xmath150 for every @xmath100 in @xmath151 . the limiting maximizers on @xmath152 $ ] are thus @xmath153 and @xmath154 , with @xmath155 . so far we have used results from @xcite but have avoided specific reference to the framework of graph limits , developed by lovsz et al , which was used to prove those results . we now need to refer directly to graph limits ; for the notation and an introduction to this material see for instance @xcite or @xcite . [ gen ] let @xmath2 be a random graph on @xmath0 vertices in one of our models . for parameter values of @xmath72 in the upper half - plane @xmath156 , the behavior of @xmath2 in the large @xmath0 limit is as follows : @xmath157 where @xmath158 is the set of maximizers of ( [ l ] ) . the assumptions of theorems 4.2 and 6.1 in @xcite are satisfied for parameter values @xmath156 . by proposition [ max ] , along the curve @xmath144 , the maximization problem ( [ l ] ) is solved at two values @xmath75 and @xmath76 ; whereas off this curve , it is solved at a unique value @xmath74 . thus in the large @xmath0 limit , along the curve @xmath144 , @xmath2 behaves like an erds - rnyi graph @xmath159 ( @xmath100 picked randomly from @xmath75 and @xmath76 ) ; whereas off this curve , @xmath2 is indistinguishable from the erds - rnyi graph @xmath160 . [ q ] fix any @xmath161 . let @xmath162 be an edge , so @xmath8 is the edge density of @xmath2 . then there exists a continuous and decreasing function @xmath163 such that @xmath164 and @xmath165 here @xmath93 and @xmath94 are defined as in the proof of proposition [ max ] : @xmath166 . as @xmath167 , @xmath168 and @xmath169 and the jump is noticeable even for relatively small values of @xmath16 . as @xmath143 is a continuous and decreasing function of @xmath15 , the inverse function @xmath163 exists and is also continuous and decreasing . we examine the effect of varying @xmath15 on the graph of @xmath81 ( and hence on the global maximizers of @xmath73 ) . first note that varying @xmath15 does not change the shape of @xmath81 . inside the v - shaped region , there are three cases . recall that @xmath107 . for @xmath170 , positive and negative areas bounded by @xmath81 equal each other , thus @xmath75 and @xmath76 are both global maximizers . for @xmath171 , the graph of @xmath81 shifts downward , negative area exceeds positive area , thus @xmath75 is the global maximizer . for @xmath172 , the graph of @xmath81 shifts upward , positive area exceeds negative area , thus @xmath76 is the global maximizer . outside the v - shaped region , there are two cases . below the lower bounding curve , @xmath81 has a unique local maximizer @xmath105 . above the upper bounding curve , @xmath81 has a unique local maximizer @xmath104 . our conclusion then follows from theorem [ gen ] . [ p ] assume that in one of our models @xmath5 is a @xmath25-star ( @xmath67 ) . for all parameter values @xmath72 , the behavior of @xmath2 in the large @xmath0 limit is as follows : @xmath157 where @xmath158 is the set of maximizers of ( [ l ] ) . this follows from related results in @xcite . we separate the parameter plane @xmath173 into upper and lower half - planes . the upper half - plane ( @xmath174 ) satisfies the assumptions of theorem 4.2 , and the lower half - plane ( @xmath175 ) satisfies the assumptions of theorem 6.4 . by similar reasoning as in theorem [ gen ] , the rest of the proof follows . .1truein real and complex analyticity are both defined in terms of convergent power series . any problem in the real analytic category may be complexified and thereby turned into a complex analytic one , and any complex analytic situation with real coefficients is obviously real analytic and can thus be treated with real analytic techniques . the following analytic implicit function theorem may be interpreted in either the real or complex setting . [ kp ] suppose that the power series @xmath176 is absolutely convergent for @xmath177 , @xmath178 . if @xmath179 and @xmath180 , then there exist @xmath181 and a power series @xmath182 such that ( [ f ] ) is absolutely convergent for @xmath183 and @xmath184 . [ ana1 ] off the end point @xmath21 , the local maximizer @xmath74 for @xmath185 ( @xmath75 and @xmath76 if inside the v - shaped region ) is an analytic function of the parameters @xmath15 and @xmath16 . a local maximizer @xmath74 for @xmath73 is a zero for @xmath81 with the additional property that @xmath81 would change sign from positive to negative across @xmath186 . fix a choice of parameters @xmath187 . set @xmath188 and @xmath189 . the function @xmath190 is thus transformed into a function @xmath191 . it is clear that @xmath190 is analytic for @xmath192 , @xmath193 , and @xmath194 . recall that @xmath73 bounded continuous , @xmath98 , and @xmath99 implies that @xmath74 can not be @xmath96 or @xmath97 . it follows that the transformed function @xmath191 has the desired domain of analyticity , and is locally absolutely convergent . as for the coefficients , @xmath195 by construction , and @xmath196 as can be seen from the proof of proposition [ max ] . in more detail , for @xmath197 , @xmath82 is positive for all @xmath100 in @xmath198 . and for @xmath199 , @xmath200 only for @xmath87 , which coincides with @xmath74 only for @xmath201 . lastly , for @xmath202 , @xmath200 only for @xmath203 and @xmath204 . three possible situations might occur . outside the v - shaped region , @xmath73 has a unique local maximizer @xmath74 , with @xmath104 if above or along the upper bounding curve , and @xmath105 if below or along the lower bounding curve . inside the v - shaped region , @xmath73 has two local maximizers @xmath75 and @xmath76 , with @xmath107 . all the conditions of theorem [ kp ] are satisfied , thus @xmath205 converges for @xmath72 close to @xmath206 . [ ana2 ] off the phase transition curve , @xmath207 ( @xmath141 or @xmath140 if inside the v - shaped region ) is an analytic function of the parameters @xmath15 and @xmath16 . it is clear that @xmath185 is analytic for @xmath208 , @xmath193 , and @xmath209 . outside the v - shaped region , @xmath73 has a unique local maximizer @xmath74 in @xmath198 , which is analytic in @xmath15 and @xmath16 by proposition [ ana1 ] . inside the v - shaped region , @xmath73 has two local maximizers @xmath75 and @xmath76 , both have values in @xmath198 and are analytic in @xmath15 and @xmath16 by proposition [ ana1 ] . below the phase transition curve , @xmath210 is given by @xmath141 , which coincides with @xmath211 along the lower bounding curve . above the phase transition curve , @xmath210 is given by @xmath140 , which coincides with @xmath211 along the upper bounding curve . our claim follows by realizing that compositions of analytic functions are analytic as long as the domains and ranges match up . [ a ] let @xmath2 be a random graph on @xmath0 vertices in one of our models . the limiting free energy density @xmath212 is an analytic function of the parameters @xmath15 and @xmath16 off the phase transition curve in the upper half - plane @xmath156 . the assumptions of theorems 4.2 and 6.1 in @xcite are satisfied for parameter values @xmath156 . our claim then follows from proposition [ ana2 ] . [ b ] assume that in one of our models @xmath5 is a @xmath25-star ( @xmath67 ) . the limiting free energy density @xmath212 is an analytic function of the parameters @xmath15 and @xmath16 off the phase transition curve . the assumptions of theorems 4.2 and 6.4 in @xcite are satisfied . our claim again follows from proposition [ ana2 ] . [ ls ] let @xmath213 ^ 2\rightarrow [ 0,1]$ ] be two symmetric integrable functions . then for every finite simple graph @xmath214 , @xmath215 .1truein _ proof of theorems [ one ] and [ two ] _ the stated analyticity is proven in theorems [ a ] and [ b ] , so we only need to examine the situation along the phase transition curve . we know from theorems [ gen ] and [ p ] that @xmath216 converges in probability to @xmath74 , off the curve . by lemma [ ls ] , @xmath217 then converges in probability to @xmath218 . as @xmath217 is uniformly bounded in @xmath0 , this implies that @xmath219 therefore @xmath220 similarly , @xmath221 by corollary [ q ] , these two first derivatives @xmath222 and @xmath223 are discontinuous across the curve ( except at the end point ) . let us now take a closer look at the behavior of @xmath224 at the critical point . recall that @xmath225 is monotonically decreasing on @xmath69 $ ] , and the unique zero is achieved at @xmath114 . take any @xmath226 . set @xmath227 . consider @xmath72 so close to @xmath21 such that @xmath228 . for every @xmath100 in @xmath69 $ ] , we then have @xmath229 . it follows that the zeros @xmath230 ( @xmath75 and @xmath76 if inside the v - shaped region ) must satisfy @xmath231 , which easily implies the continuity of @xmath232 and @xmath233 at @xmath21 . to see that the transition at the critical point is second - order , we check the second derivatives of @xmath234 in its neighborhood . off the phase transition curve , @xmath235 @xmath236 @xmath237 but as was explained in proposition [ max ] , @xmath238 converges to zero as @xmath72 approaches @xmath239 ; the desired singularity is thus justified . much of the literature on phase transitions in exponential random graph models uses techniques , such as mean - field approximations , which are mathematically uncontrolled . as such they have been useful in discovering interesting behavior but they can be misleading in detail . for instance , although phase transitions have been discovered in this way for the two - star ( @xmath5 a two - star ) @xcite and edge - triangle ( @xmath5 a triangle ) models @xcite , the approximation leads to an error in the qualitative nature of the transition , attributing phase coexistence to the full v - shaped region of figure [ v - shape ] rather than just the curve @xmath240 ; in other words it does not distinguish the local maxima in the region from the global maxima . chatterjee and diaconis @xcite gave the first rigorous proof of singular behavior in an exponential random graph model , the edge - triangle model . our paper is an extension of this important first step ; besides extending the models and parameters under control we have provided a mathematical framework of ` phases ' which we hope will be useful in motivating future mathematical work in this subject . our results show that all models with ` attraction ' ( @xmath241 ) , and also @xmath25-star models with ` repulsion ' ( @xmath242 ) exhibit a transition qualitatively like the gas / liquid transition : a first order transition corresponding to a discontinuity in density , with a second order criticial point . in theorem 7.1 of @xcite chatterjee and diaconis suggest that , quite generally , models with repulsion exhibit a transition qualitatively like the solid / fluid transition , in which one phase has nontrivial structure , as distinguished from the ` disordered ' erds - rnyi graphs , which have independent edges . we have not yet been able to extend our results to this regime , which is an important open problem . it is a pleasure to acknowledge useful discussions with persi diaconis and sourav chatterjee introducing us to the subject of random graphs and giving us early access to , and explanations of , their preprint @xcite , as well as suggestions for improving early drafts of this paper . cr also acknowledges support under nsf grant dms-0700120 . | we derive the full phase diagram for a large family of exponential random graph models , each containing a first order transition curve ending in a critical point . .4truein |
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constant width bodies , i.e. , convex bodies for which parallel supporting hyperplanes have constant distance , have a long and rich history in mathematics @xcite . due to meissner @xcite , constant width bodies in euclidean space can be characterized by _ diametrical completeness _ , that is , the property of not being properly contained in a set of the same diameter . constant width bodies also belong to a related class of _ reduced _ convex bodies introduced by heil @xcite . this means that constant width bodies do not properly contain a convex body of same minimum width . remarkably , the classes of reduced bodies and constant width bodies do not coincide , as a regular triangle in the euclidean plane shows . reduced bodies are extremal in remarkable inequalities for prescribed minimum width , as in steinhagen s inequality @xcite ( minimum inradius ) , or others that surprisingly still remain unsolved , namely , pl s problem @xcite ( minimum volume ) . while the regular simplex ( and any of its reduced subsets ) is extremal for steinhagen s , it is extremal only in the planar case for pl s problem . the reason is that while the regular triangle is reduced , this is no longer the case for the regular simplex in @xmath2 , @xmath1 . indeed , heil conjectured @xcite that a certain reduced subset of the regular simplex is extremal for pl s problem . heil also observed that some reduced body has to be extreme for pl s problem when replacing volume by quermassintegral . the existence of reduced polytopes , and the fact that smooth reduced sets are of constant width ( cf . @xcite ) , opens the door to conjecture some of them as minimizers . in full generality , any non - decreasing - inclusion functional of convex bodies with prescribed minimum width , attains its minimum at some reduced body . pl s problem restricted to constant width sets is the well - known blaschke - lebesgue problem , cf . @xcite , solved only in the planar case , where the reuleaux triangle is the minimizer of the area , and meissner s bodies are conjectured to be extremal in the three - dimensional space , see @xcite for an extended discussion . note that pl s problem has also been investigated in other geometrical settings such as minkowskian planes @xcite or spherical geometry , cf . @xcite and @xcite . reduced bodies in the euclidean space have been extensively studied in @xcite , and the concept of reducedness has been translated to finite - dimensional normed spaces @xcite . in reference to the existence of reduced polygons in the euclidean plane , lassak @xcite posed the question whether there exist reduced polytopes in euclidean @xmath0-space for @xmath1 . several authors addressed the search for reduced polytopes in finite - dimensional normed spaces @xcite . for euclidean space starting from dimension @xmath3 several classes of polytopes such as * polytopes in @xmath2 with @xmath4 vertices , @xmath4 facets , or more vertices than facets ( * ? ? ? * corollary 7 ) , * centrally symmetric polytopes ( * ? ? ? * claim 2 ) , * simple polytopes , i.e. , polytopes in @xmath2 where each vertex is incident to @xmath0 edges ( like polytopal prisms , for instance ) ( * ? ? ? * corollary 8) , * pyramids with polytopal base ( * ? ? ? * theorem 1 ) , and in particular simplices @xcite , * polytopes in @xmath5 which have a vertex @xmath6 with a strictly antipodal facet @xmath7 ( see ) such that the edges and facets incident to @xmath6 are strictly antipodal to the edges and vertices of @xmath7 , respectively , see ( * ? ? ? * theorem 2 ) , were proved to be _ not _ reduced . the theoretical results on reduced polytopes in @xmath5 in the mentioned preprint @xcite by polyanskii are accompanied with an unfortunately erroneous example , as we will show in . the purpose of the present article is to fix polyanskii s polytope and to present a reduced polytope in three - dimensional euclidean space in . the validity of our example can be checked using the algorithm provided in . throughout this paper , we work in @xmath0-dimensional euclidean space , that is , the vector space @xmath2 equipped with the inner product @xmath8 and the norm @xmath9 , where @xmath10 and @xmath11 denote two points in @xmath2 . a subset @xmath12 is said to be _ convex _ if the line segment @xmath13{\mathrel{\mathop:}=}{\left\{\lambda x+(1-\lambda)y \::\ : 0\leq\lambda\leq 1\right\}}\ ] ] is contained in @xmath14 for all choices of @xmath15 . convex compact subsets of @xmath2 having non - empty interior are called _ convex bodies_. the smallest convex superset of @xmath12 is called its _ convex hull _ @xmath16 , whereas the smallest affine subspace of @xmath2 containing @xmath14 is denoted by @xmath17 , the _ affine hull _ of @xmath14 . the _ affine dimension _ @xmath18 of @xmath14 is the dimension of its affine hull . the _ support function _ @xmath19 of @xmath14 is defined by @xmath20 for @xmath21 , the hyperplane @xmath22 is a _ supporting hyperplane _ of @xmath14 . the _ width _ of @xmath14 in direction @xmath23 , defined by @xmath24 equals the distance of the supporting hyperplanes @xmath25 multiplied by @xmath26 . the _ minimum width _ of @xmath14 is @xmath27 . a _ polytope _ is the convex hull of finitely many points . the boundary of a polytope consists of _ faces _ , i.e. , intersections of the polytope with its supporting hyperplanes . we shall refer to faces of affine dimension @xmath28 , @xmath29 , and @xmath30 as _ vertices _ , _ edges _ , and _ facets _ , respectively . faces of polytopes are lower - dimensional polytopes and shall be denoted by the list of their vertices . ( a face which is denoted in this way can be reconstructed by taking the convex hull of its vertices . ) by definition , attainment of the minimal width of a polytope @xmath31 is related to a binary relation on faces of @xmath31 called _ strict antipodality _ , see @xcite . [ def : antipodal ] let @xmath32 be a polytope . distinct faces @xmath33 , @xmath34 of @xmath31 are said to be _ strictly antipodal _ if there exists a direction @xmath23 , @xmath35 , such that @xmath36 and @xmath37 . gritzmann and klee ( * ? ? ? * ( 1.9 ) ) formulated a necessary condition on strictly antipodal pairs whose distance equals the minimum width . [ thm : width_polytope ] suppose that @xmath32 is a polytope with non - empty interior , and that @xmath33 and @xmath34 are a strictly antipodal pair of faces of @xmath31 whose distance is equal to @xmath38 . then , @xmath39 with @xmath40 when @xmath31 is centrally symmetric . by @xmath41 we shall denote the euclidean distance of a strictly antipodal facets @xmath33 , @xmath34 of @xmath31 which , additionally , satisfy . the following definition by heil @xcite is central to the present investigation . a convex body @xmath14 is said to be _ reduced _ if we have @xmath42 for all convex bodies @xmath43 . reduced polytopes can be characterized using vertex - facet distances , see ( * ? ? ? * theorem 4 ) and ( * ? ? ? * theorem 1 ) for the following result . [ thm : polytope_reduced ] a polytope @xmath32 is reduced if and only if for every vertex @xmath6 of @xmath31 , there exists a strictly antipodal facet @xmath7 of @xmath31 such that the distance between @xmath6 and @xmath44 equals @xmath38 . strongly related , there is also the following necessary condition on the orthogonal projection of a vertex onto one of its strictly antipodal facets at the correct distance , see ( * ? ? ? * lemma 2 ) . [ thm : reduced_implication ] assume that @xmath32 is a reduced polytope . then for every vertex @xmath6 of @xmath31 there exists a facet @xmath7 of @xmath31 such that @xmath45 is strictly antipodal to @xmath7 , the orthogonal projection @xmath46 of @xmath6 onto @xmath44 lies in the relative interior of @xmath7 , and the distance from @xmath6 to @xmath46 is equal to @xmath38 . in his recent arxiv preprint @xcite , polyanskii claims that he has found a reduced polytope in three - dimensional euclidean space . unfortunately , his example is invalid . in this section , we revisit polyanskii s polytope to find its flaw . the polytope @xmath31 considered by polyanskii is the convex hull of eight points @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 , @xmath52 , @xmath53 , and @xmath54 . for properly chosen parameters @xmath55 and @xmath56 , the points @xmath57 are the vertices of @xmath31 , see for an illustration . note that @xmath31 has the same combinatorial structure as johnson s solid @xmath58 . there are two types of vertices : the four vertices @xmath59 are incident to four edges each , and the remaining four vertices @xmath60 belong to five edges each . by symmetry , it is sufficient to take one vertex of each group and specify which faces should be strictly antipodal and at distance @xmath38 to the chosen vertices . polyanskii decides to pair the vertices @xmath61 and @xmath62 up with the facets @xmath63 and @xmath64 , respectively . additionally , he also sets the distances between the skew edges @xmath65 and @xmath66 , and between @xmath67 , @xmath68 equal to the minimum width of @xmath31 . therefore , the parameter @xmath69 will be this common distance . this yields the four equations formulated in the appendix of @xcite . these equations may be manipulated as follows . from @xmath70 and @xmath71 , we get @xmath72 this can be plugged into the equation @xmath73 in order to write @xmath74 as a function of @xmath75 , namely @xmath76 plugging and in @xmath77 we obtain one equation with variable @xmath75 . this equation is equivalent to the polynomial equation @xmath78 . using the relevant solution of this equation , we obtain the parameters @xmath79 if we compute the outer normal vectors of the facets @xmath80 and @xmath63 , assume their orthogonality , and substitute @xmath74 and @xmath69 according to and again , we obtain the same polynomial equation @xmath78 . in other words , provided three of polyanskii s distances are equal , the fourth distance equals the other three if and only if the orthogonal projections of the vertices @xmath59 onto their strictly antipodal faces lie on edges , i.e. , not in the relative interior of any facet . hence polyanskii s polytope is not reduced , see again . another evidence of this fact can be seen in the polyanskii s list of pairs of skew edges whose distance is larger than the presumed minimal width . in this list , pairs of edges of type @xmath81 , @xmath82 are missing , and these pairs are exactly those where the actual minimum width @xmath83 of @xmath31 is attained . \(a ) at ( axis cs:1,0,0 ) ; ( b ) at ( axis cs:-1,0,0 ) ; ( c ) at ( axis cs:0,1 , ) ; ( d ) at ( axis cs:0,-1 , ) ; ( e ) at ( axis cs:0 , , ) ; ( f ) at ( axis cs:0,- , ) ; ( g ) at ( axis cs:,0 , ) ; ( h ) at ( axis cs:-,0 , ) ; at ( a ) ; at ( b ) ; at ( c ) ; at ( d ) ; at ( e ) ; at ( f ) ; at ( g ) ; at ( h ) ; /in f / a , f / b , e / c , f / h , e / g , c / d , c / g , d / h , d / g , a / b , a / g , b / h , f / g , f / d /in e / a , e / b , e / h , c / h \(a ) at ( axis cs:1,0,0 ) ; ( b ) at ( axis cs:-1,0,0 ) ; ( c ) at ( axis cs:0,1 , ) ; ( d ) at ( axis cs:0,-1 , ) ; ( e ) at ( axis cs:0 , , ) ; ( f ) at ( axis cs:0,- , ) ; ( g ) at ( axis cs:,0 , ) ; ( h ) at ( axis cs:-,0 , ) ; at ( a ) ; at ( b ) ; at ( c ) ; at ( d ) ; at ( e ) ; at ( f ) ; at ( g ) ; at ( h ) ; /in e / h , e / c , f / h , e / g , c / d , c / h , c / g , d / h , d / g , f / g , f / d /in a / b , e / a , e / b , f / a , f / b , a / g , b / h \(a ) at ( axis cs:1,0,0 ) ; ( b ) at ( axis cs:-1,0,0 ) ; ( c ) at ( axis cs:0,1 , ) ; ( d ) at ( axis cs:0,-1 , ) ; ( e ) at ( axis cs:0 , , ) ; ( f ) at ( axis cs:0,- , ) ; ( g ) at ( axis cs:,0 , ) ; ( h ) at ( axis cs:-,0 , ) ; at ( a ) ; at ( b ) ; at ( d ) ; at ( f ) ; at ( g ) ; at ( h ) ; /in f / a , f / b , e / c , f / h , e / g , c / d , c / h , c / g , d / h , d / g , a / b , a / g , b / h , f / g , f / d /in e / a , e / b , e / h however , polyanskii s polytope can be modified in order to obtain a reduced polytope . we do this by introducing a new parameter @xmath84 and replacing each of the four vertices of degree @xmath85 in polyanskii s polytope by a line segment of length @xmath86 . consider the points @xmath87 note that we slightly changed the meaning of the parameters @xmath69 and @xmath74 . for properly chosen parameters @xmath88 the points @xmath89 are the vertices of our polytope @xmath31 . the combinatorial structure of our polytope is shown in . \(a ) at ( axis cs : , 0 , - ) ; ( b ) at ( axis cs:- , 0 , - ) ; ( c ) at ( axis cs : 0 , , ) ; ( d ) at ( axis cs : 0 , - , ) ; ( e ) at ( axis cs : , , ) ; ( f ) at ( axis cs:- , , ) ; ( g ) at ( axis cs : , - , ) ; ( h ) at ( axis cs:- , - , ) ; ( i ) at ( axis cs : , , - ) ; ( j ) at ( axis cs : , - , - ) ; ( k ) at ( axis cs:- , , - ) ; ( l ) at ( axis cs:- , - , - ) ; at ( a ) ; at ( b ) ; at ( c ) ; at ( d ) ; at ( e ) ; at ( f ) ; at ( g ) ; at ( h ) ; at ( i ) ; at ( j ) ; at ( k ) ; at ( l ) ; /in a / b , c / d , a / i , a / j , c / i , d / j , i / j , b / l , d / l , g / h , d / g , d / h , h / l , g / j , e / i , c / e , a / g , b / h /in b / k , e / f , c / f , f / k , a / e , b / f , c / k , k / l \(a ) at ( axis cs : , 0 , - ) ; ( b ) at ( axis cs:- , 0 , - ) ; ( c ) at ( axis cs : 0 , , ) ; ( d ) at ( axis cs : 0 , - , ) ; ( e ) at ( axis cs : , , ) ; ( f ) at ( axis cs:- , , ) ; ( g ) at ( axis cs : , - , ) ; ( h ) at ( axis cs:- , - , ) ; ( i ) at ( axis cs : , , - ) ; ( j ) at ( axis cs : , - , - ) ; ( k ) at ( axis cs:- , , - ) ; ( l ) at ( axis cs:- , - , - ) ; at ( a ) ; at ( b ) ; at ( c ) ; at ( d ) ; at ( e ) ; at ( f ) ; at ( g ) ; at ( h ) ; at ( i ) ; at ( j ) ; at ( k ) ; at ( l ) ; /in c / d , c / i , d / j , i / j , d / l , g / h , d / g , d / h , h / l , g / j , c / k , k / l , e / i , c / e , c / f , f / k , e / f /in a / b , a / i , a / j , a / g , a / e , b / k , b / h , b / f , b / l \(a ) at ( axis cs : , 0 , - ) ; ( b ) at ( axis cs:- , 0 , - ) ; ( c ) at ( axis cs : 0 , , ) ; ( d ) at ( axis cs : 0 , - , ) ; ( e ) at ( axis cs : , , ) ; ( f ) at ( axis cs:- , , ) ; ( g ) at ( axis cs : , - , ) ; ( h ) at ( axis cs:- , - , ) ; ( i ) at ( axis cs : , , - ) ; ( j ) at ( axis cs : , - , - ) ; ( k ) at ( axis cs:- , , - ) ; ( l ) at ( axis cs:- , - , - ) ; at ( a ) ; at ( b ) ; at ( d ) ; at ( g ) ; at ( h ) ; at ( j ) ; at ( l ) ; /in a / b , c / d , a / i , a / j , c / i , d / j , i / j , b / l , d / l , g / h , d / g , d / h , h / l , g / j , c / k , k / l , a / g , b / h /in b / k our polytope possesses the same symmetry as polyanskii s . hence , it is again sufficient to control some of the facet - vertex and edge - edge distances . in fact , we are going to solve the equations @xmath90 w.r.t . @xmath91 . here , @xmath92 are suitably chosen . by introducing the normal vectors @xmath93 where @xmath94 denotes the usual cross product of the vectors @xmath95 , these equations can be rewritten as @xmath96 now , it is easy to see that the third equation ( counting in columns ) is equivalent to @xmath97 . moreover , it is tedious to check that we can factor out @xmath98 in the first equation and @xmath99 in the fifth . hence , we are going solve the four equations @xmath100 under @xmath101 w.r.t . the remaining variables @xmath102 . note that each left - hand side of the four equations in are multivariate polynomials of degree at most @xmath103 in the four unknowns @xmath102 . numerically , we used @xmath104 , @xmath105 and @xmath106 and solved equations by newton s method starting with @xmath107 . this results in @xmath108 and the numerical residuum in the four equations is below @xmath109 . using kantorovich s theorem , see ( * ? ? ? * theorem xviii.1.6 ) , it is possible to prove that equations possess an exact root in the neighborhood of our numerical solution . using these parameters , we can check that the remaining distances are @xmath110 thus , the width of our polytope using these parameters is really @xmath111 , see . consequently , our polytope is reduced by . since the jacobian of the ( left - hand sides of ) equations w.r.t . @xmath112 is invertible at our point of interest , it follows from the implicit function theorem that we also obtain a solution for small changes of the parameters @xmath113 and @xmath114 . hence , we obtain a whole family of reduced polytopes possessing three degrees of freedom . it is quite a delicate and tedious procedure to check the reducedness of a given polytope @xmath115 . hence , we present an algorithm based on and[thm : polytope_reduced ] . it consists of two steps : 1 . compute the width of @xmath31 , compare . 2 . check whether each vertex has a strictly antipodal facet , compare . an implementation in pseudocode is given in . in step 4 of the algorithm , we denoted by @xmath116 a vector normal to the skew edges @xmath117 and @xmath118 . using and[thm : polytope_reduced ] , it is easy to check its correctness . a matlab implementation is provided at zenodo , see @xcite . * input : * polytope @xmath115 set @xmath119 set @xmath120 unmark all vertices of @xmath31 compute the strictly antipodal face @xmath121 set @xmath122 unmark all vertices of @xmath31 set @xmath123 mark vertex @xmath6 are all vertices of @xmath31 marked ? in this paper , we presented the first to our best knowledge example of a reduced polytope in three - dimensional euclidean space . as polyanskii @xcite has already pointed out , the existence of reduced polytopes in euclidean space remains open starting from dimension four . moreover , already finding a reduced polytope in three - dimensional space with different combinatorial structure than the one presented in seems to be a non - trivial task . finally , it has to be checked to which amount polyanskii s negative result ( * ? ? ? * theorem 2 ) can be generalized to higher dimensions . * acknowledgements . * we would like to thank horst martini for encouraging us in the search of reduced polytopes , and ren brandenberg and undine leopold for fruitful discussions , and alexandr polyanskii for his ideas and his talk at the conference _ discrete geometry days _ , which took place in budapest , hungary , in june 2016 . g. averkov , _ on planar convex bodies of given minkowskian thickness and least possible area _ , arch . math . ( basel ) * 84 * ( 2005 ) , no . 2 , pp . 183192 , http://dx.doi.org/10.1007/s00013-004-1152-6[doi : 10.1007/s00013 - 004 - 1152 - 6 ] . chakerian and h. groemer , _ convex bodies of constant width _ , convexity and its applications ( p. m. gruber and j. m. wills , eds . ) , birkhuser , basel , 1983 , pp . 4996 , http://dx.doi.org/10.1007/978-3-0348-5858-8[doi : 10.1007/978 - 3 - 0348 - 5858 - 8 ] . p. gritzmann and v. klee , _ inner and outer @xmath124-radii of convex bodies in finite - dimensional normed spaces _ , discrete comput * 7 * ( 1992 ) , no . 1 255280 , http://dx.doi.org/10.1007/bf02187841[doi : 10.1007/bf02187841 ] . to3em , _ reduced convex bodies in finite - dimensional normed spaces : a survey _ , results math . * 66 * ( 2014 ) , no . 3 - 4 , pp . 405426 , http://dx.doi.org/10.1007/s00025-014-0384-4[doi : 10.1007/s00025 - 014 - 0384 - 4 ] . | by correcting an example by polyanskii , we show that there exist reduced polytopes in three - dimensional euclidean space .
this partially answers the question posed by lassak @xcite on the existence of reduced polytopes in @xmath0-dimensional euclidean space for @xmath1 .
* keywords : * polytope , reducedness * msc(2010 ) : * http://www.ams.org/mathscinet/msc/msc2010.html?t=52b10[52b10 ] |
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this measurement is the first step towards a high - statistics dalitz - plot analysis of the @xmath1 decay . the latter could give insight into the controversy on the s - wave @xmath5 contribution in these decays @xcite , as well as a sensitive study of the @xmath6 violation in the neutral d meson system . knowledge of @xmath0(@xmath7)/@xmath0(@xmath8 ) ( also based on the @xmath1 dalitz analysis ) could improve our understanding of the apparent discrepancy of the measured two - body branching fractions ( @xmath9 kk , @xmath10 ) with the theoretical expectations @xcite . the accuracy of the value of @xmath0(@xmath1 ) as reported in pdg04 @xcite is poor . using a large data sample of @xmath11 decays accumulated with the belle detector , we provide a significantly improved measurement using the @xmath2 decay mode for normalization . since both decay modes involve a neutral pion and the same number of charged tracks in the final state , several sources of the systematic uncertainties are avoided in a determination of the relative branching fraction . the obtained result can then be compared to recent measurements by the cleo @xcite and babar @xcite collaborations . a detailed study of the @xmath1 decay as well as of other @xmath11 @xmath6-symmetric final states , can be used to further improve statistics for the measurement of the angle @xmath12 ( @xmath13 ) of the ckm - matrix . the belle detector is a large - solid - angle magnetic spectrometer located at the kekb @xmath4 storage rings , which collide 8.0 gev electrons with 3.5 gev positrons and produce @xmath14(4s ) at the energy of 10.58 gev . closest to the interaction point is a silicon vertex detector ( svd ) , surrounded by a 50-layer central drift chamber ( cdc ) , an array of aerogel cherenkov counters ( acc ) , a barrel - like arrangement of time - of - flight ( tof ) scintillation counters , and an electromagnetic calorimeter ( ecl ) comprised of csi ( tl ) crystals . these subdetectors are located inside a superconducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return yoke located outside the coil is instrumented to detect @xmath15 mesons and identify muons . the detector is described in detail elsewhere @xcite . for this analysis , we used a data sample of 357 fb@xmath3 accumulated with the belle detector . @xmath11 candidates are selected from @xmath16 decays where the charge of the @xmath17 tags the @xmath11 flavour : @xmath18 . @xmath19 s originate mainly from continuum . although we do not apply any topological cuts , the yield of @xmath19 s from @xmath20-meson decays is negligible ; such events are rejected by other kinematical cuts such as the strong @xmath21(@xmath19 ) requirement . @xmath11 mesons are reconstructed from combinations of two oppositely charged pions ( or a charged pion and kaon in the case of @xmath22 ) and one neutral pion . the latter is reconstructed from two @xmath13 candidates satisfying the @xmath23 mass requirement given below . + the following kinematic criteria are applied to the charged track candidates : the distance from the nominal interaction point to the point of closest approach of the track is required to be within 0.15 cm in the radial direction ( @xmath24 ) and 0.3 cm along the beam direction ( @xmath25 ) . we also require the transverse momentum of the track @xmath26 @xmath27 0.050 gev / c to suppress beam background . kaons and pions are separated by combining the responses of the acc and the tof with the @xmath28 measurement from the cdc to form a likelihood @xmath29 where @xmath30 is a pion or a kaon . charged particles are identified as pions or kaons using the likelihood ratio @xmath31 . for charged pion identification , we require @xmath32 . this requirement selects pions with an efficiency of 93% and misidentified kaons with an efficiency of 9% . for the identification of charged kaons , the requirement is @xmath33 ; in this case , the efficiency for kaon identification is 86% and the probability to misidentify a pion is 4% . + we impose conditions on the energies of the photons constituting the @xmath23 candidate ( @xmath34 @xmath27 0.060 gev ) , the two - photon invariant mass ( @xmath35 gev / c@xmath36 gev / c@xmath37 ) and the @xmath23 s momentum in the laboratory frame ( @xmath38(@xmath23 ) @xmath27 0.3 gev / c ) to suppress combinatorial @xmath23 s . + the mass difference of @xmath19 and @xmath11 candidates should satisfy the restrictions : @xmath39 gev / c@xmath40 gev / c@xmath37 and @xmath41 gev / c@xmath42 gev / c@xmath37 . the momentum of the @xmath19 in the center - of - mass frame of the @xmath43 must lie in the range : @xmath44 gev / c@xmath45 gev / c . the lower cut is applied to suppress slow fake @xmath19 s reconstructed from combinatorial background that originates from @xmath20 decays . the upper cut restricts @xmath46 to the region where the monte carlo ( mc ) distribution is in good agreement with the data ( see fig . 1 , left ) . to eliminate background from the @xmath47 decays , the following veto on @xmath48 is applied : @xmath49 gev@xmath37/c@xmath50 gev@xmath37/c@xmath51 . to obtain detection efficiencies , about 1.2@xmath52 phase - space distributed mc events have been generated for each of the two modes , processed using the geant based detector simulation @xcite and reconstructed with the same selection criteria as the data . + to obtain a more realistic @xmath11 decay model than a uniform dalitz - plot distribution , the events of both signal mc samples have been weighted using matrix elements based on decay models obtained by cleo @xcite in the framework of a 3-resonance model ( @xmath53 , @xmath54 , @xmath55 and a nonresonant contribution ) for @xmath1 and in the framework of a 7-resonance model ( @xmath53 , @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 , @xmath61 and a nonresonant contribution ) for the case of @xmath2 @xcite ( see fig . + the obtained yield is normalized to the same mc data but before detector simulation and before application of selection criteria . since for the relative branching fraction measurement only the ratio of @xmath62 and @xmath22 efficiencies is needed , we only generate events in the @xmath63 range from 3.5 to 4.3 gev / c where mc is in good agreement with the data ( see fig . 1 , left ) . the obtained efficiencies are : @xmath64 @xmath65 to describe the background shape in the @xmath66 signal region , a sample of generic mc , equivalent to @xmath67250 fb@xmath3 , has been processed with the same selection criteria as data . the contributions of @xmath68 , @xmath69 , @xmath70 , @xmath71 and @xmath72 backgrounds have been summed up . for the @xmath71 sample signal events were excluded . figure 3 shows the individual contributions to the @xmath66 distribution where also a partial data sample , corresponding to the luminosity of mc , is shown . the @xmath66 background distribution is fitted with a @xmath73 order polynomial multiplied by an error function ( kaon misidentification region ) plus a @xmath74 order polynomial ( combinatoric background ) and a small gaussian peak in the signal region ( see fig . 3 , right ) . the gaussian contribution is mainly due to combinations of a correctly reconstructed @xmath11 and a random @xmath17 candidate . + most of the background is from e@xmath75e@xmath76 @xmath71 ; the @xmath72 background is negligible and @xmath68 , @xmath69 , @xmath70 backgrounds are linearly distributed in the @xmath66 signal region . among the @xmath71 background sources @xmath77 is dominant : charged kaons are misidentified as pions and @xmath78 is typically shifted downwards by @xmath79 gev / c@xmath37 thus being well separated from the signal . + the background shape in the @xmath80 distribution is obtained using a generic mc sample . the level of background in this mode is low but still has a nontrivial structure ( see fig . 4 , right ) . the distribution has three distinctive peaks : the rightmost one is due to misidentified pions from @xmath1 , the central one has the same origin as the one in the @xmath81 case ( random @xmath17 ) , the leftmost feature originates mainly from @xmath82 where @xmath83 . the distribution is fitted by the sum of a @xmath73 order polynomial and a @xmath73 order polynomial multiplied by an error function ( left peak ) and two gaussians . the shape of the signal peak in the experimental @xmath62 invariant mass distribution ( see fig . 5 , left ) is partially fixed to the mc one @xcite : the latter was fitted with a bifurcated hyperbolic gaussian @xcite and a regular gaussian ( see fig . 1 , right ) . the obtained shape with floating normalization and @xmath84 s , together with the background shape with its floating normalization ( i.e. 5 free parameters ) is then used as the fit function for the measured @xmath66 distribution . + to fit the experimental @xmath85 invariant mass distribution we use a @xmath85 background shape with floating normalization and the sum of two bifurcated and a regular gaussian for the signal peak ( see fig . 4 , right ) . here , the parameters of the signal peak are free in the fit since the level of background is low and statistics are large enough . + the yields of signal events in each channel , as obtained from the fit , are given in table [ t7 ] : [ t7 ] .number of signal events and efficiencies [ cols="^,^,^",options="header " , ] + finally , we can compare our measurement of the ratio with the ratio obtained from the latest world average values of @xmath86 and @xmath0(@xmath87 ) @xcite . our result ( 9.71 @xmath88 0.31)% is consistent with the world average one ( 9.29 @xmath88 0.54)% and has better accuracy . by choosing the normalization mode @xmath2 we avoid many sources of systematic uncertainty including the @xmath23 detection efficiency and uncertainty in the tracking efficiency . the pid efficiency partially cancels out . using 357 fb@xmath3 of data collected with the belle detector , the first direct measurement of the relative branching fraction @xmath0(@xmath1)/ @xmath0(@xmath2 ) has been performed . our preliminary result @xmath89 is compatible with the world average @xmath90 and is more precise . the belle result differs by @xmath91 from the value recently obtained by babar @xcite : @xmath92 . the corresponding value of the absolute branching fraction @xmath0(@xmath1 ) is ( 13.69 @xmath88 0.66)@xmath93 . we thank the kekb group for the excellent operation of the accelerator , the kek cryogenics group for the efficient operation of the solenoid , and the kek computer group and the national institute of informatics for valuable computing and super - sinet network support . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china and the knowledge innovation program of the chinese academy of sciences under contract no . 10575109 and ihep - u-503 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea , the chep src program and basic research program ( grant no . r01 - 2005 - 000 - 10089 - 0 ) of the korea science and engineering foundation , and the pure basic research group program of the korea research foundation ; the polish state committee for scientific research ; the ministry of science and technology of the russian federation ; the slovenian research agency ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy . 99 ( cleo collaboration ) , phys . d * 72 * , 031102 ( 2005 ) . j.m . link _ _ ( focus collaboration ) , phys . b * 585 * , 200 ( 2004 ) . b * 379 * , 249 ( 1996 ) . s. eidelman _ et al_. , phys . b * 592 * , 1 ( 2004 ) . ( cleo collaboration ) , phys . 96 * , 081802 ( 2006 ) . ( babar collaboration ) , hep - ex/0608009 . et al . _ ( belle collaboration ) , nucl . instrum . and meth . a * 479 * , 117 ( 2002 ) . z.natkaniec _ et al . _ ( belle svd2 group ) , nucl . instr . and meth . a * 560 * , 1 ( 2006 ) . geant , r. brun _ et al . _ , geant 3.21 , cern report dd / ee/84 - 1 , 1984 . s. kopp _ ( cleo collaboration ) , phys . rev . d * 63 * , 092001 ( 2001 ) . w .- _ et al_. , j. phys . g * 33 * , 1 ( 2006 ) . ( cleo collaboration ) , hep - ex/0306048 . | we report a high - statistics measurement of the relative branching fraction @xmath0(@xmath1)/ @xmath0(@xmath2 ) .
a 357 fb@xmath3 data sample collected with the belle detector at the kekb asymmetric - energy @xmath4 collider was used for the analysis .
the relative branching fraction @xmath0(@xmath1)/ @xmath0(@xmath2 ) is determined with an accuracy comparable to the latest world average value . |
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given data @xmath1 , where @xmath2 is the response and @xmath3 is the @xmath4-dimensional covariate , the goal in many analyses is to approximate the unknown function @xmath5 by minimizing a specified loss function @xmath6 [ a common choice is @xmath0-loss , @xmath7 . in trying to estimate @xmath8 , one strategy is to make use of a large system of possibly redundant functions @xmath9 . if @xmath9 is rich enough , then it is reasonable to expect @xmath8 to be well approximated by an additive expansion of the form @xmath10 where @xmath11 are base learners parameterized by @xmath12 . to estimate @xmath8 , a joint multivariable optimization over @xmath13 may be used . but such an optimization may be computationally slow or even infeasible for large dictionaries . overfitting may also result . to circumvent this problem , iterative descent algorithms are often used . one popular method is the gradient descent algorithm described by @xcite , closely related to the method of `` matching pursuit '' used in the signal processing literature [ @xcite ] . this algorithm is applicable to a wide range of problems and loss functions , and is now widely perceived to be a generic form of boosting . for the @xmath14th step , @xmath15 , one solves @xmath16 where @xmath17 ^ 2\ ] ] identifies the closest base learner to the gradient @xmath18 in @xmath0-distance , where @xmath19 is the gradient evaluated at the current value @xmath20 , and is defined by @xmath21_{f_{m-1}(\mathbf{x}_i ) } = -l'(y_i , f_{m-1}(\mathbf{x}_i)).\ ] ] the @xmath14th update for the predictor of @xmath8 is @xmath22 where @xmath23 is a regularization ( learning ) parameter . in this paper , we study friedman s algorithm under @xmath0-loss in linear regression settings assuming an @xmath24 design matrix @xmath25 $ ] , where @xmath26 denotes the @xmath27th column . here @xmath28 represents the @xmath27th base learner ; that is , @xmath29 where @xmath30 and @xmath31 . it is well known that under @xmath0-loss the gradient simplifies to the residual @xmath32 . this is particularly attractive for a theoretical treatment as it allows one to combine the line - search and the learner - search into a single step because the @xmath0-loss function can be expressed as @xmath33 . the optimization problem becomes @xmath34 it is common practice to standardize the response by removing its mean which eliminates the issue of whether an intercept should be included as a column of @xmath35 . it is also common to standardize the columns of @xmath35 to have a mean of zero and squared - length of one . thus , throughout , we assume the data is standardized according to @xmath36 the condition @xmath37 leads to a particularly useful simplification : @xmath38 thus , the search for the most favorable direction is equivalent to determining the largest absolute value @xmath39 . we refer to @xmath40 as the _ gradient - correlation _ for @xmath27 . we shall refer to friedman s algorithm under the above settings as 2boost . algorithm [ a : l2boost ] provides a formal description of the algorithm [ we use @xmath41 for notational convenience ] . initialize @xmath42 for @xmath43 @xmath44 , where @xmath45 @xmath46 , where @xmath47 properties of stagewise algorithms similar to 2boost have been studied extensively under the assumption of an infinitesimally small regularization parameter . @xcite considered a forward stagewise algorithm @xmath48 , and showed under a convex cone condition that the least angle regression ( lar ) algorithm yields the solution path for @xmath49 , the limit of @xmath48 as @xmath50 . this shows that @xmath48 , a variant of boosting , and the lasso [ @xcite ] are related in some settings . @xcite showed in general that the solution path of @xmath49 is equivalent to the path of the monotone lasso . however , much less work has focused on stagewise algorithms assuming an arbitrary learning parameter @xmath23 . an important exception is @xcite who studied 2boost with componentwise linear least squares , the same algorithm studied here , and proved consistency for arbitrary @xmath51 under a sparsity assumption where @xmath4 can increase at an exponential rate relative to @xmath52 . as pointed out in @xcite , the @xmath48algorithm studied by @xcite bears similarities to 2boost . it is identical to algorithm [ a : l2boost ] , except for line 4 , where @xmath53 is used in place of @xmath51 and @xmath54.\ ] ] thus , @xmath48 replaces the gradient - correlation @xmath55 with the sign of the gradient - correlation @xmath56 . for infinitesimally small @xmath51 this difference appears to be inconsequential , and it is generally believed that the two limiting solution paths are equal [ @xcite ] . in general , however , for arbitrary @xmath23 , the two solution paths are different . indeed , @xcite indicated certain unique advantages possessed by 2boost . other related work includes @xcite , who described a bias - variance decomposition of the mean - squared - error of a variant of 2boost . in this paper , we investigate the properties of 2boost assuming an arbitrary learning parameter @xmath23 . during 2boost s descent along a fixed coordinate direction , a new coordinate becomes more favorable when it becomes closest to the current gradient . but when does this actually occur ? we provide an exact simple closed form expression for this quantity : the number of iterations to favorability ( theorem [ criticalpoint.theorem ] of section [ s : fixeddescent ] ) . this core identity is used to describe 2boost s solution path ( theorem [ full.path.solution.general ] ) , to introduce new tools for studying its path and to study and characterize some of the algorithm s unique properties . one of these is active set cycling , a property where the algorithm spends lengthy periods of time cycling between the same coordinates when @xmath51 is small ( section [ s : cyclingbehavior ] ) . our fixed descent identity also reveals how correlation affects 2boost s ability to select variables in highly correlated problems . we identify a _ repressible condition _ that prevents a new variable from entering the active set , even though that variable may be highly desirable ( section [ s : repressibility ] ) . using a data augmentation approach , similar to that used for calculating the elastic net [ @xcite ] , we describe a simple method for adding @xmath0-penalization to 2boost ( section [ s : elasticboost ] ) . in combination with decorrelation , this reverses the repressible condition and improves 2boost s performance in correlated problems . because 2boost is known to approximate forward stagewise algorithms for arbitrarily small @xmath51 , it is natural to expect these results to apply to such algorithms like lar and lasso , and thus our results provide a new explanation for why these algorithms may perform poorly in correlated settings and why methods like the elastic net , which makes use of @xmath0-penalization , are more adept in such settings . all proofs in this manuscript can be found in the supplemental article [ @xcite ] . to analyze 2boost we introduce the following notation useful for describing its solution path . let @xmath57 be the @xmath58 nonduplicated values in order of appearance of the selected coordinate directions @xmath59 . we refer to these ordered , nonduplicated values as _ critical directions _ of the path . for example , if @xmath60 , the critical directions are @xmath61 and @xmath62 . to formally describe the solution path we introduce the following nomenclature.=-1 [ path.def ] the descent length along a critical direction @xmath63 is denoted by @xmath64 . the critical point @xmath65 is the step number at which the descent along @xmath63 ends . thus , following step @xmath66 , the descent is along @xmath63 for a total of @xmath64 steps , ending at step @xmath65 . the set of values @xmath67 can be used to formally describe the solution path of 2boost : the algorithm begins by descending along direction @xmath68 ( the first critical direction ) for @xmath69 steps , after which it switches to a descent along direction @xmath70 ( the second critical direction ) for a total of @xmath0 steps . this continues with the last descent along @xmath71 ( the final critical direction ) for a total of @xmath72 steps . see figure [ figure1 ] for illustration of the notation . . the @xmath73 critical directions are @xmath74 with critical descent step lengths @xmath75 and critical points @xmath76 . _ a key observation is that 2boost s behavior along a given descent is deterministic except for its descent length @xmath64 ( number of steps ) . if we could determine the descent length , a quantity we show is highly amenable to analysis , then an exact description of the solution path becomes possible as 2boost can be conceptualized as collection of such fixed paths . imagine then that we are at step @xmath77 of the algorithm and that in the following step a new critical direction @xmath27 is formed . let us study the descent along @xmath27 for the next @xmath78 steps . thus , in the @xmath14th step of the descent along @xmath27 , the predictor is @xmath79 consider then algorithm [ a : incrementall2boost ] which repeatedly boosts the predictor along the @xmath27th direction for a total of @xmath80 steps . @xmath81 @xmath82 , where @xmath83 the following result states a closed form solution for the @xmath14-step predictor of algorithm [ a : incrementall2boost ] and will be crucial to our characterization of 2boost . [ incremental.operator.theorem ] @xmath84 , where @xmath85 and @xmath86 . theorem [ incremental.operator.theorem ] shows that taking a single step with learning parameter @xmath87 yields the same limit as taking @xmath14 steps with the smaller learning parameter @xmath51 . the result also sheds insight into how @xmath51 slows the descent relative to stagewise regression . notice that the @xmath14-step predictor can be written as @xmath88 the first term on the right is the predictor from a greedy stagewise step , while the second term represents the effect of slow - learning . this latter term is what slows the descent relative to a greedy step . when @xmath89 this term vanishes , and we end up with stagewise fitting , @xmath90 . theorem [ incremental.operator.theorem ] shows how to take a large boosting step in place of many small steps , but it does not indicate how many steps must be taken along @xmath27 before a new variable enters the solution path . if this were known , then the entire @xmath27-descent could be characterized in terms of a single step . to determine the descent length , suppose that 2boost has descended along @xmath27 for a total of @xmath14 steps . at step @xmath91 the algorithm must decide whether to continue along @xmath27 or to select a new direction @xmath92 . to determine when to switch directions , we introduce the following definition . [ favorable.def ] a direction @xmath92 is said to be more favorable than @xmath27 at step @xmath91 if @xmath93 and @xmath94 . thus , if @xmath92 is more favorable at @xmath91 , the descent switches to @xmath92 for step @xmath91 . to determine when @xmath92 becomes more favorable , it will be useful to have a closed form expression for @xmath95 and @xmath96 . by theorem [ incremental.operator.theorem ] , @xmath97 \\ & = & \rho_{j,1 } -\nu_{m}\rho_{k,1}r_{j , k},\end{aligned}\ ] ] where @xmath98 . setting @xmath99 yields @xmath100 . therefore , @xmath94 if and only if @xmath101 dividing throughout by @xmath102 , with a little bit of rearrangement , this becomes @xmath103 ^ 2,\ ] ] where @xmath104 . notice importantly that @xmath105 because @xmath27 is the direction with maximal gradient - correlation at the start of the descent . it is also useful to keep in mind that @xmath106 is the sample correlation of @xmath107 and @xmath28 due to , and thus @xmath108 . the following result states the number of steps taken along @xmath27 before @xmath92 becomes more favorable . [ criticalpoint.theorem ] the number of steps @xmath109 taken along @xmath27 so that @xmath92 becomes more favorable than @xmath27 at @xmath110 is the largest integer @xmath14 such that @xmath111 it follows that for @xmath112 @xmath113,\hspace*{-30pt}\ ] ] where @xmath114 is the largest integer less than or equal to @xmath115 . [ repressible.remark ] in particular , notice that @xmath116 when @xmath117 [ adopting the standard convention that @xmath118 and assuming that @xmath119 . we call @xmath117 the repressible condition . section [ s : repressibility ] will show that repressibility plays a key role in 2boost s behavior in correlated settings . when @xmath90 we obtain @xmath120 from which corresponds to greedy stagewise fitting . because this makes the @xmath90 case uninteresting , we shall hereafter assume that @xmath121 . theorem [ criticalpoint.theorem ] immediately shows that the problem of determining the next variable to enter the solution path can be recast as finding the direction requiring the fewest number of steps @xmath109 to favorability . when combined with theorem [ incremental.operator.theorem ] , this characterizes the entire descent and can be used to characterize 2boost s solution path . as before , assume that @xmath27 corresponds to the first critical direction of the path , that is , @xmath122 . by theorem [ criticalpoint.theorem ] , 2boost descends along @xmath27 for a total of @xmath123 steps , where @xmath124 and @xmath70 is the coordinate requiring the smallest number of steps to become more favorable than @xmath27 . by theorem [ incremental.operator.theorem ] , the predictor at step @xmath125 is @xmath126 applying theorem [ incremental.operator.theorem ] once again , but now using a descent along @xmath70 initialized at @xmath127 , and continuing this argument recursively , as well as using the representation for the number of steps from theorem [ criticalpoint.theorem ] , yields theorem [ full.path.solution.general ] , which presents a recursive description of 2boost s solution path . [ full.path.solution.general ] @xmath128 , where @xmath129 are determined recursively from @xmath130 , \\ l_r&=&m_{l_{r+1}}^{(r)},\qquad s_r = s_{r-1 } + l_r,\qquad s_0=0 , \\ d_j^{(r ) } & = & \frac{\rho_j^{(r)}}{\rho_{l_r}^{(r)}},\qquad \rho_{j}^{(r+1 ) } = { \mathbf{x}}_j^t({\mathbf{y}}-{\mathbf{f}}_{s_{r } } ) = \rho_{j}^{(r ) } - \nu_{l_r}\rho_{l_r}^{(r)}r_{j , l_r}.\end{aligned}\ ] ] [ step.number.tie.remark ] a technical issue arises in theorem [ full.path.solution.general ] when @xmath131 is not unique . non - uniqueness can occur due to rounding which is caused by the floor function used in the definition of @xmath109 . this is why line 1 selects the next critical value , @xmath132 , by maximizing the absolute gradient - correlation @xmath133 and not by minimizing the step number @xmath131 . this definition for @xmath132 is equivalent to the two - step solution @xmath134 [ equal.gradient.remark ] another technical issue arises when there is a tie in the absolute gradient - correlation . in line 3 of algorithm [ a : l2boost ] it may be possible for two coordinates , say @xmath92 and @xmath27 , to have equal gradient - correlations at step @xmath135 . theorem [ full.path.solution.general ] implicitly deals with such ties due to definition [ favorable.def ] . for example , suppose that the first @xmath136 steps are along @xmath27 with the tie occurring at step @xmath14 . in the language of theorem [ criticalpoint.theorem ] , because @xmath92 becomes more favorable than @xmath27 at @xmath91 , where @xmath137 , we have @xmath138 resolves the tie at @xmath14 by continuing to descend along @xmath27 , then switching to @xmath92 at step @xmath91 . although algorithm [ a : l2boost ] does not explicitly address this issue , the potential discrepancy is minor because such ties should rarely occur in practice . this is because for @xmath139 to hold , the value inside the floor function of used to define @xmath109 must be an integer ( a careful analysis of the proof of theorem [ criticalpoint.theorem ] shows why ) . a tie can occur only when this value is an integer which is numerically unlikely to occur.=-1 theorem [ full.path.solution.general ] immediately yields a recursive solution for the coefficient vector , @xmath140 . the solution path for @xmath140 is the piecewise solution @xmath141 where @xmath142 is the vector with one in coordinate @xmath63 and zero elsewhere . aside from the technical issue of ties , theorem [ full.path.solution.general ] and algorithm [ a : l2boost ] are equivalent . for convenience , we state theorem [ full.path.solution.general ] in an algorithmic form to facilitate comparison with algorithm [ a : l2boost ] ; see algorithm [ a : l2boostpath ] . computationally , algorithm [ a : l2boostpath ] improves upon algorithm [ a : l2boost ] by avoiding taking many small steps along a given descent . however , the difference is not substantial because the benefits only apply when @xmath51 is small , and as we will show later ( section [ s : cyclingbehavior ] ) , this forces the algorithm to cycle between its variables following the first descent , thus mitigating its ability to take large steps . thus , strictly speaking , the benefit of algorithm [ a : l2boostpath ] is confined primarily to the first descent . @xmath143 ; @xmath144 ; @xmath145 @xmath146 ; @xmath147 @xmath148 ; @xmath149 @xmath128 to investigate the differences between the two algorithms we analyzed the diabetes data used in @xcite . the data consists of @xmath150 patients in which the response of interest , @xmath151 , is a quantitative measure of disease progression for a patient . in total there are 64 variables , that includes 10 baseline measurements for each patient , 45 interactions and 9 quadratic terms . in order to compare results , we translated each iteration , @xmath152 , used by algorithm [ a : l2boostpath ] into its corresponding number of steps , @xmath14 . thus , while we ran algorithm [ a : l2boostpath ] for @xmath153 iterations , this translated into @xmath154 steps . as expected , this difference is primarily due to the first iteration @xmath155 which took @xmath156 steps along the first critical direction ( first panel of figure [ figure2 ] ; the rug indicates critical points , @xmath65 ) . there are other instances where algorithm [ a : l2boostpath ] took more than one step ( corresponding to the light grey tick marks on the rug ) , but these were generally steps of length 2 . the standardized gradient - correlation is plotted along the @xmath151-axis of the figure . the standardized gradient - correlation for step @xmath14 was defined as ( using the notation of algorithm [ a : l2boost ] ) @xmath157 the middle panel displays the results using algorithm [ a : l2boost ] with @xmath158 steps . clearly , the greatest gains from algorithm [ a : l2boostpath ] occur along the @xmath155 descent . one can see this most clearly from the last panel which superimposes the first two panels . against step number @xmath14 for algorithms [ a : l2boostpath ] and [ a : l2boost ] , respectively . only coordinates in the solution path are displayed ( a total of four ) . the third panel superimposes the first two panels . all analyses used @xmath159 . ] note a potential computational optimization exists in algorithm [ a : l2boostpath ] . it is possible to calculate the correlation values only once as each new variable enters the active set , then cache these values for future calculations . thus , when @xmath132 is a new variable in the active set , we calculate @xmath160 . the updated gradient - correlation is calculated efficiently by using addition and scalar multiplication using the previous gradient - correlation and the cached correlation coefficients @xmath161 this is in contrast to algorithm [ a : l2boost ] which requires a vector multiplication of dimension @xmath4 at each step @xmath14 to update the gradient - correlation : @xmath162 . [ active.set.remark ] above , when we refer to the `` active set , '' we mean the unique set of critical directions in the current solution path . this term will be used repeatedly throughout the paper . throughout the paper we illustrate different ways of utilizing @xmath109 of theorem [ criticalpoint.theorem ] to explore 2boost . so far we have confined the use of theorem [ criticalpoint.theorem ] to determining the descent length along a fixed direction , but another interesting application is determining how far a given variable is from the active set . note that although theorem [ criticalpoint.theorem ] was described in terms of an active set of only one coordinate , it applies in general , regardless of the size of the active set . thus , @xmath109 can be calculated at any step @xmath14 to determine the number of steps required for @xmath92 to become more favorable than the current direction , @xmath27 . this value represents the distance of @xmath92 to the solution path and can be used to visualize it . to demonstrate this , we applied algorithm [ a : l2boost ] to the diabetes data for @xmath163 steps and recorded @xmath109 for each of the @xmath164 variables . figure [ figure3 ] records these values . each `` jagged path '' in the figure is the trace over the 10,000 steps for a variable @xmath92 . each point on the path equals the number of steps @xmath109 to favorability relative to the current descent @xmath165 . the patterns are quite interesting . the top variables have @xmath109 values which quickly drop within the first 1000 steps . another group of variables have values which take much longer to drop , doing so somewhere between 2000 to 4000 steps , but then increase almost immediately . these variables enter the solution path but then quickly become unattractive regardless of the descent direction . of each variable @xmath92 to favorability relative to the current descent @xmath27 ( results based on algorithm [ a : l2boost ] where @xmath159 ) . for visual clarity the @xmath109 values have been smoothed using a running median smoother . ] it has become popular to visualize the solution path of forward stagewise algorithms by plotting their gradient - correlation paths and/or their coefficient paths . figure [ figure3 ] is a similar tool . a unique feature of @xmath109 is that it depends not only on the gradient - correlation ( via @xmath166 ) , but also the correlation in the @xmath167-variables ( via @xmath106 ) and the learning parameter @xmath51 . in this manner , figure [ figure3 ] offers a new tool for understanding and exploring such algorithms . it has been widely observed that decreasing the regularization parameter slows the convergence of stagewise descent algorithms . @xcite showed that the @xmath48 algorithm tracks the equiangular direction of the lar path for arbitrarily small @xmath53 . to achieve what lar does in a single step , the @xmath48 algorithm may require thousands of small steps in a direction tightly clustered around the equiangular vector , eventually ending up at nearly the same point as lar . we show that 2boost exhibits this same phenomenon . we do so by describing this property as an active set cycling phenomenon . using results from the earlier fixed descent analysis , we show in the case of an active set of two variables that 2boost systematically switches ( cycles ) between its two variables when @xmath51 is small . for an arbitrarily small @xmath51 this forces the absolute gradient - correlations for the active set variables to be nearly equal . this point of equality represents a singularity point that triggers a near - perpetual deterministic cycle between the variables , ending only when a new variable enters the active set with nearly the same absolute gradient - correlation . our insight will come from looking at theorem [ criticalpoint.theorem ] in more depth . as before , assume the algorithm has been initialized so that @xmath27 is the first critical step . previously the descent along @xmath27 was described in terms of steps , but this can be equivalently expressed in units of the `` step size '' taken . define @xmath168 recall that theorem [ incremental.operator.theorem ] showed that a single step along @xmath27 with @xmath51 replaced with @xmath169 yields the same limit as @xmath109 steps along @xmath27 using @xmath51 . we call @xmath169 the step size taken along @xmath27 . because @xmath92 becomes more favorable than @xmath27 at @xmath110 , the gradient following a step size of @xmath169 along @xmath27 satisfies @xmath170 , and in particular holds for the second critical direction , @xmath70 , which rephrased in terms of step size , is the smallest @xmath169 value , @xmath171 although inequality is strict , it becomes arbitrarily close to equality with shrinking @xmath51 . with a little bit of rearranging , implies that @xmath172 we will show @xmath173 is the step size making the absolute gradient - correlation between @xmath92 and @xmath27 equal @xmath174 converges to the smallest @xmath173 satisfying ; thus , becomes an equality in the limit . for convenience , we define @xmath175 . [ dynamic.nu.size ] let @xmath176 . then @xmath177 . furthermore , if @xmath178 and @xmath179 , then @xmath180 and @xmath181 as @xmath182 . therefore , for arbitrarily small @xmath51 , @xmath183 and @xmath27 and @xmath70 will have near - equal absolute gradient - correlations . this latter property triggers two - cycling . to see why , let us assume for the moment that the active set variables have equal absolute gradient - correlations . then by a direct application of theorem [ criticalpoint.theorem ] , one can show that the number of steps taken along @xmath70 before @xmath27 becomes more favorable is @xmath184 . thus , following the descent along @xmath27 , the algorithm switches to @xmath70 , but then immediately switches back to @xmath27 . if @xmath51 is small enough , this process is repeated , setting off a two - cycling pattern . the next result is a formal statement of these arguments . define @xmath185 for notational convenience , let @xmath186 and @xmath137 . for technical reasons we shall assume @xmath187 . recall remark [ repressible.remark ] showed that @xmath188 , the repressible condition , yields an infinite number of steps to favorability . thus , for @xmath27 to be even eligible for favorability we must have @xmath187 . [ long.descent.followed.two.cycle ] if the first two critical directions are @xmath189 and @xmath190 , then @xmath27 is favored over @xmath92 for the next step after @xmath92 if @xmath187 . theorem [ long.descent.followed.two.cycle ] assumes that @xmath190 . while this only holds in the limit , the two values should be nearly equal for arbitrarily small @xmath51 , and thus the assumption is reasonable . notice also that theorem [ long.descent.followed.two.cycle ] only shows that @xmath27 is more favorable than @xmath92 , and not that the algorithm switches to @xmath27 . however , we can see that this must be the case . for arbitrarily small @xmath51 , @xmath27 s gradient - correlation should be nearly equal to @xmath92 s , and by definition , @xmath92 has maximal absolute gradient - correlation along the second descent . indeed , the following result shows that the absolute gradient - correlations for @xmath27 and @xmath92 can be made arbitrarily close for small enough @xmath51 for any step @xmath191 following the descent along @xmath27 . the result also shows that the sign of the gradient - correlation is preserved when @xmath51 is arbitrarily small , a fact that we shall use later . [ gradient.correlation.equality ] @xmath192 as @xmath182 for each @xmath191 . combining theorems [ long.descent.followed.two.cycle ] and [ gradient.correlation.equality ] , we see that if @xmath51 is small enough , the first three critical directions of the path must be @xmath193 with critical points @xmath194 . and once the descent switches back to @xmath27 , it is clear from the same argument that the next critical direction , @xmath195 , will be @xmath92 , and so forth . we present a numerical example demonstrating two - cycling . for our example , we simulated data according to @xmath196 where @xmath197 , and @xmath198 . the first 10 coordinates of @xmath140 were set to 5 , with the remaining coordinates set to 0 . the design matrix @xmath35 was simulated by drawing its entries independently from a standard normal distribution . . top left panel details the path through the first three active variables , the remaining panels detail each active variable descent . ] figure [ figure4 ] plots the standardized gradient - correlations from algorithm [ a : l2boostpath ] using @xmath199 . as done earlier , we have converted iterations @xmath152 into step numbers @xmath14 along the @xmath167-axis . the plots show the behavior of each coordinate within an active set descent . the rug marks show each step @xmath14 for clarity , and dashed vertical lines indicate the step @xmath109 where the next step adds a new critical direction to the solution path . the top left panel shows the complete descent along the first three active variables . the remaining panels detail the coordinate behavior as the active set increases from one to three coordinates . the top right panel shows repeated selection of the @xmath68 direction shown in black . the last step along @xmath68 occurs at @xmath109 marked with the vertical dashed line , where the next step is along the @xmath70 direction shown in red . this point marks the beginning of the two - cycling behavior , which continues in the lower left panel . at each step , the algorithm systematically switches between the @xmath68 and @xmath70 directions , until an additional direction becomes more favorable . the cycling pattern is @xmath200 . the lower right panel demonstrates three - cycling behavior . here it is instructive to note that the order of selection within three - cycling is nondeterministic . in this panel the order starts as @xmath201 , but changes near @xmath202 to @xmath203 . as discussed later , nondeterministic cycling patterns are typical behavior of higher order cycling ( active sets of size greater than two ) . here we provide a formal limiting result of two - cycling . the result can be viewed as the analog of theorem [ dynamic.nu.size ] when the active set involves two variables . using a slightly modified version of 2boost we show that for arbitrarily small @xmath51 , if the algorithm cycles between its two active variables , it does so until a new variable enters the active set with the same absolute gradient - correlation . assume the active set is @xmath204 and that @xmath27 and @xmath92 are cycling according to @xmath205 . the @xmath14-step predictor for @xmath15 is @xmath206 where @xmath207 . the cycling pattern is assumed to persist for a minimum length of @xmath208 . it will simplify matters if the cycling is assumed to be initialized with strict equality of the gradient correlations : @xmath209 . with an arbitrarily small @xmath51 , this will force near equal absolute gradient - correlations at each step and by theorem [ gradient.correlation.equality ] will preserve the sign of the gradient - correlation . we assume @xmath210 it should be emphasized that the above assumptions represent a simplified version of 2boost . in practice , we would have @xmath211 where @xmath212 . however , for convenience we will not concern ourselves with this level of detail here . readers can consult @xcite for a more refined analysis . one way to ensure @xmath209 is to initialize the algorithm with the limiting predictor @xmath213 of theorem [ dynamic.nu.size ] obtained by letting along the @xmath27-descent . with a slight abuse of notation denote this initial estimator by @xmath214 . however , the fact that this specific @xmath214 is used does not play a direct role in the results . under the above assumptions , the following closed form expression for the @xmath14-step predictor under two - cycling holds . [ two.cycle.predictor ] assume that @xmath215 for @xmath216 . if @xmath217 , then for any @xmath218 satisfying @xmath219 , we have for each @xmath216 , @xmath220 , & \quad if $ m$ is odd,\vspace*{2pt } \cr { \mathbf{f}}_{0 } + v_{m}\rho_{k,1 } [ { \mathbf{x}}_k+(s-\nu r_{j , k}){\mathbf{x}}_j ] , & \quad if $ m$ is even , } \ ] ] where @xmath221 $ ] and @xmath222 . note that @xmath223 under the asserted conditions . to determine the above limit requires first determining when a new direction @xmath224 becomes more favorable . for @xmath225 to be more favorable at @xmath91 , we must have @xmath226 when @xmath14 is odd , or @xmath227 when @xmath14 is even . the following result determines the number of steps to favorability . for simplicity only the case when @xmath14 is odd is considered , but this does not affect the limiting result . [ criticalpoint.twocycle.theorem ] assume the same conditions as theorem [ two.cycle.predictor ] . then @xmath225 becomes more favorable than @xmath92 at step @xmath91 where @xmath14 is the largest odd integer @xmath228 such that @xmath229 where @xmath230 and @xmath231 clearly shares common features with . this is no coincidence . the bounds are similar in nature because both are derived by seeking the point where the absolute gradient - correlation between sets of variables are equal . in the case of two - cycling , this is the singularity point where @xmath27 , @xmath92 and @xmath225 are all equivalent in terms of absolute gradient - correlation . the following result states the limit of the predictor under two - cycling . [ dynamic step.two.cycles ] under the conditions of theorem [ two.cycle.predictor ] , the limit of @xmath232 as @xmath182 at the next critical direction @xmath233 equals @xmath234,\ ] ] where @xmath235 , @xmath179 , @xmath236 and @xmath237 . furthermore , @xmath238 , where for each @xmath225 , @xmath239 . this shows that the predictor moves along the combined direction @xmath240 taking a step size @xmath241 that makes the absolute gradient - correlation for @xmath233 equal to that of the active set @xmath204 . theorem [ dynamic step.two.cycles ] is a direct analog of theorem [ dynamic.nu.size ] to two - cycling . not surprisingly , one can easily show that this limit coincides with the lar solution . to show this , we rewrite @xmath242 in a form comparable to lar , @xmath243.\ ] ] recall that lar moves the shortest distance along the equiangular vector defined by the current active set until a new variable with equal absolute gradient - correlation is reached . the term in square brackets above is proportional to this equiangular vector . thus , since @xmath242 is obtained by moving the shortest distance along the equiangular vector such that @xmath244 have equal absolute gradient - correlation , @xmath242 must be identical to the lar solution . analysis of cycling in the general case where the active set @xmath245 is comprised of @xmath246 variables is more complex . in two - cycling we observed cycling patterns of the form @xmath247 , but when @xmath248 , 2boost s cycling patterns are often observed to be nondeterministic with no discernible pattern in the order of selected critical directions . moreover , one often observes some coordinates being selected more frequently than others . a study of @xmath249-cycling has been given by @xcite . however , the analysis assumes deterministic cycling of the form @xmath250 which is the natural extension of the two - cycling just studied . to accommodate this framework , a modified 2boost procedure involving coordinate - dependent step sizes was used . this models 2boost s cycling tendency of selecting some coordinates more frequently by using the size of a step to dictate the relative frequency of selection . under constraints to the coordinate step sizes , equivalent to solving a system of linear equations defining the equiangular vector used by lar , it was shown that the modified 2boost procedure yields the lar solution in the limit . interested readers should consult @xcite for details . now we turn our attention to the issue of correlation . we have shown that regardless of the size of the active set a new direction @xmath92 becomes more favorable than the current direction @xmath27 at step @xmath110 where @xmath109 is the smallest integer value satisfying @xmath251 using our previous notation , let @xmath173 and @xmath169 denote the left and right - hand sides of the above inequality , respectively . generally , large values of @xmath109 are designed to hinder noninformative variables from entering the solution path . if @xmath92 requires a large number of steps to become favorable , it is noninformative relative to the current gradient and therefore unattractive as a candidate . surprisingly , however , such an interpretation does not always apply in correlated problems . there are situations where @xmath92 is informative , but @xmath109 can be artificially large due to correlation . to see why , suppose that @xmath92 is an informative variable with a relatively large value of @xmath166 . now , if @xmath92 and @xmath27 are correlated , so much so that @xmath252 , then @xmath253 . hence , @xmath254 and @xmath255 due to . thus , even though @xmath92 is promising with a large gradient - correlation , it is unlikely to be selected because of its high correlation with @xmath27 . the problem is that @xmath92 becomes an unlikely candidate for selection when @xmath166 is close to @xmath106 . in fact , @xmath116 when @xmath117 so that @xmath92 can never become more favorable than @xmath27 when the two values are equal . we have already discussed the condition @xmath117 several times now , and have referred to it as the _ repressible condition_. repressibility plays an important role in correlated settings . we distinguish between two types of repressibility : weak and strong repressibility . weak repressibility occurs in the trivial case when @xmath256 . weak repressibility implies that @xmath257 . hence the gradient - correlation for @xmath92 and @xmath27 are equal in absolute value and @xmath92 , and @xmath27 are perfectly correlated . this trivial case simply reflects a numerical issue arising from the redundancy of the @xmath92 and @xmath27 columns of the @xmath35 design matrix . the stronger notion of repressibility , which we refer to as strong repressibility , is required to address the nontrivial case @xmath258 in which @xmath92 is repressed without being perfectly correlated with @xmath27 . the following definition summarizes these ideas . [ repressible.def ] we say @xmath92 has the strong repressible condition if @xmath117 and @xmath259 . we say that @xmath92 is ( strongly ) repressed by @xmath27 when this happens . on the other hand , @xmath92 has the weak repressible condition if @xmath92 and @xmath27 are perfectly correlated ( @xmath256 ) and @xmath117 . we present a numerical example of how repressibility can hinder variables from being selected . for our illustration we use example ( d ) of section 5 from @xcite . the data was simulated according to @xmath260 where @xmath261 , @xmath262 and @xmath263 . the first 15 coordinates of @xmath140 were set to 3 ; all other coordinates were 0 . the design matrix @xmath264_{100\times40}$ ] was simulated according to @xmath265 \\[-8pt ] { \mathbf{x}}_j & = & { \mathbf{z}}_3 + \tau{\boldsymbol{{{\varepsilon}}}}_j,\qquad j = 11,\ldots,15,\nonumber \\ { \mathbf{x}}_j & = & { \boldsymbol{{{\varepsilon}}}}_j,\qquad j > 15,\nonumber\end{aligned}\ ] ] where @xmath266 and @xmath267 were i.i.d.@xmath268 and @xmath269 . in this simulation , only coordinates 1 to 5 , 6 to 10 and 11 to 15 have nonzero coefficients . these @xmath167-variables are uncorrelated across a group , but share the same correlation within a group . because the within group correlation is high , but less than 1 , the simulation is ideal for exploring the effects of strong repressibility . for the first 5 coefficients from simulation : red points are iterations @xmath152 where the descent direction @xmath270 . variables 2 and 3 are never selected due to their excessively large @xmath169 step sizes : an artifact of the correlation between the 5 variables . the last panel ( bottom right ) displays @xmath271 for those iterations @xmath152 where @xmath272 . ] figure [ figure5 ] displays results from fitting algorithm [ a : l2boostpath ] for @xmath273 iterations with @xmath274 . the first 5 panels are the values @xmath275 against the iteration @xmath276 , with points colored in red indicating iterations @xmath152 where @xmath270 and @xmath27 is used generically to denote the current descent direction . notationally , the descent at iteration @xmath152 is along @xmath27 for a step size of @xmath277 , at which point @xmath225 becomes more favorable than @xmath27 and the descent switches to @xmath225 , the next critical direction . the value plotted , @xmath278 , is the step size for @xmath279 . whenever the selected coordinate is from the first group of variables ( we are referring to the red points ) one of the coordinates @xmath280 achieves a small @xmath169 value . however , coordinates @xmath281 and @xmath282 maintain very large values throughout all iterations . this is despite the fact that the two coordinates generally have large values of @xmath166 , especially during the early iterations ( see the bottom right panel ) . this suggests that 1 , 4 and 5 become active variables at some point in the solution path , whereas coordinates 2 and 3 are never selected ( indeed , this is exactly what happened ) . we can conclude that coordinates 2 and 3 are being strongly repressed by @xmath272 . interestingly , coordinate 4 also appears to be repressed at later iterations of the algorithm . observe how its @xmath166 values decrease with increasing @xmath152 ( blue line in bottom right panel ) , and that its @xmath169 values are only small at earlier iterations . thus , we can also conclude that coordinates @xmath283 eventually repress coordinate 4 as well . we note that the number of iterations @xmath273 used in the example is not very large , and if 2boost were run for a longer period of time , coordinates 2 and 3 will eventually enter the solution path ( panels 2 and 3 of figure [ figure5 ] show evidence of this already happening with @xmath169 steadily decreasing as @xmath152 increases ) . however , doing so leads to overfitting and poor test - set performance ( we provide evidence of this shortly ) . using different values of @xmath51 also did not resolve the problem . thus , similar to the lasso , we find that 2boost is unable to select entire groups of correlated variables . like the lasso this means it also will perform suboptimally in highly correlated settings . in the next section we introduce a simple way of adding @xmath0-regularization as a way to correct this deficiency . the tendency of the lasso to select only a handful of variables from among a group of correlated variables was noted in @xcite . to address this deficiency , @xcite described an optimization problem different from the classical lasso framework . rather than relying only on @xmath69-penalization , they included an additional @xmath0-regularization parameter designed to encourage a ridge - type grouping effect , and termed the resulting estimator `` the elastic net . '' specifically , for a fixed @xmath284 ( the ridge parameter ) and a fixed @xmath285 ( the lasso parameter ) , the elastic net was defined as @xmath286 to calculate the elastic net , @xcite showed that could be recast as a lasso optimization problem by replacing the original data with suitably constructed augmented values . they replaced @xmath287 @xmath288 and @xmath35 @xmath289 with augmented values @xmath290 and @xmath291 , defined as follows : @xmath292_{(n+p)\times1},\qquad { { \mathbf{x}}^*}= \frac{1}{\sqrt{1+\lambda } } \left[\matrix { { \mathbf{x}}\cr \sqrt{\lambda } { \mathbf{i } } } \right]_{(n+p)\times p } = [ { { \mathbf{x}}^*}_1,\ldots,{{\mathbf{x}}^*}_p].\hspace*{-35pt}\ ] ] the elastic net optimization can be written in terms of the augmented data by reparameterizing @xmath140 as @xmath293 . by lemma 1 of @xcite , it follows that can be expressed as @xmath294 which is an @xmath69-optimization problem that can be solved using the lasso . one explanation for why the elastic net is so successful in correlated problems is due to its decorrelation property . let @xmath295 . because the data is standardized such that @xmath296 [ recall ] , we have @xmath297 one can see that @xmath298 is a decorrelation parameter , with larger values reducing the correlation between coordinates . @xcite argued that this effect promotes a `` grouping property '' for the elastic net that overcomes the lasso s inability to select groups of correlated variables . we believe that decorrelation is an important component of the elastic net s success . however , we will argue that in addition to its role in decorrelation , @xmath298 has a surprising connection to repressibility that further explains its role in regularizing the elastic net . the argument for the elastic net follows as a special case ( the limit ) of a generalized 2boost procedure we refer to as elasticboost . the elasticboost algorithm is a modification of 2boost applied to the augmented problem . to implement elasticboost one runs 2boost on the augmented data , adding a post - processing step to rescale the coefficient solution path : see algorithm [ a : eboost ] for a precise description . for arbitrarily small @xmath51 , the solution path for elasticboost approximates the elastic net , but for general @xmath23 , elasticboost represents a novel extension of 2boost . we study the general elasticboost algorithm , for arbitrary @xmath23 , and present a detailed explanation of how @xmath298 imposes @xmath0-regularization . augment the data . set @xmath299 for @xmath300 . run algorithm [ a : l2boostpath ] for @xmath301 iterations using the augmented data . let @xmath302 denote the @xmath301-step predictor ( discard @xmath302 for @xmath303 ) . let @xmath304 denote the @xmath301-step coefficient estimate . rescale the regression estimates : @xmath305 . to study the effect @xmath298 has on elasticboost s solution path we consider in detail how @xmath298 effects @xmath306 , the number of steps to favorability [ defined as in but with @xmath287 and @xmath35 replaced by their augmented values @xmath290 and @xmath291 ] . at initialization , the gradient - correlation for @xmath308 is @xmath309 in the special case when @xmath299 , corresponding to the first descent of the algorithm , @xmath310 therefore , @xmath311 , and hence @xmath312.\ ] ] this equals the number of steps in the original ( nonaugmented ) problem but where @xmath35 is replaced with variables decorrelated by a factor of @xmath313 . for large values of @xmath298 this addresses the problem seen in figure [ figure5 ] . recall we argued that @xmath109 can became inflated due to the near equality of @xmath166 with @xmath106 . however , @xmath314 shrinks to zero with increasing @xmath298 , which keeps @xmath306 from becoming inflated . this provides one explanation for @xmath298 s role in regularization , at least for the case when @xmath298 is large . but we now suggest another theory that applies for both small and large @xmath298 . we argue that regularization is imposed not just by decorrelation , but through a combination of decorrelation and reversal of repressibility . s role is more subtle than our previous argument suggests . to show this , let us suppose that near - repressibility holds . we assume therefore that @xmath315 for some small @xmath316 . then , @xmath317}_{\mathrm{repressibility\ effect } } \\ & & \hphantom{\qquad= } \underbrace{- \log\biggl(1-\frac{r_{j , k}}{\sqrt{1+\lambda}}{\operatorname{sgn}}\biggl(r_{j , k } \biggl[\frac{1}{1+{\delta}}-\frac{1}{\sqrt{1+\lambda}}\biggr ] \biggr)\biggr)}_{\mathrm { decorrelation\ effect}}.\nonumber\end{aligned}\ ] ] the first term on the right captures the effect of repressibility . when @xmath318 is small , @xmath298 plays a crucial role in controlling its size . if @xmath319 , the expression reduces to @xmath320 which converges to @xmath321 as @xmath322 ; thus precluding @xmath92 from being selected [ keep in mind that is divided by @xmath323 , which is negative ; thus @xmath324 . on the other hand , any @xmath284 , even a relatively small value , ensures that the expression remains small even for arbitrarily small @xmath318 , thus reversing the effect of repressibility . the second term on the right of is related to decorrelation . if @xmath325 ( which holds if @xmath298 is large enough when @xmath326 , or for all @xmath284 if @xmath327 ) , the term reduces to @xmath328 which remains bounded when @xmath284 if @xmath329 . on the other hand , if @xmath330 , the term reduces to @xmath331 which remains bounded if @xmath329 and shrinks in absolute size as @xmath298 increases . taken together , these arguments show @xmath298 imposes @xmath0-regularization through a combination of decorrelation and the reversal of repressibility which applies even when @xmath298 is relatively small . these arguments apply to the first descent . the general case when @xmath332 requires a detailed analysis of @xmath333 . in general , @xmath334 we break up the analysis into two cases depending on the size of @xmath298 . suppose first that @xmath298 is small . then @xmath335 which is the ratio of gradient correlations based on the original @xmath35 without pseudo - data . if @xmath92 is a promising variable , then @xmath333 will be relatively large , and our argument from above applies . on the other hand if @xmath298 is large , then the third term in the numerator and the denominator of @xmath333 become the dominating terms and @xmath336 the growth rate of @xmath337 for the pseudo data is @xmath338 for a group of variables that are actively being explored by the algorithm . thus @xmath339 and our previous argument applies . as evidence of this , and to demonstrate the effectiveness of elasticboost , we re - analyzed using algorithm [ a : eboost ] . we used the same parameters as in figure [ figure5 ] ( @xmath273 and @xmath274 ) . we set @xmath340 . the results are displayed in figure [ figure6 ] . in contrast to figure [ figure5 ] , notice that all 5 of the first group of correlated variables achieve small @xmath341 values ( and we confirmed that all 5 variables enter the solution path ) . it is interesting to note that @xmath333 is nearly 1 for each of these variables . ) . now each of the first 5 coordinates are selected and each has @xmath333 values near one . ] ( top ) and @xmath342 ( bottom ) based on 250 independent learning samples . the distribution of coefficient estimates are displayed as boxplots ; mean values are given in red . ] to compare 2boost and elasticboost more evenly , we used 10-fold cross - validation to determine the optimal number of iterations ( for elasticboost , we used doubly - optimized cross - validation to determine both the optimal number of iterations and the optimal @xmath298 value ; the latter was found to equal @xmath343 ) . figure [ figure7 ] displays the results . the top row displays 2boost , while the bottom row is elasticboost ( fit under the optimized @xmath298 ) . the minimum mean - squared - error ( mse ) is slightly smaller for elasticboost ( 217.9 ) than 2boost ( 231.7 ) ( first panels in top and bottom rows ) . curiously , the mse is minimized using about same number of iterations for both methods ( 190 for 2boost and 169 for elasticboost ) . the middle panels display the coefficient paths . the vertical blue line indicates the mse optimized number of iterations . in the case of 2boost only 4 nonzero coefficients are identified within the optimal number of steps , whereas elasticboost finds all 15 nonzero coefficients . this can be seen more clearly in the right panels which show coefficient estimates at the optimized stopping time . not only are all 15 nonzero coefficients identified by elasticboost , but their estimated coefficient values are all roughly near the true value of 3 . in contrast , 2boost finds only 4 coefficients due to strong repressibility . its coefficient estimates are also wildly inaccurate . while this does not overly degrade prediction error performance ( as evidenced by the first panel ) , variable selection performance is seriously impacted . the entire experiment was then repeated 250 times using 250 independent learning sets . figure [ figure8 ] displays the coefficient estimates from these 250 experiments for elasticboost ( left side ) and 2boost ( right side ) as boxplots . the top panel are based on the original sample size of @xmath261 and the bottom panel use a larger sample size @xmath342 . the results confirm our previous finding : elasticboost is consistently able to group variables and outperform 2boost in terms of variable selection . finally , the left panel of figure [ figure9 ] displays the difference in test set mse for 2boost and elasticboost as a function of @xmath298 over the 250 experiments ( @xmath261 ) . negative values indicate a lower mse for elasticboost , which is generally the case for larger @xmath298 . the right panel displays the mse optimized number of iterations for 2boost compared to elasticboost . generally , elasticboost requires fewer steps as @xmath298 increases . this is interesting , because as pointed out , this generally coincides with better mse performance . a key observation is that 2boost s behavior along a fixed descent direction is fully specified with the exception of the descent length , @xmath64 . in theorem [ criticalpoint.theorem ] , we described a closed form solution for @xmath109 , the number of steps until favorability , where @xmath344 is the currently selected coordinate direction and @xmath345 is the next most favorable direction . theorem [ criticalpoint.theorem ] quantifies 2boost s descent length , thus allowing us to characterize its solution path as a series of fixed descents where the next coordinate direction , chosen from all candidates @xmath346 , is determined as that with the minimal descent length @xmath109 ( assuming no ties ) . since we choose from among all directions @xmath346 , @xmath109 , and equivalently the step length @xmath169 , can be characterized as measures to favorability , a property of each coordinate at any iteration @xmath152 . these measures are a function of @xmath51 and the ratio of gradient - correlations @xmath166 and the correlation coefficient @xmath106 relative to the currently selected direction @xmath27 . characterizing the 2boost solution path by @xmath109 provides considerable insight when examining the limiting conditions . when @xmath347 , 2boost exhibits active set cycling , a property explored in detail in section [ s : cyclingbehavior ] . we note that this condition is fundamentally a result of the optimization method which drives @xmath348 when @xmath51 is arbitrarily small . this virtually guarantees the notorious slow convergence seen with infinitesimal forward stagewise algorithms . the repressibility condition occurs in the alternative limiting condition @xmath349 . repressibility arises when the gradient correlation ratio @xmath166 equals the correlation @xmath106 . when @xmath259 , @xmath92 is said to be strongly repressed by @xmath27 , and while descending along @xmath27 , the absolute gradient - correlation for @xmath92 can never be equal to or surpass the absolute gradient - correlation for @xmath27 . strong repressibility plays a crucial role in correlated settings , hindering variables from being actively selected . adding @xmath0 regularization reverses repressibility and substantially improves variable selection for elasticboost , an 2boost implementation involving the data augmentation framework used by the elastic net . | we consider @xmath0boosting , a special case of friedman s generic boosting algorithm applied to linear regression under @xmath0-loss .
we study @xmath0boosting for an arbitrary regularization parameter and derive an exact closed form expression for the number of steps taken along a fixed coordinate direction .
this relationship is used to describe @xmath0boosting s solution path , to describe new tools for studying its path , and to characterize some of the algorithm s unique properties , including active set cycling , a property where the algorithm spends lengthy periods of time cycling between the same coordinates when the regularization parameter is arbitrarily small .
our fixed descent analysis also reveals a _ repressible condition _ that limits the effectiveness of @xmath0boosting in correlated problems by preventing desirable variables from entering the solution path . as a simple remedy ,
a data augmentation method similar to that used for the elastic net is used to introduce @xmath0-penalization and is shown , in combination with decorrelation , to reverse the repressible condition and circumvents @xmath0boosting s deficiencies in correlated problems . in itself , this presents a new explanation for why the elastic net is successful in correlated problems and why methods like lar and lasso can perform poorly in such settings . . |
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one of the most common ways to investigate the properties of a dynamical system is to study how it responds to controlled external perturbations . the response of a system to a weak perturbing field is related to its equilibrium fluctuations by the celebrated fluctuation dissipation relation @xcite . the response provides a direct measure of system dynamics and fluctuations . in a time - domain response measurement one uses a series of impulsive perturbations ( fig . [ fig1 ] ) and records some property of the system as a function of their time - delays . impulsive perturbations make it possible to study the free dynamical evolution of the system during the time delays unmasked by the time profile of the perturbing field . furthermore , the joint dependence on several time delays can be used to separate the contributions of different dynamical pathways . due to its dependence on multiple time delays this method is termed multidimensional . the response of a system is typically measured by the expectation value of some operator @xcite . this is a linear functional of the system probability density ( or the density matrix for quantum systems ) . multidimensional response have had considerable success in nonlinear spectroscopy , due to the ability to control and shape optical fields . applications range from spin dynamics in nmr @xcite , vibrational dynamics of proteins in infrared systems and electronic energy transfer in photosynthetic complexes as probed by visible pulses @xcite . these span a broad range of timescales from milliseconds to femtoseconds . interestingly , there exist non - linear functionals of probability densities which have interesting physical interpretations . one such quantity is the von - newman entropy @xmath0 . a related quantum nonlinear measure called the concurrence serves as a measure of quantum entanglement @xcite . the kullback - leibler distance ( kld ) or relative entropy , @xmath1 , which compares one probability distribution to another , is a nonlinear measure that had been found useful in many applications . this paper aims at developing multidimensional measures based on the kld . numerous applications of the kld differ in the probability distributions involved . the ratio of the probability of a stochastic path and its reverse at a steady state has been connected to a change of entropy @xcite . for an externally driven system a similar quantity was found to be related to the work done on the system @xcite . as a result , the kld which compares the path distribution to a distribution of reversed paths is a measure of the lack of reversibility of a thermodynamical process . for distributions in phase - space ( as opposed to path - space ) , the kld between the density of a driven system and the density of a reversed process @xcite , or the distance between the driven density and the corresponding equilibrium density @xcite ( for the same value of parameters ) were shown to be bounded by dissipated work in the process . the transfer of information through a stochastic resonance is quantified by the kld between the probability distributions with and without the external input @xcite . the ability of neuronal networks to retain information about past events was characterized by the fisher - information @xcite , which is closely related to the kld between a distribution and the one obtained from it by a small perturbation . we shall examine the response of a system to impulsive perturbations which drive it out of a stationary ( steady state or equilibrium ) state . the kld between the distribution before and after the perturbation does not correspond to an entropy , or work . however , since it compares the perturbed and unperturbed densities , it characterize how `` easy '' it is to drive the system away from its initial state . in ordinary response theory , one compared the expectation values of some operator taken over the perturbed and unperturbed probability density . this depends on the specific properties of the observed operator . the kld is a more robust measure for the effect of the impulsive perturbation on the probability density . by expanding the kld in the perturbation strength we obtain a hierarchy of kullback - leibler response functions ( klrf ) . these differ qualitatively from the hierarchy of ordinary response functions ( orf ) , since they are nonlinear in the probability density . the klrf serve as a new type of measures characterizing the dynamics and encoding different information than the orf . for example , the second order klrf , which we connect to the fisher - information , is found to exhibit qualitatively different dependence on the time delays , depending on whether the system is perturbed out of a steady state or out of thermal equilibrium . this is in contrast to the corresponding orf . the fluctuation dissipation relation @xcite , which is linear in the density matrix , can also distinguish between systems driven out - of - equilibrium and out of a steady state . the kld offer a different window into this aspect . the structure of the paper is as follows . in sec . [ formalsec ] we describe the multidimensional measures and present the two heirarchies of orf and klrf response functions . these are then calculated using a formal perturbation theory in the coupling strength to the external perturbation . in sec . [ overdampedsec ] we show that for systems undergoing overdamped stochastic dynamics the non - linear klrf are naturally described using a combination of the stochastic dynamics and its dual dynamics . in sec . [ mastersec ] we extend the results of sec . [ overdampedsec ] to discrete markovian systems with a finite number of states . our results are discussed in sec . [ discsec ] . we consider a system initially at a stationary state ( either equilibrium or a steady state ) , which is perturbed by a series of short pulses , as depicted in fig . [ fig1 ] . and subjected to impulsive perturbations . the @xmath2th pulse is centered at @xmath3 and its strength is denoted by @xmath4 . @xmath5 are the time intervals between successive pulses . [ fig1 ] ] the probability distribution describing the driven system at time @xmath6 , @xmath7 , depends parametrically on @xmath8 , the strength of the @xmath9th pulse , as well as on the time differences between the pulses @xmath10 with @xmath11 . ordinary response theory focuses on the expectation value of some observable @xmath12,\ ] ] and its dependence on the parameters @xmath13 . the lowest ordinary response functions ( orf ) are @xmath14 and so forth . the time differences in eqs . ( [ defr2])-([defr4 ] ) can be expanded in terms of the time delays between the pulses , @xmath15 . @xmath16 are used to investigate various properties of the unperturbed dynamics , such as the existence of excited modes , and the relaxation back to a steady state . here , we focus on different , but closely related quantity . instead of studying an expectation value of an observable , we focus on a quantity that compares the perturbed and unperturbed probability distributions . the kld , also known as the relative entropy , is defined as @xmath17 the kld vanishes when the two distributions are equal @xmath18 , and is positive otherwise , @xmath19 for @xmath20 @xcite . note that the kld is not a true distance since @xmath21 and it does not satisfy the triangle inequality . the kld measures the dissimilarity between two distributions . it had found many applications in the field of information theory @xcite . for instance , the mutual information between two random variables @xmath22 is @xmath23 , where @xmath24 is the joint distribution while @xmath25 ( @xmath26 ) is the marginal distribution of @xmath27 ( @xmath28 ) . in the present application the kullback - leibler distance is a measure for the deviation of the system from its initial state . in a manner similar to the definition of the orf , we define a klrf hierarchy by taking derivatives of the kld with respect to pulse strengths , and displaying them with respect to the time delays @xmath29 all the derivatives are calculated at @xmath30 , and we have used the relation @xmath31 . this is also true for all other @xmath32 derivatives in the following . to keep the notation simple we will not state this explicitly . higher order klrf are defined similarly . it is important to note that the orf are linear in @xmath33 whereas the kld are nonlinear . we thus expect the kld to carry qualitatively different information about the dynamics . the second derivative ( [ second ] ) , known as the fisher memory ( or information ) matrix , plays an important role in information theory , since the cramer-rao inequality means that it is a measure of the minimum error in estimating the value of a parameter of a distribution @xcite . the fisher information has been used recently to analyze the survival of information in stochastic networks @xcite . conservation of probability implies that the first klrf vanishes @xmath34 the second derivative , the fisher memory matrix , is given by @xmath35 a straightforward calculation allows to recast the third order derivative of @xmath36 in terms of products of lower order derivatives @xmath37 in what follows the derivatives will be calculated perturbatively in @xmath38 . it is important to note that the @xmath39th derivative of @xmath40 has contributions from interaction with at most @xmath39 pulses . the contribution from the linear component , which interacts with @xmath39 pulses , has the same structure of the perturbation theory for observables ( which is also linear ) . however , since @xmath40 is a non linear function of @xmath33 the @xmath39th derivative contains a non - linear contribution which is a product of lower order contributions for @xmath33 . the klrf encode qualitatively new information about the system dynamics in comparison to the orf . the time evolution of the probability distribution is given by @xmath41 this formal equation is quite general , and can describe either hamiltonian ( unitary ) or stochastic dynamics where the operator @xmath42 will accordingly be the liouville , or the fokker - planck operator . for a system subjected to a time dependent weak perturbation we can write @xmath43 where we assume that the unperturbed system is time independent and is intially in a steady state , @xmath44 , so that @xmath45 . we consider an impulsive perturbation of the form @xmath46 where @xmath47 describes the action of a pulse on the probability distribution , and @xmath8 is the overall strength of the @xmath9th pulse . using these definitions , the state of the system at time @xmath6 can be expanded as a power series in the number of interactions with the pulses @xmath48 the partial corrections for the density , @xmath49 , appearing in eq . ( [ pseries ] ) , contain all the information necessary for computing both the klrf and orf . they are given by @xmath50 here @xmath51 $ ] is the free propagator of the unperturbed system . conservation of probability requires that @xmath52 , which , in turn , means that @xmath53 . ( [ pseries])-([defs2 ] ) can be used to calculate the logarithmic derivatives , which then determine the klrfs . we will only need the first two logarithmic derivatives , which are given by @xmath54 and @xmath55 to calculate the orf , we substitute eq . ( [ pseries ] ) in eq . ( [ avga ] ) , resulting in @xmath56 it is interesting to compare the expressions of the klrf the orf . we calculate @xmath57 and @xmath16 for @xmath58 perturbing pulses . to leading order , we find @xmath59 at the next order , we compare the fisher information to the second order orf , @xmath60 and @xmath61 the non - diagonal elements of @xmath62 depend on the two delay times . expressions for the third order response functions are given in app . [ thirdorderformal ] . both @xmath16 and @xmath57 depend on the same set of @xmath58 time intervals with some important differences . @xmath63 vanishes , while the linear response @xmath64 does not . @xmath65 and @xmath66 have a different structure : @xmath66 can be calculated from the second order correction to the density ( or @xmath67 ) while @xmath65 is determined from a product of @xmath68 s describing the first order interaction with different pulses . this difference reflects the non - linear dependence of the klrf on @xmath33 , and also applies to higher orders . a comment is now in order regarding our choice of the kld ( [ defkld ] ) . we have chosen to use @xmath69 as the measure for the effect of the perturbations . @xmath70 would have been equally suitable . however , as discussed in app . [ hamiltonianapp ] , the leading order of both klds in the strength of the perturbation , i.e. their fisher informations , coincide . therefore all the following results pertaining to the fisher information would hold for either choice . in the following we use the formal results of sec . [ formalsec ] to calculate the leading order orf and klrf for a system undergoing overdamped stochastic dynamics . we show that the fisher information is related to a forward - backward stochastic process . the backward part is driven by the @xmath44-dual process , which will be simply referred to as the dual in what follows . the fisher information is found to exhibit qualitatively different properties for systems perturbed from equilibrium , or from a steady state . we also use the eigenfunctions and eigenvalues of the dynamics to derive explicit expressions for several low order orf and klrf . in stochastic dynamics the probability density plays the role of a reduced density matrix , which depends on a few collective coordinates . in this reduced description the entropy @xmath71 typically increases with time . this should be contrasted with a description which includes all the degrees of freedom , where the dynamics is unitary and the entropy does not change in time . for completeness unitary dynamics is discussed in app . [ hamiltonianapp ] . the fisher information can be represented in terms of the dual stochastic dynamics . this interesting property reflects its non - linear dependence on @xmath33 . we examine a stochastic dynamics of several variables @xmath72 , given by @xmath73 here we use the ito stochastic calculus . ) . both methods are equally viable as long as they are used in a consistent manner . details can be found is ref . the noise terms are assumed to be gaussian with @xmath74 with @xmath75 a symmetric positive definite matrix . while for many systems this matrix does not depend on the coordinate , @xmath76 , this assumption will not be used in what follows . equation ( [ stochasticoriginal ] ) is equivalent to the fokker - planck equation @xmath77 in what follows we present the dual dynamics , which can be loosely thought as the time reversed dynamics : it have the same steady state , but with reversed steady state current . we consider the current density @xmath78 the fokker - planck equation can be written in terms of the current , @xmath79 the steady - state is the solution of @xmath80 we write @xmath81 which defines @xmath82 . for systems at equilibrium @xmath82 is simply the potential . however , this is not the case for general steady states . the steady state current can be written as @xmath83 after some algebra , the generator of the stochastic dynamics can be written in terms of the steady state density and currents @xcite @xmath84 the dual dynamics is given by @xmath85 with @xmath86 a straightforward calculation gives @xmath87 with @xmath88 it is a simple matter to verify that the dual dynamics has the same steady state as the original one , but the steady state currents have opposite signs . it can be simulated by integrating the ito stochastic equation @xmath89 the dual dynamics reverses the non conservative forces in the system . this relates the joint probability to go from one place to another in the original dynamics to the joint probability of the reversed sequence of events in the dual dynamics @xcite @xmath90 the left hand side of eq . ( [ conddual ] ) is the joint steady state probability to first the system at @xmath27 , and at @xmath91 after a time @xmath92 . the right hand side is the joint probability of the reversed sequence of events , but for a modified dynamics . when this modified dynamics is the dual these joint probabilities become equal . we next turn to discuss the system s response to a series of impulsive perturbation . we assume that the perturbation is of the form @xmath93 with @xmath94 as a potential field perturbing the system . using eq . ( [ defs1 ] ) we have @xmath95 where @xmath96 with the help of equation ( [ conddual ] ) , we obtain @xmath97 we now have all the tools needed to compare the orf and klrf for overdamped stochastic dynamics . the leading order response function is given by @xmath98 while @xmath99 . at the next order , we have @xmath100 and @xmath101 b({\bf x}_0 ) \rho_0 ( { \bf x}_0 ) . \label{r2deriv}\ ] ] some insight into the structure of different response functions can be gained by representing them as ensemble averages over stochastic trajectories . the first order response function can be simulated directly using stochastic trajectories of the original dynamics . the appearance of a derivative of the conditional probability complicates the direct simulation of @xmath102 . it may be possible to circumvent this difficulty using the finite field method , where one combined simulations with and without a finite , but small perturbation @xcite . @xmath103 can be simulated with trajectories which follow the original dynamics for time @xmath104 and then the dual dynamics for time @xmath105 . systems at equilibrium are self dual , allowing to substitute @xmath106 in eq . ( [ q12usingdual ] ) . as a result the fisher information only depends on a single time variable @xmath107 . this is in contrast to systems which are perturbed from a nonequilibrium steady state , whose fisher information is a two dimensional function of @xmath92 and @xmath105 . the fisher information is therefore qualitatively different for systems which are perturbed out of a steady state , or out of an equilibrium state . for the self - dual case @xmath103 has the same structure as @xmath64 , up to a replacement of @xmath94 with @xmath108 . however , in the general , non self - dual case , the structure of @xmath109 is manifestly different than that of @xmath64 and @xmath66 . an alternative approach for the calculation of the fisher information , as well as higher order klrf , uses eigenfunction expansions for the density @xmath33 . it will be sufficient to examine a simple one dimensional model where @xmath110 is a fokker - planck operator and the perturbation is given by eq . ( [ la ] ) . the propagation of the unperturbed system can be described in terms of the eigenvalues and eigenfunctions of @xmath110 . the right eigenfunctions satisfy @xmath111 similarly , the left eigenfunctions , @xmath112 , satisfy @xmath113 . it is assumed that the right and left eigenfunctions constitute a byorthogonal system , that is @xmath114 any probability density can be expanded in terms of right eigenfunctions , @xmath115 with @xmath116 we consider systems perturbed out of thermal equilibrium , with a probability density @xmath117 . in this case the left eigenvalues are simply related to the right eigenvalues . for variable @xmath27 which is even with respect to time reversal , this relation takes the form @xcite @xmath118 as a result , one can use only the left eigenfunctions , which in this case satisfy the orthogonality condition @xmath119 we note that @xmath120 and @xmath121 correspond to the equilibrium distribution . our goal is to calculate the response of the system to a perturbation with a coordinate dependent operator @xmath94 . several types of integrals appear repeatedly in the calculation , and it will be convenient to introduce an appropriate notation . one comes from the need to decompose the probability distribution into eigenstates after each interaction with a pulse @xmath122 = -\int dx \frac{\partial q_m}{\partial x } \frac{\partial a}{\partial x } q_n(x ) \rho_0(x),\label{defb}\ ] ] where integration by parts was used in the second equality . ( we assume that @xmath123 falls of fast enough to eliminate boundary terms . ) the calculation of the response also involves an evaluation of the average of an observable , which , in the current setting , leads to integrals of the form @xmath124 it is straightforward to calculate @xmath68 and @xmath67 with the help of eq . ( [ defb ] ) . we find @xmath125 and @xmath126 we now calculate the first few orf and klrf . the first order response functions is @xmath127 similarly , the second order response function is given by @xmath128 @xmath66 should be compared to the klrf of the same order , namely the fisher information @xmath129},\ ] ] which is calculated with the help of eqs . ( [ fisher2 ] ) and ( [ orthleft ] ) . again , the fisher information of systems perturbed out of equilibrium depends only on @xmath130 . for a system initially in a steady state the fisher information is @xmath131 - \lambda ( m ) t_2 } \int dx \rho_0^{-1 } ( x ) \rho_n ( x ) \rho_m ( x ) . \label{geneigenq}\ ] ] however , since the relation eq . ( [ lrrelation ] ) does not hold in this case , the integral in eq . ( [ geneigenq ] ) does not vanish for @xmath132 , and the fisher information becomes a two dimensional function of @xmath92 and @xmath105 . @xmath133 and @xmath134 are calculated in app . [ thirdordereigen ] , where it is shown that @xmath133 is a three dimensional function of all its time variables . this qualitatively different signature of systems initially at steady state vs equilibrium is unique to the fisher information , due to its quadratic dependence on @xmath135 . it does not apply to higher order quantities , such as @xmath136 . the eigenfunctions for a simple example , of an harmonic oscillator with an exponential perturbation , are presented in app . [ exampleosc ] . the description of the klrf in terms of a combination of the regular stochastic dynamics and its dual holds also for markovian systems with a finite number of states . below we derive a simple expression for the fisher information of a stochastic jump process in terms of the dual dynamics of the original process . consider a system with a finite number of states , undergoing a markovian stochastic jump process described by the master equation @xmath137 @xmath138 is a vector of probabilities to find the system in its states and @xmath139 is the transition rate matrix @xcite . its off diagonal elements are positive , @xmath140 for @xmath141 , and express the rate of transitions from state @xmath58 to @xmath9 , given that the system is at @xmath58 . the diagonal elements satisfy @xmath142 . we further assume that there exist a unique steady state , @xmath143 , satisfying @xmath144 and that at this steady state there is a non vanishing probability to find the system at each of the states @xmath9 , @xmath145 . the master equation is one of the simplest models for irreversible stochastic dynamics . below we briefly describe some of its relevant properties , such as the backward equation , and its dual dynamics . one can define an evolution operator @xmath146 which satisfy the equation of motion @xmath147 with the initial condition @xmath148 . for this model a dynamical variable @xmath149 is a vector with a value corresponding to each state of the system . its expectation value is given by @xmath150 instead of calculating this average by propagating @xmath138 , one can define a time dependent dynamical variables , @xmath151 , such that @xmath152 . this is the analogue of the scrdinger and heisenberg pictures in quantum mechanics . the equation of motion for @xmath151 is easily shown to be @xmath153 this equation is known as the backward equation , which turns to be related to the dual dynamics . ( the backward equation is also written as a function of an _ initial _ rather than a final time . ) @xmath154 and @xmath155 have the same eigenvalues . however , the roles of the right and left eigenvectors are interchanged and @xmath151 decays to a uniform vector with at long times . let us now define a diagonal matrix @xmath156 , using @xmath157 . the dual evolution is then defined as @xmath158 the fact that @xmath159 is built from the steady state of @xmath154 , @xmath160 , guaranties that @xmath161 is a physically reasonable rate matrix , that is , that is satisfies @xmath162 . the dual dynamics describes a physically allowed process which has the same steady state as the original process it was derived from . however , at this steady state the dual currents have opposite signs compared to the steady state currents of the original dynamics . a process is self - dual , that is , @xmath163 , if and only if it satisfies detailed balance . self duality is therefore related to being in thermal equilibrium . consider a system subjected to several impulsive perturbations @xmath164 where @xmath165 corresponds to some physical perturbation @xmath166 . here the free evolution is given by @xmath167 similarly the evolution during an impulsive perturbation can be described by @xmath168 where @xmath169 is arbitrarily small . we expand the exponent in eq . ( [ impulse ] ) , and collect all terms of the same order in @xmath170 . the expression for the fisher information , following eq . ( [ fisher2 ] ) is @xmath171 this expression can be simplified by writing one of the propagators in terms of the dual process , using the relation @xmath172 after some algebraic manipulations , we find @xmath173 in eq . ( [ fisherdual ] ) we have defined @xmath174 . @xmath175 is not the dual of @xmath165 since @xmath159 is not composed of the eigenvector of @xmath165 which corresponds to a vanishing eigenvalues . ( @xmath159 is composed of the eigenvalue of @xmath176 . ) equation ( [ fisherdual ] ) is the analogue of eq . ( [ q12usingdual ] ) for discrete systems . it demonstrates that for this model the fisher information can be simulated using a combination of the ordinary process and its dual . however one must also include a ( possibly artificial ) dual perturbation , @xmath175 , which can nevertheless be computed from the known physical one . as before , equation ( [ fisherdual ] ) shows that the fisher information , for self - dual systems , is a function of @xmath130 . as a side remark , the model considered here has only even degrees of freedom under time reversal . more general models may also include odd degrees of freedom , such as momenta . in that case one may speculate that there would be anti - self - dual systems whose dual turns out to be the time reversed dynamics . this would lead to a fisher information depending on @xmath92 alone , as is the case for unitary dynamics . in this work we have studied a system driven from its steady state by a sequence of impulsive perturbations . we have defined a new set of measures for the response of the system to the perturbation , the klrf , which are given by the series expansion of the kullback - leibler distance between the perturbed and unperturbed probability distributions . at each order the klrf and orf depend on the same time differences between the pulses . however , there are important differences stemming from the nonlinear dependence of the klrf on @xmath33 . the expression for the klrf , for instance eqs . ( [ fisher2 ] ) and ( [ q123 ] ) reveal quantities which can be simulated using several trajectories which end at the same point . we have shown that a simpler , but equivalent description exists . it uses the dual dynamics which allows to `` run some of the trajectories backward '' . this description is especially appealing for the fisher information . instead of viewing the fisher information as composed over sum of pair of trajectories joined at their end point one can view it as an average of contributions of a single forward - backward trajectory . another difference between the klrf and the orf has to do with the appearance of derivatives of conditional probabilities , which for _ deterministic _ systems would correspond to groups of very close trajectories . these first appear in @xmath67 , see for instance eqs . ( [ r2deriv ] ) for @xmath66 . the nonlinear character of the klrf means that such terms appear in comparatively higher order of the perturbation theory . for example , @xmath67 contributes to @xmath66 but not to the fisher information . instead it first contributes to @xmath133 . we have demonstrated that the fisher information behaves in a qualitatively different way depending on whether the system is perturbed from equilibrium or from an out - of - equilibrium steady state . for the classically stochastic systems considered here we have seen that in the former case @xmath177 . that is , the fisher information is a one dimensional function of the two time delays . this qualitative difference results from the self - duality of equilibrium dynamics , which is another expression of the principle of detailed balance . @xmath178 does not show a similar reduction of dimension [ see eq . ( [ eigenq3 ] ) ] . this property is special to the fisher information . we have focused on the properties of the fisher information for overdamped stochastic dynamics . do other types of systems exhibit similar behavior ? in app . [ hamiltonianapp ] we consider deterministic hamiltonian systems . in that case the unitary dynamics results in a fisher information which depends only on the time @xmath92 . it will be of interest to study how ( not overdamped ) stochastic systems bridge between the overdamped and unitary limits . we showed that the klrf can serve as a useful measure characterizing the system s dynamics . they encode information which differs from the information encoded in the orf . this is demonstrated by the ability of the fisher information to distinguish between systems perturbed out of equilibrium or out of a non - equilibrium steady state . we expect other useful properties of the klrf to be revealed by further studies . one of us ( s. m. ) wishes to thank eran mukamel for most useful discussions . the support of the national science foundation ( grant no . che-0745892 ) and the national institutes of health ( grant no . gm-59230 ) is gratefully acknowledged . for completeness , in this appendix we discuss the application of nonlinear response theory to deterministic hamiltonian systems . we use simple examples to clarify the relation between the response of a system and the kullback - leibler distance . the hamiltonian of the system is assumed to be of the form @xmath179 , where @xmath166 is the perturbation . we start with some general comments stemming from the fact that the dynamics in phase space is an incompressible flow . the state of a classical system is described by its phase space density @xmath33 . it is important to note that this is the full probability density , which includes information on all the degrees of freedom . let us denote the propagator of the classical trajectories by @xmath180 , so that @xmath181 where @xmath182 denotes a phase space point ( all coordinates and momenta ) . similarly , the propagator for the probability distribution is denoted by @xmath183 . liouville theorem tells us that phase space volumes do not change in time . as a result , there is a simple relation between the phase space density at different times , @xmath184 the density is simply transported with the dynamics in phase space . the density at @xmath185 at time @xmath6 is equal to the density at @xmath182 at time @xmath186 . this property of the phase space dynamics have an interesting consequence . any integral whose integrand depends locally on @xmath33 alone is time independent , since the values of the integrand are just transported around by the dynamics . an interesting example is the entropy function @xmath187 let us change the integration variable to @xmath188 , which is just the phase space point which would flow to @xmath182 after a time @xmath6 . liouville theorem assures us that the jacobian of the transformation is unity and therefore @xmath189 where we have used eq . ( [ transport ] ) . it is clear that this entropy does not depend on time . ( [ constent ] ) is a result of the unitary evolution in phase space . it connects with all the intricate problems related to the emergence of macroscopic irreversibility out of microscopic reversible dynamics . this deep problem is beyond the scope of the current paper . such problems are circumvented when one uses a reduced probability density , whose dynamics is irreversible to begin with . unitary dynamics lead to an interesting result for the reverse kullback - leibler distance @xmath190 we have seen that the first term is a constant of the motion . assuming a hamiltonian system intially in equilibrium @xmath191 . we have @xmath192 where @xmath193 is the free energy of the initial equilibrium state . @xmath194 is therefore linear in @xmath33 , and thus it is equivalent to a calculation of response functions ! the general comments above raise two interesting points . first , the fact that terms such as @xmath195 are constant means that their expansion in powers of pulse strengths @xmath8 turns out to have only the constant term , all other terms in the expansion must vanish . the second point of interest has to do with the relation between the kullback - leibler distance and its reverse . generally , @xmath196 . we are interested in systems pertubed by a series of pulses and compare the initial and final distributions . in that case we can use unitarity to change variables from the phase space points at the final time to the points at the initial time which are connected to it by the dynamics . this gives @xmath197 here @xmath198 , that is , @xmath199 is the density that would evolve to the equilibrium density under the influence of the pulses . it is not equal to @xmath200 , which tells us that the distance and its reverse are not equal . while in general the distance and its reverse are not equal , when the distance between the distributions is small , its easy to show that their leading order expansion in @xmath201 is the same . loosely speaking @xmath202 we can deduce that the fisher information , which is determined by this leading order , could be obtained from a calculation of response for systems with unitary evolution . ( since it could also be obtained from the reverse distance . ) our last general point is also related to unitarity , but has to do with the fisher information . according to eq . ( [ fisher2 ] ) the fisher information is built from two partial densities . ( more accurately , density differences . ) these evolve with respect to the same hamiltonian @xmath203 between interactions with the pulses . as a result , the integral in eq . ( [ fisher2 ] ) do not depend on @xmath105 , for the same reason that caused the entropy @xmath204 to be time independent . we find that @xmath205 as a result of the unitarity of the phase space dynamics . similar independence on the final time interval would also appear for higher order terms in the expansion of the kullback - leibler distance . the simplest system that can serve as an example is a single harmonic oscillator @xmath206 we take the perturbation to be @xmath207 for this system it is trivial to solve for the free evolution @xmath208 this relation can be inverted , expressing @xmath209 in terms of @xmath210 @xmath211 eq . ( [ reverseosc ] ) will be useful when one propagates probability distributions in time . the equilibrium distribution of the harmonic oscillator is a gaussian @xmath212 we also note that @xmath213 is the linear order correction for the density just after interaction with one pulse . to calculate @xmath68 we need to propagate eq . ( [ tempdrho ] ) . with the help of eqs . ( [ transport ] ) and ( [ reverseosc ] ) we find @xmath214 } \rho_0 ( q , p).\ ] ] in the derivation we have used the fact that @xmath44 is invariant under the evolution with respect to @xmath203 . to calculate @xmath67 we operate with @xmath215 on @xmath68 , and then propagate the resulting correction for the density for a time interval @xmath105 . a straightforward calculation leads to @xmath216 + \frac{1}{q_0 m \omega } \sin \omega t_1 \right ) \right\ } \\ & \times & \exp \left[-\frac{q}{q_0 } \left ( \cos \omega t_2 + \cos \omega ( t_1+t_2)\right ) + \frac{p}{m \omega q_0}\left(\sin \omega t_2 + \sin \omega ( t_1+t_2 ) \right)\right ] \rho_0 \nonumber\end{aligned}\ ] ] we now turn to calculate the first two orf , using eqs . ( [ formalr1 ] ) and ( [ formalr2 ] ) . the calculation of @xmath64 is cumbersome but straightforward . @xmath217.\ ] ] the calculation of @xmath66 is more involved , and we only include the final result @xmath218 \\ \times \exp \left [ \frac{1}{m \beta \omega^2 q_0 ^ 2 } \left\ { \frac{3}{2 } + \cos \omega t_1 + \cos \omega t_2 + \cos \omega ( t_1+t_2)\right\}\right].\end{gathered}\ ] ] we would like to compare these response functions to the klrf , and in particular to the fisher information , which can be calculated using eq . ( [ fisher2 ] ) . the calculation can be simplified by using the classical coordinates right after the second interaction with the pulse as integration variables . due to unitarity , the fisher information only depends on the time difference @xmath92 , see also the discussion in the previous subsection . we get @xmath219 \exp \left [ \frac{1}{\beta m \omega^2 q_0 ^ 2 } \left ( 1+\cos \omega t_1\right)\right].\ ] ] this expression seems similar to the first order response function . one can indeed show that they are related by @xmath220 it will be interesting to check whether this expression could be generalized to any hamiltonian system ( with unitary dynamics ) . in the main text we have calculated the orf and klrf to second order . in the appendix we present the third order quantities , @xmath134 , and @xmath133 . following the calculations performed in sec . [ formalsec ] , the third order response functions are given by @xmath221 and @xmath222 \right . \\ \left . -2 \rho_0^{-2}(x ) { \cal s}^{(1)}(x;t_1+t_2+t_3 ) { \cal s}^{(1)}(x ; t_2+t_3 ) { \cal s}^{(1)}(x ; t_3 ) \right\},\label{q123}\end{gathered}\ ] ] where @xmath223 here we present expressions for the third order orf and klrf in term of the eigenvalues and eigenfunctions of the stochastic dynamics . at high orders the nonlinear character of the contributions to the kullback - leibler distance result in integrals with products of several eigenfunctions . here we will only need the one with three eigenfunctions @xmath224 the third order orf is given by @xmath225 this orf should be compared to the third order klrf , which is calculated using eq . ( [ q123 ] ) . we find @xmath226 } \right . + e^{-\lambda(n ) \left [ t_1 + t_2 \right ] } e^{-\lambda ( m ) \left [ t_2 + 2 t_3\right ] } + e^{-\lambda ( n ) t_1 } e^{-\lambda ( m ) \left [ t_2 + 2 t_3\right ] } \right\ } \\ - 2 \sum_{nml } { \cal b}_{n0 } { \cal b}_{m0 } { \cal b}_{l0 } { \cal j}_{nml } e^{-\lambda ( n ) \left [ t_1+t_2 + t_3\right ] } e^{-\lambda ( m ) \left [ t_2 + t_3\right ] } e^{-\lambda ( l ) t_3}. \label{eigenq3}\end{gathered}\ ] ] the expression for @xmath133 , presented in eq . ( [ eigenq3 ] ) , has three terms in which the orthogonality condition ( [ orthleft ] ) has been used , pointing to a reduction of dimension in the time dependence of this specific term . however , the time combinations in these terms are all different . in addition , the fourth term in eq . ( [ eigenq3 ] ) clearly depends on all its time variables . we conclude that in contract to the fisher information , the higher order klrf depend on all their time variables . the reduction of dimension is therefore specific for the fisher information . it results from the fact that it is built out of a single product of two density corrections . in this appendix we consider a simple example of an overdamped harmonic oscillator with a potential @xmath227 , with a perturbing potential @xmath228 . for this system it is possible to write explicit expressions for the eigenvalues and eigenfunctions of the fokker - planck operator , as well as to perform several of the integral , defined in sec . [ overdampedsec ] . in this case @xmath229 \\ \hat{\cal l}_a \rho & = & - \frac{\alpha}{q_0 \gamma } \frac{\partial}{\partial q } \left [ e^{-q / q_0 } \rho \right].\end{aligned}\ ] ] this model has been studied extensively @xcite . the equilibrium density is given by @xmath230 the left eigenfunctions , and the eigenvalues , of this model are @xmath231 here , @xmath232 are the hermit polynomials . for instance , @xmath235 one can substitute eq . ( [ hn2 ] ) for the hermit polynomial , and use integration by parts . this leads to @xmath236 this is a gaussian integral which is easily evaluated to give @xmath237 one can also easily calculate the integrals @xmath238 by the same technique . one finds @xmath239 comparing eqs . ( [ osccalc ] ) and ( [ osccalb0 ] ) , we see that for this model @xmath240 as a result , there is a simple relation between the fisher information and the first order response function , @xmath241 this relation is a special result for this model , and is not expected to hold for other systems . one can also obtain explicit results for @xmath242 with non vanishing indices . however , the calculation and the result are quite combersome , and are omitted . we were not able to calculate @xmath243 explicitly . | by subjecting a dynamical system to a series of short pulses and varying several time delays we can obtain multidimensional characteristic measures of the system .
multidimensional kullback - leibler response function ( klrf ) , which are based on the kullback - leibler distance between the initial and final states , are defined .
we compare the klrf , which are nonlinear in the probability density , with ordinary response functions ( orf ) obtained from the expectation value of a dynamical variable , which are linear .
we show that the klrf encode different level of information regarding the system s dynamics . for overdamped stochastic dynamics
two dimensional klrf shows a qualitatively different variation with the time delays between pulses , depending on whether the system is initially in a steady state , or in thermal equilibrium . |
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the strong couplings among soft qcd gluons manifest themselves in a variety of complex long - distance phenomena . most of them are thoroughly entwined with the vacuum state , as illustrated by such prominent examples as quark confinement , spontaneous chiral symmetry breaking , vacuum tunneling processes , the ensuing @xmath0 structure as well as large gluon condensates . despite the apparent diversity of these and other effects , however , the essence of the underlying dynamics is often expected to involve just a few soft gluonic modes . the quest for these long - wavelength excitations began shortly after the inception of qcd and has inspired the development of various kinds of vacuum models , based e.g. on glueball condensation @xcite , gaussian stochastic processes @xcite , gluonic domains @xcite and instanton @xcite as well as meron @xcite ensembles . over the last decade , lattice simulations increasingly assisted in the search for predominant infrared ( ir ) gluon fields , mostly by means of numerical `` filtering '' @xcite and gauge - fixing @xcite techniques . these simulations are now beginning to generate quantitative insights into the role of instantons and their size distribution @xcite , and into the classic confinement scenarios based on ( gauge - projected abelian ) monopole @xcite or center vortex @xcite condensation . nevertheless , at present no mechanism involving soft vacuum gluons can be uniquely or systematically related to qcd , and many crucial questions regarding the underlying fields , their stability , gauge - independent physical interpretation , mutual interactions , relations to other vacuum fields etc . remain unanswered . analytical progress in this realm has been particularly slowed by the inevitable gauge dependence of the generally rather complex classical gluon field configurations on which most of the existing proposals are based . similarly , approaches which reformulate non - abelian gauge theory in terms of gauge - independent loop variables @xcite or resolve the gauge constraints explicitly ( e.g. in coulomb gauge @xcite ) , are often technically too involved for direct practical applications . in the present paper , we circumvent such complications by developing an approach in which not the contributions of single gauge fields but rather those of gauge invariant classes are treated jointly . these classes gather contributions from dominant gauge field orbits to low - energy yang - mills amplitudes and thus represent collective gluonic ir degrees of freedom . technically , they are the saddle points of a soft mode action for gauge invariant matrix fields and therefore also provide the principal input for a systematic saddle point expansion of soft yang - mills amplitudes . manifest gauge invariance is maintained throughout all calculations by working in the hamiltonian formulation of yang - mills theory in the `` coordinate '' schrdinger representation and by making use of explicit gauge projection operators . the schrdinger picture is adopted mainly because it restricts gauge transformations to a fixed reference time , thereby effectively decoupling them from the dynamical time evolution , and because it often renders the impact of topological gluon properties particularly transparent ( even without recourse to the semi - classical approximation ) @xcite . for the reasons already alluded to , we will focus on gluonic effects and work in pure gauge theory without quarks . the individual ir degrees of freedom , i.e. the solutions of the saddle point equations , turn out to comprise a diverse range of specific features . additionally , they have several important properties in common , including stability against scale transformations ( an indispensable prerequisite for the saddle point expansion which is ensured by a virial theorem ) and characteristic topological properties inherited from the gauge group . the topology will turn out to be particularly useful for establishing relations between specific saddle - point families and the instanton and meron solutions of the classical yang - mills equation . our approach maintains explicitly traceable links between the soft collective fields and the underlying gluon fields . the resulting ir dynamics is at an intermediate level of complexity , somewhere inbetween the microscopic theory itself and effective theories ( e.g. for polyakov loops ) which just share the symmetries of the fundamental dynamics while coupling parameters have to be fitted to experimental ( or lattice ) data . our soft - mode lagrangian , in contrast , follows uniquely from the adopted vacuum wave functional and combines a reasonable amount of transparency with accessibility to essentially analytical treatment . the paper is organized as follows : in sec . [ voa ] we recapitulate the definition of the vacuum overlap amplitude in the schrdinger picture . we then implement a gauge - projected vacuum wave functional on the basis of the gaussian approximation and rewrite the overlap in terms of a bare action which had previously emerged in a variational context . in sec . [ sge ] , we take advantage of known 1-loop results to integrate out the hard - mode contributions to the bare action perturbatively . by means of a controlled derivative expansion , the renormalized soft - mode action density is then transformed it into a local lagrangian which lends itself to direct analytical treatment . in sec . [ spex ] we build on these results by establishing the ir - sensitive saddle point expansion for the functional integral over the soft modes and by deriving the saddle point equations whose nontrivial solutions constitute the new gluonic ir degrees of freedom . important generic properties of these ir variables are established in sec . [ sps ] , including their scale stability due to a virial theorem , three topological quantum numbers and a lower bound of bogomolnyi type on their action . in sec . [ solncl ] , several classes of the more symmetric and most important saddle point solutions are found explicitly . they comprise topological soliton solutions of hedgehog type , which are related to classical solutions of the yang - mills equation , and solutions which carry different types of topological information and seem to have no obvious counterparts in classical yang - mills theory . one of the most interesting solution classes consists of solitonic links , twisted links and knots . those emerge from a generalization of faddeev - niemi theory which turns out to be embedded in our soft - mode lagrangian . in sec . [ qspi ] we classify all hedgehog soliton solutions , find their most important representatives explicitly , and establish the role of the regular solutions as mainly summarizing contributions from instanton and meron gauge orbits to the vacuum overlap . in sec . [ suc ] , finally , we collect our principal results , comment on evaluating the contributions of the gluonic ir degrees of freedom to relevant amplitudes , and suggest directions for future work . the vacuum overlap amplitude of @xmath1 yang - mills theory ( without matter fields ) in the schrdinger coordinate representation @xcite reads @xmath2 \psi_{0}\left [ \vec{a},t_{-}\right ] .\label{zprime}%\ ] ] the vacuum wave functional ( vwf ) @xmath3 depends on half of the canonical variables , i.e. on the static gauge fields @xmath4 . its gauge invariance , like that of any other physical state and wave functional , is dictated by gau law . this crucial requirement can be imposed on a given functional by simply projecting out its gauge - singlet component , i.e. by integration over the ( compact ) gauge group @xcite . starting from an approximate and therefore generally gauge dependent wave functional @xmath5 , one then obtains the associated vwf @xmath6 = \sum_{n}e^{iq\theta}\int d\mu\left ( u^{\left ( q\right ) } \right ) \psi_{0}\left [ \vec{a}^{u^{\left ( q\right ) } } \right ] = : \int du\psi_{0}\left [ \vec{a}^{u}\right ] \label{ginvvwf}%\ ] ] ( @xmath7 is the invariant haar measure of the gauge group , @xmath8 is the homotopy degree or winding number of the group element @xmath9 , and @xmath0 is the vacuum angle ) for which gau law is manifest . the vacuum energy has been set to zero . after interchanging the order of integration over gauge fields and gauge group in eq . ( [ zprime ] ) , it becomes obvious that a gauge group volume can be factored out of @xmath10 , i.e. @xmath11 \psi_{0}\left [ \vec{a}^{u_{-}}\right ] = : z\int du_{-},\ ] ] since the @xmath12-integral is gauge invariant . in fact , the group volume @xmath13 is left over when the two un - normalized gauge projectors in the matrix element @xmath10 are multiplied into one . the integrand of the remaining integral over the gauge group is naturally rewritten as a boltzmann factor , i.e. @xmath14 \right ) , \label{z}%\ ] ] which defines the 3-dimensional euclidean bare action @xmath15 as a functional of the `` relative '' gauge orientation @xmath16 only . owing to the gauge invariance of the gluon field measure , @xmath15 is gauge invariant as well and takes the explicit form @xmath17 = -\ln\int d\vec{a}\psi_{0}^{\ast}\left [ \vec { a}^{u}\right ] \psi_{0}\left [ \vec{a}\right ] .\label{gammab}%\ ] ] this action describes the dynamical correlations which the gauge projection of the functional @xmath5 in eq . ( [ ginvvwf ] ) has generated . hence it would become trivial , i.e. @xmath18-independent , if @xmath5 were gauge invariant by itself . more specifically , @xmath19 $ ] gathers all those contributions to @xmath20 whose approximate vacua @xmath5 at @xmath21 differ by the relative gauge orientation @xmath18 . the variable @xmath18 thus represents the contributions of a specifically weighted ensemble of all gluon field orbits to the vacuum overlap and is gauge invariant by construction . to proceed in an analytically tractable fashion , we now adopt the standard gaussian approximation @xmath22 = \exp\left [ -\frac{1}% { 2}\text { } \int d^{3}x\int d^{3}ya_{i}^{a}\left ( \vec{x}\right ) g^{-1ab}\left ( \vec{x}-\vec{y}\right ) a_{i}^{b}\left ( \vec{y}\right ) \right ] \label{ga}%\ ] ] for the unprojected vwf , which was discussed in ref . @xcite . a minimal gauge - invariant extension of the exponent in eq . ( [ ga ] ) characterizes the ground state of yang - mills theory in 2 + 1 dimensions @xcite . ] which has the decisive advantage of allowing integrals over @xmath12 to be done exactly . as expected from a ground state wave functional , @xmath23 has no nodes . it describes a `` squeezed '' state , i.e. an oscillator - type extension of the unstable coherent gluon states @xcite and thus the simplest natural candidate for the vacuum functional . in fact , eq . ( [ ga ] ) turns into the exact ground state for @xmath24 gauge theory ( up to color factors ) if the `` covariance '' @xmath25 is taken to be the inverse of the static vector propagator . several additional properties indicate that gaussian vwfs with suitably adapted covariances capture crucial features of the yang - mills dynamics as well . indeed , with an appropriate choice for @xmath25 ( see below ) the wave functional ( [ ga ] ) becomes exact at high momenta and incorporates asymptotic freedom . moreover , it is known from variational analyses that gaussian vwfs generate a dynamical mass gap and possibly confinement @xcite . ( mass generation and most other features of 2 + 1 dimensional compact photodynamics are also reproduced @xcite . ) additional support for the gaussian approximation will emerge from our results below . after specializing the expression ( [ gammab ] ) for the bare action to @xmath23 , the functional integral over the gluon fields becomes gaussian and can readily be carried out . the result takes the form of a 3-dimensional , bilocal nonlinear sigma model @xcite , @xmath17 = \frac{1}{2g_{b}^{2}}\int d^{3}x\int d^{3}% yl_{i}^{a}\left ( \vec{x}\right ) d^{ab}\left ( \vec{x}-\vec{y}\right ) l_{i}^{b}\left ( \vec{y}\right ) .\label{effact}%\ ] ] ( above we have omitted a term of higher order in the small bare coupling @xmath26 which vanishes at the saddle points in which we will be interested below . ) the @xmath18-dependence enters @xmath15 both via the one - forms@xmath27 i.e. the lie - algebra valued , left - invariant maurer - cartan `` currents '' ( with real components @xmath28 ) , and through higher - order corrections to the bilocal operator @xmath29 ^{ab}\simeq\frac{1}% { 2}g^{-1}\delta^{ab}+ ... \ ] ] where @xmath30 , @xmath31 and @xmath32 . the above reformulation of the yang - mills vacuum overlap on the basis of a gauge - invariant gaussian vwf was employed in ref . @xcite as the starting point for a variational approach . alternatively , it can be obtained from a saddle point evaluation of the functional integral @xcite in eq . ( [ gammab ] ) which becomes exact for the gaussian vwf . since center elements of @xmath1 act trivially on the gauge fields . we refrain from making this restriction manifest here , although it could be implemented in the action ( [ effact ] ) and may become relevant , e.g. , for the discussion of center vortices . ] although the nonlinear sigma model ( [ effact ] ) is easier to handle than the original yang - mills theory , its exact non - perturbative treatment remains beyond analytical reach @xcite . nevertheless , the parametric enhancement of the action ( [ effact ] ) by the factor @xmath33 suggests that a useful approximation may be obtained from a saddle point expansion of the functional integral ( [ z ] ) . in order to render this approximation practical , however , one has to deal with the nonlocality of the bare action ( [ effact ] ) which encumbers the identification and evaluation of the saddle points . we will show in the following section that this can be efficiently accomplished by combining a renormalization group evolution of the bare action ( which removes the explicit uv modes ) with a subsequent derivative expansion to transform the ir dynamics into an approximately local soft - mode action . for the reasons outlined in the introduction , we are mainly interested in soft yang - mills amplitudes with external momenta @xmath34 smaller than a typical hadronic scale @xmath35 ( the lowest glueball mass , for example ) . this restricted focus permits us to recast the bare action ( [ effact ] ) into a form which only retains soft field modes explicitly and which can be systematically approximated by a local lagrangian . the present section describes the derivation and some useful features of this soft - mode action . since the action ( [ effact ] ) incorporates asymptotic freedom ( for proper choices of @xmath36 , see below ) , the bare coupling @xmath26 is small at the large cutoff scale @xmath37 where the theory is originally defined . the hard modes of the @xmath18 field with momenta @xmath38 can therefore be integrated out of the functional integral ( [ z ] ) perturbatively , down to values of the infrared scale @xmath35 where the renormalized coupling @xmath39 ceases to be much smaller than unity . in practice , this may be done for instance by wilson s momentum - shell technique @xcite , after factorizing @xmath18 into contributions from high- and low - frequency modes . to one - loop order , the resulting renormalization of the action just amounts to the replacement of the bare coupling @xmath26 by the running coupling @xmath39 . this was confirmed in ref . @xcite where the one - loop coupling was obtained as @xmath40 ( for @xmath41 at @xmath42 ) . the scaling behavior of @xmath39 makes asymptotic freedom explicit and reaffirms that the gaussian vwf reproduces the qualitative uv behavior of yang - mills theory . in fact , eq . ( [ g(mu ) ] ) equals the one - loop yang - mills coupling up to a small correction factor @xmath43 which arises from the absence of transverse gluons and could be avoided by introducing an anisotropic component for @xmath25 @xcite . the one - loop integration over the high - momentum modes was found to be reliable down to @xmath44 gev @xcite which provides a useful benchmark for numerical estimates . in the valid range of @xmath35 values ( [ effact ] ) turns into the renormalized soft - mode action @xmath45 = \frac{1}{4g^{2}\left ( \mu\right ) } \int d^{3}x\int d^{3}yl_{<,i}^{a}\left ( \vec{x}\right ) g^{-1}\left ( \vec{x}% -\vec{y}\right ) l_{<,i}^{b}\left ( \vec{y}\right ) \label{gamless}%\ ] ] where the subscript @xmath46 indicates that @xmath18 contains only @xmath47 modes . the action ( [ gamless ] ) is still nonlocal . however , this nonlocality is substantially weaker than in the bare action ( [ effact ] ) since the soft @xmath18 fields in the integrand vary too little to resolve details of @xmath25 over distances smaller than @xmath48 . this observation can be turned into a controlled , _ approximation scheme for the soft - mode action ( [ gamless ] ) by exploiting the fact that the gradients of @xmath49 are bounded by the ir gluon mass scale , @xmath50 indeed , this bound suggests to expand the nonlocality of @xmath25 into derivatives @xmath51 which will act upon @xmath49 after partial integration . using the isotropy of @xmath52 ( as mentioned above , an anisotropic component could be allowed in principle and would lead to somewhat more general expressions ) , one has @xmath53 \delta^{3}\left ( \vec{x}-\vec{y}\right ) . \label{gexp}%\ ] ] the dimensionless constants @xmath54 encode the low - momentum behavior of @xmath25 and could , e.g. , be determined variationally . for our present purposes , however , it will be sufficient to adopt the standard expression @xmath55 which approximates the solution of schwinger - dyson equations and variational estimates @xcite and incorporates both asymptotic freedom and a dynamical mass gap . the corresponding @xmath54 can be read off directly from the fourier transform @xmath56 ( @xmath57 is a mcdonald function @xcite ) , i.e. @xmath58 @xmath59 @xmath60 etc . the combination of the above results leads to the intended reformulation of the nonlocal dynamics ( [ effact ] ) . as anticipated , the bilocal action density for the soft modes in eq . ( [ gamless ] ) becomes a ( `` quasi''- ) local lagrangian @xmath61 and the action takes the familiar form @xmath45 = \int d^{3}x\mathcal{l}\left ( \vec{x}\right ) .\label{gamsoft}%\ ] ] the lagrangian is an expansion into powers of @xmath62 and therefore belongs to the class of generalized nonlinear sigma models . when expressed in terms of the cartan - maurer currents @xmath63 , it reads ) could immediately be generalized to arbitrary @xmath25 by restoring the original @xmath54 dependence from eq . ( [ gexp]).]@xmath64 we have omitted total derivatives @xmath65 from the higher - order terms of @xmath66 since they do not affect the field equations . nevertheless , they may generate non - vanishing surface terms due to infinite - action configurations which are generally irrelevant for the saddle point expansion . to lowest order,@xmath67 + ... \right\ } .\label{sfterm}%\ ] ] the gradient expansion in eq . ( [ efflagr ] ) is controlled by increasing powers of the parametrically small @xmath68 . for practical purposes it can therefore be reliably truncated , at an order which is determined by the desired accuracy of the approximation to the exact action . below , we will be interested in specific field configurations ( saddle points ) which we are going to find explicitly . for those one may directly check _ a posteriori _ whether the full action is sufficiently well reproduced and , if not , include contributions from higher gradients . based on such tests and a virial theorem ( see sec . [ virth ] ) , we found the truncation at @xmath69 to yield a generally sufficient approximation ( at the few percent level ) to the full action ( [ gamless ] ) . the first term in the lagrangian ( [ efflagr ] ) has the standard form of a 3-dimensional nonlinear @xmath70-model ( or principal chiral model ) . the second one , with four derivatives acting on @xmath49 s , is reminiscent of a similar term in the skyrme model @xcite . however , the identity@xmath71 ^{2}+\partial_{i}l_{j}\partial_{j}l_{i}\label{commplus}%\ ] ] shows that the four - derivative contribution to eq . ( [ efflagr ] ) contains , besides the commutator or skyrme term , a part which qualitatively alters the character of the euler - lagrange equations . while the commutator generates only second - order terms to the field equations ( see below ) , the piece without equivalent in the skyrme model leads to additional fourth - order terms which allow for new types of solutions . ( several families of topological soliton solutions from the lagrangian ( [ efflagr ] ) will nevertheless turn out to resemble static skyrmions .- field arises from yang - mills theory even without quarks , and that its physical interpretation ( representing gauge fields of a fixed relative color orientation ) is completely different from that in the skyrme model where @xmath18 represents fluctuations around the chiral order parameter . ] ) we end this section by emphasizing that the construction of an analogous gradient expansion in terms of the original gauge fields @xmath12 would require the ( residual ) gauge freedom to be completely fixed and thus give rise to all the associated conceptual and calculational complications . ( otherwise a `` soft '' gauge field could just be turned into a `` hard '' one by a suitable gauge transformation and would spoil the `` convergence '' of the derivative expansion . ) the locality and structural simplicity of the soft - mode lagrangian ( [ efflagr ] ) can therefore be regarded as a benefit of reformulating the dynamics in terms of the gauge invariant @xmath18 field variables . our next task will be to set up the saddle point ( or , more specifically , steepest descent ) expansion of the functional integral over the soft modes , @xmath72 \right ) , \label{zsoft}%\ ] ] where the vacuum overlap @xmath20 serves as the prototype for similar integrals in other soft amplitudes . this expansion is based on the ir saddle point fields @xmath73 , i.e. the local minima of the soft - mode action ( [ gamsoft ] ) . the search for these minima is simplified by the fact that all finite - action @xmath18-field configurations , including most saddle points , fall into disjoint topological classes ( cf . [ top ] and below ) . since fields which carry different topological charges - for now summarily denoted by @xmath8 - can not be continuously deformed into each other , the local variation of the action may be performed in each topological sector separately . this amounts to solving the saddle point equations@xmath74 } { \delta u_{<}\left ( \vec { x}\right ) } \right| _ { u_{<}=\bar{u}_{i}^{\left ( q\right ) } } = 0\label{spaeq}%\ ] ] at fixed @xmath8 . to leading order , the saddle - point expansion for the vacuum overlap @xmath20 can then be assembled by summing the contributions from the solutions @xmath75 of eq . ( [ spaeq ] ) , i.e. @xmath76 \exp\left ( -\gamma\left [ \bar{u}_{i}^{\left ( q\right ) } \right ] \right ) , \label{zspa}%\ ] ] where the pre - exponential factors @xmath77 are generated by zero - mode contributions which typically arise if continuous symmetries of the action are broken by the solutions . the sum over the saddle points , labeled by @xmath78 , symbolically includes integrals with the appropriate measure when the saddle points come in continuous families . functional integrals for more complex amplitudes , including the gluonic green functions , receive contributions from the same saddle points and are obtained by differentiating @xmath20 with respect to suitably implemented sources . higher - order corrections to the approximation ( [ zspa ] ) can be systematically calculated from the ( nondegenerate ) fluctuations around the solutions @xmath75 . the reliability of the leading - order approximation ( [ zspa ] ) increases with the action values of the saddle points because @xmath79 \gg1 $ ] prevents the saddle point contributions from being rendered insignificant by the fluctuations of @xmath80 around them . in our case , compliance with this criterion is reinforced by the overall factor @xmath81 ( for @xmath82 gev ) in the lagrangian ( [ efflagr ] ) which parametrically enhances the action . ) since @xmath83 . ] for the explicit solution of the saddle - point equation ( [ spaeq ] ) , as well as for part of our general analysis below , it is practically necessary to adopt a parametrization of the @xmath1 group elements @xmath18 which allows to work directly with their @xmath84 independent degrees of freedom . we will use the exponential representation @xmath85 \label{uparam}%\ ] ] for this purpose , where the coefficient vector of the lie algebra generators is decomposed into its direction , specified by the vector field @xmath86 with @xmath87 ( which parametrizes the coset @xmath88 where @xmath89 is the cartan subgroup of the gauge group ) , and its length * * @xmath90 . for simplicity , we will also specialize our following discussion to @xmath91 . the soft - mode lagrangian ( [ efflagr ] ) can then be rewritten in terms of the unit vector field @xmath86 and the spin-0 field @xmath90 as @xmath92 where , for the reasons discussed in the paragraph below eq . ( [ sfterm ] ) , all terms of the gradient expansion with up to four derivatives on the @xmath18 fields are retained . the two - derivative part @xmath93 , i.e. the standard nonlinear @xmath70-model , becomes@xmath94\ ] ] while the four - derivative contributions turn into@xmath95 .\end{aligned}\ ] ] the general expressions above show that @xmath96 , as required for the existence of the functional integral ( [ zsoft ] ) . the same remains true if higher orders of the derivative expansion ( [ efflagr ] ) are included ( as long as @xmath97 , of course ) . for the analysis of some generic saddle point solution properties , including their behavior under scale transformations ( cf . sec . [ sps ] ) , it will prove useful that both @xmath93 and @xmath98 are even individually nonnegative . by varying the action ( [ gamsoft ] ) of the lagrangian ( [ l24d ] ) with respect to @xmath90 and @xmath86 , the saddle point equation ( [ spaeq ] ) turns into a nonlinear system of four coupled partial differential equations . for @xmath90 one directly obtains@xmath99 } { \delta\phi\left ( \vec{x}\right ) } & = \partial^{4}\phi-2\partial^{2}% \phi\left ( \partial_{i}\hat{n}^{a}\right ) ^{2}-4\partial_{i}\phi\partial _ { i}\partial_{k}\hat{n}^{a}\partial_{k}\hat{n}^{a}-4\cos\phi\left ( \partial_{i}\partial_{j}\phi\partial_{i}\hat{n}^{a}+\partial_{j}\phi \partial^{2}\hat{n}^{a}+\partial_{i}\phi\partial_{i}\partial_{j}\hat{n}% ^{a}\right ) \partial_{j}\hat{n}^{a}\nonumber\\ & + 2\sin\phi\left [ \left ( \partial_{i}\phi\partial_{i}\hat{n}^{a}\right ) ^{2}-\left ( 1-\cos\phi\right ) \left ( \partial_{i}\hat{n}^{a}\partial _ { j}\hat{n}^{a}\right ) ^{2}-2\partial_{i}\hat{n}^{a}\partial_{i}\partial ^{2}\hat{n}^{a}-\left ( \partial_{i}\partial_{j}\hat{n}^{a}\right ) ^{2}+\left ( \partial_{i}\hat{n}^{a}\right ) ^{2}\left ( \partial_{j}\hat { n}^{b}\right ) ^{2}\right ] \nonumber\\ & -\sin\phi\cos\phi\left ( \partial_{i}\hat{n}^{a}\right ) ^{2}\left ( \partial_{j}\hat{n}^{b}\right ) ^{2}-\sin\phi\left ( \delta^{ab}+\hat{n}% ^{a}\hat{n}^{b}\right ) \partial^{2}\hat{n}^{a}\partial^{2}\hat{n}^{b}% -2\mu^{2}\left [ \partial^{2}\phi-\sin\phi\left ( \partial_{i}\hat{n}% ^{a}\right ) ^{2}\right ] = 0.\label{omeq}%\end{aligned}\ ] ] the three equations for the components of the @xmath100 field , on the other hand , have to be derived by a constrained variation whose task it is to preserve the unit length of @xmath100 . as a consequence , they can be cast into the succinct form @xmath101 } { \delta\hat{n}^{a}\left ( \vec{x}\right ) } = 0\label{neq}%\ ] ] where the projection operator ensures that the action is only affected by those variations @xmath102 which maintain orthogonality to @xmath103 . the evaluation of the functional derivative with respect to @xmath104 yields @xmath105 } { \delta\hat{n}^{a}\left ( \vec{x}\right ) } & = -\left ( 1-\cos\phi\right ) \left ( \partial^{2}\hat{n}^{a}\hat{n}^{c}\partial^{2}\hat{n}^{c}\right ) + \partial_{i}\left\ { \sin\phi\partial^{2}\phi\partial_{i}\hat{n}^{a}\right . -\sin\phi\partial_{i}\phi\partial^{2}\hat{n}^{a}-2\cos\phi\partial_{i}% \phi\partial_{j}\phi\partial_{j}\hat{n}^{a}\nonumber\\ & + \cos\phi\partial_{i}\phi\partial_{k}\phi\partial_{k}\hat{n}^{a}+\sin \phi\partial_{i}\partial_{k}\phi\partial_{k}\hat{n}^{a}+\sin\phi\partial _ { k}\phi\partial_{i}\partial_{k}\hat{n}^{a}+\sin\phi\partial_{i}\phi\left ( \partial^{2}\hat{n}^{a}-\hat{n}^{a}\hat{n}^{c}\partial^{2}\hat{n}^{c}\right ) \nonumber\\ & + \left ( 1-\cos\phi\right ) \left [ \partial_{i}\partial^{2}\hat{n}% ^{a}-\hat{n}^{c}\partial_{i}\hat{n}^{a}\partial^{2}\hat{n}^{c}-\hat{n}% ^{a}\partial_{i}\hat{n}^{c}\partial^{2}\hat{n}^{c}-\hat{n}^{a}\hat{n}% ^{c}\partial_{i}\partial^{2}\hat{n}^{c}-\left ( \partial_{j}\phi\right ) ^{2}\partial_{i}\hat{n}^{a}-2\mu^{2}\partial_{i}\hat{n}^{a}\right ] \nonumber\\ & -\sin^{2}\phi\partial_{i}\hat{n}^{a}\left ( \partial_{i}\hat{n}^{c}\right ) ^{2}-2\left ( 1-\cos\phi\right ) ^{2}\left [ \partial_{i}\hat{n}^{a}\left ( \partial_{j}\hat{n}^{c}\right ) ^{2}-\partial_{i}\hat{n}^{c}\partial_{j}% \hat{n}^{c}\partial_{j}\hat{n}^{a}\right ] \left . { } \right\ } .\end{aligned}\ ] ] the saddle point equations ( [ omeq ] ) and ( [ neq ] ) are independent of the coupling @xmath106 because it enters the action only through the overall factor @xmath107 . in general , their solutions have to be found numerically . however , we will demonstrate below that several nontrivial analytical solutions exist and that further important solution classes with a rather high degree of symmetry can be obtained by solving substantially simplified field equations . moreover , essential qualitative solution properties can be derived without solving the saddle point equations explicitly ( cf . [ sps ] ) . each topological charge sector contains at least one action minimum , for example , i.e. one solution of eqs . ( [ omeq ] ) and ( [ neq ] ) . the requirement of finding and including all of them would render the saddle point expansion practically useless . fortunately , however , this turns out to be unnecessary . below we will establish lower bounds on the action of the saddle points which are monotonically increasing functions of their topological charges . hence contributions from saddle points in high topological charge sectors can generally be ignored . before actually solving the saddle - point equations ( [ omeq ] ) and ( [ neq ] ) in secs . [ solncl ] and [ qspi ] , it will be useful to obtain a few general insights into the topological structure and stability properties of the solutions . this is the objective of the present section . in order to analyze the scaling behavior of the extended saddle point solutions and to establish the underlying virial theorem , we define the scaled fields @xmath108 for real @xmath109 and note that @xmath35 is the only mass scale in the field equations ( [ omeq ] ) and ( [ neq ] ) . this immediately implies that the solutions of the saddle point equations with scaled parameter @xmath110 can be obtained by rescaling the original solutions @xmath111 to @xmath112 . in the following , however , we will keep @xmath35 fixed . the scale - transformed extended solutions then cease to solve the field equations , and this simple observation together with two basic properties of the lagrangian ( [ l24d ] ) can be turned into a virial theorem . the first step towards its derivation consists in establishing the relation between the actions of scaled and unscaled fields ( as long as they stay finite ) , which can be read off from the 2- and 4-derivative parts of the lagrangian ( [ l24d ] ) separately : @xmath113 = \gamma_{2d}\left ( \lambda\right ) + \gamma_{4d}\left ( \lambda\right ) = \frac{1}{\lambda}\gamma_{2d}\left ( 1\right ) + \lambda \gamma_{4d}\left ( 1\right ) .\label{gamlam}%\ ] ] the second relevant property of the action based on eq . ( [ l24d ] ) is its strict positivity for extended , i.e. not translationally invariant field configurations ( cf . [ spex ] ) , @xmath114 the remaining step is to specialize the fields under consideration to the saddle point solutions . since those extremize the action under arbitrary small variations - which of course include infinitesimal scale transformations - one immediately has @xmath115 and consequently the virial theorem @xmath116 eq . ( [ vth ] ) clearly exhibits the crucial role of the four - derivative terms @xmath117 in guaranteeing the stability of the saddle point solutions : for @xmath118 the remaining nonlinear @xmath70-model action would be minimized by sending @xmath119 ( cf . ( [ gamlam ] ) ) , i.e. by the scale collapse of the solutions which derrick s theorem predicts @xcite . our truncation of the gradient expansion ( [ efflagr ] ) at @xmath120 therefore turns out to be the minimal approximation which can support localized , stable soliton solutions . ) one might therefore suspect that their contributions could destabilize the extended solutions . however , this would just indicate an incorrect truncation of the gradient expansion . indeed , if a localized solution is small enough to be significantly affected by higher - derivative terms , those would have to be added to the field equations in the first place . ( a somewhat analogous situation is encountered in one - loop coleman - weinberg potentials : their physically relevant local minima are stable only as long as fluctuations remain small , i.e. as long as the underlying truncation of the loop expansion is justified @xcite . ) ] furthermore , @xmath121 implies that the scaling extrema are indeed action minima and that solutions with @xmath122 are points of inflection . finally , it is worth emphasizing that the coexistence of terms with different numbers of derivatives in the lagrangian ( [ efflagr ] ) , and thus ultimately the virial theorem ( [ vth ] ) and the existence of stable solutions , is brought about by the mass scale @xmath35 which reflects the short - wavelength quantum fluctuations integrated out in sec . this situation is reminiscent of the classical instanton solutions to the yang - mills equation whose typical size scale must likewise be generated by quantum fluctuations . to summarize , we have established a virial theorem which ensures that the extended solutions of eqs . ( [ omeq ] ) and ( [ neq ] ) are stable against scale transformations . in our context this is an indispensable property since unstable solutions would prohibit a useful saddle point expansion . as a side benefit , the virial theorem also provides stringent tests for the numerical solutions of the saddle point equations . as a three - dimensional principal chiral model with stabilizing higher - derivative terms , the soft - mode lagrangian ( [ l24d ] ) allows for topological soliton solutions . in the present section we discuss three topological invariants or charges which such solutions and more general continuous fields @xmath18 may carry @xcite . the most fundamental topological classification arises from the fact that all @xmath18-fields with a finite action @xmath123 based on any truncation of the lagrangian ( [ efflagr ] ) have to approach the same constant at @xmath124 . as a higher - dimensional analog of the stereographic projection , this compactifies their domain to a three - sphere @xmath125 . all @xmath18 s with finite @xmath123 therefore describe continuous maps from @xmath125 into the `` topologically active '' part of the group manifold . for @xmath1 the latter is the trivially embedded subgroup @xmath126 . the resulting maps @xmath127 fall into disjoint homotopy classes , the elements of the third homotopy group @xmath128 , which are characterized by a topological degree or charge@xmath129 = \frac{1}{24\pi^{2}}\int d^{3}x\varepsilon_{ijk}tr\left\ { u^{\dagger}\partial_{i}uu^{\dagger}\partial_{j}uu^{\dagger}\partial _ { k}u\right\ } .\label{q}%\ ] ] ( the integrand can be shown to be a total derivative , as expected for a topological `` charge density '' . ) for finite - action fields @xmath130 . in terms of the @xmath131 parametrization ( [ uparam ] ) for @xmath132 , ( [ q ] ) reduces to@xmath133 = \frac{1}{2^{4}\pi^{2}}\int d^{3}x\left ( \cos\phi-1\right ) \varepsilon_{ijk}\varepsilon^{abc}\partial_{i}\phi\hat { n}^{a}\partial_{j}\hat{n}^{b}\partial_{k}\hat{n}^{c}.\label{qomn}%\ ] ] two additional topological quantum numbers of @xmath18 are carried solely by its @xmath100-field component . the first is the homotopy degree of the maps @xmath86 from the space boundary @xmath134 ( where @xmath135 ) into the unit sphere @xmath136 on which @xmath86 takes values . continuous maps of this sort are classified by the elements of the homotopy group @xmath137 . ( the same topology characterizes the magnetic charge of wu - yang monopoles @xcite . ) an explicit integral representation of their degree is @xmath138 = \frac{1}{8\pi}\int_{\partial r^{3}}d\sigma _ { i}\varepsilon_{ijk}\varepsilon^{abc}\hat{n}^{a}\partial_{j}\hat{n}% ^{b}\partial_{k}\hat{n}^{c}\label{qm}%\ ] ] where the integral extends over the closed surface @xmath139 at @xmath140 . as expected , @xmath141 is @xmath90-independent . the third topological invariant owes its existence to the fact that all @xmath100 fields of finite action are required to approach a constant unit vector at spacial infinity . as above , this requirement compactifies @xmath142 into @xmath125 and thereby turns @xmath143 into continuous maps @xmath144 . such maps carry a hopf charge @xmath145 which labels the elements of the homotopy group @xmath146 , i.e. the hopf bundle @xcite . an explicit integral representation for @xmath145 can be constructed by means of the local isomorphism between the nonlinear @xmath147 and @xmath148 fields which expresses @xmath86 in terms of a complex 2-component field @xmath149 with unit modulus @xmath150 as @xmath151 . then one has @xmath152 = \frac{1}{4\pi^{2}}\int d^{3}x\varepsilon_{ijk}\left ( \chi^{\dagger}\partial_{i}\chi\right ) \left ( \partial_{j}\chi^{\dagger}\partial_{k}\chi\right ) .\label{qh}%\ ] ] ( a _ local _ integral representation for @xmath145 directly in terms of the @xmath100-field does not exist . ) in contrast to the topological charges @xmath8 and @xmath141 which are of winding - number type , the hopf invariant @xmath145 is a linking number . the underlying topological structure enables and classifies the link and knot solutions to be encountered in sec . [ fn ] . finally , we recall that the @xmath18 field topology - as summarized in the conserved topological quantum numbers @xmath8 , @xmath141 and @xmath145 - characterizes not only the saddle point solutions but a much larger class of continuous field configurations with finite and in some cases even infinite ( see below ) action . the distribution of the solutions to eqs . ( [ omeq ] ) and ( [ neq ] ) over a denumerably infinite set of topological charge sectors allows for the existence of more saddle points than could be handled in practical applications of the expansion ( [ zspa ] ) . hence additional criteria are required to select the most relevant saddle points in a controlled fashion . such criteria will be established below , in the form of action bounds which are monotonically increasing functions of the absolute topological charge values . these bounds imply that contributions from saddle points with higher topological quantum numbers to functional integrals are increasingly suppressed by the boltzmann factor @xmath153 and can be systematically neglected . in the present section we establish the action bound for fields which carry finite values of @xmath8 . a similar bound for fields with nonvanishing hopf charge will be obtained in sec . [ fn ] . the lower bound on the action of the lagrangian ( [ l24d ] ) for any field @xmath18 with a well - defined homotopy degree ( [ q ] ) can be derived from the lie - algebra valued expression@xmath154 where @xmath155 and @xmath156 are the components of the maurer - cartan one - form ( [ l ] ) . with the help of the maurer - cartan identity@xmath157\ ] ] one obtains for its square@xmath158 which obeys the basic inequality @xmath159 ( recall from eq . ( [ l ] ) that the @xmath156 are expanded into anti - hermitean generators . ) after specializing the bound ( [ trmm ] ) to @xmath160 and @xmath161 , in eq . ( [ gexp ] ) could be accomodated by modifying the values of @xmath162 and @xmath163 and would result in a different factor in front of @xmath164 in the bogomolnyi bound . ] multiplying by @xmath165 , integrating over @xmath166 and using the integral representation ( [ q ] ) for @xmath8 , one arrives at@xmath167 .\label{prebnd}%\ ] ] the more stringent of these inequalities results from the lower ( upper ) sign on the right - hand side if @xmath168 ( @xmath169 ) . their left - hand side amounts to the action which is produced by the first two terms ) in @xmath123 . after specializing to @xmath160 and @xmath170 , the obvious identity @xmath171 then results in @xmath172 = -\frac{\mu}{2g^{2}\left ( \mu\right ) } \int d^{3}xtr\left\ { l_{i}% l_{i}-\frac{1}{2\mu^{2}}l_{i}\partial^{2}l_{i}\right\ } \geq\mp\frac{12\pi ^{2}}{g^{2}\left ( \mu\right ) } q\left [ u\right ] + \int d^{3}x\delta \mathcal{l}\left ( \vec{x}\right ) .$ ] ] of the lagrangian ( [ efflagr ] ) . hence the expressions ( [ prebnd ] ) for both signs combine into an inequality of bogomolnyi type,@xmath173 \geq\frac{12\pi^{2}}{g^{2}\left ( \mu\right ) } \left| q\left [ u\right ] \right| .\label{bb}%\ ] ] this is the desired lower bound on the action of any sufficiently smooth @xmath18 field with well - defined degree @xmath8 . it is saturated by those fields which obey the bogomolnyi - type equation @xmath174 where the lower ( upper ) sign again refers to @xmath168 ( @xmath169 ) . the equations ( [ beq ] ) can be considered as analogs of the self-(anti)-duality equations of yang - mills theory and have the interesting consequence that the rescaled maurer - cartan currents @xmath175 of minimal - action fields in any @xmath8-sector become generators of the @xmath176 lie algebra , @xmath177 = i\varepsilon_{ijk}\tilde{l}_{k}.\ ] ] translationally invariant solutions ( cf . [ classvac ] ) , for example , have @xmath178 and therefore trivially saturate the bound in the @xmath179 sector . it remains to be seen whether nontrivial solutions in sectors with larger @xmath164 exist as well can only be saturated on a hyperspherical domain @xmath180 , for example @xcite . ] the large factor multiplying @xmath164 in the inequality ( [ bb ] ) indicates that contributions from saddle points with higher @xmath181 are strongly suppressed . in fact , they seem safely negligible in most amplitudes which receive nonvanishing contributions from the @xmath179 sector . however , one should keep in mind that even contributions with extremely small `` fugacities '' can sometimes have an important qualitative impact on the partition function . a case in point are the decisive monopole contributions in the 2 + 1 dimensional yang - mills - higgs model @xcite . the physical interpretation of the @xmath182 solutions and their analogs in yang - mills theory will be discussed in sec . in the following sections we are going to solve the four saddle point equations ( [ omeq ] ) and ( [ neq ] ) explicitly . as already mentioned , our main focus will be on solutions with a relatively high amount of symmetry since their typically smaller action values enhance their contributions to the saddle point expansion . besides playing a predominant role in most amplitudes , these solutions can often be obtained either analytically or with moderate numerical effort . the simplest solutions of the saddle point equations ( [ omeq ] ) and ( [ neq ] ) are the @xmath166-independent matrices@xmath183 = const.\ ] ] where @xmath184 and @xmath185 are both constant . these solutions form the complete vacuum manifold of the dynamics ( [ l24d ] ) , i.e. the set of all fields which attain the absolute action minimum@xmath186 = 0.\ ] ] due to a redundancy in the parametrization ( [ uparam ] ) , the subset of vacua in the center of the gauge group is completely @xmath86-independent : @xmath187 in addition , those are the only vacua which do not break the global @xmath188 symmetry of the lagrangian ( [ l24d ] ) spontaneously . a glance at the integral representation ( [ qomn ] ) shows that none of the @xmath189 carries topological charges @xmath190 . any constant vector @xmath86 solves the saddle point equation ( [ neq ] ) identically and reduces the other one , eq . ( [ omeq ] ) for @xmath90 , to the linear field equation@xmath191 obviously , the solutions of this equation constitute families of new saddle points which differ by an additive constant and have degenerate action values . ( this trivially ensures periodicity in @xmath90 . ) alternatively , eq . ( [ omeqconstn ] ) and the action of its solutions can be derived from the reduced lagrangian , its contributions to @xmath20 remain nontrivial since the haar measure generates interactions among the @xmath90 modes.]@xmath192 .\label{lconstn}%\ ] ] the presence of the 4th - order term in eq . ( [ omeqconstn ] ) allows for solution types which have no equivalent in skyrme models . ( indeed , the commutator or skyrme term ( cf . eq . ( [ commplus ] ) ) alone leads to a laplace equation for @xmath90 , without a mass scale and with only constant regular finite - action solutions on @xmath142 . ) a glance at the integrals ( [ qomn ] ) - ( [ qh ] ) shows that solutions with constant @xmath100 do not carry any topological quantum numbers , i.e. @xmath193 all solutions of the linear field equation ( [ omeqconstn ] ) can be constructed by standard green function techniques . the perhaps most straightforward approach is to fold the static klein - gordon propagator@xmath194 with a `` scalar potential '' @xmath195 which is defined both to be a solution of the laplace equation , @xmath196 , and to act as the inhomogeneity of the static klein - gordon equation @xmath197 the field equation ( [ omeqconstn ] ) is recovered from eq . ( [ inhomkg ] ) by applying the laplacian to both sides . since the potential @xmath195 plays the role of a source for the @xmath90 field , inversion of the klein - gordon operator immediately yields the general solution @xmath198 of course , the regular finite - action solutions form but a small subset of those comprised in eq . ( [ gensolconstn ] ) . spherically symmetric solutions can be obtained more directly by restricting the angular dependence of @xmath90 , i.e. by substituting the ansatz @xmath199 with @xmath200 into the general equation ( [ omeqconstn ] ) and ignoring for the moment potential singularities at the origin . this yields the radial equation @xmath201 ( radial derivatives @xmath202 are denoted by a prime ) whose four linearly independent solutions can be found analytically : @xmath203 the subset of regular finite - action solutions is therefore of the form@xmath204 the associated potential@xmath205 shows that the expression ( [ phineqc ] ) in fact solves a generalization of the homogeneous field equation ( [ omeqconstn ] ) , with an additional delta - function singularity at the origin . ( [ phineqc ] ) is therefore a solution of eq . ( [ omeqconstn ] ) everywhere except at @xmath206 and , strictly speaking , one of its green functions . a representative of this solution class is drawn in fig . 1 . after insertion into eq . ( [ gamsoft ] ) , based on the lagrangian ( [ l24d ] ) , and use of the virial theorem ( [ vth ] ) one finds the solutions ( [ phineqc ] ) to have the action @xmath207 = \frac{2\pi\mu}{g^{2}\left ( \mu\right ) } \int_{0}^{\infty}dr\left ( r\phi^{\left ( \hat{n}=c\right ) \prime}\right ) ^{2}=\frac{\pi}{\sqrt{2}% } \frac{c_{2}^{2}}{g^{2}\left ( \mu\right ) } .\ ] ] this action is not subject to topological bounds and reaches the absolute minimum @xmath208 for @xmath209 where the constant-@xmath100 solutions turn into the translationally invariant vacua of sec . [ classvac ] . ( in contrast to the center elements ( [ omck ] ) , however , the value of @xmath90 remains unrestricted here . ) due to their partly very small action values , the constant-@xmath100 solutions may have a strong impact on the saddle point expansion which should be explored in detail by studying their contributions to suitable amplitudes . in addition to the translationally invariant saddle points of sec . [ classvac ] , there are other and less trivial solutions of the field equations ( [ omeq ] ) and ( [ neq ] ) with constant @xmath90 fields . among them , the most intriguing class has the general form @xmath210 which satisfies eq . ( [ omeq ] ) identically and carries no topological charge @xmath8 . in fact , the nontrivial topology of @xmath211 ( as that of any other constant-@xmath90 field configuration ) has to reside exclusively in the @xmath212-dependence of its @xmath100 field , whose dynamics is governed by the ( @xmath213-independent ) equation @xmath214 = 0.\label{knoteq}%\ ] ] the field equation ( [ knoteq ] ) follows from the general saddle point equation ( [ neq ] ) by substituting @xmath215 and simultaneously plays the role of a ( static ) continuity equation for the conserved @xmath216 current . alternatively , it can be obtained by directly varying the reduced lagrangian@xmath217 \label{ln}%\ ] ] which follows from eq . ( [ l24d ] ) after specialization to @xmath215 and reproduces the action ( [ gamsoft ] ) for the @xmath218 . remarkably , the lagrangian ( [ ln ] ) is a generalization of the static skyrme - faddeev - niemi ( sfn ) model @xcite @xmath219 in contrast to the sfn model , however , which was postulated on the basis of qualitative symmetry and renormalization group arguments @xcite , our lagrangian ( [ ln ] ) follows uniquely from the yang - mills dynamics and the gaussian approximation to the vacuum wave functional . all coefficients are therefore fixed in terms of the ir scale @xmath35 and the coupling @xmath220 , i.e. eq . ( [ ln ] ) does not contain free parameters . particular solutions of equation ( [ knoteq ] ) are @xmath221 , which belong to the class of translationally invariant vacua ( cf . sec . [ constn ] ) , and @xmath222 ( except at @xmath223 ) which is an example from the `` hedgehog '' solution family whose detailed discussion will be the subject of the following sections . the @xmath224 hedgehog has infinite action ( since @xmath225 develops a monopole - type singularity at @xmath226 cf . [ shs ] ) and its lagrangian reduces exactly to the faddeev - niemi form ( [ fnl ] ) . ( [ knoteq ] ) does probably also have cylindrically symmetric vortex solutions which are analogs of the `` baby skyrmion '' solutions @xcite in similar models . the most interesting and many - faceted solution classes of the field equation ( [ knoteq ] ) , however , are expected to be twists , linked loops and knots made of closed fluxtubes . indeed , an intriguing variety of such topological soliton solutions was found numerically for the sfn part ( [ fnl ] ) of the lagrangian ( [ ln ] ) in refs . these solutions generally lack axial symmetry and carry a finite hopf charge in the sub - model ( [ ln ] ) , it might be possible to `` unwind '' and thus destabilize them by excitations into the @xmath90 direction of the complete theory . this possibility deserves further investigation . ] moreover , their number and complexity increases strongly with the value of @xmath228 . as in the higher-@xmath164 solution sectors discussed previously , a practically useful saddle point expansion thus requires an effective means for selecting the relevant contributions in a controlled fashion . as anticipated in sec . [ bogbnd ] , such a means can be provided by establishing that the action @xmath229 based on the lagrangian ( [ ln ] ) is bounded from below by a monotonically increasing function of @xmath230 . actually , this just requires a straightforward adaptation of a known bound on the sfn action @xcite . one combines the obvious inequalities @xmath231 & \geq\frac{\mu}{g^{2}\left ( \mu\right ) } \int d^{3}x\left [ \left ( \partial_{i}\hat{n}^{a}\right ) ^{2}+\frac{1}{\mu^{2}}\left ( \varepsilon^{abc}\partial_{i}\hat{n}^{b}% \partial_{j}\hat{n}^{c}\right ) ^{2}\right ] \\ & \geq\frac{2}{g^{2}\left ( \mu\right ) } \left [ \int d^{3}x\left ( \partial_{i}\hat{n}^{a}\right ) ^{2}\right ] ^{1/2}\left [ \int d^{3}x\left ( \varepsilon^{abc}\hat{n}^{a}\partial_{i}\hat{n}^{b}\partial_{j}\hat{n}% ^{c}\right ) ^{2}\right ] ^{1/2}%\end{aligned}\ ] ] ( the first one holds because the omitted term in the lagrangian ( [ ln ] ) is manifestly non - negative ; the second one is a consequence of @xmath232 for any real @xmath233 ) with the sobolev - type inequality @xcite@xmath234 \right| ^{3/2}\leq\frac{1}{2^{6}\sqrt { 2}3^{3/4}\pi^{4}}\int d^{3}x\sqrt{\left ( \varepsilon^{abc}\hat{n}% ^{a}\partial_{i}\hat{n}^{b}\partial_{j}\hat{n}^{c}\right ) ^{2}}\int d^{3}x\left ( \varepsilon^{abc}\hat{n}^{a}\partial_{i}\hat{n}^{b}\partial _ { j}\hat{n}^{c}\right ) ^{2}\label{soi}%\ ] ] and a simple inequality due to ward @xcite , @xmath235 ^{2}\geq2\left ( \varepsilon^{abc}\hat{n}^{a}\partial_{i}\hat{n}^{b}\partial_{j}\hat{n}% ^{c}\right ) ^{2},\ ] ] to end up with the bound@xmath236 \geq\frac{2^{9/2}3^{3/8}\pi^{2}}% { g^{2}\left ( \mu\right ) } \left| q_{h}\left [ \hat{n}\right ] \right| ^{3/4}.\label{vkb}%\ ] ] ( @xmath237 ) this bound is rather rough and could probably be made more stringent by incorporating the 3rd term of the lagrangian ( [ ln ] ) and by refining the sobolev bound ( [ soi ] ) which is expected to remain valid with about half of the factor on its right - hand side @xcite . the probably most important lesson of the present section is a new physical interpretation for faddeev - niemi - type knot solutions . in our framework , they reemerge as gauge invariant ir degrees of freedom which represent gluon field ensembles with a nonvanishing `` collective '' hopf charge in the vacuum overlap and other amplitudes . this new interpretation may actually put the tentative identification of knot solutions with glueballs , advocated as a natural generalization of the fluxtube picture for quark - antiquark mesons in refs . @xcite , on a more solid basis . indeed , the original association of refs . @xcite is obscured by the interpretation of the @xmath100 field as a gauge - dependent local color direction . our @xmath100 field , on the other hand , is manifestly gauge invariant . field is that one does not have to deal with unwanted , colored goldstone bosons if the @xmath147 symmetry of the lagrangians ( [ l24d ] ) and ( [ ln ] ) is spontaneously broken . ] moreover , glueballs are anyhow natural candidates for gluonic ir degrees of freedom in the @xmath179 sector , so that their ( perhaps partial or indirect ) appearance in the saddle point solution spectrum would not be unexpected . an additional advantage of our new framework for the knot dynamics is that it allows the investigation of potential relations between the solutions of eq . ( [ knoteq ] ) and specific yang - mills fields which may play important roles in the vacuum , including e.g. topologically nontrivial pure - gauge fields in a non - linear maximally abelian gauge @xcite and center vortices @xcite . if such relations exist , they could perhaps be qualitatively traced by analytical methods . a full quantitative survey of the knot solution sector , however , will require a devoted numerical effort . ) is impeded by the fact that the obvious anstze tend to fail . a few numerical knot solutions with small @xmath230 are probably sufficient for the saddle point expansion , however , since contributions to physical amplitudes are severely limited by the stringent action bound ( [ vkb ] ) , especially when @xmath238 solutions contribute as well . ] the existence of skyrmions @xcite in nonlinear @xmath70-models with higher - derivative interactions suggests that our field equations ( [ omeq ] ) and ( [ neq ] ) have topological soliton solutions of `` hedgehog '' type , @xmath239 ( @xmath240 , @xmath241 ) , as well . their @xmath242 `` grand spin '' symmetry characterizes the invariance of the corresponding @xmath18 fields under simultaneous spatial and internal rotations and implies a substantial simplification of their dynamics . ( for larger gauge groups @xmath1 with @xmath243 , the three components of @xmath244 form the part of the @xmath86 field which parametrizes the trivially embedded @xmath242 subgroup . ) in the present section , we discuss general properties of the hedgehog fields and derive their reduced lagrangian and saddle point equation . in the subsequent sections [ anhh ] and [ qspi ] , we find the most important solution classes explicitly and determine their physical interpretation . the principal topological characteristic of the hedgehog configurations ( [ hh ] ) is their @xmath245 winding number @xmath8 . for fields of the form ( [ hh ] ) , its integral representation ( [ qomn ] ) reduces to @xmath246 & = \frac{1}{2\pi}\int _ { 0}^{\infty}\phi^{\left ( hh\right ) \prime}\left ( \cos\phi^{\left ( hh\right ) } -1\right ) dr\nonumber\\ & = \frac{1}{2\pi}\left [ \sin\phi^{\left ( hh\right ) } \left ( \infty\right ) -\sin\phi^{\left ( hh\right ) } \left ( 0\right ) + \phi^{\left ( hh\right ) } \left ( 0\right ) -\phi^{\left ( hh\right ) } \left ( \infty\right ) \right ] .\label{hq}%\end{aligned}\ ] ] ( as expected from a topological invariant , it depends only on the boundary values of the @xmath90 field . ) in sec . [ top ] we established that finite action fields carry integer values of @xmath8 , and eq . ( [ hq ] ) confirms this explicitly . indeed , the parametrization ( [ uparam ] ) for hedgehog fields ( [ hh ] ) implies that well - defined @xmath18 fields necessitate the boundary condition @xmath247 and that finite - action fields must additionally satisfy @xmath248 ( see below ) where @xmath249 and thus @xmath250 are integers . nevertheless , it is instructive to consider the more general boundary conditions @xmath251 = \frac{n - m}{2}\label{q - hh}%\ ] ] ( @xmath252 integer ) which admit infinite - action fields with half - integer winding numbers as well ( for either @xmath253 or @xmath254 odd ) . we will show in sec . [ qspi ] that hedgehog solutions to the saddle point equations ( [ omeq ] ) , ( [ neq ] ) under the boundary conditions ( [ q - hh ] ) , both with finite and infinite action , can indeed be found . the hedgehog solutions with @xmath255 will be of particular importance since they probably dominate all @xmath190 contributions to the saddle point expansion . this follows from the bound ( [ bb ] ) and from skyrme - model type arguments @xcite which suggest that the minimal - action solutions in the @xmath256 sectors are hedgehogs . in addition , all hedgehog fields carry one unit of a second topological quantum number , the monopole - type charge @xmath257 . this becomes explicit when evaluating the ( @xmath90 independent ) integral representation ( [ qm ] ) for @xmath141 with @xmath222 : @xmath258 = \frac{1}{4\pi}\int d\sigma_{i}\frac{\hat{x}_{i}}{r^{2}}=1.\ ] ] obviously , @xmath259 is independent of the boundary conditions for @xmath90 and therefore of @xmath8 . the field with @xmath260 ( the `` anti - monopole '' ) is obtained by replacing @xmath261 by @xmath262 , which corresponds to @xmath263 for fixed @xmath90 . ( [ qh ] ) reveals , finally , that the hopf charge of all hedgehog configurations vanishes . the dynamics of the hedgehog fields is governed by the soft - mode lagrangian ( [ l24d ] ) . since @xmath244 is an identical solution of the general field equation ( [ neq ] ) for @xmath100 , eq . ( [ l24d ] ) can be directly specialized to @xmath222 . hence the integration over angles becomes trivial and the hedgehog action turns into@xmath264 = \int_{0}^{\infty } dr\mathcal{l}^{\left ( hh\right ) } \left ( r\right)\ ] ] where @xmath265 is a @xmath266-dependent radial lagrangian . after substituting the ansatz ( [ hh ] ) into the full lagrangian ( [ l24d ] ) , dropping total derivatives , suppressing the superscript of @xmath267 and again denoting radial derivatives @xmath202 by a prime , one arrives at the explicit expression @xmath268 .\label{lrad}%\ ] ] all terms in @xmath265 are nonnegative . this has the consequence that each of them must vanish individually at any absolute action minimum . the complete set of hedgehog vacua is therefore @xmath269 and forms a subset of the translationally invariant center elements ( [ omck ] ) . as anticipated , any finite - action solution of the form ( [ hh ] ) has to approach one of these constant minima when @xmath270 . the constant solutions @xmath271 , on the other hand , are maxima of the action . a representative of this type was already encountered in sec . [ fn ] . the radial equation for @xmath267 can be derived by inserting the ansatz ( [ hh ] ) into the general field equation ( [ omeq ] ) or , more directly , by varying the radial lagrangian ( [ lrad ] ) . either way , the result is@xmath272 i.e. an ordinary nonlinear differential equation of fourth order and of fuchsian type . the associated boundary value problem can be solved numerically with rather moderate computer resources . the exploration of the full solution space is aided by two discrete symmetries of eq . ( [ eqrad ] ) which imply that any solution @xmath273 gives rise to the additional solutions @xmath274 and @xmath275 . the former is a consequence of @xmath172 = \gamma\left [ u^{\dagger}\right ] $ ] while the latter simply reflects the periodicity in the angular variable @xmath90 . not surprisingly , the field equation ( [ eqrad ] ) comprises the gribov equation @xcite . it consists of the terms proportional to @xmath276 which originate from the nonlinear-@xmath70-model part of the lagrangian ( [ lrad ] ) and dominate when @xmath35 becomes the largest scale and/or when the higher derivatives become small .- independent terms of eq . ( [ eqrad ] ) are not negligible . this makes stable soliton solutions possible . ] the analogy between the nonlinear potential term @xmath277 and a one - dimensional pendulum in a gravitational field , with the logarithmic variable @xmath278 . ] often used to characterize the solution spectrum of the gribov equation , therefore applies to eq . ( [ eqrad ] ) as well . the stable ( unstable ) equilibrium positions of the `` pendulum '' are @xmath279 ( @xmath280 ) , modulo a multiple of @xmath281 which represents additional full turns . it will be shown in sec . [ qspi ] that this analogy suffices to understand the qualitative behavior of all numerical solutions . the @xmath131 parametrization ( [ uparam ] ) of the @xmath18 field implies that regular solutions @xmath267 of the radial hedgehog equation ( [ eqrad ] ) approach a multiple of @xmath281 at the origin . their small-@xmath282 behavior can therefore be determined analytically , either by expanding the nonlinearity of eq . ( [ eqrad ] ) into powers of small deviations @xmath283 from the constant action minima @xmath284 or by expanding the @xmath282 dependence of the full solution into a frobenius series . similarly , finite - action solutions can be obtained for @xmath270 by asymptotically expanding around the hedgehog vacua . inside their regions of validity , these expansions provide useful insights into the qualitative behavior of the hedgehog solutions as well as quantitative checks on the numerical solutions to be found in sec . [ qspi ] . we start by deriving the solutions of the linearized hedgehog equation and the corresponding power series expansion around the origin . inserting the ansatz @xmath285 into the radial saddle point equation ( [ eqrad ] ) and retaining only terms up to first order in @xmath286 , one arrives at the fourth - order linear differential equation @xmath287 which can be solved analytically by standard techniques . the general solution is a superposition of four linearly independent base solutions @xmath288 , @xmath289 whose real , dimensionless coefficients @xmath290 remain undetermined and have to be specified by imposing initial or boundary conditions . the base solutions @xmath291 are @xmath292 the requirements of regularity and uniqueness on the solutions at the origin dictate two of the boundary conditions . the first one , @xmath293 , implies @xmath294 and thus ensures uniqueness at @xmath223 while the second one , @xmath295 , is then imposed by the behavior of the base solutions ( [ pbas ] ) . accordingly , the small-@xmath282 behavior of the general regular solution is restricted to @xmath296 where the constants @xmath297 are linear combinations of the @xmath298 whose values can e.g. be specified by providing initial data for @xmath299 and @xmath300 . all higher - order coefficients of the expansion are then fixed . alternatively , one can obtain the solutions of the full , nonlinear saddle point equation ( [ eqrad ] ) towards @xmath301 by analytical continuation into a frobenius series . a somewhat tedious calculation yields @xmath302 r^{7}+o\left ( r^{9}\right ) \label{sersoln}%\ ] ] where the coefficients @xmath303 and @xmath304 are again left to be determined by initial conditions . even - order derivatives of @xmath90 ( or equivalently the coefficients @xmath305 ) vanish at @xmath223 while those of odd order , @xmath306 , can be expressed in terms of @xmath303 and @xmath304 . a comparison between eqs . ( [ linsoln ] ) and ( [ sersoln ] ) shows that the solutions of the exact radial equation start to differ from those of the linearized equation at @xmath307 . hence the series solution ( [ sersoln ] ) permits a more accurate check of the numerical solutions over a larger radial interval . analogous expansions around the constant action minima @xmath308 exist asymptotically , i.e. towards @xmath270 , for all finite - action solutions . infinite - action solutions of eq . ( [ eqrad ] ) , finally , can be linearized around the constant solutions @xmath309 which they approach at one or both ends of the radial domain . the resulting equation for @xmath286 differs from eq . ( [ lineq ] ) only in the sign of the @xmath286 term . its solutions are linear combinations of generalized hypergeometric functions . we now turn to the numerical solution of the hedgehog saddle point equation ( [ eqrad ] ) . due to the periodicity in @xmath90 , the considered range of boundary values can be limited without loss of generality to @xmath310 0,2\pi\right ] $ ] . regularity at the origin then further specifies @xmath311 and imposes @xmath312 ( see sec . [ anhh ] ) . the value of the topological charge @xmath8 fixes a third boundary condition , @xmath313 owing to eq . ( [ hq ] ) . hence all regular hedgehog solutions in a given @xmath8-sector can be found by just varying the value of a fourth boundary condition . in the following , we use the initial slope @xmath314 for this purpose . at the end of the section , we will also find irregular solutions with @xmath315 . we begin our exploration of the hedgehog solution space by searching for the regular finite - action solutions of eq . ( [ eqrad ] ) which , as established in sec . [ top ] , carry integer values of @xmath8 . the numerical analysis shows ( and the pendulum analogy in sec . [ shs ] will explain ) that only one solution of this type exists in each @xmath8 sector . in the simplest case , @xmath179 , this is just the translationally invariant vacuum solution @xmath316 . for @xmath317 we find the prototypical nontrivial hedgehog solution , depicted in fig . 2 . in order to clarify its physical interpretation , we note that it shares the @xmath245 homotopy classification , encoded in the topological charge @xmath318 $ ] of the relative gauge orientation @xmath319 , with the yang - mills instanton @xcite . of course , both also share the saddle point property , as the instanton minimizes the classical euclidean yang - mills action in the @xmath317 sector . in order to trace their association further , we inspect the relative gauge orientation @xmath320 of a yang - mills instanton with size @xmath321 . it is of hedgehog form as well , and its @xmath166-dependence in ( euclidean ) temporal gauge is known to be @xcite @xmath322 in the parametrization ( [ hh ] ) . for a direct comparison with our @xmath317 solution , we have included @xmath323 with @xmath324 as the dashed curve in fig . 2 ( and adapted it to our periodicity interval convention by adding @xmath281 ) . the radial dependence of both can be seen to be surprisingly similar . this implies that the dominant contributions from all @xmath317 gauge field orbits to the vacuum overlap have a relative gauge orientation close to that of the yang - mills instanton and indicates that the @xmath317 hedgehog solution primarily summarizes contributions from the instanton orbit . ( of course , one would not expect exact agreement since our solutions contain scale - symmetry breaking quantum corrections and contributions from other @xmath317 gauge fields as well . ) accordingly , and generalizing the above findings to multi - instanton solutions , we will refer to the unique regular finite - action solution of eq . ( [ eqrad ] ) with integer @xmath8 as the `` @xmath8-instanton class '' . our hedgehog saddle point equation ( [ eqrad ] ) and its instanton class solutions derive from the gradient - expanded soft - mode lagrangian ( [ efflagr ] ) . it is instructive to compare this approach with a variational estimate of one - instanton contributions to the bare action ( [ gammab ] ) in ref . @xcite . by approximately minimizing the bare action with one - parameter families of trial functions similar to the instanton profile ( [ omin ] ) and using qualitative scaling properties , it was argued in ref . @xcite that radiative corrections can stabilize the instanton size . our exact saddle point solutions make the dynamical size stabilization manifest . we have already traced the underlying mechanism to the virial theorem ( [ vth ] ) which is independent of most specific features of the soft - mode dynamics and thus overcomes the chronic infrared instability of dilute instanton gases @xcite in a rather generic way . for @xmath325 gev , the size @xmath326 and the average instanton size might be related to an effective ir fixed point of the type considered in ref . ] of the 1-instanton class solution agrees inside errors with the results of instanton liquid model @xcite and lattice @xcite simulations . it also assures that the two leading terms of the gradient expansion ( [ gexp ] ) yield a sufficiently accurate approximation to the instanton action ( cf . the comments below eq . ( [ sfterm ] ) ) . our 1-instanton class profile function @xmath327 is rather similar to the one found by approximately minimizing the bare action ( [ gammab ] ) variationally @xcite . this indicates that the bulk of the instanton s physics and size distribution is generated by soft modes , as one would intuitively expect . our approach therefore provides a well - adapted and efficient framework for the treatment of these and other vacuum fields . in contrast to variational approaches , furthermore , it allows to systematically find _ all _ saddle points exactly ( including those which are not of hedgehog form ) . already in the hedgehog sector , for example , we will find solutions with more complex and unprecedented shapes than eq . ( [ omin ] ) . since there is little guidance for the choice of suitable trial functions in these and other cases , such solutions would be difficult to find variationally . according to eq . ( [ q - hh ] ) , all monotonic hedgehog solutions with @xmath168 ( @xmath169 ) have negative ( positive ) slopes @xmath328 at the origin . the anti - instanton class with @xmath329 , in particular , results from changing the sign of the instanton boundary value , @xmath330 , and can be obtained without further calculation : it simply results from the combined action of the two symmetry transformations @xmath331 and @xmath332 on the instanton class solution . hence the @xmath255 instanton classes have degenerate action values , precisely as their yang - mills counterparts . multi - instanton class solutions are characterized by an integer topological charge @xmath333 . the modulus @xmath334 of their ( negative ) initial slope grows monotonically with @xmath8 , i.e. @xmath335 as in the 1-instanton case , the action - degenerate multi - antiinstanton classes with @xmath336 can be constructed by flipping the sign of the corresponding multi - instanton classes and adding @xmath337 . the initial slopes of the multi-(anti)instanton solutions are therefore related by @xmath338 the @xmath339 hedgehog solutions correspond to special arrangements of the underlying yang - mills multi-(anti)instantons . in fact , the relative gauge orientation @xmath319 of a multi - instanton configuration is of hedgehog type only if all individual ( anti)instantons are centered at the origin . this raises the question whether @xmath340 yang - mills instanton solutions with separated individual positions are at least approximately represented by other solutions of the saddle point equations ( [ omeq ] ) and ( [ neq ] ) . experience from skyrme - type models , whose analogous @xmath339 skyrmion solutions are well approximated by rational @xcite or harmonic @xcite maps , might suggest that similar types of non - hedgehog field configurations approximate higher-@xmath8 solutions of eqs . ( [ omeq ] ) and ( [ neq ] ) as well . the action of the 1-instanton class solution is large ( @xmath341 at@xmath342 gev ) , in analogy with the large action of typical yang - mills instantons , and its direct impact on the saddle point expansion is therefore small . ) and hence does not solve the bogomolnyi - type equation ( [ beq ] ) . this is in constrast to the yang - mills instanton which is the absolute minimum of the euclidean yang - mills action in the @xmath317 sector and therefore self - dual . ] moreover , the action bound ( [ bb ] ) implies that higher-@xmath8 instanton classes should be irrelevant for most amplitudes , with potentially important exceptions as mentioned in sec . [ bogbnd ] . instanton `` liquid '' vacuum models ( ilms ) @xcite are built on the same premise and suggest that physically far more relevant contributions originate instead from ensembles of instantons and anti - instantons with equal average densities . it would be important to determine whether contributions of this sort are approximately represented by nontrivial @xmath179 solutions of eqs . ( [ omeq ] ) and ( [ neq ] ) as well . in any case , our above results imply that they can not be of hedgehog type . we now extend our search to hedgehog solutions with infinite action . although their relevance for the saddle point expansion is not obvious , one might speculate that their infinite - action suppression could be overcome by some additional mechanism ( see below ) . our chief motivation for discussing them here , however , derives from their association with the infinite - action meron solutions @xcite of the classical ( euclidean ) yang - mills equation . hedgehog solutions with infinite action are far more the rule than the exception . in fact , all regular ( @xmath311 ) solutions of eq . ( [ eqrad ] ) with initial slopes @xmath343 inbetween the discrete set of instanton - class values @xmath344 have infinite action since they approach one of the constant fields @xmath345 towards spacial infinity . the latter carry a nonzero action density ( cf . ( [ lrad ] ) ) and furthermore imply that the corresponding , asymptotic @xmath18 fields @xmath346 remain angle - dependent . exactly the same behavior characterizes the relative gauge orientations @xmath319 of yang - mills merons in temporal gauge , which are of hedgehog form as well . furthermore , eq . ( [ phihhinf ] ) shows that solutions with @xmath347 carry half - integer topological charge @xmath8 , again as the yang - mills merons . in analogy with the instanton - class solutions of the previous section , we will therefore call these solutions `` @xmath348-meron classes '' . the profile function @xmath349 of a typical 1-meron class solution with @xmath350 is drawn in fig . a direct comparison with the corresponding profile of the yang - mills meron in temporal gauge , @xmath351 ( where @xmath0 is the step function ) , is complicated by the fact that the classical meron is pointlike while our solutions incorporate quantum effects which break dilatation symmetry and stabilize their size at a finite value . such effects are expected to smoothen the singularity of the yang - mills meron , too , and probably cause our solution @xmath349 to become non - monotonic by overshooting in the transition region . we therefore draw the yang - mills meron profile in fig . 3 ( dashed curve ) with the finite size @xmath352 of the instanton class solutions ( and adapt it to our periodicity interval convention ) . our nomenclature for multi - meron classes with @xmath353 includes only solutions with half - integer topological charge because solutions with even `` meron number '' and correspondingly integer @xmath8 coincide with the @xmath8-instanton classes . this is expected since the relative gauge orientation @xmath354 of the underlying yang - mills multi - merons is of hedgehog type only if all individual merons sit on top of each other . such configurations , when carrying integer overall values of @xmath8 , coalesce into the corresponding yang - mills instantons and those are represented by the @xmath8-instanton class solutions in our framework . the behavior of the multi - meron class solutions with @xmath355 is qualitatively rather similar to that of the 1-meron class , although size and strength scales may differ substantially . as an example , fig . 4 shows a typical 3-meron class solution . an important general property of all solutions with half - integer @xmath8 is that they come in families of continuously varying sizes . as already alluded to , this is because their size depends on a second mass scale @xmath356 ( in addition to @xmath35 ) and because solutions for all values of @xmath357 in the finite intervals@xmath358 ( where @xmath359 , i.e. @xmath360 , and @xmath361 are implied ) can be found . as in the instanton sector , multi - anti - meron classes with negative @xmath8 are obtained from the positive-@xmath8 solutions by changing their sign and adding @xmath337 . the variable mass scale @xmath357 in the meron sectors implies that there are infinitely more meron than instanton class solutions . this makes it tempting to speculate that the meron `` entropy '' contributions to the weight function of functional integrals over @xmath18 might be able to overcome the infinite - action suppression . if so , it would shed new light on the physical interpretation not only of our solutions but also of the yang - mills merons themselves , whose potential role remains controversial . furthermore , it would suggest a modified saddle point expansion in which action and entropy are minimized jointly . these issues deserve further investigation . above we have classified all regular hedgehog solutions , i.e. those which satisfy the initial condition @xmath311 . we are now going to examine the remaining solution classes of the radial field equation ( [ eqrad ] ) . its members share the alternative initial condition @xmath315 , may display a rather complex spacial structure and are characterized by a monopole - type singularity at the origin , i.e. they solve eq . ( [ eqrad ] ) everywhere except at @xmath206 . in order to understand the qualitative behavior of both regular and irregular hedgehog solution classes from a common perspective , it is useful to elaborate on the analogy between the hedgehog equation ( [ eqrad ] ) and the pendulum equation which was mentioned in sec . [ hhtype ] . according to this analogy , the instanton classes correspond to exactly @xmath8 full turns of the pendulum , where the sign of @xmath8 indicates the direction of the rotation . the pendulum mass starts in the unstable equilibrium position at time @xmath362 with just enough initial speed @xmath363 to finally end up there again for @xmath364 . this analogy implies , in particular , that there is exactly one regular hedgehog solution for each integer @xmath8 and that the constant @xmath365 is the only regular solution with @xmath179 . the meron class solutions start from the unstable equilibrium position as well . however , their initial velocity @xmath343 is insufficient for completing all turns in full . the last turn remains uncompleted , i.e. the pendulum swings back , oscillates around and finally settles into the stable equilibrium position . hence all meron class solutions have half - integer @xmath8 and are non - monotonic . for the irregular solutions , on the other hand , the pendulum starts at the stable equilibrium position @xmath279 . when not provided with sufficient initial speed @xmath343 to complete a full turn , it just performs damped oscillations around @xmath279 . the corresponding solution has @xmath179 and is depicted in fig . 5 . when @xmath366 is sufficiently large , however , the pendulum can perform @xmath8 full turns before settling into the stable equilibrium position . as a consequence , all these solutions have integer @xmath190 and infinite action ( cf . eq . ( [ lrad ] ) ) . pursuing the analogy further , one would also expect irregular solutions which have an initial value @xmath299 exactly as needed to end up at the unstable equilibrium position when @xmath270 ( after possibly completing a number of full turns ) . such configurations would carry a half - integer topologically charge @xmath8 and a finite action . for obvious reasons they turn out to be highly sensitive to variations of the initial condition , however , and therefore difficult to establish numerically . in contrast to the instanton and meron classes , the singular hedgehog solutions do not seem to have obvious analogs among the solutions of the classical yang - mills equation . as opposed to the instanton ( meron ) classes , furthermore , the irregular integer-@xmath8 ( half - integer-@xmath8 ) solutions have infinite ( finite ) action . this allows for nontrivial hedgehog solutions with @xmath179 , which are necessarily irregular at the origin . the physical interpretation of all irregular hedgehog solutions and their relevance for the saddle point expansion remain to be clarified . the main results of this paper are a practicable saddle point expansion for the yang - mills vacuum overlap amplitude in terms of gauge invariant , local matrix fields and the identification of new gluonic ir degrees of freedom in this framework . after adopting a gauge - projected gaussian approximation to the vacuum wave functional , the ir degrees of freedom can be obtained explicitly as the saddle points of a soft - mode action which gather contributions from dominant gluon field families to soft yang - mills amplitudes and thus represent collective properties of the yang - mills dynamics . since their gauge invariant definition makes no reference to specific amplitudes , furthermore , the ir degrees of freedom are universal . they provide both new structural insights into the organization of the low - energy yang - mills dynamics and the principal input for a systematic saddle point expansion of soft amplitudes . our survey of the saddle point solution space uncovered a diverse spectrum of ir degrees of freedom which carry several topological charges with associated bogomolnyi - type action bounds and obey a virial theorem which guarantees their scale stability . solutions with a relatively high degree of symmetry were obtained either analytically or with modest numerical effort . since solutions of this type are generally characterized by small action values and hence play a dominant role in the saddle point expansion , we have investigated their properties in some detail . besides translationally invariant vacua and analytical solutions with a fixed relative gauge orientation , we have found topological solitons of hedgehog , ( vortex ) link and knot types . some of the ir degrees of freedom have a transparent physical interpretation directly in terms of the underlying gluon fields . the contributions from the gauge orbits of yang - mills instanton and merons , in particular , are gathered by saddle point fields of hedgehog type which share their ( integer or half - integer ) topological charge and represent vacuum tunneling processes in the hamiltonian formulation of non - abelian gauge theory . although our saddle point solutions contain quantum effects and potentially relevant contributions from other gauge fields , those in the instanton class turn out to reproduce the relative gauge orientation between the in- and out - vacua of the yang - mills instanton rather closely . the finite extent of our meron solution classes , on the other hand , is generated by quantum effects which smoothen the singularities of the classical , pointlike yang - mills merons . nevertheless , our meron classes turn out to share the infinite action of their yang - mills counterparts . among all those ir fields which carry one unit of topological ( instanton ) charge , the single ( anti- ) instanton classes are expected to attain the minimal action value . as a consequence of the action bound , they should therefore dominate the saddle point expansion in all topological charge sectors . similar configurations emerged in a variational treatment along with qualitative arguments in favor of their size stabilization . in our approach , this stabilization is manifest in the exact instanton class solutions themselves . in fact , their size turns out to be fixed at about twice the inverse ir gluon mass scale and agrees inside errors with instanton liquid model and lattice results . the underlying virial theorem and the soft gluon mass generation therefore provide new insight into the mechanism by which the chronic infrared diseases of dilute yang - mills instanton gases are overcome . the sizes of the meron class solutions with half - integer topological charge turn out to be of a more complex origin . besides the dynamical gluon mass they depend on a second , variable mass scale which is encoded in a boundary condition . hence meron classes exist within large and continuous size ranges and consequently form a far more extensive solution family than the instanton classes . this opens up the hypothetical possibility for their entropy to overcome their infinite action suppression in functional integrals . such a mechanism would not only help to clarify the physical impact of our solutions but also shed new light on the still controversial role of the yang - mills merons themselves . in addition to the instanton and meron classes , finally , there exists a third class of hedgehog solutions which contains a monopole - type singularity at the origin . these irregular solutions can carry half - integer and integer ( including zero ) topological charges as well , and generally have infinite action . hence their potential physical relevance seems to depend on the existence of additional mechanisms which could both smoothen their singularity and overcome their infinite - action suppression . several other remarkable families of ir degrees of freedom turn out to be represented by topological solitons of faddeev - niemi type , i.e. by ( potentially twisted ) links and knots . in fact , our saddle point dynamics contains a specific generalization of the faddeev - niemi lagrangian and shows explicitly how it is embedded in the gaussian approximation to the yang - mills vacuum wave functional . this puts faddeev - niemi theory into a new perspective , as the effective dynamics of dominant sets of gauge field orbits with a collective hopf charge , and provides the underlying unit - vector field with a manifestly gauge invariant meaning . the latter would well accord with the tentative interpretation of knot solutions as glueballs by faddeev , niemi and coworkers . this and other interpretations could be tested in our framework by directly evaluating the impact of the knot saddle points on suitable amplitudes , e.g. on glueball correlation functions . more generally , the saddle point expansion allows the systematic calculation of contributions from all relevant ir degrees of freedom to functional integrals which represent soft yang - mills amplitudes . first calculations of this type , focusing on fundamental vacuum properties including gluon condensates and the topological susceptibility , are underway . in addition , our approach makes it possible to analyze the gauge field content of any ir variable individually by applying standard functional techniques to the integrals over gluon fields with which they are associated . investigations of this sort would not only provide further structural insight into specific ir degrees of freedom and their physical role but may also shed new light on the dynamical mechanisms by which soft gauge fields organize themselves into collective degrees of freedom . the diverse topological properties of the ir variables demonstrate that the gauge - projected gaussian wave functional not only captures the homotopy structure of the gauge group but also implements `` derivative '' topologies which further characterize the saddle point families . due to the typical robustness of such topological properties , the related results are expected to remain at least qualitatively valid beyond the gaussian approximation . moreover , the saddle point expansion engenders the means to test this expectation quantitatively , by mapping out limitations of the underlying vacuum wave functional in comparison with lattice data . extensions of our framework to suitable supersymmetric gauge theories would even permit analytical tests of this sort , e.g. by tracing vestiges of the monopole - based confinement mechanism in the vacuum functional . the insights gained from such investigations may also provide specific guidance for the development of improved collective - mode actions and consequently generate systematic corrections to the ir variables . our ir saddle point expansion can be extended in several directions . a first important task would be a more exhaustive survey of the saddle - point solution space which should encompass potentially relevant approximate solutions . the extension to qcd proper requires the generalization to the gauge group @xmath367 and the implementation of quarks into the gaussian wave functional , both of which pose no conceptual problems . most topological properties , in particular , reside in the trivially embedded @xmath368 subgroup of the full color group and will remain unchanged . a sufficiently complete treatment of the quark - gluon interactions and their impact on the effective action , however , appears to be more challenging . our approach opens up a variety of directions for future research . besides those already mentioned , it would for example be interesting to explore relations between the gauge - invariant ir degrees of freedom and gauge - dependent gluonic structures ( monopoles , vortices , branes etc . ) and amplitudes ( e.g. the 2-dimensional nonlocal gluon condensate and green functions ) which appear in gauge - fixed formulations . another useful endeavor would be the calculation of those higher - dimensional vacuum condensates which provide the principal input for the operator product expansion of glueball correlators . duality sum rules could then link the contributions from different ir degrees of freedom with the low - lying glueball spectrum @xcite , e.g. as a precursor and complement to a direct saddle - point evaluation of glueball correlation functions . financial support by the conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) of brazil as well as the hospitality and a visiting scientist grant of the ect * in trento ( italy ) are gratefully acknowledged . 1 . an example of the @xmath369 solution class . as in all following figures , we display the solution at @xmath370 gev . other values of @xmath35 can immediately be accommodated by scaling the @xmath282-axis . our conventions for the periodicity interval of @xmath90 restrict its initial value to @xmath371 0,2\pi\right ] $ ] . 2 . the canonical 1-instanton class solution with @xmath317 . 3 . a typical meron - 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wesley , reading , ma , usa ( 1995 ) . faddeev , _ quantization of solitons _ , institute for advanced study preprint ias-75-qs70 ; in _ einstein and several contemporary tendencies in the field theory of elementary particles , relativity , quanta and cosmology _ , vol . m. pantaleo and f. de finis ) , 185 ( johnson reprint , 1979 ) . c. michael and p.s . spencer , phys . d * 52 * , 4691 ( 1995 ) ; t. degrand , a. hasenfratz and t.g . kovacs , nucl . * b 505 * , 417 ( 1997 ) ; ph . de forcrand , m. garcia perez and i .- o . stamatescu , nucl . phys . * b 499 * , 409 ( 1997 ) ; m.j . teper , phys . d * 58 * , 014505 ( 1998 ) . | we set up a new calculational framework for the yang - mills vacuum transition amplitude in the schrdinger representation . after integrating out hard - mode contributions perturbatively ,
we perform a gauge invariant gradient expansion of the ensuing soft mode action which renders a subsequent saddle point expansion for the vacuum overlap manageable .
the standard `` squeezed '' approximation for the vacuum wave functional then allows for an essentially analytical treatment of physical amplitudes .
moreover , it leads to the identification of dominant and gauge invariant classes of gauge field orbits which play the role of gluonic infrared ( ir ) degrees of freedom . those emerge as a rich variety of ( mostly solitonic ) solutions to the saddle point equations which are characterized by a common relative gauge orientation of the underlying gluon fields .
we discuss their scale stability , guaranteed by a virial theorem , and other general properties including topological quantum numbers and action bounds .
we then find important saddle point solutions explicitly and examine their physical impact .
some of them are related to tunneling solutions of the classical yang - mills equation , i.e. to instantons and merons , while others appear to play unprecedented roles .
a remarkable new class of ir degrees of freedom comprises vortex and knot solutions of faddeev - niemi type , potentially related to glueballs . |
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covering theorems are known to be among some of the fundamental tools of measure theory . they reflect the geometry of the space and are commonly used to establish connections between local and global behavior of measures . covering theorems and their applications have been studied for example in @xcite and @xcite . there are several types of covering results , all with the same purpose : from an arbitrary cover of a set in a metric space , one extracts a subcover as disjointed as possible . we will consider more specifically here the so - called besicovitch covering property ( bcp ) which originates from the work of besicovitch ( @xcite , @xcite , see also @xcite , @xcite , @xcite ) in connection with the theory of differentiation of measures . see subsection [ subsection : bcp ] for a more detailed presentation of the besicovitch covering property and its applications . the geometric setting in which we are interested is the setting of carnot groups equipped with so - called homogeneous distances , and more specifically here the heseinberg groups @xmath0 . our main result in this paper , theorem [ thm : main ] , is the fact that bcp holds for the homogeneous distances on @xmath0 whose unit ball centered at the origin coincides with an euclidean ball centered at the origin . these distances are bi - lipschitz equivalent to any other homogeneous distance on @xmath0 and in particular to the commonly used cygan - kornyi ( also usually called kornyi or gauge ) and carnot - carathodory distances . recall that two distances @xmath1 and @xmath2 are said to be bi - lipschitz equivalent if there exists @xmath3 such that @xmath4 . cygan - kornyi and carnot - carathodory distances are known not to satisfy bcp ( @xcite , @xcite , @xcite ) . we indeed stress that the validity of bcp depends strongly on the distance the space is endowed with , and more specifically on the geometry of its balls . to put some more evidence on this fact and to put our result in perspective , we also prove in the present paper two criteria that imply the non - validity of bcp . they give two large families of homogeneous distances on @xmath0 that do not satisfy bcp and show that in some sense our example for which bcp holds is sharp . see section [ section : distancesw / obcp ] , theorem [ thm : nobcpingoingcorners ] and theorem [ thm : nobcpoutgoingcorners ] . as a matter of fact , our first criterion applies to the cygan - kornyi and to the carnot - carathodory distance , thus giving also new geometric proofs of the failure of bcp for these distances . it also applies to the so - called box - distance ( the terminology might not be standard although this distance is a standard homogeneous distance on @xmath0 , see ) thus proving the non - validity of bcp for this latter homogeneous distance as well . going back to the distances considered in the present paper and for which we prove that bcp holds , hebisch and sikora showed in @xcite that in any carnot group , there are homogeneous distances whose unit ball centered at the origin coincides with an euclidean ball centered at the origin with a small enough radius . in the specific case of the heisenberg groups , these distances are related to the cygan - kornyi distance . they can indeed be expressed in terms of the quadratic mean of the cygan - kornyi distance ( at least for some specific value of the radius of the euclidean ball which coincides with the unit ball centered at the origin ) together with the pseudo - distance on @xmath0 given by the euclidean distance between horizontal components . these distances have been previously considered in the literature . lee and naor proved in @xcite that these metrics are of negative type on @xmath0 . recall that a metric space @xmath5 is said to be of negative type if @xmath6 is isometric to a subset of a hilbert space . combined with the work of cheeger and kleiner @xcite about weak notion of differentiability for maps from @xmath0 into @xmath7 , which leads in particular to the fact that @xmath0 equipped with a homogeneous distance does not admit a bi - lipschitz embedding into @xmath7 , this provides a counterexample to the goemans - linial conjecture in theoretical computer science , which was the motivation for these papers . let us remark that the cygan - kornyi distance is not of negative type on @xmath0 . we refer to subsection [ subsection : heisenberg ] for the precise definition of our distances and their connection with the cygan - kornyi distance and the distances of negative type considered in @xcite . let @xmath5 be a metric space . when speaking of a ball @xmath8 in @xmath9 , it will be understood in this paper that @xmath8 is a closed ball and that it comes with a fixed center and radius ( although these in general are not uniquely determined by @xmath8 as a set ) . thus @xmath10 for some @xmath11 and some @xmath12 where @xmath13 . one says that the besicovitch covering property ( bcp ) holds for the distance @xmath1 on @xmath9 if there exists an integer @xmath14 with the following property . let @xmath15 be a bounded subset of @xmath5 and let @xmath16 be a family of balls in @xmath5 such that each point of @xmath15 is the center of some ball of @xmath16 . then there is a subfamily @xmath17 whose balls cover @xmath15 and such that every point in @xmath9 belongs to at most @xmath18 balls of @xmath19 , that is , @xmath20 where @xmath21 denotes the characteristic function of a set @xmath15 . when equipped with a homogeneous distance , the heisenberg groups turn out to be doubling metric spaces . recall that this means that there exists an integer @xmath22 such that each ball with radius @xmath12 can be covered with less than @xmath23 balls with radius @xmath24 . when @xmath5 is a doubling metric space , bcp turns out to be equivalent to a covering property , strictly weaker in general , that we call the weak besicovitch covering property ( w - bcp ) ( the terminology might not be standard ) and with which we shall work in this paper . first , let us fix some more terminology with the following definition . [ besicovitch balls ] we say that a family @xmath16 of balls in @xmath5 is a _ family of besicovitch balls _ if @xmath25 is a finite family of balls such that @xmath26 for all @xmath8 , @xmath27 , @xmath28 , and for which @xmath29 . one says that the weak besicovitch covering property ( w - bcp ) holds for the distance @xmath1 on @xmath9 if there exists an integer @xmath14 such that @xmath30 for every family @xmath16 of besicovitch balls in @xmath5 . the validity of bcp implies the validity of w - bcp . we stress that there exists metric spaces for which w - bcp holds although bcp is not satisfied . however , when the metric is doubling , both covering properties turn out to be equivalent as stated in characterization [ characterization : equivalentbcp ] below . this characterization can be proved following the arguments of the proof of theorem 2.7 in @xcite . [ characterization : equivalentbcp ] let @xmath5 be a doubling metric space . then bcp holds for the distance @xmath1 on @xmath9 if and only if w - bcp holds for the distance @xmath1 on @xmath9 . as already said , covering theorems and especially the besicovitch covering property and the weak besicovitch covering property play an important role in many situations in measure theory , regularity and differentiation of measures , as well as in many problems in harmonic analysis . this is particularly well illustrated by the connection between w - bcp and the so - called differentiation theorem . the validity of bcp in the euclidean space is due to besicovitch and was a key tool in his proof of the fact that the differentiation theorem holds for each locally finite borel measure on @xmath31 ( @xcite , @xcite , see also @xcite , @xcite ) . moreover , as emphasized in theorem [ thm : diff ] , the validity of w - bcp turns actually out to be equivalent to the validity of the differentiation theorem for each locally finite borel measure as shown in @xcite . * preiss ) [ thm : diff ] let @xmath5 be a complete separable metric space . then the differentiation theorem holds for each locally finite borel measure @xmath32 on @xmath5 , that is , @xmath33 for @xmath32-almost every @xmath11 and for each @xmath34 if and only if @xmath35 where , for each @xmath36 , w - bcp holds for family of balls centered on @xmath37 with radii less than @xmath38 for some @xmath39 . as already stressed , the fact that bcp holds in a metric space depends strongly on the distance with which the space is endowed . on the one hand , with very mild assumptions on the metric space ( namely , as soon as there exists an accumulation point ) , one can indeed always construct bi - lipschitz equivalent distances as close as we want from the original distance and for which bcp is not satisfied , as shown in the following result . [ thm : destroybcp ] let @xmath5 be a metric space . assume that there exists an accumulation point in @xmath5 . let @xmath40 . then there exists a distance @xmath2 on @xmath9 such that @xmath41 and for which w - bcp , and hence bcp , do not hold . a slightly different version of this result is stated in theorem 3 of @xcite . for sake of completeness , we give in section [ section : destroybcp ] a construction of such a distance as stated in theorem [ thm : destroybcp ] . on the other hand , the question whether a metric space can be remetrized so that bcp holds is in general significantly more delicate . as already explained , the main result of the present paper , theorem [ thm : main ] , is a positive answer to this question for the heisenberg groups equipped with ad - hoc homogeneous distances , namely those whose unit ball at the origin coincides with an euclidean ball with a small enough radius . as a set we identify the heisenberg group @xmath0 with @xmath42 and we equip it as a topological space with the euclidean topology . we choose the following convention for the group law @xmath43 where @xmath44 , @xmath45 , @xmath46 and @xmath47 belong to @xmath48 , @xmath49 and @xmath50 belong to @xmath51 and @xmath52 denotes the usual scalar product in @xmath31 . this corresponds to a choice of exponential and homogeneous coordinates . the one parameter family of dilations on @xmath0 is given by @xmath53 where @xmath54 these dilations are group automorphisms . [ def : homogeneousdistance ] a distance @xmath1 on @xmath0 is said to be _ homogeneous _ if the following properties are satisfied . first , it induces the euclidean topology on @xmath0 . second , it is left invariant , that is , @xmath55 for all @xmath56 , @xmath57 , @xmath58 . and third , it is one - homogeneous with respect to the dilations , that is , @xmath59 for all @xmath56 , @xmath60 and all @xmath61 . it turns out that homogeneous distances on @xmath0 do exist in abundance and make it a doubling metric space . it is also well known that any two homogeneous distances are bi - lipschitz equivalent . see for example @xcite for more details about the heisenberg groups and more generally carnot groups . the ( family of ) homogeneous distance(s ) we consider in this paper can be defined in the following way . for @xmath62 , we denote by @xmath63 the euclidean ball in @xmath64 centered at the origin with radius @xmath65 , that is , @xmath66 where @xmath67 denotes the euclidean norm in @xmath68 and we set @xmath69 hebisch and sikora proved in @xcite that if @xmath62 is small enough , then @xmath70 actually defines a distance on @xmath0 . more generally this holds true in any carnot group starting from the set @xmath63 given by the euclidean ball centered at the origin with radius @xmath62 small enough , where one identifies in the usual way the group with some @xmath71 where @xmath72 is its topological dimension . it then follows from the very definition that @xmath70 turns out to be the homogeneous distance on @xmath0 for which the unit ball centered at the origin coincides with the euclidean ball with radius @xmath65 centered at the origin . the geometric description of arbitrary balls that can then be deduced from the unit ball centered at the origin via dilations and left - translations is actually of crucial importance for understanding the reasons why bcp eventually holds for these distances . on the other hand , it is particularly convenient to note that in the specific case of the heisenberg groups , one also has a fairly simple analytic expression for such distances whose unit ball at the origin is given by an euclidean ball centered at the origin . this will actually be technically extensively used in our proof of theorem [ thm : main ] . this also gives the explicit connection with the cygan - kornyi distance and the distances of negative type considered by lee and naor in @xcite . set @xmath73 for @xmath74 . then one has @xmath75 see section [ section : preliminaries ] . first , note that @xmath76 is a left - invariant pseudo - distance on @xmath0 that is one - homogeneous with respect to the dilations . next , when @xmath77 , @xmath78 is nothing but the cygan - kornyi norm which is well known to be a natural gauge in @xmath0 . it can actually be checked by direct computations that @xmath79 satisfies the triangle inequality for any @xmath80 and hence defines a homogeneous distance on @xmath0 . this was first proved by cygan in @xcite when @xmath77 . one then recovers from the analytic expression that @xmath70 actually defines a homogeneous distance on @xmath0 for any @xmath80 , giving also an explicit range of values of @xmath65 in @xmath0 for which this fact holds and was first observed in @xcite for general carnot groups and for small enough values of @xmath65 . for any @xmath81 , @xmath70 defines a homogeneous distance on @xmath0 . note that there might be other values of @xmath82 such that @xmath70 defines a homogeneous distance on @xmath0 . these distances turn out to be those considered by lee and naor in @xcite . the authors actually proved in @xcite that @xmath83 is of negative type in @xmath0 to provide a counterexample to the so - called goemans - linial conjecture . let us mention that it can easily be checked that the proof in @xcite extend to the distances @xmath70 for all @xmath81 . let us now state our main result . [ thm : main ] let @xmath62 be such that @xmath70 defines a homogeneous distance on @xmath0 . then bcp holds for the homogeneous distance @xmath70 on @xmath0 . for technical and notational simplicity , we will focus our attention on the first heisenberg group @xmath84 . we shall point out briefly in section [ section : hn ] the non - essential modifications needed to make our arguments work in any heisenberg group @xmath0 . the rest of the paper is organized as follows . in section [ section : preliminaries ] we fix some conventions about @xmath85 and the distance @xmath70 and state three technical lemmas on which the proof of theorem [ thm : main ] is based . the proof of these lemmas is given in sections [ section : prooflemma : axis ] and [ section : prooflemma : comparisonincone ] . section [ section : proofmainthm ] is devoted to the proof of theorem [ thm : main ] itself . in section [ section : distancesw / obcp ] we prove two criteria , theorem [ thm : nobcpingoingcorners ] and theorem [ thm : nobcpoutgoingcorners ] , for homogeneous distances on @xmath85 that imply that bcp does not hold . theorem [ thm : destroybcp ] is proved in section [ section : destroybcp ] . as already stressed we will focus our attention in sections [ section : preliminaries ] to [ section : distancesw / obcp ] on the first heisenberg group @xmath84 for technical and notational simplicity . the modifications needed to handle the case of @xmath0 for any @xmath86 will be indicated in section [ section : hn ] . we first fix some conventions and notations . next , we will conclude this section with the statement of the main lemmas on which the proof of theorem [ thm : main ] will be based . recall that we identify the heisenberg group @xmath85 with @xmath87 equipped with the group law given in and we equip it with the euclidean topology . we define the projection @xmath88 by @xmath89 when considering a specific point @xmath90 , we shall usually denote by @xmath91 its coordinates and we set @xmath92 from now on in this section , as well as in sections [ section : proofmainthm ] , [ section : prooflemma : axis ] and [ section : prooflemma : comparisonincone ] , we fix some @xmath93 such that @xmath94 as given in defines a homogeneous distance on @xmath85 . thus all metric notions and properties will be understood in these sections relatively to this fixed distance @xmath94 . in particular we shall denote the closed balls with center @xmath90 and radius @xmath12 by @xmath95 without further explicit reference to the distance @xmath94 with respect to which they are defined . remembering , we have the following properties . for @xmath96 , we have @xmath97 and @xmath98 from which we get @xmath99 for a point @xmath90 , we shall set @xmath100 using left - translations , we have the following properties for any two points @xmath101 , @xmath102 and @xmath103 where @xmath104 and @xmath105 by definition of the group law . note that if @xmath106 then @xmath107 . let us point out that balls in @xmath108 are convex in the euclidean sense when identifying @xmath85 with @xmath87 with our choosen coordinates . indeed , the unit ball centered at the origin is by definition the euclidean ball with radius @xmath65 in @xmath109 and thus is euclidean convex . next , dilations are linear maps and left - translations ( see ) are affine maps , hence @xmath110 is also an eucliden convex set in @xmath109 . this will be of crucial use for some of our arguments in the sequel and we state it below as a proposition for further reference . [ prop : ball_convex ] balls in @xmath108 are convex in the euclidean sense when identifying @xmath85 with @xmath87 with our choosen coordinates . we shall also use the following isometries of @xmath108 . first , rotations around the @xmath49-axis are defined by @xmath111 for some angle @xmath112 . next , the reflection @xmath113 is defined by @xmath114 using , one can easily check that these maps are isometries of @xmath108 . we state now the main lemmas on which the proof of theorem [ thm : main ] will be based . for @xmath115 , @xmath116 and @xmath117 , we set ( see figure [ fig1 ] ) @xmath118 [ fig : p ] .,title="fig:",height=226 ] [ fig : c_theta ] .,title="fig:",height=226 ] [ lemma : x : axis0 ] there exists @xmath119 , which depends only on @xmath120 , such that for all @xmath121 , there exists @xmath122 such that for all @xmath123 and for all @xmath124 , the following holds . let @xmath125 and @xmath126 be such that @xmath127 and @xmath128 . then at most one of these two points belongs to @xmath129 . for @xmath116 and @xmath117 , we set ( see figure [ fig2](a ) ) @xmath130 [ fig : t ] and @xmath131.,title="fig:",height=264 ] [ fig : c ] and @xmath131.,title="fig:",height=226 ] [ lemma : z : axis0 ] there exists @xmath132 and @xmath133 , depending only on @xmath120 , such that for all @xmath134 and all @xmath135 , the following holds . let @xmath125 and @xmath126 be such that @xmath127 and @xmath128 . then at most one of these two points belongs to @xmath136 . these two lemmas will be proved in section [ section : prooflemma : axis ] . for @xmath137 , we set ( see figure [ fig2](b ) ) @xmath138 [ lemma : comparisonincone ] there exists @xmath139 , which depends only on @xmath120 , such that for all @xmath140 the following holds . let @xmath90 and @xmath141 be such that @xmath142 then we have @xmath143 and @xmath144 this lemma will be proved in section [ section : prooflemma : comparisonincone ] . this section is devoted to the proof of theorem [ thm : main ] . recall that we consider here the case @xmath145 equipped with a homogeneous distance @xmath94 as defined in ( see section [ section : hn ] for the general case @xmath0 , @xmath86 ) . recall also that due to characterization [ characterization : equivalentbcp ] , theorem [ thm : main ] will follow if we find an integer @xmath14 such that @xmath30 for every family @xmath16 of besicovitch balls . see definition [ besicovitch balls ] for the definition of a family of besicovitch balls . we first reduce the proof to the case of some specific families of besicovitch balls . in what follows , when considering families of points @xmath146 we shall simplify the notations and set @xmath147 , @xmath148 and @xmath149 . recall that @xmath150 is defined in . [ lemma : reduction ] let @xmath151 and let @xmath152 be a family of besicovitch balls . then there exists a finite family of points @xmath146 such that @xmath153 is a family of besicovitch balls with the following properties . for every point @xmath154 in the family , we have @xmath155 and @xmath156 let @xmath157 be a family of besicovitch balls where @xmath158 . take @xmath159 . set @xmath160 remembering that left - translations are isometries and that , by convention , we set @xmath149 , we get that @xmath161 and @xmath162 hence @xmath163 is a family of besicovitch balls . since balls are euclidean convex ( see proposition [ prop : ball_convex ] ) and since @xmath164 for all @xmath165 , there are at most two balls in @xmath166 with their center on the @xmath49-axis . next , up to replacing the family @xmath146 by @xmath167 ( see for the definition of the reflection @xmath113 ) and up to re - indexing the points , one can find @xmath168 points @xmath169 that satisfy , such that @xmath170 ( see for the definition of the projection @xmath171 ) , and with @xmath172 . finally , up to a rotation around the @xmath49-axis ( see for the definition of rotations ) and up to re - indexing the points , we get by the pigeonhole principle that there exists an integer @xmath173 such that @xmath174 and such that @xmath154 satisfies for all @xmath175 . then the family @xmath176 gives the conclusion . we are now ready to conclude the proof of theorem [ thm : main ] using lemma [ lemma : x : axis0 ] , lemma [ lemma : z : axis0 ] and lemma [ lemma : comparisonincone ] . * _ proof of theorem [ thm : main ] . _ * we fix some values of @xmath177 , @xmath116 , and @xmath117 so that the conclusions of lemma [ lemma : x : axis0 ] , lemma [ lemma : z : axis0 ] and lemma [ lemma : comparisonincone ] hold . next , we fix some @xmath178 large enough so that @xmath179 , \;|z_p|<b,\ ; |y_p|<x_p \tan \theta\ } \subset u(0,r)\ ] ] and @xmath180,\ ; \rho_p < b\ } \subset u(0,r),\ ] ] where @xmath181 denotes the open ball with center 0 and radius @xmath182 in @xmath183 . such an @xmath182 exists since in the above two inclusions , the sets on the left are bounded . as a consequence , we have @xmath184 and @xmath185 recall for the definition of @xmath186 and for the definition of @xmath136 . let us now consider a family of besicovitch balls @xmath187 where , as defined by convention , we have @xmath149 and where the centers @xmath154 satisfy and . noting that the family @xmath188 also satisfies the same properties for all @xmath189 , one can assume with no loss of generality that @xmath190 up to a dilation by a factor @xmath191 . let @xmath192 and @xmath193 be defined as @xmath194 and @xmath195 we will bound @xmath196 in terms of the constants @xmath72 , @xmath9 , @xmath197 and @xmath198 . we re - index the points so that @xmath199 let @xmath200 be such that @xmath201 . by choice of @xmath168 and by definition of @xmath72 and @xmath9 , we have @xmath202 let @xmath203 be a large enough integer such that @xmath204 . then we have @xmath205 . indeed , otherwise we would get from in lemma [ lemma : comparisonincone ] that @xmath206 and hence @xmath207 . then , by choice of @xmath182 ( remember ) , @xmath208 and @xmath209 would be distinct points in @xmath136 which contradicts lemma [ lemma : z : axis0 ] . let @xmath210 be a large enough integer such that @xmath211 . then we have @xmath212 . indeed , otherwise we would get from in lemma [ lemma : comparisonincone ] that @xmath213 and hence @xmath214 . then , by choice of @xmath182 ( remember ) , @xmath215 and @xmath216 would be distinct points in @xmath186 which contradicts lemma [ lemma : x : axis0 ] . all together we get the following bound on @xmath217 , @xmath218 combining this with in lemma [ lemma : reduction ] , we get the following bound on the cardinality of arbitrary families @xmath152 of besicovitch balls , @xmath219 which concludes the proof of theorem [ thm : main ] . this section is devoted to the proof of lemma [ lemma : x : axis0 ] and lemma [ lemma : z : axis0 ] . we begin with a remark that will be technically useful . given @xmath90 and @xmath141 , we set @xmath220 recall that , following , we have @xmath221 by convention . [ a(p , q ) ] we have @xmath222 if and only if @xmath223 recalling , we have @xmath224 combining this with , which gives @xmath225 we get the conclusion . [ lemma : pp ] there exist constants @xmath226 and @xmath227 , depending only on @xmath120 , such that , for all @xmath228 , all @xmath116 and @xmath117 such that @xmath229 , we have @xmath230 for all @xmath231 . by , we always have @xmath232 . on the other hand , we can bound from above @xmath233 using that @xmath234 , since @xmath235 , and that @xmath236 if @xmath231 ( see for the definition of @xmath129 ) . namely , we have @xmath237 for @xmath238 , @xmath117 and @xmath115 , we set ( see figure [ fig3](a ) ) @xmath239 [ fig : r_in_p ] and @xmath240.,title="fig:",height=245 ] [ fig : r ] and @xmath240.,title="fig:",height=207 ] [ lemma : x : axis ] there exists @xmath119 , which depends only on @xmath120 , such that for all @xmath241 , there exists @xmath242 such that for all @xmath123 and for all @xmath124 , we have @xmath243 for all @xmath244 and all @xmath245 $ ] . take @xmath228 , @xmath246 , @xmath244 , @xmath247 and consider @xmath248 . by lemma [ a(p , q ) ] , showing that @xmath249 is equivalent to prove that @xmath250 is negative . since @xmath251 , we have @xmath252 note that all terms in the last inequality are positive except @xmath253 , since both @xmath254 and @xmath255 are positive . we now use the conditions @xmath256 , @xmath257 , @xmath258 , @xmath259 , @xmath260 , @xmath261 and @xmath262 , since @xmath263 , to get @xmath264 we consider now separately the case @xmath265 and @xmath266 . for @xmath265 , we bound using lemma [ lemma : pp ] @xmath267 hence @xmath268 as @xmath255 goes to infinity . thus , choosing @xmath198 small enough so that @xmath269 , we get that @xmath270 provided @xmath255 is large enough . for @xmath266 , we use once again lemma [ lemma : pp ] and get @xmath271 hence @xmath272 as @xmath255 goes to infinity . thus , choosing @xmath198 small enough so that @xmath273 , we get that @xmath270 provided @xmath255 is large enough . all together we have showed that one can find @xmath274 , depending only on @xmath65 , and for all @xmath275 , some @xmath242 , such that for @xmath123 and @xmath261 and for all @xmath276 , we have @xmath277 since @xmath278 is euclidean convex by proposition [ prop : ball_convex ] , we conclude the proof noting that @xmath279 , for @xmath280 $ ] , is in the euclidean convex hull of @xmath281 and @xmath282 . _ * proof of lemma [ lemma : x : axis0 ] . * _ let @xmath119 be given by lemma [ lemma : x : axis ] . let @xmath283 and let @xmath242 be given by lemma [ lemma : x : axis ] . let @xmath123 and @xmath124 . let @xmath90 and @xmath141 be such that @xmath127 and @xmath284 . let us assume with no loss of generality that @xmath285 . then , if both @xmath56 and @xmath57 were in @xmath186 , by lemma [ lemma : x : axis ] we would have @xmath286 since @xmath287 $ ] . but this would contradict the assumptions . [ lemma : tt ] let @xmath288 and @xmath117 . then for all @xmath289 , we have @xmath290 let @xmath289 ( see for the definition of @xmath136 ) . since @xmath291 and @xmath292 , we have ( recall ) @xmath293 for @xmath294 and @xmath117 , we set ( see figure [ fig3](b ) ) @xmath295 [ lemma : z : axis ] there exists @xmath132 and @xmath133 , depending only on @xmath120 , such that for all @xmath134 and all @xmath135 , we have @xmath296 for all @xmath297 and all @xmath298 $ ] . take @xmath246 , @xmath297 , @xmath299 and consider @xmath300 . by lemma [ a(p , q ) ] , showing that @xmath301 is equivalent to prove that @xmath250 is negative . since @xmath302 , we have @xmath303 note that all terms in the last inequality are positive except @xmath304 , assuming both @xmath254 and @xmath305 negative . we bound using lemma [ lemma : tt ] and using that the absolute value of each of the first two components of @xmath56 and @xmath57 is smaller than @xmath197 , @xmath306 where in the last inequality we assumed that @xmath197 is small enough , @xmath307 for some @xmath308 which depends only on @xmath65 . we consider now separately the case @xmath309 and @xmath310 . for @xmath309 , we need @xmath311 which is true as soon as @xmath312 . for @xmath310 , we need @xmath313 which is true as soon as @xmath314 is large enough . all together we showed that one can find @xmath315 and @xmath316 , depending only on @xmath65 , such that , for all @xmath134 and @xmath135 and all @xmath289 , we have @xmath317 recall that the set @xmath318 is euclidean convex by proposition [ prop : ball_convex ] . therefore we conclude the proof since @xmath319 , for @xmath320 $ ] , is in the euclidean convex hull of @xmath321 and @xmath322 . * _ proof of lemma [ lemma : z : axis0 ] . _ * let @xmath315 and @xmath316 be given by lemma [ lemma : z : axis ] . let @xmath134 and @xmath135 . let @xmath90 and @xmath323 be such that @xmath127 and @xmath284 . assume with no loss of generality that @xmath324 . then , if both @xmath56 and @xmath57 were in @xmath136 , by lemma [ lemma : z : axis ] we would have @xmath325 since @xmath326 $ ] . but this would contradict the assumptions . this section is devoted to the proof of lemma [ lemma : comparisonincone ] . we first fix some notations . for @xmath327 , we set @xmath328 . for @xmath329 , @xmath90 and @xmath330 , let @xmath331 denote the two dimensional euclidean half cone in @xmath332 contained in the plane @xmath333 with vertex @xmath334 , axis the half line starting at @xmath334 and passing through @xmath335 and aperture @xmath336 . see figure [ fig4](a ) . for @xmath329 , @xmath90 and @xmath330 , let @xmath337 denote the two dimensional euclidean equilateral quadrilateral contained in the plane @xmath333 with vertices @xmath334 , @xmath338 , @xmath339 and @xmath340 . note that it is the euclidean convex hull in @xmath332 of these four points . see figure [ fig4](b ) . [ fig : c_thetha_p_z ] and @xmath341.,title="fig:",height=188 ] ( -37,66)@xmath342 ( -47,63.5)@xmath343 ( -115,117)@xmath56 ( -106.56,114)@xmath343 ( -94,46)@xmath198 [ fig : q ] and @xmath341.,title="fig:",height=207 ] ( -166,90)@xmath342 ( -15,86)@xmath344 ( -19,77)@xmath343 ( -95,126)@xmath345 ( -86.5,111)@xmath343 ( -104,84)@xmath335 ( -86.5,77)@xmath343 ( -95,29)@xmath346 ( -86.5,42.5)@xmath343 recall for the definition of @xmath150 . note that @xmath347 if and only if @xmath348 . we have the following properties , @xmath349 and @xmath350 for @xmath351 , we have @xmath352 this follows from elementary geometry noting that the angle between the half lines starting at @xmath345 and passing through @xmath334 and @xmath353 respectively is larger than @xmath354 . [ lemma : sev1 ] there exists @xmath355 , which depends only on @xmath120 , such that @xmath356 for all @xmath357 , all @xmath358 and all @xmath327 such that @xmath359 . recalling proposition [ prop : ball_convex ] , we only need to prove that the vertices @xmath334 , @xmath345 , @xmath346 and @xmath353 of @xmath337 belong to @xmath360 . we have @xmath359 and , recalling and , @xmath361 hence @xmath362 that is , recalling , @xmath363 . similarly we have @xmath364 hence @xmath365 . next , let us prove that @xmath366 . set @xmath367 we need to prove that @xmath368 . we have @xmath369 where the last inequality follows from the fact that @xmath370 which implies in particular that @xmath371 hence we get that @xmath372 choosing @xmath373 small enough so that @xmath374 for all @xmath357 , we get the conclusion . the fact that @xmath375 is proved in a similar way . * _ proof of lemma [ lemma : comparisonincone]_. * let @xmath376 where @xmath377 is given by lemma [ lemma : sev1 ] . let @xmath378 and let @xmath90 and @xmath141 satisfying , , and . let us first prove . assume by contradiction that @xmath379 . then @xmath380 . hence @xmath381 according to lemma [ lemma : sev1 ] . on the other hand , it follows from , , , that @xmath382 and hence @xmath383 which contradicts . thus we have @xmath384 and thus @xmath385 . it follows from , and that @xmath386 . finally we get from lemma [ lemma : sev1 ] that @xmath387 and then follows from . in this section we prove two criteria which imply the non - validity of bcp . this shows that in some sense our example of homogeneous distance @xmath70 for which bcp holds is sharp . roughly speaking the first criterion applies to homogeneous distances whose unit sphere centered at the origin has either inward cone - like singularities in the euclidean sense at the poles ( i.e. , at the intersection of the sphere with the @xmath49-axis ) or is flat at the poles with @xmath388 curvature in the euclidean sense . the second one applies to homogeneous distances whose unit sphere at the origin has outward cone - like singularities in the euclidean sense at the poles . note that the unit sphere centered at the origin of our distance @xmath70 is smooth with positive curvature in the euclidean sense . let @xmath1 be a homogeneous distance on @xmath85 and let @xmath8 denote the closed unit ball centered at the origin in @xmath389 . in this subsection we shall most of the time identify @xmath85 with @xmath87 equipped with its usual differential structure . for @xmath125 , @xmath390 , @xmath391 , and @xmath392 , let @xmath393 denote the euclidean half - cone in @xmath85 , identified with @xmath87 , with vertex @xmath56 , axis @xmath394 and opening @xmath395 . we say that @xmath396 , @xmath391 , _ points out of @xmath8 at @xmath397 _ if there exists an open neighbourhood @xmath398 of @xmath56 and some @xmath392 such that @xmath399 let @xmath400 denote the left translation defined by @xmath401 . we consider it as an affine map from @xmath85 , identified with @xmath87 , to @xmath87 whose differential , in the usual euclidean sense in @xmath87 is thus a constant linear map and will be denoted by @xmath402 . let @xmath403 be defined by @xmath404 . for @xmath405 , @xmath391 , and @xmath406 , let @xmath407 denote the set of points @xmath408 such that @xmath409 points out of @xmath8 at @xmath410 and let @xmath411 denote the set of points @xmath412 such that @xmath413 for some @xmath414 such that @xmath415 ( here @xmath416 denotes the euclidean norm in @xmath87 ) . [ thm : nobcpingoingcorners ] assume that there exists @xmath417 , @xmath391 , and @xmath418 such that @xmath419 for all @xmath420 . then bcp does not hold in @xmath389 . we first construct a sequence of points @xmath421 in @xmath422 such that @xmath423 for all @xmath424 and @xmath425 points out of @xmath8 at @xmath426 for all @xmath86 and all @xmath427 . note that if @xmath412 then there exists @xmath428 such that @xmath429 points out of @xmath8 at @xmath410 for all @xmath430 such that @xmath431 ( note that the set of vectors that points out of @xmath8 at some point @xmath397 is open ) . let us start choosing some @xmath432 . by induction assume that @xmath433 have already been chosen . let @xmath434 where each @xmath435 is associated to @xmath436 as above . then we choose @xmath437 . we have @xmath438 for some @xmath61 and some @xmath439 such that @xmath440 . hence , by choice of @xmath441 and of the @xmath442 s , we have that @xmath443 points out of @xmath8 at @xmath426 for all @xmath444 as wanted . next , we claim that if @xmath445 , @xmath446 are such that @xmath447 and that @xmath448 points out of @xmath8 at @xmath410 , then there exists @xmath449 such that @xmath450 for all @xmath451 . indeed the curve @xmath452 is a smooth curve starting at @xmath410 and whose tangent vector at @xmath453 is given by @xmath448 . since this vector points out of @xmath8 at @xmath410 , it follows that @xmath454 for all @xmath189 small enough and hence @xmath455 as wanted . then it follows that for all @xmath86 , one can find @xmath456 such that for all @xmath457 and all @xmath458 , one has @xmath459 then we set @xmath460 and by induction it follows that we can construct a decreasing sequence @xmath461 so that @xmath462 for all @xmath86 and all @xmath458 . for @xmath424 , we set @xmath463 . by construction we have @xmath464 for all @xmath465 and @xmath424 such that @xmath466 . it follows that @xmath467 is a family of besicovitch balls for any finite set @xmath468 and hence bcp does not hold . let us give some examples of homogeneous distances for which the criterion given in theorem [ thm : nobcpingoingcorners ] applies . a first class of examples is given by rotationally invariant homogeneous distances @xmath1 that satisfy that there exists @xmath469 such that @xmath470 and such that @xmath471 by rotationally invariant distances , we mean distances for which rotations @xmath472 , @xmath473 , are isometries ( see for the definition of @xmath472 ) . indeed , consider @xmath474 and , for @xmath475 , set @xmath476 then consider @xmath477 . by rotational and left invariance ( which implies in particular that @xmath478 for all @xmath141 ) , one has @xmath445 . on the other hand , since @xmath479 , any vector with a positive third coordinate points out of @xmath8 at @xmath410 . in particular @xmath480 points out of @xmath8 at @xmath410 . hence @xmath481 . this class of examples includes the so - called box - distance @xmath482 defined by @xmath483 with @xmath484 for which the fact that bcp does not hold was not known . it also includes the carnot - carathodory distance and hence this gives a new proof of the non - validity of bcp for this distance . see @xcite for a previous and different proof . other examples of homogeneous distances @xmath1 for which the criterion given in theorem [ thm : nobcpingoingcorners ] applies can be obtained in the following way . assume that @xmath8 , respectively @xmath422 , can be described as @xmath485 , respectively @xmath486 , for some @xmath487 real valued function @xmath488 in a neighbourhood of a point @xmath397 . then the outward normal to @xmath422 at some point @xmath445 is given in a neighbourhood of @xmath56 by @xmath489 ( here it is still understood that we identify @xmath85 with @xmath87 and @xmath490 denotes the usual gradient in @xmath87 ) . then theorem [ thm : nobcpingoingcorners ] applies if one can find a vector @xmath491 , @xmath391 , such that for all @xmath492 small enough , the following holds . there exists @xmath408 such that @xmath413 for some @xmath414 such that @xmath415 and such that @xmath410 lies in a neighbourhood of @xmath56 and @xmath493 where @xmath52 denotes the usual scalar product in @xmath87 . a particular example is given when @xmath8 , respectively @xmath422 , can be described near the north pole ( intersection of @xmath422 with the positive @xmath49-axis ) as the subgraph @xmath494 , respectively the graph @xmath495 , of a @xmath496 function @xmath497 whose first and second order partial derivatives vanish at the origin . indeed , in that case one can choose for example @xmath498 and for a fixed @xmath499 , one looks for some @xmath481 of the form @xmath500 for some @xmath189 . then @xmath501 lies near the north pole for @xmath189 small and we have @xmath502 that is equivalent to @xmath503 when @xmath189 is small enough . hence @xmath504 . this argument applies to the cygan - kornyi distance @xmath505 , and more generally to @xmath506 for all values of @xmath62 such that @xmath506 defines a distance , thus in particular for all values of @xmath507 . recall from that @xmath508 where @xmath509 and that @xmath505 is the cygan - kornyi distance . hence theorem [ thm : nobcpingoingcorners ] gives in particular a new geometric proof of the fact that bcp does not hold for the cygan - kornyi distance on @xmath85 , see @xcite and @xcite for previous analytic proofs . let @xmath1 be a homogeneous distance on @xmath85 and let @xmath8 denote the closed unit ball centered at the origin in @xmath389 . set @xmath510 . [ thm : nobcpoutgoingcorners ] assume that there exists two sequences of points @xmath511 and @xmath512 and some @xmath116 and @xmath513 such that @xmath514 then bcp does not hold in @xmath389 . the geometric meaning of the above assumptions is the following . in some vertical plane ( here we take the @xmath515-plane for simplicity ) one can find two sequences of points @xmath516 and @xmath517 , each one of them on a different side of the @xmath49-axis . such points are on the unit sphere centered at the origin and are converging to the north pole . the slope between @xmath517 and @xmath516 is assumed to be bounded away from zero . we further assume that at the north pole the intersection of the sphere and the @xmath515-plane can be written both as graph @xmath518 and @xmath519 . see figure [ out : cone ] . -plane and the unit sphere at the origin of the distance @xmath520 when @xmath521 and @xmath77.,title="fig:",height=151 ] ( -86,102)@xmath516 ( -142,109.4)@xmath517 theorem [ thm : nobcpoutgoingcorners ] applies in particular if the intersection of @xmath8 with the @xmath515-plane can be described near the north pole as @xmath522 for some function @xmath488 of class @xmath487 on @xmath523 such that @xmath524 and @xmath525 exist and are finite with @xmath526 . this is for instance the case of the following distances built from the cygan - kornyi distance , and more generally from the distances @xmath506 , and given by @xmath527 with @xmath528 for some @xmath529 . see for the definition of @xmath530 and @xmath531 . figure [ out : cone ] is exactly the intersection of the @xmath515-plane and the unit sphere at the origin when @xmath521 and @xmath77 . note that it follows in particular that the @xmath532-sum of the pseudo - distance @xmath533 with the distance @xmath506 does not satisfy bcp in contrast with their @xmath534-sum which is a multiple of the distance @xmath70 . _ * proof of theorem [ thm : nobcpoutgoingcorners ] . * _ by induction , we construct a sequence of points @xmath535 such that @xmath536 for all @xmath537 , where @xmath538 , and such that @xmath539 for all @xmath537 and all @xmath540 . then , we will have @xmath541 for all @xmath542 and @xmath537 such that @xmath543 , so that @xmath544 is a family of besicovitch balls for any finite set @xmath468 . hence bcp does not hold . we start from a point @xmath545 with @xmath546 . next assume that @xmath547 have been constructed and choose @xmath548 large enough so that @xmath549 we set @xmath550 note that @xmath551 since @xmath552 . we have @xmath553 by choice of @xmath548 ( see ) . we also have @xmath554 hence it remains to check that @xmath555 and that @xmath556 for @xmath540 . using dilation , left translation and the assumption @xmath557 , it follows that @xmath558 hence , taking into account the fact that @xmath559 and that @xmath560 , to prove that @xmath555 and that @xmath556 for @xmath540 , we only need to check that @xmath561 , which follows from , and that @xmath562 . using the fact that @xmath563 , and , we have @xmath564 which gives the conclusion . the case of @xmath0 for @xmath86 arbitrary can be easily handled similarly to the case of @xmath85 adopting the following convention . for @xmath565 , we set @xmath566 where @xmath567 , @xmath568 and @xmath569 . note that this is different from the more standard presentation adopted in the introduction ( section [ section : introduction ] ) . to avoid any confusion , the explicit correspondance between theses two conventions is the following . if @xmath570 , @xmath571 and @xmath330 denote the exponential and homogeneous coordinates of @xmath572 as in , by denoting @xmath566 with @xmath567 , @xmath568 and @xmath569 , we mean @xmath573 , @xmath574 and @xmath575 . it follows that @xmath576 should be replaced by @xmath577 and @xmath578 by @xmath579 where @xmath580 denotes the euclidean norm in @xmath581 . in particular , we get @xmath582 and setting @xmath583 and @xmath584 one can easily check that lemma [ lemma : x : axis0 ] and lemma [ lemma : z : axis0 ] hold true in @xmath0 with essentially the same proofs . lemma [ lemma : comparisonincone ] and its proof extend to the case of @xmath0 setting @xmath585 and considering the analogue of the sets @xmath331 and @xmath337 ( introduced in section [ section : prooflemma : comparisonincone ] ) defined in the following way . the set @xmath331 is now defined as the @xmath586-dimensional euclidean half cone contained in the hyperplane @xmath587 with vertex @xmath588 , axis the half line starting at @xmath334 and passing through @xmath335 and aperture @xmath336 . the set @xmath337 is defined as the @xmath586-dimensional euclidean convex hull in the hyperplane @xmath587 of @xmath334 , @xmath589 and the @xmath590-dimensional euclidean ball @xmath591 . here @xmath171 denotes the obvious analogue of the map defined in , @xmath592 , @xmath593 . this section is devoted to the proof of theorem [ thm : destroybcp ] . the construction is inspired by the construction given by the first - named author in theorem 1.6 of @xcite where it is proved that there exist translation - invariant distances on @xmath51 that are bi - lipschitz equivalent to the euclidean distance but that do not satisfy bcp . let @xmath5 be a metric space . assume that @xmath594 is an accumulation point in @xmath5 and let @xmath595 be a sequence of distinct points in @xmath9 such that @xmath596 for all @xmath86 and such that @xmath597 . set @xmath598 next , we will prove that @xmath594 is an isolated point of @xmath606 for all @xmath607 . more precisely , by definition of @xmath2 , we have , for all @xmath607 , @xmath608 hence @xmath609 for all @xmath607 . on the other hand , we will prove in lemma [ lem : isolatedpoint ] that @xmath610 for all @xmath607 . then let us extract a subsequence @xmath611 starting at @xmath612 in such a way that @xmath613 for all @xmath614 and all @xmath615 . it follows from that @xmath616 for all @xmath614 and all @xmath615 ( remember that the sequence @xmath617 is assumed to be decreasing ) . by definition of @xmath198 , one has @xmath620 for all @xmath621 and @xmath622 . it follows that @xmath623 note that since @xmath1 is a distance , one indeed has @xmath624 which follows from one side from the triangle inequality and for the other side from the fact that one can consider @xmath625 , @xmath626 and @xmath627 , so that @xmath628 . we get from lemma [ lem : equivalentdist ] that if @xmath633 then @xmath634 and hence @xmath635 . since @xmath636 , one has @xmath637 . to prove the triangle inequality , let us consider @xmath44 , @xmath45 and @xmath49 in @xmath9 and two arbitrary chains of points @xmath638 , @xmath639 . since @xmath640 is a chain of points from @xmath44 to @xmath45 , one has @xmath641 and hence @xmath642 first , we claim that @xmath648 for all @xmath649 provided @xmath441 is small enough . indeed , otherwise , with no loss of generality , we would have @xmath650 and @xmath651 , and hence @xmath652 which implies that @xmath653 on the other hand , together with the triangle inequality would give @xmath654 which is impossible as soon as @xmath655 . indeed , first , if @xmath657 for some @xmath658 , then we must have @xmath659 . otherwise , since @xmath617 is decreasing , we would have @xmath660 . hence we would get @xmath661 which is impossible as soon as @xmath662 . * acknowledgement . * the authors are very grateful to jeremy tyson for useful discussions and in particular for pointing out the link between the distances @xmath70 and the distances of negative type considered by lee and naor . the second author would like to thank for its hospitality the department of mathematics and statistics of the university of jyvskyl where part of this work was done . | our main result is a positive answer to the question whether one can find homogeneous distances on the heisenberg groups that have the besicovitch covering property ( bcp ) .
this property is well known to be one of the fundamental tools of measure theory , with strong connections with the theory of differentiation of measures .
we prove that bcp is satisfied by the homogeneous distances whose unit ball centered at the origin coincides with an euclidean ball .
such homogeneous distances do exist on any carnot group by a result of hebisch and sikora . in the heisenberg groups ,
they are related to the cygan - kornyi ( also called kornyi ) distance .
they were considered in particular by lee and naor to provide a counterexample to the goemans - linial conjecture in theoretical computer science .
to put our result in perspective , we also prove two geometric criteria that imply the non - validity of bcp , showing that in some sense our example is sharp .
our first criterion applies in particular to commonly used homogeneous distances on the heisenberg groups , such as the cygan - kornyi and carnot - carathodory distances that are already known not to satisfy bcp . to put a different perspective on these results and for sake of completeness
, we also give a proof of the fact , noticed by d. preiss , that in a general metric space , one can always construct a bi - lipschitz equivalent distance that does not satisfy bcp . |
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auroral kilometric radiation ( akr ) bursts exhibit a wide variety of fine structure as seen on frequency - time spectra . the cyclotron maser instability ( cmi ) @xcite is widely assumed to be the basic plasma mechanism responsible for the emission . this mechanism originally assumed a loss - cone electron velocity distribution function , but in situ observations in the acceleration region have shown that a horseshoe or crescent distribution is more accurate @xcite . the horseshoe distribution , which arises naturally for electron beams in the presence of inhomogeneous magnetic fields @xcite , provides a robust and efficient free energy source for the cmi mechanism , as shown both by model calculations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and in laboratory experiments ( e.g. , * ? ? ? this mechanism has been applied not only to terrestrial akr emission , but to many astrophysical environments in which the requisite conditions ( low density , beamed electrons , inhomogeneous magnetic fields ) are thought to be present , e.g. planetary magnetospheres @xcite , stellar magnetospheres @xcite , and even relativistic jets in active galaxies @xcite . detailed physical models for the rich variety of observed spectral signatures are still an active area of research . akr fine structure exhibits many morphologies on time - frequency spectra , including slowing drifting or nearly stationary features lasting tens of seconds @xcite , drifting features which may be interacting with each other @xcite , and periodically modulated or banded emission @xcite . although some akr bursts appear to be broadband emission extending over 100 s of kilohertz @xcite , the total power is dominated by highly time - variable fine structures , especially during periods of enhanced bursts @xcite , suggesting that perhaps all akr radiation is a superposition of narrowband , short duration fine structures . @xcite first suggested that drifting akr fine structure may be due to localized sources rising and falling along auroral magnetic field lines , with emitted frequency equal to the local electron cyclotron frequency . @xcite suggested that the emission is triggered by electrostatic waves which drift along the field at the local ion - acoustic speed . an early detailed model which attempts to explain akr fine structure is the tuned cavity model of @xcite in which the source region acts as a waveguide with sharp density boundaries . the source emits radiation in normal modes analogous to an optical laser , resulting in narrowband emission which drifts as the wave packet propagates into regions of varying width . a related idea is that of @xcite in which the cavity boundary is oscillating quasi - periodically . under these conditions , broadband radio photons will be stochastically accelerated , resulting in quasi - monochromatic discrete tones , whose frequencies slowly drift with changes in the cavity geometry . a difficulty with these models is that they require special conditions ( e.g. sharp density boundaries , oscillating walls ) which are not supported by observations @xcite . several authors have reproduced akr fine structure using electromagnetic particle simulations to model the akr source region . @xcite used a one - dimensional electromagnetic particle - in - cell ( pic ) code to model the cyclotron maser instability in a plasma with an inhomogeneous magnetic field . they found that radiation is emitted in individual packets which combine to form drifting features that both rise and fall , consistent with some types of akr fine structure . both x- and o - mode drifting fine structures are created , consistent with observations @xcite but not predicted in feedback models such as @xcite . a weakness of their model is that predicted drift rates are generally much higher than observed drifts . @xcite used a 2-dimensional pic code with model parameters based on in situ fast spacecraft measurements of the akr source region . the simulated akr is most strongly amplified longitudinally and consists entirely of short - timescale fine structures . the predicted bandwidth ( @xmath1 ) is somewhat larger than at least some observed akr fine structure . more recently , pottelette and colleagues have investigated the possible connection between akr fine structure and electron holes @xcite and also tri - polar structures @xcite . akr radiation from small packets ( elementary radiation sources ) associated with the holes are thought to be strongly amplified as the packets slow down and are reflected from the field - aligned electric field . since the scale size of electron holes is very small ( a few debye lengths , @xcite ) , the observed narrow bandwidth of akr fine structures is easily accounted for . however , since neither the observed speeds or direction of electron holes ( 1000 - 2500 km s@xmath2 always downward , @xcite ) are representative of fine structure frequency drifts , there is not a straightforward relationship between the observed properties of electron holes and the dynamics of the akr fine structure . in this paper , we present evidence that a particular type of akr fine structure called striped or striated akr @xcite is triggered by ion solitary structures ( ion holes ) . in section 2 we present new observations of sakr bursts , including new bandwidth and angular beamwidth measurements . in section 3 we calculate the change in cmi gain for a density depleted region with a pre - existing horseshoe velocity distribution when perturbed by a passing ion hole . we compare the observed properties of sakr with those calculated by radiation generated from the cmi instability in an ion hole . in section 4 we discuss some inferred properties of ion holes , such as lifetimes , probability of occurrence as a function of location , and radial dependence of speed , that can not be inferred from single spacecraft in situ measurements . striated akr ( sakr ) consists of trains of narrowband drifting bursts with negative slopes in the range -2 to -20 khz s@xmath2 , corresponding to upward - traveling sources with speeds between 100 - 1,000 km@xmath3 . figure [ fig - dynspec1 ] shows a typical dynamical spectrum of sakr bursts observed with the cluster wideband ( wbd ) plasma wave instrument @xcite on the four cluster spacecraft . the bursts were observed in the 125 - 135 khz band between 16:14:20 - 16:14:35 ut on 31 august 2002 . although the spectra are similar on all four spacecraft , there are clear differences in individual striations , indicative of angular beaming on the scale of the projected spacecraft separations , as shown in the inset at lower left . also , individual bursts have very narrow bandwidths and are nearly always observed in groups with spacing between bursts of 30 - 300 ms . these characteristics are described in more detail below . the occurrence probability of sakr emission is quite low at frequencies above 100 khz : we detect sakr bursts in less than 1% of all wbd spectra observed when the spacecraft was above 30magnetic latitude ( note that below this latitude , there is often shadowing by the earth s plasmasphere . ) the occurrence probabilities were computed by dividing the number of dynamic spectra ( length 52 sec ) for which sakr emission was clearly detected by the total number of spectra over a one year interval ( july 2002 to 2003 ) . this overestimates the actual occurrence probability for sakr since we do not correct for the fractional time within each spectrum that sakr is present . there is a strong inverse correlation with observing frequency as shown figure [ fig - histo - freq ] . there are only few detections in several hundred hours of data in the 500 - 510 khz band ( detection probability @xmath4 ) , while @xmath5% in the 125 - 125 khz band . @xcite also studied sakr emission using the plasma wave instrument ( pwi ) receiver on the polar spacecraft . they also found an inverse correlation with observing frequency , with a 6% detection rate in the 90 khz band ( fig . [ fig - histo - freq ] , shaded bar ) and comparable probabilities and overlapping frequencies . by contrast , non - striated akr emission is frequently detected , especially during sub - storm onset . the occurrence probability of akr is nearly 30% for geomagnetic index k@xmath6 in the range 1@xmath7k@xmath8 3 , with the highest occurrence frequency in the winter polar regions @xcite . sakr bursts have negatively sloped , nearly linear morphologies , at least within the normally sampled bandwidth ( 9.5 khz ) of the wbd receivers . the majority of the slopes range between -2 khz s@xmath2 and -8 khz s@xmath2 . assuming the emission frequency is identified with the local electron gyrofrequency , the speed of the stimulator as a function of frequency and slope can be written @xmath9 where v is the stimulator speed ( km@xmath10 , _ @xmath11 _ is the observed slope of the sakr burst ( khz s@xmath12 , @xmath13 is the observed frequency ( khz ) , and we have assumed the source moves upward along a dipolar magnetic field line at magnetic latitude @xmath14 = 70 . figure [ fig - histo - slope ] shows a histogram of the observed slopes of 650 sakr burst events observed in the 125 - 135 khz band along with the derived trigger speeds ( top x axis ) . the mean slope is -5.6 khz s@xmath2 , with more than 90% of the slopes in the range -2 to -8 khz s@xmath2 . the corresponding trigger speeds , using equation [ eq - speed ] , are in the range 76 303 km s@xmath2 , with a mean value 213 km s@xmath2 . these results are similar to those reported by @xcite . figure [ fig - dyn - spect2 ] shows a group of sakr bursts detected on one spacecraft while operating the wbd instrument in a wider bandwidth ( 77 khz ) mode . over this wider bandwidth , the striations are not linear , but rather have a curved , frequency - dependent slope . the curve can be fit by assuming a source radiating at the local electron cyclotron frequency and moving with constant speed along a field line in a dipolar magnetic field . the overlaid white line shows the expected trace for a source moving upward at a constant velocity of 300 km@xmath3 . sakr bursts with similar frequency - dependent behavior are seen in plate 1 of @xcite . the altitude range of the sakr locations are shown on the right y - axis . for this example , individual bursts originated near 6500 km altitude and moved upward at 300 km@xmath3 to an altitude near 8100 km , implying a lifetime of several seconds . we have examined several other sakr events with wide bandwidth and have found that all exhibit similar lifetimes and have nearly constant speed , although the speed varies with epoch . sakr bursts are almost always detected superposed on broader band akr emission . this can be clearly seen in both figure [ fig - dynspec1 ] and [ fig - dyn - spect2 ] , as well as in plate 1 of @xcite . this appears to be a universal characteristic of sakr , except in cases where the sakr intensity is so low that underlying broadband emission may have been undetected . background akr emission coeval with sakr bursts is consistent with the hypothesis that ion holes trigger the sakr bursts ( section 3.2 ) since the cmi gain in the source region is significant even in the absence of the ion holes . one of the most unusual features of sakr bursts is their extraordinarily narrow bandwidth . the top panel of figure [ fig - bw ] shows a single isolated sakr burst . the middle panel shows the same feature , but after de - trending by subtraction of a ` chirp ' signal of best - fit constant negative slope . the lower panel shows the full width at half maximum of a gaussian fit along the frequency axis of the de - trended signal after summing in time intervals of 37 ms each . the resulting bandwidths , in the range 15 22 hz , are much narrower than previously reported either observationally ( e.g. @xcite ) or resulting from model calculation of akr emission ( e.g. @xcite ) although @xcite also reported akr bandwidths as small as 5 hz . ( in the latter paper , the bursts do not appear to be sakr emission ) . other sakr bursts we have examined have bandwidths ranging from 15 to 40 hz . the narrow bandwidth implies a small source extent along the z ( b - field ) direction , viz , @xmath15^{\frac{4}{3}}\left [ { \frac{\delta f}{10~hz } } \right]\ ] ] where @xmath16z is the source extent along the b field and @xmath16f is the observed bandwidth . for @xmath17 khz and @xmath16f = 20 hz we obtain @xmath16z = 0.76 km . this is considerably smaller than the lateral extent of the akr source region and indicates the trigger for sakr must have a dimension along the magnetic field of order 1 km . the cluster spacecraft array has the unique ability to simultaneously sample the flux density of individual akr bursts at four widely separated points in space . by comparing the flux density on pairs of spacecraft , we can estimate the average angular beam size of individual bursts . the beaming probability is defined using the following algorithm . for each pair of spacecraft during a given observation , we calculate the projected angular separation between spacecraft as seen from a location situated above the magnetic pole of the hemisphere being observed , at a height corresponding to the electron cyclotron frequency at the center frequency of each observing band ( e.g. , 2.35 r@xmath18 for the 125 - 135 khz band ) . this constitutes an average akr location in that band without regard to location on the auroral oval . we next correct for differential propagation delay by shifting the waveform data from each spacecraft to the distance of the nearest spacecraft . we then divide the time - frequency spectrum from each spacecraft pair ( 52 sec duration for 125 - 135 khz band ) into data ` cells ' 19 msec x 53 hz in size . this window was chosen to match the observed bandwidth of sakr ( cf . section 2.4 ) but the beaming results are not very sensitive to the data window size . we computed the angular beam size with data windows factors of two smaller and larger , and the results did not differ significantly . the intensities are normalized to correct for differing distances between individual spacecraft and the akr source . we then omit from further analysis all data cells whose intensities are below a threshold , arbitrarily chosen to be 10 db below the maximum flux density for that spectrum . finally , we compare intensities in each pair of data windows , assigning a weight 1 to pairs for which the intensities are within 10 db of each other , and 0 otherwise . the overall beaming probability for each angular separation interval is the sum of the beaming weights divided by all cell pairs . note that this scheme is susceptible to overestimation of the beaming probability , since it is possible that independent akr sources will illuminate separate spacecraft at the same frequency and time interval . this ` confusion ' problem is smaller for sakr emission since it is often clear from the burst morphology on a time - frequency spectrum that only one source contributes to a given data cell at one time , whereas with normal akr there are very often several intersecting sources which contribute to a given data cell . in figure [ fig - angbeam ] we plot the beaming probability versus angular separation for 651 sakr bursts in the 125 - 135 khz and 250 - 260 khz bands . we have fitted a gaussian function to the 125 khz band observations using a least - squares fitting algorithm . the resulting full width at half maximum ( fwhm ) angular size is _ @xmath19 _ = 5.0 ( solid angle @xmath20 = 0.006 sr ) . this is surprisingly small compared with most previously published observations of akr beam size ( e.g. @xcite ) , who reported beaming solid angles of 4.6 sr and 3.3 sr at 178 and 100 khz respectively . they made angular beaming estimates by comparing time - averaged spectra observed using two satellites ( hawkeye and imp-6 ) which simultaneously observed akr bursts while the spacecraft were both over the same polar region . since the time resolution used for the hawkeye - imp6 spectrum comparison ( several minutes ) far exceeds the time - scale of individual akr bursts , their measured angular beam is actually a measure of the ensemble - averaged sky distribution of akr bursts over a several minute time - scale rather than the angular beam size of individual akr emission sources . this is the confusion problem mentioned above . an important unanswered question is the 2-dimensional structure of the sakr burst angular beam pattern : it is asymmetric , or perhaps a hollow cone as suggested by @xcite ? since the spatial frequency coverage of the cluster spacecraft array is often nearly one - dimensional at high magnetic latitude , it is difficult to analyze the 2-dimensional structure of individual bursts . the angular beaming probability plot shown in figure [ fig - angbeam ] includes measurements over a range of baseline orientations . however , a preliminary analysis of beaming probability grouped by baseline orientation did not reveal any obvious trends . we are presently analyzing a much a larger dataset consisting of a large variety of akr emission and wider range of baseline orientations to investigate the 2-dimensional structure of the akr angular beam . the angular beaming observations , combined with measured flux density , allow a direct estimate of the average intrinsic power of individual sakr bursts . for intense sakr bursts , the observed square of the electric field intensity is @xmath21 at a source - spacecraft distance @xmath22 = 10r@xmath18 . converting to flux density , we obtain @xmath23 where @xmath24 = 377 ohms is the impedance of free space . this flux density range is about one hundred times smaller than the flux density of intense akr bursts reported by @xcite . this is consistent with the conjecture that intense akr bursts are the sum of many spatially distinct elementary radiation sources @xcite the power emitted at the akr source is the isotropic power corrected by the angular beamsize of an individual sakr burst @xmath25 where _ @xmath26 _ @xmath27 20 - 50 hz is the bandwidth of a burst , and _ @xmath20 _ @xmath27 0.006 sr , is the solid angle of the emission beam . using these values , the resulting power is p @xmath27 1 10 w , much smaller than the previous estimates of p @xmath27 10@xmath28 - 10@xmath29 w for a single elementary radiator @xcite . ion solitary structures , also known as ion holes , are small - scale ( @xmath301 km ) regions of negative electrostatic potential associated with upgoing ion beams . they are seen in spacecraft electric field measurements as symmetric bipolar parallel electric field structures with amplitudes 10 500 mv m@xmath2 and timescales 3 - 10 ms . they were first detected in s3 - 3 spacecraft observations @xcite and have been subsequently been studied using in situ measurements in the acceleration region by several authors ( e.g. @xcite ) . ion holes travel upward at speeds between 75 - 300 km@xmath3 @xcite ( although @xcite argue for somewhat higher speeds , in the range 550 - 1100 km@xmath3 ) . the width of the waves increase with amplitude @xcite which is inconsistent with small amplitude 1-d soliton models , but which supports a bgk - type generation mode @xcite . akr radiation arises from wave growth resulting from the interaction of a radiation field with an electron velocity distribution having a positive slope in the direction perpendicular to the magnetic field ( @xmath31/@xmath32v@xmath33 0 ) . the condition for waves of angular frequency @xmath34 and wave normal angle @xmath19 to resonate with electrons with angular frequency @xmath35 and velocity @xmath36 is given by @xcite @xmath37 where @xmath38 is the electron cyclotron frequency , @xmath39 is an integer , and @xmath40 is the lorentz factor of the relativistic electrons . the cyclotron maser instability ( cmi ) is the case @xmath41=1 . if we assume that the electrons are mildly relativistic , as is observed in the akr source region ( e@xmath42 10 kev , so @xmath431.01 ) , we can expand the lorentz factor to obtain the equation for a circle in velocity space @xmath44 where the resonant circle s center is on the horizontal axis displaced by @xmath45 and the radius of the resonant circle is @xmath46^{\frac{1}{2}}\ ] ] analysis of recent fast observations of akr emission in the source region @xcite provides evidence , based on wave polarization , that the akr k - vector direction at the source is nearly perpendicular to the b field ( @xmath47 ) . the e - field of the wave is polarized perpendicular to the ambient magnetic field which indicates that the wave is purely x - mode @xcite . in the following , we explicitly assume @xmath48 , so that the radius of the resonance circle becomes @xmath49 where @xmath50 @xmath51 . the growth rate of the cmi mechanism is given by calculating the imaginary part of the angular frequency , @xmath52 where the @xmath53 integral is performed on the closed circular path given by equation 4 . the electron velocity distribution in the upward current acceleration region has been measured _ in situ _ by both viking @xcite and fast @xcite spacecraft . it consists of an incomplete shell or ` horseshoe ' shape in velocity space . the density of cold electrons ( e @xmath54 1 kev ) in this region is much smaller than the hot electron population which comprises the horseshoe component @xcite . in this paper we assume that all electrons are in the hot component . we have modeled the observed velocity distribution using a simple analytic functional form @xmath55\ ] ] where _ @xmath16 _ is the horseshoe width , @xmath56 the horseshoe radius and the loss - cone function @xmath57 @xmath58 where @xmath59 is a dimensionless scaling factor and @xmath60 is the characteristic opening angle of the loss cone . the model velocity distribution function is shown in figure [ fig - dist-2x1 ] along with a measured velocity distribution function from fast @xcite . , and inside an ion hole ( @xmath61 . black circle is the cmi resonant circle . the @xmath62direction is the horizontal axis , earthward to left . bottom panel : weighted partial derivative @xmath31/@xmath32v@xmath63 outside an ion hole ( @xmath64 , and inside an ion hole ( @xmath65 . the arrows indicate the parts of the growth line integral ( equation 9 ) enhanced inside the hole . ] ) . the inset at lower right shows a simple spherical model of an ion hole with inward electric field . ( b ) calculated parallel velocity vs. distance from an ion hole for an electron passing through an ion hole with an impact parameter 1 km and initial velocities 20,000 km s@xmath2 ( dotted line ) and 30,000 km@xmath3 ( solid line ) . the assumed hole speed is 300 km@xmath3 . ] the range of possible interactions between background electrons and ion holes is very complex and varied @xcite . these include trapping of electrons between ion holes , excitation and trapping of high - frequency langmuir waves from electron streams resulting from ion hole collisions , and modification of the ion holes via the ponderomotive force . in this paper we will tacitly assume that the ion holes are well - spaced and non - interacting , and that the only significant effect on the electron population is a transient speed decrease as the electrons traverse the hole s negative potential well . this assumption is motivated by the relatively simple structure of sakr bursts and by the success in using this assumption to explain all significant observed properties of sakr bursts . other more complicated forms of akr fine structure seen on dynamical spectra may well involve one or more of the complex interactions mentioned above . since the electrons are magnetized ( gyro - radius r@xmath6650 m , much smaller than the ion hole parallel scale size r@xmath671 km ) , the ion s hole s effect on the perpendicular velocity component is negligible ( a small * * e**x*b * drift , v@xmath68 40 m s@xmath12 . however , the parallel velocity component will experience a significant decrease as the electrons are repelled by the negative potential well of the hole . to determine the magnitude of the effect , we have modeled the observed electric field structure of an ion hole using the analytic form @xmath69 figure [ fig - hole - efield]a shows an observed e - field of an ion hole along with a plot of the model e - field with @xmath70 = 500 mv m@xmath2 , and @xmath71 ms . we used this model to calculate the velocities of electrons as they traverse the hole for a variety of impact parameters and initial speeds . for initial parallel speeds of 20,000 and 30,000 km@xmath3 , the speed of an electron traversing an ion hole at a minimum distance of 1 km from the center is shown in figure [ fig - hole - efield]b . as expected , the parallel velocity briefly decreases , so that the horseshoe velocity distribution is ` squeezed ' on the parallel ( horizontal ) axis , as shown in figure [ fig - elec - dist-4by1]b . the speed decrease is proportional to @xmath72 as expected by a simple energy conservation argument : for an electric field e and ion hole diameter l@xmath73 , the potential well of the hole is approximately @xmath74e@xmath75l@xmath73 . the electron loses kinetic energy as it traverses the potential well , so that energy conservation requires @xmath76 solving for the velocity change , @xmath77 for an ion hole electric field e @xmath27 300 mv m@xmath2 , diameter l@xmath78 @xmath27 2 km , and @xmath56 = 20,000 km@xmath3 , the expected speed decrease is @xmath79 5,000 km@xmath3 , in good agreement with the exact calculation . in order to calculate the power gain in the perturbed region , we need an estimate of the convective growth length @xmath80 where @xmath81 is the group velocity of the wave and @xmath82 is the growth rate . the group velocity is very sensitive to the ratio of plasma to gyro - frequency and the detailed velocity distribution function . as a first approximation , we have used the cold plasma dispersion relations to estimate the group velocity @xmath83 for frequencies near the r - mode cut - off frequency . we find that @xmath84 is of order 300 - 1000 km s@xmath2 , similar to the estimate used by @xcite . for a maximum growth rate @xmath85 5000 s@xmath2 ( at @xmath86 = @xmath87 khz ) , the corresponding convective growth length is l@xmath88200 m . hence , there are approximately 5 - 10 e - folding lengths in the region of the iss , assuming a perpendicular physical scale of order 1 - 2 km . this results in a maximum power gain of e@xmath89 43 - 87 db . the power gain required to amplify the background radiation to observed levels of normal ( wideband ) akr was estimated by @xcite to be e@xmath90 , which is at the high end of our calculation . however , as discussed above , since sakr emission is highly beamed , the required power is a factor of @xmath27100 smaller , so the requisite power gain is closer to e@xmath91= 65 db . this is comfortably within the range predicted by an ion hole trigger . figure [ fig - cmi - growth ] illustrates the growth rate and power gain both inside and exterior to an ion hole for a range of relativistic electron energies and plasma densities . figure [ fig - cmi - growth]@xmath92 show the growth rate and power gain respectively for a horseshoe velocity distribution with a radius @xmath93 ( e = 5.7 kev ) and @xmath94 = 0.1 , 0.2 and 0.5 @xmath95 , while figure [ fig - cmi - growth]@xmath96 shows the growth rate and gain for @xmath94 = 0.2 @xmath95 and a range of radii @xmath97 = 0.10 , 0.15 , and 0.20 ( e = 2.5 , 5.7 , 10.2 kev ) . these values are typical of the values of electron density and energy observed in the auroral density cavity @xcite . there are several noteworthy features of these plots : 1 . the range of electron densities which result in substantial cmi gain is very limited . the maximum density is constrained by the condition that the r - mode cutoff frequency is less than the electron cyclotron frequency . this can by expressed by the inequality @xmath98 ^{1/2}\ ] ] where @xmath99 and @xmath100 are the electron plasma and cyclotron frequencies respectively , and @xmath101 is the lorentz factor . this inequality can also be written @xmath102 where @xmath103 is the rest - mass energy of the electron ( 511 kev ) . for example , for @xmath104 = 5 kev and @xmath13= 125 khz , we find @xmath105 . on the other hand , for very low densities ( @xmath106 ) , the cmi gain decreases dramatically ( cf . fig . [ fig - cmi - growth]@xmath107 ) 2 . the gain is a sharply peaked function of frequency . for example , for @xmath94 = 0.3 @xmath95 , @xmath108 5.7 kev ( fig . [ fig - cmi - growth]@xmath109 ) , a dynamic range of 30 db near the peak ( approximately that observed on dynamic spectra of sakr bursts ) corresponds to a fractional bandwidth @xmath110 0.0005 , or @xmath111 50 hz at @xmath13 = 125 khz , in good agreement with observed bandwidths ( section 2.4 ) . these extremely narrow bandwidths are characteristic of cmi gain curves with partial - ring velocity distributions @xcite . 3 . even outside the ion hole there is substantial cmi gain under some favorable conditions ( e.g. 80 db for the middle plot of figure [ fig - cmi - growth]@xmath65 . this may be the explanation for the background akr seen in figure [ fig - dyn - spect2 ] and discussed in section 2.3 . given the rather small volume of an ion hole , it is reasonable to ask if there is sufficient free energy available from the resonant electrons to power the observed sakr emission . as shown in section 2.5 , the observed flux levels and angular beamwidths of sakr emission correspond to radiated powers @xmath112 w. using a an ion hole scale size @xmath113 1 km and speed @xmath114 , and mean electron density and energy @xmath115 , @xmath116 = 5 kev respectively , the ratio of radiated power to the electron kinetic energy traversing a hole per unit time is @xmath117 this range of conversion efficiencies is similar to previous models of akr emission ( e.g. , * ? ? ? * ) and indicates that while a single ion hole extracts relatively little energy from the ambient electron population , a train of hundreds of holes traversing the same region could account for a significant modification of the electron distribution function . if sakr bursts are stimulated by upward traveling ion holes in the acceleration region of the magnetosphere , then observations of these bursts provides a new technique to study characteristics of ion holes such as lifetimes and relative number and speed as a function of altitude , that are impossible to measure with single _ in situ _ spacecraft . 1 . the 77 khz bandwidth observations of sakr bursts indicate that ion holes propagate upward for more than 1,000 km , implying lifetimes of a few seconds . this is much longer than estimates of solitary wave lifetimes derived from pic simulations of solitary waves generated by the two - stream instability @xcite , which have lifetimes @xmath118 5 - 75 ms . the majority of ion hole speeds derived from sakr observations are in the range 75 - 400 km s@xmath2 ( fig . [ fig - histo - slope ] ) . these speeds are in very good agreement with _ in situ _ measurements of ion hole speeds from polar ( 75 - 300 km s@xmath2 ) at altitudes between 5500 km - 7000 km @xcite . however , @xcite finds somewhat higher ion hole speeds ( 550 - 1100 km s@xmath2 ) based on data from the fast satellite at altitudes between 3000 km - 4000 km . it is possible that at speeds above @xmath119600 km s@xmath2 ( slopes @xmath12020 khz s@xmath2 ) sakr bursts , especially in closely spaced groups , would be undetected since they would merge into quasi - continuous emission on dynamic spectra . 3 . fits to sakr bursts assuming a constant speed source , ( fig . [ fig - dyn - spect2 ] ) indicate that ion holes propagate at nearly constant speed for their entire lifetime . some numerical simulations @xcite show a significant change in ion hole speed as they evolve . this is not supported by our observations . sakr bursts , and hence ion holes , are much more common at higher altitude , being more than one hundred times as common at 10,000 km ( @xmath121 90 khz ) than at 3,200 km ( @xmath121 500 khz ) altitude , assuming ambient conditions favorable to the generation of sakr are not dissimilar in this altitude range . since sakr bursts are almost always detected in groups with typical spacing 30 - 300 ms . , ( section 2.6 ) , this likely also applies to ion holes . both laboratory plasma experiments @xcite and observations of ion holes in the magnetosphere ( e.g. , * ? ? ? * ) show trains of ion holes with spacings similar to the sakr bursts . finally , the uniformity of sakr intensity and bandwidth over a large frequency range ( fig.[fig - dyn - spect2 ] ) implies that there is little evolution of the electric field intensity or spatial structure of ion holes over their lifetime . this paper summarizes the observed properties of a distinct form of auroral kilometric radiation fine structure called striated akr , first described by @xcite . we present new observational results using the wbd instrument on cluster which characterize the bandwidth and angular beamsize of individual sakr sources , as well as derived properties such as intrinsic power and speed along the magnetic field . assuming the frequency of sakr bursts can be identified with the local electron cyclotron frequency , the speed and direction of sakr sources calculated from their observed drift rates are very similar to those observed for ion solitary structures ( ion holes ) in the upward current region of the magnetosphere . hence , we investigated whether sakr bursts could be the result of enhancement of the cyclotron maser instability by ion holes . using observed electric field signatures of ion holes in this region , we calculate the perturbation caused by the passage of an ion hole on a horseshoe electron velocity distribution in a dilute ( @xmath122 @xmath95 ) plasma . the cyclotron maser instability is strongly enhanced inside the ion hole , with power gain exceeding 100 db in a narrow frequency range just above the x - mode cutoff frequency . these characteristics are in excellent agreement with the observed bandwidth , speed , direction , and flux density of sakr bursts . alternative suggestions involving other types of solitary structures to explain akr fine structure , such as electron holes @xcite or tri - polar structures @xcite , are not consistent with sakr properties , although they may be important in other types of akr fine structures . if sakr bursts are in fact triggered by ion holes , a number of derived properties of ion holes can be deduced which would be difficult or impossible to obtain using _ in situ _ measurements . these include average lifetimes ( a few seconds ) , evolution of propagation speed ( nearly constant over lifetime of hole ) , and relative numbers versus location ( much more common high in the acceleration region than near the base ) . gurnett , d. and r. anderson , the kilometric radio emission spectrum : relationship to auroral acceleration processes , _ physics of auroral arc formation _ , geophysical monograph series , vol . 25 . akasofu and j.r . kan . washington dc : american geophysical union , 341 - 350 , 1981 . pritchett p. l. , r. strangeway , r. ergun , and c. carlson . , generation and propagation of cyclotron maser emissions in the finite auroral kilometric radiation source cavity , _ j. geophys . res_. , vol . 107 , a12 , doi:10.1029/2002ja009403 , 2002 . | we describe the statistical properties of narrowband drifting auroral kilometric radiation ( striated akr ) based on observations from the cluster wideband receiver during 2002 - 2005 .
we show that the observed characteristics , including frequency drift rate and direction , narrow bandwidth , observed intensity , and beaming angular sizes are all consistent with triggering by upward traveling ion solitary structures ( ` ion holes ' ) .
we calculate the expected perturbation of a horseshoe electron distribution function by an ion hole by integrating the resonance condition for a cyclotron maser instability ( cmi ) using the perturbed velocity distribution .
we find that the cmi growth rate can be strongly enhanced as the horseshoe velocity distribution contracts inside the passing ion hole , resulting in a power gain increase greater than 100 db .
the gain curve is sharply peaked just above the r - mode cut - off frequency , with an effective bandwidth @xmath050 hz , consistent with the observed bandwidth of striated akr emission .
ion holes are observed _ in situ _ in the acceleration region moving upward with spatial scales and speeds consistent with the observed bandwidth and slopes of sakr bursts .
hence , we suggest that sakr bursts are a remote sensor of ion holes and can be used to determine the frequency of occurrence , locations in the acceleration region , and lifetimes of these structures . |
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graphene@xcite , composed carbon atoms arranged in a two - dimensional honeycomb lattice , is the building block for all graphitic materials from the 0d fullerenes to 1d nanotubes and also the common 3d graphite . however , following peierls and landau s arguments , the two - dimensional lattice is unstable and can not exist at any finite temperature . therefore , graphene is often used as a toy model and viewed as an academic material until its recent discovery in laboratory@xcite . since graphene has promising application potentials in nanoscale electronic devices , it is important to study how the band structure changes with respect to the finite transverse width and also the edge topology . in addition , we also expect the electron - electron interaction will play a crucial role for physics properties in low - energy limit . therefore , we are motivated to investigate the ground state properties in graphene nanoribbon ( gnr ) here . with lithography techniques , gnr with width smaller than 20 nm has been fabricated successfully@xcite . however , the edge roughness seems to post a tough challenge on both theoretical and experimental sides . a breakthrough comes from chemical approach@xcite recently , that can produce gnr down to 10 nm in the controlled way . moreover , these gnrs have remarkably smooth edges and make the fast electronics at molecular scale possible@xcite . in this work , we concentrate on the gnr with armchair edges since the open boundaries give rise to peculiar flat bands . note that the edges are hydrogenated so that the dangling @xmath0 bonds are saturated and only the @xmath1 bands remain active in low energy@xcite . by combining analytic weak - coupling analysis , numerical density matrix renormalization - group ( dmrg ) method , and the first - principles calculations , it was shown@xcite that armchair gnr exhibits novel carrier - mediated ferromagnetism upon appropriate doping . even though only @xmath1 bands are active in low - energy , in appropriate doping regimes , the armchair edges give rise to both itinerant bloch and localized wannier orbitals . these localized orbitals are direct consequences of quantum interferences in armchair gnr and form flat bands with zero velocity . the carrier - mediated ferromagnetism can thus be understood in two steps : electronic correlations in the flat band generate intrinsic magnetic moments first , then the itinerant bloch electrons mediate ferromagnetic exchange coupling among them . here we use a variational wave function approach and try to understand how the flat - band ferromagnetism evolves with the interaction strength . we start with the hubbard model to describe the armchair gnr , @xmath2 + u\sum_{{\bf r } } n_{\uparrow}({\bf r})n_{\downarrow}({\bf r}).\end{aligned}\ ] ] where @xmath3 is the hopping amplitude , @xmath4 is the repulsive on - site interaction , @xmath5 is the spin index , @xmath6 , and @xmath7 is taken only for nearest neighbor bonds . rough estimates from experiments give @xmath8 ev and @xmath9-@xmath10 ev , putting the ratio @xmath11 to be of order one . before diving into the numerical details , it is insightful to highlight the peculiar flat - band orbitals first . for the armchair gnr , its transverse width can be labeled by an integer @xmath12 , i.e. the size of the unit cell for each sublattice . writing down the wave function on different sublattices as the two - componenet spinor , it is easy to check that , for symmetric armchair gnr ( odd @xmath12 ) , one can construct a localized wave function at @xmath13 , @xmath14,\end{aligned}\ ] ] and show it is indeed an eigenstate of the hopping hamiltonian . applying translational invariance to shift the wannier orbital , the flat band emerges at the end . this is the one - dimensional analog of the landau level degeneracy for two - dimensional electrons in magnetic field . it is interesting that the edge topology in 1d quenches the kinetic energy without the necessity to couple to external magnetic field . furthermore , we would like to point out that the flat - band ferromagnetism in armchair gnr is not the same by mielke - tasaki@xcite mechanism since there is no direct overlap between these flat - band orbitals . the ferromagnetism is actually mediated by other dispersive bands that couple the magnetic moments in the flat band . on the purpose of seeing the polarization for the ground state , we introduce the following trial wave function , @xmath15 \left[\prod_{k_{s m \downarrow } } a_{s m \downarrow}^{\dag}(k_{s m \downarrow } ) \right ] valence bands ( @xmath16 ) and the conducting bands ( @xmath17 ) separately and @xmath18 is the band index . thus , there are @xmath19 number of bands totally . the lower bounds for @xmath20 are decided by their band structures ( which are either @xmath21 or @xmath22 ) , while the upper bounds are the fitting parameters to minimize the variational energy , reducing the problem to look for minimum in the @xmath23-dimensional ( one constraint from the total particle conservation ) parameter space . for convenience , we introduce the filling factor , @xmath24 for each band as the variational parameters . since the filling factor denotes the fraction of the filled states in the band , summing @xmath25 over all bands gives the number of particles in one unit cell , i.e. @xmath26 , where @xmath27 is the average particle number per site . meanwhile , the total spin of the ground state , @xmath28 , can also be expressed in terms of the filling factors easily . the kinetic energy per unit cell is obtained by integrating the energy spectra to the desired fillings . the integration is fundamental and can be expressed in terms of the elliptic integrals , @xmath29 + \sum_{s\alpha}st\nu_{s_{f\alpha}},\end{aligned}\ ] ] where @xmath30 for @xmath31 is the elliptic integral of the second kind . the magnitude of the transverse momentum is @xmath32 and @xmath33 . for clarity , we separate the kinetic energy from the dispersive bands and the flat bands in above . thus , the summation of the band index @xmath34 does not include the flat bands . it shall be clear that the kinetic energy of the flat band ( denoted by @xmath35 ) is rather trivial since it is directly proportional to filling factor multiplied by the energy @xmath36 . for dispersive bands , the initial momentum @xmath37 starts from either @xmath21 or @xmath1 , depending where the spectrum minimum sits in the 1d brillouin zone . the corresponding filled momentum @xmath38 is either @xmath39 or @xmath40 . similarly , we can also compute the interaction energy per unit cell , @xmath41 \nu_{sm\uparrow } \nu_{s'm'\downarrow},\end{aligned}\ ] ] where the effective repulsive interaction @xmath42 between the bands @xmath43 and @xmath44 is the summation inside the bracket in above . the reason why the effective repulsion does not carry the explicit dependence on @xmath45 is due to the particle - hole symmetry in the armchair gnr . the analytic expressions for the kinetic and interaction energies are nice but the search for minima in the global parameter space can only be done by numerical approach due to the number of all filling factors are large . the numerical search is carried out extensively at interaction strength from @xmath46 to @xmath47 . as an illustrating example , we present our numerical results at @xmath48 and @xmath49 in fig . [ fig : s ] for @xmath50 armchair gnr . for wider gnr , the numerical results are qualitatively the same . meanwhile , to avoid being trapped in local minima , we randomly choose the initial searching points and select the final configurations with lowest energy . spin polarization in each bands at different electron number per site @xmath51 for the ( a ) @xmath48 and ( b ) @xmath49 cases . the flat - band regime ( @xmath52 ) is between the yellow dashed lines and the gray open circle denotes twice of the total spin per unit cell @xmath53 . spin polarization from the valence bands are shown in red , green , blue open circles , corresponding to the flat band , the lower ( @xmath54 ) and the upper ( @xmath55 ) dispersive bands . the spin polarization for the conduction bands are mirror images of these around the symmetric axis @xmath56 . for clarity of the figure , they are not shown here.,title="fig : " ] spin polarization in each bands at different electron number per site @xmath51 for the ( a ) @xmath48 and ( b ) @xmath49 cases . the flat - band regime ( @xmath52 ) is between the yellow dashed lines and the gray open circle denotes twice of the total spin per unit cell @xmath53 . spin polarization from the valence bands are shown in red , green , blue open circles , corresponding to the flat band , the lower ( @xmath54 ) and the upper ( @xmath55 ) dispersive bands . the spin polarization for the conduction bands are mirror images of these around the symmetric axis @xmath56 . for clarity of the figure , they are not shown here.,title="fig : " ] for the relatively weak interaction strength @xmath48 , the influence of the band structure is still manifest . first of all , the particle - hole symmetry is manifest in the total spin @xmath57 . note that the finite spin polarization near @xmath58 and @xmath59 is due to the van hove singularity and is not our main focus here . it is rather interesting that the ferromagnetic phase coincide with the flat - band regime rather nicely . in fact , the total spin per unit cell @xmath57 mainly comes from the flat - band orbitals . furthermore , since the upper dispersive band ( @xmath55 in light blue color ) intersects with the flat band , it is also polarized in the flat - band regime . on the other hand , the lower dispersive band ( @xmath54 in green color ) is separated by a finite gap and does not participate in the ferromagnetism . these numerical findings agree with field - theory approach , which shows that the exchange coupling between the flat - band moments are mediated by the itinerant carriers in the intersecting dispersive bands . as the interaction strength increases to @xmath49 , the competition between the kinetic and the interaction energies become highly non - trivial even at the mean - field level . the ferromagnetic regime due to van hove singularity expands a little bit as expected from stoner criterion . it is rather remarkable that the ferromagnetic phase is still confined to the flat - band regime rather well . but , since the interaction strength is large enough , both dispersive bands contribute despite the minor gap in one of them . the intersecting band greatly enhances the ferromagnetic correlations and accounts for roughly one - third of the total spin density . this implies that the itinerant carriers no longer play the secondary role as messengers to line up the localized moments in the flat band . instead , they also develop ferromagnetic correlations and contribute to the total spin density . the role of the dispersive band with a minor gap is peculiar . it participates the ferromagnetic phase in the first - half of the flat - band regime and drops out in the second - half . it is not yet clear whether this asymmetrical behavior is an artifact of the simple wave function we chose , or it hints for something real in realistic armchair gnr . we show that the armchair edges of gnr quenches the kinetic energy of the itinerant carriers and leads to the flat - band orbitals at finite doping . this is a beautiful one - dimensional analogy of the 2d landau levels except the magnetic field is not needed here . following the hints from the field - theory analysis , we study magnetic properties of the ground state via variational wave function approach and found the flat - band ferromagnetism in armchair gnr . we acknowledge support from the national science council of taiwan through grants nsc-95 - 2112-m-007 - 009 and nsc-96 - 2112-m-007 - 004 and also from the national center for theoretical sciences in taiwan . | we study the electronic correlation effects in armchair graphene nanoribbons that have been recently proposed to be the building blocks of spin qubits .
the armchair edges give rise to peculiar quantum interferences and lead to quenched kinetic energy of the itinerant carriers at appropriate doping level .
this is a beautiful one - dimensional analogy of the landau - level formation in two dimensions except the magnetic field is not needed here . combining the techniques of effective field theory and variational wave function approach
, we found that the ground state exhibits a new type of flat - band ferromagnetism that hasnt been found before . at the end
, we address practical issues about realization of this novel magnetic state in experiments . |
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where the lower part of the classical instability strip intersects the main sequence , three distinct classes of multiperiodically pulsating variables can be found . the @xmath9 doradus stars pulsate in gravity modes of high radial order and have periods of the order of one day ( kaye et al . the @xmath5 scuti stars have periods of the order of a few hours ( breger 1979 ) and are thus pressure and gravity mode pulsators of low radial order . the fastest pulsations in this domain in the hr diagram are however excited in the rapidly oscillating ap ( roap ) stars ( kurtz 1982 , kurtz & martinez 2000 ) , with typical periods around 10 minutes , indicating pressure modes of high radial overtones . the photometric semi - amplitudes associated with these pulsations are in most cases only a few mmag , which makes them difficult to detect . the roap stars are also remarkable due to their spectral peculiarities , since they are pulsating representatives of cool magnetic ap stars of the srcreu subtype . the magnetic fields of ap stars cause elemental segregation on the stellar surface . in other words , the chemical elements are arranged in patches on the stellar surface ( see , e.g. , kochukhov et al . 2004 or lueftinger et al . 2003 for well - documented examples ) . this causes modulation of the mean apparent brightness of the star with the rotation period . as the chemical elements show some alignment with the magnetic poles , the rotational light curves show a single - wave structure if only one magnetic pole is seen and a double - wave variation if both magnetic poles come into view during a rotation cycle . the strong magnetic fields present in the roap stars are usually not aligned with the stellar rotation axis . since the pulsation axis does however coincide with the magnetic axis , the roap stars are oblique pulsators ( kurtz 1982 ) . this means that the pulsation modes excited in the roap stars are seen at different aspects during the rotation cycle , which , for nonradial modes , causes amplitude modulation over the rotation period ( which can then be inferred ) and allows us to put constraints on the geometry of the pulsator . the oblique pulsator model ( opm ) , in its simplest form , then predicts that ( except for special geometric orientations of the axes ) dipole ( @xmath2 ) pulsation modes will be split into equally spaced triplets . quadrupole ( @xmath10 ) modes will give rise to equally spaced frequency quintuplets , where the spacing of consecutive multiplet components is exactly the stellar rotational frequency . the magnetic field of the roap stars affects the pulsations in two additional ways . firstly , the pulsation frequencies are shifted with respect to their unperturbed value ( cunha & gough 2000 ) and secondly , the pulsation modes are distorted so that a single spherical harmonic can no longer describe them fully ( e.g. , kurtz , kanaan & martinez 1993 , takata & shibahashi 1995 ) . this is observationally manifested by the presence of additional multiplet components surrounding the first - order singlets , triplets or quintuplets , which are spaced by integer multiples of the rotation frequency . for a long time , the predominant observing method for studying the pulsations of roap stars was time - resolved high - speed photometry , yielding interesting results on the pulsational behaviour , geometry and asteroseismology of these stars ( e.g. , see matthews , kurtz & martinez 1999 ) . the most recent observational advances however came from time - resolved spectroscopy : since the vertical wavelengths of the pulsation modes are comparable to the size of the line - forming regions in the atmospheres of these stars , the vertical structure of the atmospheres can be resolved ( ryabchikova et al . 2002 , kurtz , elkin & mathys 2003 ) . spectroscopy is also more sensitive to the detection of roap star pulsations than photometry . due to the vertical stratification of chemical elements in their atmospheres ( e.g. ryabchikova et al . 2002 ) the pulsational radial velocity amplitudes of some spectral lines ( most notably rare earth elements ) can reach several kilometres per second . the most extreme example of such high radial velocity amplitudes is hd 99563 ( xy crt ) . this star was photometrically discovered to pulsate by dorokhova & dorokhov ( 1998 ) and was confirmed by handler & paunzen ( 1999 ) . elkin , kurtz & mathys ( 2005 ) discovered pulsational radial velocity variations with semi - amplitudes of up 5 for some euii and tmii lines in their time - resolved spectroscopy of this star . such high amplitudes are capable of yielding information on the structure of the atmospheres of ap stars with the best possible signal to noise . however , one important piece of information that spectroscopy can not supply at this point is detailed knowledge of the stellar pulsation spectrum . the reason is that the largest telescopes are necessary for obtaining spectra of the required time resolution and signal to noise . however , observing time on these telescopes is sparse . therefore , lengthy photometric measurements of these stars on small telescopes are still required to decipher the pulsational spectra fully . to this end , we included hd 99563 as a secondary target in a multi - site campaign originally devoted to the roap star hd 122970 ( handler et al.2002 ) to be observed at times when the latter star was not yet accessible . however , hd 99563 turned out to be quite interesting , which is why we continued to observe it after the original campaign was finished . we also acquired some mean - light observations of hd 99563 in an attempt to determine its rotation period . in this paper , we report the results of these measurements . our multi - site time - series photometric observations were obtained with seven different telescopes at four observatories : the 0.75 m t6 automatic photometric telescope at fairborn observatory ( fapt ) in arizona , the 0.5 m , 0.75 m , 1.0 m and 1.9 m telescopes at the south african astronomical observatory ( saao ) , the 0.6 m reflector at siding spring observatory ( sso ) in australia and the 0.9 m telescope at observatorio de sierra nevada ( osn ) in spain . whereas the latter observations consisted of strmgren @xmath11 photometry , the other telescopes used the b filter only . the measurements are summarised in table 1 . .journal of the observations of hd 99563 ; @xmath12 is the number of 40-second data bins obtained in the respective night and @xmath13 is the rms scatter of the residual light curves after prewhitening and low - frequency filtering ; the remaining columns are self - explanatory . [ cols="<,^,^,^,^,^,^ " , ] from the values in table 5 it can be seen that the mode of hd 99563 is dominated by the dipole component , which is reasonable as the magnetic fields in ap stars are predominantly dipoles . the contribution of the @xmath4 and @xmath10 terms to the mode are fairly small , but the @xmath3 component does have some influence . table 6 shows that the observations are quite well reproduced by our fit . it would be interesting to see whether the theory by saio & gautschy ( 2004 ) is capable of reproducing our results . we can now check how well our model reproduces the observed pulsational amplitudes and phases over the stellar rotation cycle . to this end , we subdivided the time series into pieces some @xmath14 pulsation cycles long and determined the amplitudes and phases for the 1557.653 @xmath0hz variation within these subsets . the spherical harmonic decomposition method by kurtz ( 1992 ) yields a fit to the amplitude / phase behaviour over the rotation cycle , and we show it together with the data in fig . 6 . we can see that both magnetic ( pulsation ) poles are actually in view for approximately the same amount of time ; the relative fractions are 53 and 47 per cent . this is also the reason why it was so difficult to determine the rotation period of the star ; the pulsations as well as the mean light variations are almost symmetrical along the rotation period . the fitted curves reproduce the observations fairly well , with one exception : the pulsation amplitude is overestimated for the pole that is for a shorter time in the line of sight , whereas the amplitude is underestimated for the other pole . the harmonic frequencies can not be made responsible for this , because they do not affect the first - order amplitudes analysed here . consequently , we believe that the poor fits near the pulsation amplitude maxima could be due to presently undetected additional multiplet components of the pulsation mode ; the amount of over- and under - fitting can be fully explained with the noise level in our residual amplitude spectrum . our photometric multisite observations of the roap star hd 99563 resulted in the detection of a frequency quintuplet that is due to a single distorted dipole pulsation mode . the splitting within this frequency quintuplet together with our mean light observations and published magnetic measurements allowed us to determine the stellar rotation period as @xmath1 d. within the errors , the mean light extrema occur in phase with pulsation amplitude maximum , suggesting that the abundance spots on hd 99563 should be fairly concentric around its magnetic poles . to our knowledge , hd 99563 is only the fourth roap star where both magnetic poles become visible throughout a rotation cycle ( the others are hr 3831 , e.g. , see kurtz et al . ( 1993 ) , hd 6532 , e.g. , kurtz et al.(1996 ) , and hd 80316 e.g. , kurtz et al . ( 1997 ) ) , and only the third whose pulsational mode distortion has been quantified . in this context it is interesting to note that the observational features of hd 99563 are in many aspects similar to another roap star , the well - studied hr 3831 ( e.g. , kurtz et al . 1993 , kochukhov 2004 ) : their geometrical orientations , rotation period and effective temperatures are alike . only the pulsation period of hd 99563 is some 10 per cent shorter , and the phases its first - order combination frequencies are more consistent with them being harmonics . it would thus be very interesting to compare these two objects in detail . this would however require more in - depth studies of hd 99563 . its basic parameters are still poorly known ; the radius we inferred is based on the assumption that the visual companion is physical , which needs to be checked . if hd 99563 a and b were a physical pair , then the separation of the components is too wide for motion around a common centre of mass to be determined in a reasonable period of time . however , common proper motion may be detectable within a few years . the most efficient way to pin down the radius of the roap star would perhaps come from a combination of magnetic and polarisation measurements ( landolfi et al . 1997 ) . with that , the inclination @xmath15 and the magnetic obliquity @xmath16 can be constrained , and given the accurate rotation period and @xmath17 already available , a fairly accurate radius could be obtained . there are several other possibilities that make hd 99563 interesting for future observations . as already argued by elkin et al . ( 2005 ) , the star is a very attractive target for pulsational radial velocity measurements . given the high amplitude of the mean light variations of the star and its favourable geometry , doppler imaging of its surface should also be within reach . finally , we have evidence that we have not yet deciphered the full pulsational content of the star s light curves . another multi - site campaign , aiming at obtaining @xmath18 h of observations on 1-metre - class telescopes would therefore also be justifiable . as we pointed out , hd 99563b is located within the @xmath5 scuti instability strip and should therefore also be tested for pulsations , with more suitable means than ours . the austrian fonds zur frderung der wissenschaftlichen forschung partially supported this work under grants s7303-ast and s7304-ast . we are grateful to don kurtz for supplying results prior to publication , for helpful discussions and for comments on a draft version of this paper . we also thank hiromoto shibahashi for his accurate referee s report . | we undertook a time - series photometric multi - site campaign for the rapidly oscillating ap star hd 99563 and also acquired mean light observations over two seasons .
the pulsations of the star , that show flatter light maxima than minima , can be described with a frequency quintuplet centred on 1557.653 @xmath0hz and some first harmonics of these .
the amplitude of the pulsation is modulated with the rotation period of the star that we determine with @xmath1 d from the analysis of the stellar pulsation spectrum and of the mean light data .
we break the distorted oscillation mode up into its pure spherical harmonic components and find it is dominated by the @xmath2 pulsation , and also has a notable @xmath3 contribution , with weak @xmath4 and 2 components .
the geometrical configuration of the star allows one to see both pulsation poles for about the same amount of time ; hd 99563 is only the fourth roap star for which both pulsation poles are seen and only the third where the distortion of the pulsation modes was modelled .
we point out that hd 99563 is very similar to the well - studied roap star hr 3831 .
finally , we note that the visual companion of hd 99563 is located in the @xmath5 scuti instability strip and may thus show pulsation .
we show that if the companion was physical , the roap star would be a 2.03 @xmath6 object , seen at a rotational inclination of @xmath7 , which then predicts a magnetic obliquity @xmath8 .
stars : variables : other stars : oscillations techniques : photometric stars : individual : hd 99563 stars : individual : xy crt |
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supermassive black holes with masses @xmath8 are ubiquitous in the nuclei of local galaxies of moderate to high mass ( e.g. , * ? ? ? it is now well established that most of the total mass in black holes in the nearby universe was accreted in luminous episodes with high eddington rates ( e.g. , * ? ? ? * ; * ? ? ? * ) , with the growth for massive ( @xmath9 ) black holes occurring predominantly at @xmath10 ( e.g. , * ? ? ? * ; * ? ? ? these rapidly accreting black holes are most readily identified as bright optical quasars with characteristic broad ( @xmath11 km s@xmath12 ) emission lines , and luminous continuum emission that can dominate the light from the host galaxy , particularly at ultraviolet and optical wavelengths ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? optical quasars thus provide powerful tools for tracing the rapid growth of black holes over cosmic time ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , it is increasingly clear that a significant fraction of the quasar population does not show characteristic blue continua or broad lines because their nuclear emission regions are obscured . key evidence for the existence of obscured ( type 2 ) quasars comes from synthesis models of the cosmic x - ray background ( e.g. , * ? ? ? * ; * ? ? ? * ) , as well as direct identification of these objects through various observational techniques . these include selection of luminous quasars with only narrow optical lines @xcite or relatively weak x - ray emission @xcite , detection of powerful radio galaxies lacking strong nuclear optical continua or broad lines ( e.g. , * ? ? ? * ; * ? ? ? * ) , and detection of x - ray sources that are optically faint ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , have hard x - ray spectra ( e.g. , * ? ? ? * ) , or have radio bright , optically weak counterparts ( e.g. , * ? ? ? * ) . with the launch of the _ spitzer space telescope _ , large numbers of obscured quasars can now be efficiently identified based on their characteristic ( roughly power - law ) spectral energy distributions ( seds ) at mid - infrared ( mid - ir ) wavelengths ( @xmath2324 ) . because mid - ir emission is less strongly affected by dust extinction than optical and ultraviolet light , obscured quasars can appear similar to their unobscured counterparts in the mid - ir , but have optical emission characteristic of their host galaxies . a number of studies using mid - ir colors ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * hereafter h07 ) , sed fitting @xcite , or selecting objects based on similarities to mid - ir quasar templates ( e.g. , * ? ? * ) have been successful in identifying large numbers of dust - obscured quasars , indicating that a large fraction , and possibly a majority of rapid black hole growth is obscured by dust . these large new samples enable detailed statistical studies that can explore the role of obscured quasars in galaxy and black hole evolution . at present there are a number of possible physical scenarios for obscured quasars ; in the simplest `` unified models '' , obscuration is attributed to a broadly axisymmetric `` torus '' of dust that is part of the central engine , so obscuration is entirely an orientation effect ( e.g. , * ? ? ? * ; * ? ? ? alternatively , obscuration may not be due to a central `` torus '' but to larger dust structures such as those predicted during major mergers of galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and obscured quasars may represent an early evolutionary phase when the growing black hole can not produce a high enough accretion luminosity to expel the surrounding material ( e.g. , * ? ? ? * ; * ? ? ? observations have revealed evidence for obscuration by a `` torus '' in some cases and by galactic - scale structures in others ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and while there are examples of obscured quasars that show clear signs of radiative feedback on interstellar gas , it is unclear whether they are driving the galaxy - scale outflows invoked in evolutionary models @xcite . thus the physical nature of obscured quasars remains poorly understood , and analyses with large samples of mid - ir selected quasars will be essential for a more complete understanding of rapidly growing , obscured black holes . one particularly powerful observational tool is spatial clustering , which allows us to measure the masses of the dark matter halos in which quasars reside . clustering studies of unobscured quasars have shown that the masses of quasar host halos are remarkably constant with cosmic time , with @xmath13 @xmath14 @xmath15 over the large redshift range @xmath16 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this lack of variation in halo mass implies that the bias factor ( clustering relative to the underlying dark matter ) is an increasing function of redshift , since the dark matter is more weakly clustered earlier in cosmic time . the characteristic @xmath17 provides a strong constraint on models of quasar fueling by the major mergers of gas - rich galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) , secular instabilities ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or accretion of recycled cold gas from evolved stars @xcite , and may be related to quasars role in regulating star formation and the emergence of the red galaxy population in halos of roughly similar mass @xmath7@xmath18@xmath19 @xmath14 @xmath15 ( e.g. , @xcite ) . despite the power of clustering measurements in understanding quasar populations , little is known about the clustering of obscured quasars . some measurements of lower - luminosity agns indicate no significant difference between obscured and unobscured sources @xcite . however , these agns likely have different physical drivers compared to powerful quasars ( e.g. , * ? ? ? * ) . for obscured quasars at high luminosities ( @xmath20 erg s@xmath12 ) and high redshift ( @xmath21 ) , the clustering has remained largely unexplored . in this paper we present the first measurement of the clustering of mid - ir selected obscured quasars and make direct comparisons to their unobscured counterparts . we use a large sample of quasars ( both obscured and unobscured ) in the redshift range @xmath0 selected on the basis of irac colors by , using data from the 9 deg@xmath1 botes multiwavelength survey . we also employ a sample of @xmath2250,000 galaxies with good estimates of photometric redshift , and measure the two - point cross - correlation between quasars and galaxies . we utilize a novel method developed by ( * ? ? ? * hereafter m09 ) to derive the projected real - space projected cross - correlation function , making use of the full probability distributions for the photometric redshifts . throughout this paper we assume a cosmology with @xmath22 and @xmath23 . for direct comparison with other works , we assume @xmath24 km s@xmath12 mpc@xmath12 ( except for comoving distances and dark matter halo masses , which are explicitly given in terms of @xmath25 km s@xmath12 mpc@xmath26 ) . in order to easily compare to estimated halo masses in other recent works on quasar clustering ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , we assume a normalization for the matter power spectrum of @xmath27 . photometry is presented in vega magnitudes . all quoted uncertainties are @xmath28 ( 68% confidence ) . the 9 deg@xmath29 survey region in botes covered by the noao deep wide - field survey ( ndwfs ; * ? ? ? * ) is unique among extragalactic multiwavelength surveys in its wide field and uniform coverage using space- and ground - based observatories . extensive optical spectroscopy makes this field especially well suited for studying the statistical properties of a large number of agns ( c. kochanek et al . 2011 , in preparation ) . further details of the botes data set have been presented in previous papers ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? redshifts for this study come from the agn and galaxy evolution survey ( ages ; kochanek et al . 2011 , in preparation ) which used the hectospec multifiber spectrograph on the mmt @xcite . we use ages data release 2 ( dr 2 ) , which includes all the ages spectra taken in 20042006 . details of the agn redshifts are given in and @xcite . optical photometry from ndwfs was used for the selection of ages targets and to derive optical colors and fluxes for ages sources . ndwfs images were obtained with the mosaic-1 camera on the 4-m mayall telescope at kitt peak national observatory , with 50% completeness limits of 26.7 , 25.0 , and 24.9 mag , in the @xmath30 , @xmath31 , and @xmath32 bands , respectively . photometry is derived using sextractor @xcite . mid - infrared observations are taken from the _ spitzer _ irac shallow survey ( iss ; * ? ? ? * ) , and _ spitzer _ deep wide - field survey ( sdwfs ; * ? ? ? * ) . iss covers the full ages field in all four irac bands ( 3.6 , 4.5 , 5.8 , and 8 @xmath33 m ) , with @xmath34 flux limits of 6.4 , 8.8 , 51 and 50 @xmath33jy respectively . the irac photometry for iss is described in detail in @xcite . the more recent sdwfs exposures extend these limits to 3.5 , 5.3 , 30 , and 30 @xmath33jy , respectively . as discussed below , the quasar sample was selected using iss data , while the galaxy sample for cross - correlation is selected from the full sdwfs data set . in computing bolometric luminosities for the quasars , we also make use of 24 flux measurements available from the multiband imaging photometer for _ spitzer _ ( mips ) gto observations ( irs gto team , j. houck ( pi ) , and m. rieke ) of the botes field . significant fluxes ( @xmath35 ) were obtained for 97% of the quasars in our sample that lie in the region covered by mips . our primary analysis is the two - point cross - correlation between mid - ir selected quasars and galaxies . in this section we give details of the the quasar ( both obscured and unobscured ) and galaxy samples . the quasars are taken from the sample of luminous mid - ir selected agns presented by . quasars are identified on the basis of their colors in the mid - ir as observed by _ irac , using the color - color criterion of @xcite ( figure [ fstern]a ) , and are selected such that their best estimates of redshift are at @xmath36 . to the relatively shallow flux limits of the irac shallow survey , the agn sample is highly complete and suffers little contamination from star - forming galaxies ( as discussed in detail in 7 of ; see also @xcite ) . we note that while refers to the sample as `` agns '' , their bolometric luminosities are estimated to be in the range @xmath37@xmath38 erg s@xmath12 , corresponding roughly to an x - ray luminosity range @xmath39 erg s@xmath12 @xcite . such high luminosity agns are typically referred to as `` quasars '' in the literature , so to avoid confusion with studies of lower - luminosity active galaxies , here we refer to our sample as `` quasars '' . showed that at the iss flux limits , the ir - selected quasars show a bimodal distribution in optical to mid - ir color . the selection boundary at @xmath40=6.1 $ ] can be interpreted as dividing quasars into unobscured ( optically bright and so `` blue '' in @xmath40 $ ] ) and obscured ( optically faint and so `` red '' in @xmath40 $ ] ) subsets ( figure [ fstern]b ) . for the purposes of this study these objects will be referred to as `` qso-1s '' and `` obs - qsos '' , respectively ; the reader is reminded that the selection is based not on optical spectroscopy but only on optical to mid - ir color . this selection yields samples of 839 qso-1s and 640 obs - qsos at @xmath36 . a detailed study of the optical colors , morphologies , and average x - ray spectra of these objects is given in . to briefly summarize , found that the qso-1s have blue optical colors , point - like optical morphologies , and soft x - ray spectra characteristic of unobscured quasars , while the obs - qsos had redder optical colors , extended optical morphologies , hard x - ray spectra and high @xmath41 characteristic of obscured quasars . the sample does not include all obscured quasars , as sources with very large extinction may fall below the ir flux limits of the survey or move out of the @xcite selection region ( as shown in figure 1 of ; see also @xcite ) . the typical absorbing column for the obs - qso sample is estimated to be @xmath42@xmath43 @xmath44 . we expect the obs - qsos to suffer little contamination from bright star - forming galaxies . used an x - ray stacking analysis and constraints from deeper surveys to estimate the possible contamination , and concluded that the contamination is at most @xmath230% , and likely significantly smaller ( @xmath4510% ) . for our spatial correlation analysis , we limit the ir - selected quasar sample to the redshift range @xmath0 , to maximize overlap with the normal galaxies in the field ( [ galaxy ] ) . we also include only objects in regions of good optical photometry and away from bright stars . these criteria yield 563 qso-1s and 361 obs - qsos . finally , we restrict the qso-1 sample to those spectroscopically identified as broad - line agns , to ensure that they unambiguously represent a sample of unobscured quasars and to enable clean tests of photo-@xmath6 errors ( see [ qsophotoz ] ) . of the full sample of qso-1s all redshifts , the vast majority ( 80% ) have optical spectra from ages and 96% of these are classified as broad - line agns at @xmath46 , supporting their selection as unobscured quasars . we limit the qso-1 sample to the 445 that have accurate optical spectroscopic redshifts in the range @xmath0 and clear broad emission line features . ( in a sense this is conservative ; we verify that including the 20% of objects with only photo-@xmath6s has no significant effect on the clustering results . ) based on these selection criteria , our qso-1 sample is essentially equivalent to other type 1 quasar samples selected purely on optical photometric colors and/or spectroscopy ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , since the vast majority of spectroscopic type 1 quasars show agn - like mid - ir colors @xcite . the positions on the sky of the final samples of qso-1s and obs - qsos are shown in figure [ fagnsky](a ) , and their distribution in redshift is given in figure [ fz ] . the obs - qsos are ( by definition ) optically faint , and so few ( only 7% ) are bright enough to obtain good redshifts from mmt optical spectroscopy . ages targeted objects down to a flux limit of @xmath47 for sources that are optically extended , which is the case for almost all the obs - qsos . therefore the vast majority of the obs - qso sample has only photometric estimates of redshift , derived using an artificial neural net technique @xcite . uncertainties on photo-@xmath6s using this technique for optically - bright quasars are typically @xmath48 . however the errors are more difficult to estimate for optically - faint obs - qsos , for which there are few spectroscopic redshifts for comparison . photo-@xmath6 uncertainties were discussed at length by , with the conclusion that typical uncertainties are at most @xmath49 and are likely smaller . figure [ fspz ] shows the photo-@xmath6s and spec-@xmath6s for the handful of obs - qsos with spectroscopic redshifts , as well as those for the qso-1s for comparison . the impact of photo-@xmath6 errors in the present clustering analysis are addressed in detail in [ qsophotoz ] . as discussed in [ qsophotoz ] , random errors in the photo-@xmath6s can only tend to _ decrease _ the observed clustering amplitude , so we expect the present analysis to provide a robust lower limit on the clustering of the obs - qsos . since the primary aim of this analysis is to compare the clustering of quasars with and without obscuration by dust , it is imperative that the samples are otherwise matched in key properties such as redshift and luminosity . we show in figure [ fz ] that the redshift distributions of the two samples are similar , and we obtain bolometric luminosities ( @xmath50 ) for the quasars by scaling from the rest - frame 8 luminosity . we compute the flux at rest - frame 8 by extrapolating between the fluxes at 8 and 24 in the observed frame , and use this flux to obtain the monochromatic luminosity @xmath51 at 8 . we then multiply by a luminosity - dependent bolometric correction from @xcite , which ranges from factors of @xmath28 to 11 , in order to obtain @xmath50 . visual inspection of the _ spitzer _ data shows that essentially all of the quasars have broadly power - law seds at these wavelengths , indicating the rest - frame 8 emission is indeed dominated by the agn . we note that 49 quasars lie outside the region covered the mips 24 observations , while 26 ( @xmath23% ) of those inside the mips area are not detected at 24 . for these 75 objects , we use the estimates of @xmath50 derived from the rest - frame 2 luminosity as in 4.6 of . in general , the @xmath50 derived from the rest - frame 2 luminosity as used in ( which did not make use of the 24 data ) broadly matches that obtain from the extrapolated 8 flux . however , the median @xmath50 obtained from 2 is smaller for the obs - qsos than for the qso-1s by @xmath20.15 dex , primarily because the obs - qsos have somewhat redder mid - ir seds consistent with the nuclear emission being reddened by dust ( e.g. , * ? ? ? the distributions in @xmath50 are almost identical for the qso-1 and obs - qso samples , as shown in the top panel of figure [ fstern](b ) . the median and dispersion in @xmath52 ( erg s@xmath12 ) is ( 45.86 , 0.37 ) and ( 45.83 , 0.39 ) for qso-1s and obs - qsos , respectively , indicating that the two samples are very well matched in bolometric luminosity . for completeness , we note that if we use the @xmath50 estimates derived from rest - frame 2 in and restrict our analysis to qso-1 and obs - qso samples that are matched in @xmath50 , this has a negligible effect on the clustering results . the sample of 256,124 galaxies is selected from the deeper sdwfs irac observations , with a flux limit @xmath53 < 18.6 $ ] . the galaxies are selected to have best estimates of photometric redshift between 0.5 and 2 , with an average photo-@xmath6 of @xmath54 . the sample includes an optical magnitude cut of @xmath55 to restrict it to optical fluxes for which the photo-@xmath6s are well - calibrated . to eliminate powerful agns , we have also excluded any object detected in 5 ks _ chandra _ x - ray observations @xcite or with 5@xmath56 sdwfs detections in all four irac bands and colors in the @xcite agn selection region . the exclusion of agns from the galaxy sample removes only 6,979 objects and has negligible effect on the results . the distribution on the sky of the 256,124 galaxies are shown in figure [ fagnsky](b ) , and their distribution in photometric redshift is shown in figure [ fz ] . photometric redshifts are obtained using an updated version of the @xcite algorithm , which is based on template fitting to the optical - ir seds . the sed fitting produces a redshift probability density function ( pdf ) for each object , where @xmath57 represents the probability that the object lies at redshift @xmath6 . ( note that the neural net used for the quasar photo-@xmath6s does not produce an equivalent estimate of the pdf . thus for the quasars we use the best value for the redshift , as discussed in [ corranal ] . ) @xmath57 is normalized such that @xmath58 . for most galaxies the pdf is roughly gaussian in shape , although often with a broader tail toward higher redshift . the typical redshift uncertainties are @xmath59 , and only a small fraction ( @xmath20.6% ) of galaxies show multiple significant peaks in the pdf at different redshifts . typical galaxy pdfs are shown in figure [ fpdf ] . in addition to the observed galaxy catalog , the correlation analysis requires a reference sample of objects with random sky positions , in order to compare the observed quasar - galaxy pair counts with the number expected for an uncorrelated distribution . we use a catalog of @xmath60 random `` galaxies '' that are assigned to random positions in the regions of good photometry , reflecting the spatial selection function for the sdwfs galaxies . in this section we outline our methods for measuring the spatial cross - correlation between quasars and galaxies , the autocorrelation of the galaxies , and the absolute bias and characteristic dark matter halo masses . to measure the spatial clustering of quasars , we can in principle derive the autocorrelation of the quasars themselves , or measure their _ cross_-correlation with a sample of other objects ( specifically , normal galaxies ) at the same redshifts . our quasar sample is too small to obtain sufficiently good measurements of their autocorrelation function . however , cross - correlation with galaxies ( of which there are @xmath61 times as many objects in the botes data set ) allows far greater statistical power . further , cross - correlation requires knowledge only of the selection function for the galaxies , which is generally better constrained than that for agns . cross - correlations of agns with galaxies have proved an effective technique in a number of previous studies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? for the present analysis , the uncertainties in the galaxy photo-@xmath6s restrict our ability to perform a full three - dimensional clustering analysis . however , making use of the quasar redshifts and the galaxy photo-@xmath6 information , we can derive a projected spatial correlation function ( @xmath3 , with @xmath31 in comoving @xmath14 mpc ) that has both higher signal - to - noise , and a more straightforward physical interpretation than , for example , the purely angular correlation function @xmath62 . the two - point correlation function @xmath63 is defined as the probability above poisson of finding a galaxy in a volume element @xmath64 at a physical separation @xmath65 from another randomly chosen galaxy , such that @xmath66dv,\ ] ] where @xmath67 is the mean space density of the galaxies in the sample . the projected correlation function @xmath3 is defined as the integral of @xmath63 along the line of sight , @xmath68 where @xmath31 and @xmath69 are the projected comoving separations between galaxies in the directions perpendicular and parallel , respectively , to the mean line of sight from the observer to the two galaxies . by integrating along the line of sight , we eliminate redshift - space distortions owing to the peculiar motions of galaxies , which distort the line - of - sight distances measured from redshifts . @xmath3 has been used to measure correlations in a number of surveys , for example sdss @xcite , 2slaq @xcite , deep2 @xcite , botes @xcite , cosmos @xcite and goods @xcite . in the range of separations @xmath70 @xmath14 mpc , @xmath63 for galaxies and quasars is roughly observed to be a power - law , @xmath71 for sufficiently large @xmath72 such that we average over all line - of - sight peculiar velocities , @xmath3 can be directly related to @xmath63 ( for a power law parameterization ) by @xmath73}{\gamma(\gamma/2)}. \label{eqnwpplaw}\ ] ] we use equation ( [ eqnwpplaw ] ) to obtain power - law parameters for the observed correlation functions , to facilitate straightforward comparisons to other works . however , we note that a number of recent studies have shown evidence for separate terms in the correlation function owing to pairs of galaxies found within a single dark matter halo ( the `` one - halo '' term ) , and from pairs in which each galaxy is in a different halo ( the two - halo term ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? a halo occupation distribution ( hod ) analysis accounting for both the one- and two - halo terms can provide valuable constraints on the distribution of objects within their dark matter halos , however a full hod calculation is beyond the scope of the present analysis . to measure @xmath3 for the quasar - galaxy cross - correlation , we employ the method developed by . this technique makes use of the full photo-@xmath6 pdf for every galaxy , to weight quasar - galaxy pairs based on the probability of their being associated in redshift space . we describe the formalism briefly here , and refer the reader to for further details . for a set of spectroscopic quasars all at the same comoving distance @xmath74 from the observer , the angular cross - correlation between the ( spectroscopic ) quasars and ( photometric ) galaxies can be expressed in terms of the physical transverse comoving distance by ( e.g. , * ? ? ? * ) : @xmath75 where @xmath31 is the projected comoving distance for a given angular separation @xmath76 , such that @xmath77 . @xmath78 and @xmath79 are the total numbers of photometric galaxies and random galaxies , respectively , and @xmath80 and @xmath81 are the number of quasar - galaxy and quasar - random pairs in each bin of @xmath31 . defining the radial distribution function for the full galaxy sample as @xmath82 , where @xmath83 , and assuming that @xmath82 varies slowly at the redshifts of interest , then the angular correlation function @xmath84 is related to the projected real space correlation function @xmath3 by @xmath85 ( for a derivation see 3.2 of @xcite ) . as discussed in detail in , we can generalize the analysis such that the contribution to @xmath3 is calculated individually for each quasar - galaxy pair , with @xmath86 defined as the average value of the radial pdf @xmath87 for each photometric object @xmath88 , in a window of size @xmath89 around the comoving distance to each spectroscopic object @xmath90 . we use @xmath91 @xmath14 mpc to effectively eliminate redshift space distortions , although the results are insensitive to the details of this choice . in this case of weighting by pairs , we obtain , as in equation ( 13 ) of : @xmath92 where @xmath93 we refer the reader to 2 of for a detailed derivation and discussion of these equations . use equation ( [ eqnwp ] ) to compute the cross - correlation between spectroscopic and photometric quasars from sdss in the relatively narrow redshift bin @xmath94 , corresponding to comoving distances @xmath95 @xmath14 mpc ; they obtain a cross - correlation length @xmath96 @xmath14 mpc , assuming @xmath97 . ( note that do not derive @xmath97 , they assume it purely to describe their method . higher values of gamma are typically obtained in the recent literature , and we obtain @xmath98 in the present work . ) our quasar sample spans a comparatively larger range in redshift ( @xmath99 , corresponding to @xmath100 @xmath14 mpc ) . we evaluate equation ( [ eqnwp ] ) by calculating the @xmath101 term individually for each quasar . that is , for each quasar and each bin in separation @xmath31 , we sum the redshift weights @xmath102 for galaxies in the given range of distance from the quasar , and divide by the number of random galaxies in the same distance range ( note that this implies @xmath103 in equation [ eqnwp ] ) . the advantage of this procedure is that it consists of a simple sum and accounts exactly for the comoving distance to each quasar . however , the calculation is limited by shot noise on small scales where we have small numbers of quasar - galaxy and quasar - random pairs . to check that this does not significantly affect the results , we also divide the quasar sample into bins of width @xmath104 ( over which the comoving distance variations are small enough that there is little mixing between bins in @xmath31 ) , and calculate the @xmath105 term for all the @xmath106 quasars in the bin . we then average the @xmath3 values for the different bins to obtain a mean @xmath3 over the redshift range of interest . the resulting clustering amplitude differs by @xmath10710% ( and in the majority of cases , @xmath45 a few percent ) compared to evaluating equation ( [ eqnwp ] ) treating each quasar separately . the choice of method does not affect any of our conclusions , but to account for these differences we conservatively include an additional 10% systematic uncertainty on the measurement of the clustering amplitude . finally , we emphasize that we are averaging @xmath3 over the whole redshift range of @xmath0 . the validity of this procedure depends on the fact that the observed @xmath3 varies slowly in the redshift range of interest , which we verify explicitly in [ variationz ] . to estimate dark matter halo masses for the quasars , we calculate the relative bias between quasars and galaxies from which we derive the absolute bias of the quasars relative to dark matter . as discussed below , calculation of absolute bias ( and thus halo mass ) requires a measurement of the autocorrelation function of the sdwfs galaxies . the large sample size enables us to derive the clustering of the galaxies accurately from the angular autocorrelation function @xmath108 alone . although we expect the photometric redshifts for the sdwfs galaxies to be well - constrained ( as discussed in [ galaxy ] ) , by using the angular correlation function we minimize any uncertainties relating to individual galaxy photo-@xmath6s for this part of the analysis . the resulting clustering measured for the galaxies has much smaller uncertainties than that for the quasar - galaxy cross - correlation . to save computation time , for the galaxy autocorrelation analysis we use a significantly smaller random catalog with only @xmath109 random `` galaxies '' . this likely introduces some additional shot noise into the calculation of @xmath108 , however since the resulting uncertainties are still far smaller than those for the quasar - galaxy cross - correlation , they are more than sufficient for the present analysis . we calculate the angular autocorrelation function @xmath108 using the @xcite estimator : @xmath110 where @xmath111 , @xmath112 , and @xmath113 are the number of data - data , data - random , and random - random galaxy pairs , respectively , at a separation @xmath76 , where each term is scaled according to the total numbers of quasars , galaxies , and randoms . the galaxy autocorrelation varies with redshift , owing to the evolution of large scale structure , and because the use of a flux - limited sample means we select more luminous galaxies at higher @xmath6 . this will affect the measurements of relative bias between quasars and galaxies , since the redshift distribution of the quasars peaks at higher @xmath6 than that for the galaxies and so relatively higher-@xmath6 galaxies dominate the cross - correlation signal . to account for this in our measurement of galaxy autocorrelation , we randomly select galaxies based on the overlap of the pdfs with the quasars in comoving distance ( in the formalism of [ crosscorr ] this is @xmath86 for each galaxy , averaged all quasars ) . we select the galaxies so their distribution in redshift is equivalent to the _ weighted _ distribution for all galaxies ( weighted by @xmath114 ) . the redshift distribution of this galaxy sample is shown in figure [ fgalselect ] . we use this smaller galaxy sample to calculate the angular autocorrelation of sdwfs galaxies . in fields of finite size , estimators of the correlation function based on pair counts are subject to the integral constraint , which can be expressed as @xcite @xmath115 where @xmath116 is the angle between the solid angle elements @xmath117 and @xmath118 and the integrals are over the survey area . if the number density fluctuations in the volume are small , and the angular correlations are smaller than the variance within the volume , then to first order the correlation function is simply biased low by a constant equal to the fractional variance of the number counts . a straightforward way to remove this bias is to add to the observed @xmath108 the term @xmath119 where @xmath120 is the area of the survey region . the value of @xmath121 , where @xmath122 is an estimate of the mean number of galaxies per unit area , is the contribution of clustering to the variance of the galaxy number counts @xcite . evaluating equation ( [ eqnint ] ) for the botes survey area and the typical slope of the @xmath108 for the objects considered here , we obtain @xmath123 . we estimate @xmath124 by interpolating the observed @xmath108 , then add @xmath125 to @xmath108 before performing model fits . for the projected real - space correlation functions @xmath3 ( which is ultimately derived from individual estimates of @xmath108 , as in equation [ eqnccorr ] ) , we perform an approximate correction for the integral constraint . we determine the value of @xmath3 at the physical scale ( typically 0.51 @xmath14 mpc ) corresponding to 1 for each quasar , and add the average of these estimates ( multiplied by 0.03 ) to the observed @xmath3 . these corrections increase the observed clustering amplitude by @xmath210% , but have little effect on our overall conclusions . ideally , uncertainties in @xmath3 and @xmath108 would be determined by calculating the correlation function for various random realizations of mock ir - selected quasar and galaxy samples , for example by populating dark matter @xmath126-body simulations . in the absence of such mock catalog , we instead determine uncertainties in @xmath3 directly from the data through bootstrap resampling . in a standard bootstrap analysis , the survey volume is divided into @xmath127 subvolumes , and these subvolumes are drawn randomly ( with replacement ) for inclusion in the calculation of the correlation function . owing to the relatively small size of the field compared to large surveys such as sdss or 2df , we are only able to divide the field into a small number of subvolumes ( we choose @xmath128 ) . the width of one subvolume corresponds to @xmath250 @xmath14 mpc at @xmath129 , so that correlations between the subvolumes should be relatively weak . ( we verify explicitly that using a larger @xmath130 has no significant effect on the results . ) for each bootstrap sample draw a total of @xmath131 subvolumes ( with replacement ) , which has been shown to best approximate the intrinsic uncertainties in the clustering amplitude @xcite . we then re - calculate @xmath3 including only the subvolumes in the bootstrap sample . for the calculations of @xmath3 we use 10,000 bootstrap samples , for which the uncertainties at each scale converge to better than 1% . ( to save computing time , we limit the analysis to 2000 bootstrap samples for the angular correlation analyses , for which the uncertainties converge to within @xmath21.5% . ) this bootstrap technique works well for the galaxy autocorrelation , for which we have a large number of objects and the uncertainty is dominated by the clustering of the sample rather than counting statistics . however , for the quasar - galaxy cross - correlation the bootstrap analysis results in very small errors that are significantly smaller than the observed scatter between points . this appears to be caused by the fact that , owing to the small quasar samples of only a few hundred objects , the uncertainties are dominated by shot noise that is not fully characterized by randomly selecting entire subvolumes . to account for the shot noise , we therefore take the sets of @xmath131 bootstrap subvolumes and randomly draw from them ( with replacement ) a sample of objects ( quasars or galaxies ) equal in size to the parent sample ; only pairs including these objects are used in resulting cross - correlation calculation . this procedure yields a good estimate of the shot noise ( the resulting @xmath132 ) while also accounting for covariance due to the large - scale structure . when fitting power - law models to the observed correlation functions , we compute parameters by minimizing @xmath133 , taking into account covariance between different bins in @xmath31 . from the bootstrap analysis , we can estimate the covariance matrix @xmath134 by @xmath135 \ , \end{aligned}\ ] ] where @xmath136 and @xmath137 are the projected correlation function derived for the @xmath138-th bootstrap samples , @xmath126 is the total number of bootstrap samples , and @xmath3 is the correlation function for the full sample . this formalism is equally valid for bins of angular separation @xmath76 in calculations of @xmath108 . the 1@xmath56 uncertainty in each bin in @xmath31 is the square root of the diagonal component of this matrix ( @xmath139 ) . taking into account covariance , @xmath133 is defined as @xmath140 where @xmath141 is the inverse of the covariance matrix @xmath134 . we determine best - fit parameters by minimizing @xmath133 , and derive 1@xmath56 errors in each parameter by the range for which @xmath142 . as a check , we also estimate parameter uncertainties by calculating best - fit parameters for each of the bootstrap samples and calculating the variance between them ; this obtains almost identical estimates of the errors . further , we note that if we use only the diagonal terms in the covariance matrix in determining @xmath133 , the variation in the best - fit parameters is significantly smaller than the statistical uncertainties , indicating that the precise details of the covariance matrix are relatively unimportant . we also note that while in principle the sdwfs field is large enough to enable measurements of clustering up to @xmath750 @xmath14 mpc at @xmath143 , we limit the analysis to scales @xmath144 @xmath14 mpc , because of edge effects that skew the correlation function on large scales but have minimal effect on smaller scales . an investigation of this effect is given in the appendix . for the projected real - space quasar - galaxy cross - correlation analysis , we fit power - law models of @xmath3 using equation ( [ eqnwpplaw ] ) . we also fit power laws to the angular correlation functions ( both galaxy autocorrelations and quasar - galaxy cross - correlations ) , using the simple expression @xmath145 for meaningful comparison to other clustering measurements obtained using samples with different distributions in redshift , we wish to convert the observed parameters @xmath146 and @xmath147 to the real - space @xmath148 and @xmath149 as defined in equation ( [ eqnplaw ] ) . inverting limber s equation , the conversion between these parameters can be computed analytically ( here we follow 4.2 of @xcite ; for the full derivation see @xcite ) : @xmath150\left[\int^{\infty}_0({\rm d}n_2/{\rm d}z){\rm d}z\right ] } r_{0}^\gamma \label{eqndeproj}\end{aligned}\ ] ] where @xmath151\right)/\gamma(0.5\gamma)$ ] , @xmath152 is the gamma function , @xmath153 is the radial comoving distance , @xmath154 are the redshift distributions of the samples ( for an autocorrelation @xmath155 ) , and @xmath156 . the hubble parameter @xmath157 can be found via @xmath158 . \label{eqnhz}\ ] ] equation ( [ eqndeproj ] ) assumes no evolution with redshift in the clustering of the sample ( equivalent to the implicit assumption made in fitting @xmath3 with equation [ eqnwpplaw ] ) . for each angular correlation analysis , we derive @xmath146 and @xmath147 from the observed @xmath108 and then obtain the corresponding @xmath149 and @xmath148 from equations ( [ eqndelta ] ) and ( [ eqndeproj ] ) . the masses of the dark matter halos in which galaxies and quasars reside are reflected in their absolute clustering bias relative to the dark matter distribution . to determine absolute bias ( following e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) we first calculate the two - point autocorrelation of dark matter as a function of redshift . we use the halofit code of @xcite to determine the nonlinear - dimensionless power spectrum @xmath159 of the dark matter assuming our standard cosmology , and the slope of the initial fluctuation power spectrum , @xmath160 . the fourier transform of the @xmath159 gives us the real - space correlation function @xmath63 , which we then integrate to @xmath161 @xmath14 mpc following equation ( [ eqnwpint ] ) to obtain the dark matter projected correlation function @xmath162 . the uncertainty in the dm power spectrum obtained from halofit is @xmath163% ; this corresponds to a systematic uncertainty @xmath70.05 dex in @xmath17 , but does not impact the relative halo masses of the different subsamples . to derive quasar absolute bias from the projected real - space correlation function , we average the @xmath164 over the redshift distribution of the sample , weighted by the overlap with the galaxy pdfs . the overlap of each quasar with the galaxy pdfs is given by @xmath165 and the corresponding @xmath3 for the dark matter is given by @xmath166 where @xmath167 is the quasar redshift . the redshift distributions for the qso-1s and obs - qsos are essentially identical ( the resulting @xmath168 values for the two samples differ by @xmath452% on all scales ) so for simplicity we use the same @xmath164 ( defined for the qso-1s ) for both sets of quasars . we obtain the bias by calculating the average ratio between the best - fit power - law model and @xmath164 over the range of scales of 110 @xmath14 mpc , for which @xmath164 corresponds closely to a power law and is dominated by the two - halo term . the observed clustering amplitude relative to the dark matter corresponds to @xmath169 , where @xmath170 and @xmath171 are the absolute linear biases of the quasars and sdwfs galaxies , respectively . to measure @xmath171 from the galaxy autocorrelation function ( or @xmath169 from the quasar - galaxy angular cross - correlation , described in [ qsoang ] ) , we require an estimate of the corresponding @xmath108 of the dark matter . to obtain @xmath108 we use limber s equation to project the power spectrum @xmath172 into the angular correlation @xcite . specifically , we perform a monte carlo integration of equation ( a6 ) of @xcite to obtain @xmath108 for the dark matter . the key parameter in this equation is @xmath173 where @xmath174 is the redshift distribution of the galaxies . we calculate @xmath174 from the sum of the pdfs of the galaxies for which we perform the autocorrelation . in deriving the dark matter @xmath108 for the quasar - galaxy cross - correlation , we replace @xmath175 with @xmath176 where @xmath177 is the distribution of quasar redshifts . for each angular correlation analysis we compute the average ratio between the best - fit power law model and the dark matter @xmath108 on scales 110 , where @xmath108 is dominated by the two - halo term . this ratio yields @xmath178 for galaxy autocorrelations or @xmath179 for quasar - galaxy cross - correlations . finally , we use @xmath170 and @xmath171 to estimate the characteristic mass of the dark matter halos hosting each subset of galaxies or quasars . @xcite derive a relation between dark - matter halo mass and large - scale bias that agrees well with the results of cosmological simulations . we use eqn . ( 8) of @xcite to convert @xmath180 to @xmath17 for the mean redshift of each subset of objects . if we use a different relation between @xmath181 and @xmath17 @xcite , we obtain estimates for @xmath17 that are similar , although slightly larger by 0.20.3 dex ; these differences do not significantly affect our conclusions . we note that to estimate @xmath17 , we have performed fits to the observed @xmath3 on scales of 0.312 @xmath14 mpc . in principle the dark matter and galaxy correlation functions can have somewhat different shapes such that the bias depends on the range of scales considered . if we limit the fits on scales 112 @xmath14 mpc , the results change by @xmath1075% , but with slightly larger uncertainties . we also note that our estimates of @xmath17 are relatively insensitive to our choice of @xmath182 . if we change @xmath182 from 0.84 to 0.8 ( as favored by the more recent wmap cosmology , e.g. * ? ? ? * ) our @xmath17 estimates for quasars and galaxies increase by @xmath183 dex . in this section we discuss the results of the correlation analysis and the characteristic dark matter halo masses for galaxies and quasars . we first calculate the cross - correlation of the full qso-1 and obs - qso samples with sdwfs galaxies . the resulting @xmath3 values and best - fit models are shown in figure [ fcorr ] , and fit parameters are given in table [ tabcorr ] . for both sets of the quasars the observed real - space projected cross - correlation is highly significant on all scales from 0.112 @xmath14 mpc , and the power law fits return @xmath184 , similar to many previous correlation function measurements for quasars ( e.g. , * ? ? ? * ; * ? ? ? * ) and galaxies ( e.g. , * ? ? ? * ; * ? ? ? the best - fit parameters are @xmath185 @xmath14 mpc , @xmath186 for the qso-1s , and @xmath187 @xmath14 mpc , @xmath188 for the obs - qsos . the results indicate that the cross - correlation of the obs - qsos with galaxies is somewhat stronger than that for the qso-1s . the corresponding values of @xmath179 are given in table [ tabcorr ] . as a check , we also perform power law fits to @xmath3 but leaving the slope fixed to @xmath189 , which corresponds to the slope of the @xmath3 for the dark matter . this also yields a significant difference in the clustering amplitude , although somewhat smaller , with @xmath190 and @xmath191 @xmath14 mpc for the qso-1s and obs - qsos , respectively . ( note that the formal uncertainties in @xmath148 here are smaller than for the above results because they do not account for covariance with @xmath149 . ) to obtain the absolute bias of sdwfs galaxies ( @xmath171 ) in order to extract the quasar bias @xmath170 from the cross - correlation results , we next derive the autocorrelation of sdwfs galaxies for the sample described in [ galauto ] . the observed @xmath108 is shown in figure [ fgalang ] , along with the correlation function for dark matter , calculated as discussed in [ absbias ] . fit parameters are given in table [ tabcorr ] . the power - law model fits well on the chosen scales of 112 , although there is a clear excess corresponding to the one - halo term at @xmath192 , as is common in galaxy autocorrelation measurements ( e.g. , * ? ? ? * ; * ? ? ? the best - fit power law parameters are @xmath193 and @xmath194 , and the ratio of the best - fit power law to the dark matter @xmath108 yields @xmath195 or @xmath196 . this accurate value for @xmath171 allows us to estimate @xmath170 for both types of quasars , based on the cross - correlation measurements . we obtain @xmath197 and @xmath198 , for qso-1s and obs - qsos , respectively . converting these to dark matter halo masses using the prescription of @xcite as described in [ absbias ] , we arrive at @xmath199)}= 12.7^{+0.4}_{-0.6}$ ] and @xmath5 for qso-1s and obs - qsos , respectively . the difference is marginally significant ( @xmath200 , although as we discuss below , the obs - qso clustering may represent only a robust lower limit ) . for direct comparison with other studies that directly measure the quasar autocorrelation , it is useful to present the quasar clustering in terms of effective power law parameters for their autocorrelation . assuming linear bias , the quasar autocorrelation can be inferred from the cross - correlation by @xmath201 ( e.g. , * ? ? ? we can therefore use the power law fits to the quasar - galaxy cross - correlation and galaxy autocorrelation to derive an effective @xmath148 and @xmath149 for the quasar autocorrelation . this yields @xmath202 @xmath14 mpc and @xmath203 for the qso-1s and @xmath204 @xmath14 mpc and @xmath205 for the obs - qsos . the autocorrelation amplitude and @xmath17 for qso-1s are in excellent agreement with previous estimates for unobscured quasars , while the best - fit amplitude for obs - qsos is higher than most previous measurements of quasar clustering . we compare these results to previous work and discuss possible interpretations in [ discussion ] . lcccccccc + qso-1 & 445 & 1.27 & @xmath206 & @xmath207 & 1.1 & @xmath208 & @xmath209 & @xmath210 + obs - qso & 361 & 1.24 & @xmath211 & @xmath212 & 1.2 & @xmath213 & @xmath214 & @xmath215 + + qso-1 & 445 & 1.27 & @xmath216 & 1.8 & 1.1 & @xmath217 & @xmath218 & @xmath219 + qso-1 ( photo-@xmath6 ) & 445 & 1.27 & @xmath220 & 1.8 & 1.2 & @xmath221 & @xmath222 & @xmath223 + obs - qso & 361 & 1.24 & @xmath224 & 1.8 & 1.2 & @xmath225 & @xmath226 & @xmath227 + + galaxies & 151256 & 1.10 & @xmath228 & @xmath229 & 1.1 & @xmath230 & @xmath229 & @xmath231 + qso-1 & 445 & 1.27 & @xmath232 & 1.8 & 1.1 & @xmath233 & @xmath234 & @xmath235 + obs - qso & 361 & 1.24 & @xmath236 & 1.8 & 1.2 & @xmath237 & @xmath238 & @xmath239 anumber of objects include in the correlation analysis . for quasar - galaxy cross - correlation , we use the full sample of 256,124 galaxies ( for @xmath3 calculations ) or 151,256 galaxies ( for @xmath108 calculations ) . bmedian redshift for the objects included in the correlation analysis . cuncertainties in the dm power spectrum introduce an additional systematic error of @xmath75% in @xmath180 ( and corresponding @xmath70.05 dex in @xmath17 ) . further systematic errors in @xmath17 of @xmath70.2 dex are caused by uncertainty in @xmath182 and in the conversion from @xmath180 to @xmath240 , as discussed in [ absbias ] . however , these do not significantly effect the _ relative _ halo masses , so these uncertainties is not included here . note that for fits with fixed @xmath149 , uncertainties on @xmath148 , bias , and @xmath17 do not account for covariance with @xmath149 and thus somewhat underestimate the error on the clustering amplitude . dreal - space projected cross - correlation between quasars and galaxies , calculated as described in [ corranal ] . for all @xmath3 calculations , error estimates for @xmath148 , bias , and @xmath240 include a 10% systematic uncertainty on the amplitude as described in [ crosscorr ] . eangular galaxy autocorrelation and quasar - galaxy cross - correlation , calculated as described in [ corranal ] . in this section we perform several tests to verify the validity of the clustering analysis outlined in [ corranal ] . we first calculate the quasar - galaxy cross - correlation using a simple angular correlation function , minimizing dependence on the photometric redshifts . we then check for variation in the observed clustering over the redshift range of interest and confirm that any variation is relatively weak . finally , we estimate the effects of uncertainties on the photometric redshifts on the observed real - space clustering amplitude for the obs - qsos . these checks confirm that our projected correlation analysis provides a robust estimate of the quasar - galaxy cross - correlation . we first calculate the cross - correlation of quasars and sdwfs galaxies using a simple angular clustering analysis , and check whether the corresponding absolute bias is consistent with that derived from the more sophisticated @xmath3 calculation . to calculate the @xmath108 we use an estimator corresponding to equation ( [ eqnxidef1a ] ) but for cross - correlations : @xmath241 where each term is scaled according to the total numbers of galaxies and randoms . to maximize the signal - to - noise ratio by cross - correlating objects associated in redshift space , the galaxies include only the redshift - matched sdwfs sample of 151,256 objects described in [ galauto ] . uncertainties are estimated using bootstrap resampling as described in [ uncertainties ] . we fit the observed cross - correlation with a a power law as described in [ plawang ] . owing to the limited statistics which provide only very weak constraints on the power law slope , we fix @xmath242 ( corresponding to real - space @xmath243 ) . the resulting cross - correlations and scaled dark matter fits are shown in figure [ fqsoang ] , and fit parameters are given in table [ tabcorr ] . the estimates of @xmath148 and @xmath179 are in broad agreement between the two estimators , although as may be expected , the statistical uncertainties for the angular correlation analysis are larger ( by @xmath750% ) than for the real - space analysis with fixed @xmath149 . given that the absolute bias derived from the projected correlation function corresponds broadly to the bias from the noisier , but simpler angular cross - correlation , we conclude that there are no significant systematic effects that skew our estimate of @xmath3 . our calculation of the real - space quasar - galaxy correlation function over the redshift range @xmath0 requires that @xmath3 varies only slowly between these redshifts , as discussed in [ crosscorr ] . if the objects reside in similar halos at all redshifts , then we may expect @xmath148 to change slowly ; simulations suggest that the typical @xmath148 for the autocorrelation of dm halos of mass @xmath7@xmath18@xmath19 @xmath14 @xmath15 should change by @xmath244 @xmath14 mpc between @xmath245 and 2 ( see figure 10 of * ? ? ? * ) . to test explicitly the redshift variation for the clustering in our sample , we re - derived @xmath3 using the method outlined in [ corranal ] but selecting quasars over smaller redshift bins of @xmath246 and @xmath247 uncertainties are calculated using the bootstrap method as for the full quasar samples , and dark matter and power - law fits are again performed over the range of separations @xmath248 @xmath14 mpc . we evaluate @xmath249 as in [ absbias ] , but only including the quasars in the redshift ranges of interest . owing to larger statistical errors and for simple comparison to the results over the full redshift range , in the power law fits we fix @xmath149 to 1.8 . the resulting @xmath3 for the separate redshift bins and the power law fits are shown in figure [ fwpz ] . for the qso-1s we obtain @xmath250 @xmath14 mpc and @xmath251 @xmath14 mpc for the low- and high - redshift bins , respectively , and for the obs - qsos we correspondingly obtain @xmath252 @xmath14 mpc and @xmath253 @xmath14 mpc . the measured quasar - galaxy cross - correlation should be largely independent between the two redshift bins . although the quasars are cross - correlated against the same galaxy sample in each bin , the galaxy samples will be weighted toward higher and lower redshifts in the high- and low-@xmath6 bins , respectively . for the high and low redshift bins , the best - fit @xmath148 values bracket those for the full redshift samples , and are broadly consistent within the uncertainties . interestingly , the best - fit clustering amplitude for the obs - qsos increases with redshift while it decreases for the obs - qsos ; however , given the uncertainties we decline to speculate on any possible difference in redshift evolution between the two subsets . overall , the results in the different redshift bins confirm that any variation in the observed @xmath3 is sufficiently weak over the redshift range of interest , so that the method outlined in [ crosscorr ] should provide a reasonable estimate of the average clustering amplitude over the full redshift range . the primary uncertainty in our estimate of @xmath3 for the obs - qsos is the lack of accurate ( that is , spectroscopic ) redshifts and difficulty in estimating the photo-@xmath6 uncertainties from the neural net calculations . as described in [ corranal ] , in calculating @xmath3 for the obs - qso galaxy cross - correlation , we simply assume that obs - qsos lie exactly at the best redshifts output by the neural net estimator . any uncertainties in the photo-@xmath6s or systematic offsets from the true redshifts could therefore affect the resulting clustering measurement . the fact that we obtain very similar estimates of @xmath170 for the obs - qsos from a simple angular cross - correlation analysis as from our @xmath3 calculation suggests that uncertainties on individual photo-@xmath6s do not strongly affect our estimates of the quasar bias , as long as the overall distribution in redshifts for the obs - qsos is correct . however , it is possible that very large discrepancies from the true photo-@xmath6s , or any systematic shift in the redshift distribution , could affect both the estimate of @xmath3 and the real - space clustering parameters derived from @xmath108 . to precisely explore the effect of these errors , we take advantage of the fact that we have an equivalent sample of objects ( the qso-1s ) that have a similar redshift distribution and for which the redshifts are known precisely from spectroscopy . we can therefore adjust the redshifts of the qso-1s and re - calculate @xmath3 to determine how uncertainties or systematic shifts affect the observed correlation amplitude . as a simple first test , we calculate the @xmath3 for the qso-1s using the photo-@xmath6s ( as shown in figure [ fspz ] ) rather than spectroscopic redshifts . figure [ fagn5pz ] shows that the resulting @xmath3 differs little from that obtained using spectroscopic redshifts ; the clustering amplitude for a power - law fit with fixed @xmath189 is lower by 12% . note that if we allow @xmath149 to float , the average bias for the photo-@xmath6 sample is actually larger by @xmath210% ( owing to a slightly flatter slope ) but well within the statistical uncertainties . we conclude that for the qso-1s , photo-@xmath6 errors do not have a significant effect on our measurements of the clustering . to explore photo-@xmath6 errors in more detail , we systematically test the effects of gaussian random errors in the quasar redshifts . for each quasar , we shift the best estimates of redshift ( spec-@xmath6s for qso-1s and photo-@xmath6s for obs - qsos ) by offsets @xmath254 selected from a gaussian random distribution with dispersion @xmath255 . using these new redshifts we recalculate @xmath3 , using the full formalism described in [ corranal ] . we perform the calculation ten times for each of several values of @xmath255 from 0.02 up to 0.2 ( which smears the redshifts across most of the redshift range of interest ) . to ensure that this step does not artificially smear out the redshift distribution beyond the range probed by the galaxies , we require that the random redshifts lie between @xmath0 ; any random redshift that lies outside this range is discarded and a new redshift is selected from the random distribution . for each trial we obtain the relative bias by calculating the mean ratio of @xmath3 , on scales 110 @xmath14 mpc , relative to the @xmath3 for the best estimates of redshift . we then average the ten trials at each @xmath256 to obtain a relation between relative bias and @xmath256 , shown in figure [ ftest_fake_box](a ) . as may be expected , figure [ ftest_fake_box](_a _ ) shows that shifting the qso-1 redshifts from their true values causes a decrease in the cross - correlation amplitude , as the quasars are preferentially correlated with galaxies that are not actually associated in redshift space . we find a monotonic decrease in relative bias with @xmath256 , from @xmath257 for @xmath258 to @xmath259 for @xmath260 . repeating this calculation for the obs - qsos reveals a very similar trend . the decrease in bias with @xmath255 shown in figure [ ftest_fake_box](_a _ ) indicates that such errors would affect the measurements of the clustering amplitude by at most @xmath720% . while the above analysis suggests that random errors in the obs - qso photo-@xmath6s do not strongly affect the observed clustering amplitude , it is also possible that systematic uncertainties in the photo-@xmath6 ( consistent over- or under - estimates of the redshift ) could significantly alter the observed bias . to test this , we shift the redshifts of the quasars as discussed in [ randomerrors ] , but in place of random shifts , we compress all redshifts toward one end or the other of the @xmath0 range . ( this procedure allows us to test the effects of systematic shifts in redshift while keeping the same overall redshift range . ) the shift in redshift is defined by a redshift scaling parameter @xmath261 , such that @xmath262 as an additional check we also perform a simple linear offset @xmath263 of the redshifts , allowing the redshifts to move outside the selection range of @xmath0 . as in [ randomerrors ] , we use these new redshifts to recalculate @xmath3 via the full formalism described in [ corranal ] , and determine the relative bias on scales 110 @xmath14 mpc . relative bias versus @xmath261 and @xmath264 are shown in figures [ ftest_fake_box](b ) and ( c ) . for the qso-1s , the peak of the observed clustering amplitude is very close to @xmath265 , while shifting the redshifts down or up systematically decreases the bias . the obs - qsos show a similar peak near @xmath265 , indicating that the obs - qso photo-@xmath6s are not systematically offset higher or lower than the true redshifts by a large factor . we note that for the obs - qsos a slight shift to higher redshifts ( @xmath104 ) increases the clustering by a small amount ( @xmath28% ) . finally , we emphasize that any possible low - redshift contaminants , will serve only to decrease the observed clustering signal , as they will be completely uncorrelated in angular space with the higher - redshift sdwfs galaxies that lie in entirely separate large - scale structures . therefore the observed @xmath3 represents a robust lower limit to the clustering amplitude for the obs - qsos . we have used the ir - selected quasar sample of to measure the clustering amplitude and to estimate characteristic dark matter halo masses for roughly equivalent samples of unobscured and obscured quasars . we obtain highly significant detections of the clustering for both samples , with marginally stronger clustering for the obs - qsos . in this section , we compare our results for qso-1s to previous results on unobscured quasar clustering , speculate on physical explanations for possible stronger clustering for obs - qsos , and discuss prospects for future studies with the next generation of observatories . we compare our observed absolute bias for the mid - ir selected quasars to the clustering of optically - selected ( type 1 ) quasars , which has been well established by a number of works . among the most precise measurements to date are studies that have used data from the 2df and sdss surveys . recently , @xcite measured the evolution of the quasar 3-d autocorrelation function based on spectroscopic quasars from sdss data release 5 , and compared to previous results from spectroscopic samples from the 2qz @xcite and 2slaq @xcite , as well as the clustering of photometrically - selected quasars from sdss @xcite . figure [ fcompare](a ) shows the redshift evolution of linear bias for type 1 quasars from these studies , taken from figure 12 of @xcite . where appropriate , we have converted the bias to our adopted cosmology using the formalism in the appendix of @xcite , assuming a shape factor @xmath266 for simplicity . figure [ fcompare](b ) shows the corresponding estimates of characteristic @xmath17 derived from the linear bias using the prescription of @xcite . the linear bias and halo masses for the qso-1s and obs - qsos are shown for comparison . it is readily apparent from figure [ fcompare](a ) that the observed bias of type 1 quasars increases with redshift , as discussed in [ intro ] . ( these results are also consistent with a number of other quasar clustering studies using other data see e.g. , figure 15 of @xcite . ) the dashed curves in figure [ fcompare](a ) show the increase in bias with redshift for halos of constant mass , clearly showing that at all redshifts the qso-1s reside in dark matter halos of roughly a few @xmath267 @xmath14 @xmath15 . our measurement for the qso-1s is in excellent agreement with the evolution in linear bias and roughly constant @xmath17 observed in previous measurements of type 1 quasar clustering . for the obs - qsos , the best - fit bias is marginally larger ( @xmath268 ) , corresponding to a factor of roughly four difference in the characteristic @xmath17 between the qso-1s and obs - qsos ( figure [ fcompare]b ) . as discussed above , random errors in the photo-@xmath6s can only decrease the observed clustering amplitude and the inferred @xmath17 . thus our measurement of obs - qso clustering represents a lower limit , and it is possible that the true obs - qso bias is somewhat higher ( although the results of [ verification ] suggest the true bias may be larger by at most @xmath720% ) . based on this analysis we can make the robust conclusion that the obs - qsos are _ at least _ as strongly clustered as their qso-1 counterparts . stronger clustering for obscured quasars would have significant implications for physical models of the obscured quasar population . in terms of unified models , a difference in clustering between obscured and unobscured quasars would rule out the simplest picture in which obscuration is purely an orientation effect , but may be consistent with more complicated scenarios where the effective covering fraction changes with environment . alternatively , if obscuration is caused by large ( @xmath7kpc scale ) structures , then the processes that drive these asymmetries ( e.g. , mergers , disk instabilities , accretion of cold gas from the surrounding halo ) may be more common in halos of larger mass . indeed , given that some fraction of quasars might naturally be expected to be obscured by a `` unified''-model torus , any observed differences in clustering may reflect even stronger intrinsic dependence of large - scale obscuration and environment . an intriguing scenario for obscured quasars is that they represent an early evolutionary phase of rapid black hole growth before a `` blowout '' of the obscuring material from the central regions of the galaxy and the emergence of an unobscured quasar ( e.g. , figure 1 of * ? ? ? quasars tend to radiate at large fraction of the eddington rate ( @xcite ; although see @xcite ) , so that the similar @xmath50 for qso-1s and obs - qsos would imply that they host black holes of similar masses . any correlation ( e.g. , * ? ? ? * ; * ? ? ? * ) , even if indirect ( e.g. , * ? ? ? * ) , between the final masses of black holes and those of their host halos would thus suggest that our obscured and unobscured quasars would have the same @xmath17 , as long as their black holes are near their final masses . however , if obscured quasars are in an earlier phase of rapid growth and so are in the process of `` catching up '' to their final mass ( e.g. , * ? ? ? * ) , then they would have a larger @xmath17 compared to unobscured quasars with the same @xmath269 . in light of recent debate as to whether black holes generally grow before or after their hosts ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , this scenario would imply that black hole growth lags behind that of the host halo . in any physical picture , a significant difference in clustering between obscured and unobscured quasars would also imply a difference in accretion duty cycles ( or equivalently , lifetime ) . qso-1s and obs - qsos are found in roughly equal numbers , but the abundance of dark matter halos drops rapidly with mass ( e.g. , * ? ? ? * ) , thus implying that if one type of quasars are found in larger halos then they must be longer - lived . with our current results , we are able to rule out any model in which obscured quasars are substantially _ less _ strongly clustered or have _ shorter _ lifetimes than their unobscured counterparts . with more accurate future measurements , detailed studies of halo masses and lifetimes for obscured and unobscured quasars could place powerful constraints on evolutionary scenarios such as those described above . our results demonstrate the potential for studying the clustering of obscured quasars in extragalactic multiwavelength surveys , and the marginally significant difference in clustering we observe for obscured and unobscured quasars provides strong motivation for more precise measurements in the future . the two main avenues for progress are improvements in redshift accuracy and selection of larger samples for better statistical accuracy . upcoming sensitive , wide - field multi - object spectrographs will enable efficient measurements of redshift for large numbers of optically - faint sources and so improve calibrations of obscured quasar photo-@xmath6s , or with large enough samples , enable fully 3-d clustering studies . in addition , we will soon have the capability to detect many thousands of obscured quasars based on very wide - field observations in the mid - ir with the _ wide - field infrared survey explorer _ @xcite and in x - rays with erosita @xcite or the _ wide - field x - ray telescope _ @xcite . these data sets will allow us to measure obscured quasar clustering with statistical precision that is comparable to current measurements of unobscured quasars . we have used data from the botes wide - field multiwavelength survey to measure the two - point spatial cross - correlation between unobscured ( qso-1 ) and obscured ( obs - qso ) mid - ir selected quasars in the redshift range @xmath0 . the qso-1s exhibit clustering corresponding to a typical @xmath270 @xmath14 @xmath15 , similar to previous studies of optically - selected quasar clustering . we robustly determine that the obs - qsos are clustered _ at least _ as strongly as the qso-1s , with a marginally stronger signal corresponding to host halos of mass @xmath271 @xmath14 @xmath15 ; the true clustering amplitude could be up to @xmath720% larger owing to photo-@xmath6 uncertainties for the obs - qsos that can decrease the observed correlation amplitude . our results motivate more accurate measurements of obscured quasar clustering with larger quasar samples and more accurate redshifts . if future studies confirm that obscured quasars are more strongly clustered than their their unobscured counterparts , this would rule out the simplest `` unified '' models and may provide evidence for scenarios in which rapid obscured accretion represents an evolutionary phase in the growth of galaxies and their central black holes . we thank our colleagues on the ndwfs , ages , sdwfs , and xbotes teams . we thank the anonymous referee for helpful comments that improved the paper , and philip hopkins and peder norberg for productive discussions . the noao deep wide - field survey , and the research of a.d . and b.t.j . are supported by noao , which is operated by the association of universities for research in astronomy ( aura ) , inc . under a cooperative agreement with the national science foundation . this paper would not have been possible without the efforts of the _ chandra _ , _ spitzer _ , kpno , and mmt support staff . optical spectroscopy discussed in this paper was obtained at the mmt observatory , a joint facility of the smithsonian institution and the university of arizona . the first _ spitzer _ mips survey of the botes region was obtained using gto time provided by the _ spitzer _ infrared spectrograph team ( pi : james houck ) and by m. rieke . we thank the collaborators in that work for access to the 24 micron catalog generated from those data . r.c.h . was supported by an stfc postdoctoral fellowship and an sao postdoctoral fellowship , and a.d.m . was generously funded by the nasa adap program under grant nnx08aj28 g . d.m.a . is grateful to the royal society and philip leverhulme prize for their generous support . r.j.a . was supported by the nasa postdoctoral program , administered by oak ridge associated universities through a contract with nasa . the large area of the botes survey allows us to measure galaxies at relatively large physical separations ; 1 degree corresponds to 50 @xmath14 mpc at @xmath272 . however , when we use equation ( [ eqnwp ] ) to calculate the projected real - space correlation function on large scales , we find that the @xmath3 flattens on scales @xmath27320 @xmath14 mpc , corresponding to tens of arcmin , and then becomes negative at @xmath274 @xmath14 mpc ( figure [ fwprand ] ) . this behavior is not observed for quasar clustering from other , wider - field surveys ( e.g. * ? ? ? * ; * ? ? ? * ) , for which the correlation function continues to decrease on larger scales . while the integral constraint require that the correlation function becomes negative on some scales , in galaxy auto - correlation surveys this generally only happens at @xmath275 200 @xmath14 mpc ( e.g. , * ? ? ? one possibility is that the observed behavior is due edge effects arising from the finite geometry of the sdwfs field , which are not taken into account by the simple @xmath276 estimator in the formalism ( e.g. , * ? ? ? * ) . to test this possibility , we re - performed the correlation analysis described in [ corranal ] , after randomizing the positions of the quasars on the sky within the area of good sdwfs photometry . we performed 10 separate random trials , for which the cross - correlations are shown in figure [ fwprand ] along with the @xmath3 values for the qso-1s and obs - qsos . it is clear from figure [ fwprand ] that on scales @xmath10710 @xmath14 mpc , the projected cross - correlation between the random quasars and galaxies is small compared to the @xmath3 for the real quasar sample . however , on scales @xmath27720 @xmath14 mpc , both the real and random samples show an increase in @xmath3 which eventually becomes negative around @xmath278 @xmath14 mpc . the quantities of interest in this paper ( i.e. absolute bias and dark matter halo mass ) can be measured by studying the correlations on scales @xmath4512 @xmath14 mpc , where the artifacts are small and have negligible impact on the fits to the correlation function . for this paper we therefore limit the correlation analyses to those scales . , b. t. & dey , a. 1999 , in asp conf . ser . 191 : photometric redshifts and the detection of high redshift galaxies , ed . r. weymann , l. storrie - lombardi , m. sawicki , & r. brunner ( san francisco : asp ) , 111 | we present the first measurement of the spatial clustering of mid - infrared selected obscured and unobscured quasars , using a sample in the redshift range @xmath0 selected from the 9 deg@xmath1 botes multiwavelength survey . recently
the _ spitzer space telescope _ and x - ray observations have revealed large populations of obscured quasars that have been inferred from models of the x - ray background and supermassive black hole evolution . to date , little is known about obscured quasar clustering , which allows us to measure the masses of their host dark matter halos and explore their role in the cosmic evolution of black holes and galaxies . in this study
we use a sample of 806 mid - infrared selected quasars and @xmath2250,000 galaxies to calculate the projected quasar - galaxy cross - correlation function @xmath3 .
the observed clustering yields characteristic dark matter halo masses of @xmath4 ) = 12.7^{+0.4}_{-0.6}$ ] and @xmath5 for unobscured quasars ( qso-1s ) and obscured quasars ( obs - qsos ) , respectively .
the results for qso-1s are in excellent agreement with previous measurements for optically - selected quasars , while we conclude that the obs - qsos are _ at least _ as strongly clustered as the qso-1s .
we test for the effects of photometric redshift errors on the optically - faint obs - qsos , and find that our method yields a robust lower limit on the clustering ; photo-@xmath6 errors may cause us to underestimate the clustering amplitude of the obs - qsos by at most @xmath720% .
we compare our results to previous studies , and speculate on physical implications of stronger clustering for obscured quasars . |
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the full sky has not been surveyed in space ( imaging ) and time ( variability ) at hard x - ray energies . yet the hard x - ray ( hx ) band , defined here as 10 - 600 kev , is key to some of the most fundamental phenomena and objects in astrophysics : the nature and ubiquity of active galactic nuclei ( agn ) , most of which are likely to be heavily obscured ; the nature and number of black holes ; the central engines in gamma - ray bursts ( grbs ) and the study of grbs as probes of massive star formation in the early universe ; and the temporal measurement of extremes : from khz qpos to sgrs for neutron stars , and microquasars to blazars for black holes . a concept study was conducted for the energetic x - ray imaging survey telescope ( exist ) as one of the new mission concepts selected in 1994 ( grindlay et al 1995 ) . however the rapid pace of discovery in the hx domain in the past 2 years , coupled with the promise of a likely 2 - 10 kev imaging sky survey abrixas2 ( see http://www.aip.de/cgi-bin/w3-msql/groups/xray/abrixas/index.html ) in c.2002 - 2004 and the recent selection of swift ( see http://swift.gsfc.nasa.gov/ ) which will include a 10 - 100 kev partial sky survey ( to 1 mcrab ) in c.2003 - 2006 , have prompted a much more ambitious plan . a dedicated hx survey mission is needed with full sky coverage each orbit and 0.05 mcrab all - sky sensitivity in the 10 - 100 kev band ( comparable to abrixas2 ) and extending into the 100 - 600 kev band with 0.5 mcrab sensitivity . such a mission would require very large total detector area and large telescope field of view . these needs could be met very effectively by a very large coded aperture telescope array fixed ( zenith pointing ) on the international space station ( iss ) , and so exist - iss was recommended by the nasa gamma - ray program working group ( grapwg ) as a high priority mission for the coming decade . this mission concept has now been included in the nasa strategic plan formulated in galveston as a post-2007 candidate mission . in this paper we summarize the science goals and briefly present the mission concept of exist - iss . details will be presented in forthcoming papers , and are partially available on the exist website ( http://hea-www.harvard.edu/exist/exist.html ) . exist would pursue two key scientific goals : a very deep hx imaging and spectral survey , and a very sensitive hx all - sky variability survey and grb spectroscopy mission . these survey ( * s * ) and variability ( * v * ) goals can be achieved by carrying out several primary objectives : + + it is becoming increasingly clear that most of the accretion luminosity of the universe is due to obscured agn , and that these objects are very likely the dominant sources for the cosmic x - ray ( and hx ) diffuse background ( e.g. fabian 1999 ) . no sky survey has yet been carried out to measure the distribution of these objects in luminosity , redshift , and broad - band spectra in the hx band where , as is becoming increasingly clear from bepposax ( e.g. vignati et al 1999 ) , they are brightest . exist would detect at least 3000 seyfert 2s and conduct a sensitive search for type 2 qsos . spectra and variability would be measured , and detailed followup could then be carried out with the narrow - field focussing hx telescope , hxt , on constellation - x ( harrison et al 1999 ) as well as ir studies . + + the study of black holes , from x - ray binaries to agn , in the hx band allows their ubiquitous comptonizing coronae to be measured . the relative contributions of non - thermal jets at high requires broad band coverage to 511 kev , as does the transition to adafs at lower values . hx spectral variations vs. broad - band flux can test the underlying similarities in accretion onto bhs in binaries vs. agn . + + x - ray novae ( xn ) appear to be predominantly bh systems , so their unbiased detection and sub - arcmin locations , which allow optical / ir identifications , can provide a direct measure of the bh binary content ( and xn recurrence time ) of the galaxy . xn containing neutron stars can be isolated by their usual bursting activity ( thermonuclear flashes ) , and since they may solve the birth rate problem for millisecond pulsars ( yi and grindlay 1998 ) , their statistics must be established . a deep hx survey of the galactic plane can also measure the population of galactic bhs not in binaries , since they could be detected as highly cutoff hard sources projected onto giant molecular clouds . compared to ism accretion onto isolated nss , for which a few candidates have been found , bhs should be much more readily detectable due to their intrinsically harder spectra and ( much ) lower expected space velocities , v , and larger mass m ( bondi accretion depending on m@xmath0/v@xmath6 ) . + + type ii sne are expected to disperse 10@xmath7 of 44 , with the total a sensitive probe of the mass cut and ns formation . with a 87y mean - life for decay into @xmath5sc which produces narrow lines at 68 and 78 kev , obscured sne can be detected throughout the entire galaxy for 300y given the 10@xmath8 photons cm2sec line sensitivity and 2 kev energy resolution ( at 70 kev ) possible for exist . thus the likely detection of cas a ( iyudin et al 1994 ) can be extended to more distant but similarly ( or greater ) obscured sn to constrain the sn rate in the galaxy . the all - sky imaging of exist would extend the central - radian galactic survey planned for integral to the entire galaxy . + + since at least the `` long '' grbs located with bepposax are at cosmological redshifts , and have apparent luminosities spanning at least a factor of 100 , it is clear that even the apparently lower luminosity grbs currently detected by bepposax could be detected with batse out to z 4 and that the factor 5 increase in sensitivity with swift will push this back to z 5 - 15 ( lamb and reichart 1999 ) . the additional factor of 4 increase in sensitivity for exist would allow grb detection and sub - arcmin locations for z 15 - 20 and thus allow the likely epoch of pop iii star formation to be probed if indeed grbs are associated with collapsars ( e.g. woosley 1993 ) produced by the collapse of massive stars . the high throughput and spectral resolution for exist would enable high time resolution spectra which can test internal shock models for grbs . + + only 3 sgr sources are known in the galaxy and 1 in the lmc . since a typical 0.1sec sgr burst spike can be imaged ( 5@xmath9 ) by exist for a peak flux of 200 mcrab in the 10 - 30 kev band , the typical bursts from the newly discovered sgr1627 - 41 ( woods et al 1999 ) with peak flux 2 10@xmath8 ergcm2sec would be detected out to 200 kpc . hence the brightest `` normal '' sgr bursts are detectable out to 3 mpc and the rare giant outbursts ( e.g. march 5 , 1979 event ) out to 40 mpc . thus the population and physics of sgrs , and thus their association with magnetars and young snr , can be studied throughout the local group and the rare super - outbursts beyond virgo . + + the cosmic ir background ( cirb ) over 1 - 100@xmath10 is poorly measured ( if at all ) and yet can constrain galaxy formation and the luminosity evolution of the universe ( complementing * s1 * above ) . as reviewed by catanese and weekes ( 1999 ) , observing spectral breaks ( from @xmath11 absorption ) for blazars in the band 0.01 - 100 tev can measure the cirb out to z 1 _ if _ the intrinsic spectrum is known . since the @xmath12-ray spectra of the detected ( low z ) blazars are well described by synchrotron - self compton ( ssc ) models , for which the hard x - ray ( 100kev ) synchrotron peak is scattered to the tev range , the hx spectra can provide both the required underlying spectra and time - dependent light curves for all objects ( variable ! ) to be observed with glast and high - sensitivity ground - based tev telescopes ( e.g. veritas ) . + + the success of batse as a hx monitor of bright accreting pulsars in the galaxy ( cf . bildsten et al 1997 ) , in which spin histories and accretion torques were derived for a significant sample , can be greatly extended with exist : the very much larger reservoir of be systems can be explored , and wind vs. disk - fed accretion studied in detail . the wide - field hx imaging and monitoring capability will also allow a new survey for pulsars and axps in highly obscured regions of the disk , complementing * s4 * above . + + the rms variability generally and qpo phenomena appear more pronounced above 10kev for x - ray binaries containing both bh and ns accretors , suggesting the comptonizing corona is directly involved . thus qpos and hx spectral variations can allow study of the poorly - understood accretion disk coronae , with extension to the agn case . although the wide - field increases backgrounds , and thus effective modulation , the very large area ( 1m@xmath0 ) of hx imaging area on any given source means that multiple 100 mcrab lmxbs could be simultaneously measured for qpos with 10% rms amplitude in the poorly explored 10 - 30 kev band . to achieve the desired 0.05 mcrab sensitivity full sky up to 100 kev ( and beyond ) requires a very large area array of wide - field coded aperture ( or other modulation ) telescopes . the very small field of view ( 10 ) of true focussing ( e.g. multi - layer ) hx telescopes precludes their use for all sky imaging and monitoring surveys . exist - iss would take the coded aperture concept to a practical limit , with 8 telescopes each with 1m@xmath0 in effective detector area and 4040 in field of view ( fov ) . the individual fovs are offset by 20 for a combined fov of 60x40deg , or 2sr . by orienting the 160axis perpendicular to the orbit vector , the full sky can be imaged each orbit if the telescope array is fixed - pointed at the local zenith . this gravity - gradient type orientation , and the large spatial area of the telescope array , are ideally matched for the iss , which provides a long mounting structure ( main truss ) conveniently oriented perpendicular to the motion , as depicted on the exist website . the sensitivity would yield 10@xmath13 agns full sky , thus setting a confusion limit resolution requirement ( 1/40 `` beam '' ) of 5 . with this coded mask pixel size , high energy occulting masks ( 5 mm , w ) can be constructed with 2.5 mm pixel size for minimal collimation . the mask shadow is then recorded by tiled arrays of cdznte ( czt ) detectors with effective pixel sizes of 1.3 mm , yielding a compact ( 1.3 m ) mask - detector spacing . the czt detectors would likely be 20 mm square thick ( for 20% efficiency at 500 kev ) and read out by flip - chip bonded asics ( e.g. bloser et al 1999 , harrison et al 1999 ) . the 8-telescope array is continuously scanning ( sources on the orbital plane drift across the 40fov in 10min ; correspondingly longer exposures / orbit near the poles ) , with each photon time - tagged and aspect corrected ( 10 ) so that iss pointing errors or flexure are inconsequential over the large fov . source positions are centroided to 1for 5@xmath9 detections . the resulting sky coverage is remarkably uniform with 25% variation in exposure full sky over the 2mo precession period of the iss orbit . more details of the current mission concept are given in grindlay et al ( 2000 ) , and will be further developed in the implementation study being conducted by the exist science working group ( exswg ) . bildsten , l. et al , _ apjs _ , * 113*,367 ( 1997 ) . + bloser , p. , grindlay , j. , narita , t. and jenkins , j. , _ proc . spie _ , * 3765 * , 388 ( 1999 ) . + catanese , m. and weekes , t. , _ pasp _ , * 111 * , 1193 ( 1999 ) . + fabian , a. _ mnras _ , * 308 * , l39 ( 1999 ) . + grindlay , j.e . et al , _ proc . spie _ , * 2518 * , 202 ( 1995 ) . + grindlay , j.e . et al , _ proc . staif-2000 _ , in preparation ( 2000 ) . + harrison , f.a . et al , _ proc . , * 3765 * , 104 ( 1999 ) . + iyudin , a.f . et al , _ a&a _ , * 284 * , l1 ( 1994 ) . + lamb , d.q . and reichart , d.e . , _ apj _ , submitted ( astro - ph/9909002 ) ( 1999 ) . + vignati , p. et al , _ a&a _ , * 349 * , 57l ( 1999 ) . + woods , p.m. et al , _ apj _ , * 519 * , l139 ( 1999 ) . + woosley , s.e . _ apj _ , * 405 * , 273 ( 1993 ) . + yi , i. and grindlay , j.e . , _ apj _ , * 505 * , 828 ( 1998 ) . | a deep all - sky imaging hard x - ray survey and wide - field monitor is needed to extend soft ( rosat ) and medium ( abrixas2 ) x - ray surveys into the 10 - 100 kev band ( and beyond ) at comparable sensitivity ( 0.05 mcrab ) .
this would enable discovery and study of 3000 obscured agn , which probably dominate the hard x - ray background ; detailed study of spectra and variability of accreting black holes and a census of bhs in the galaxy ; gamma - ray bursts and associated massive star formation ( popiii ) at very high redshift and soft gamma - ray repeaters throughout the local group ; and a full galactic survey for obscured supernova remnants .
the energetic x - ray imaging survey telescope ( exist ) is a proposed array of 8 1m@xmath0 coded aperture telescopes fixed on the international space station ( iss ) with 60x40deg field of view which images the full sky each 90 min orbit .
exist has been included in the most recent nasa strategic plan as a candidate mission for the next decade .
an overview of the science goals and mission concept is presented . /cm2secergs
@xmath1 s@xmath2 2ergs @xmath1 2cm@xmath0 cm2sec @xmath1 s@xmath2 60x40deg160@xmath4 40 0x6deg20@xmath4 6 44@xmath5ti |
You are an expert at summarizing long articles. Proceed to summarize the following text:
within the standard model framework , the strong interaction is described by quantum chromodynamics ( qcd ) , which suggests the existence of the unconventional hadrons , such as glueballs , hybrid states and multiquark states . the establishment of such states remains one of the main interests in experimental particle physics . decays of the @xmath4 particle are ideal for the study of the hadron spectroscopy and the searching for the unconventional hadrons . in the decays of the @xmath4 particle , several observations in the mass region 1.8 gev / c@xmath7 - 1.9 gev / c@xmath7 have been presented in different experiments@xcite@xcite , such as the @xmath8@xcite@xcite , @xmath9@xcite@xmath10@xcite , @xmath11@xcite@xmath10@xcite and @xmath12@xcite . recently , using a sample of @xmath13 @xmath4 events@xcite collected with besiii detector@xcite at bepcii@xcite , the decay of @xmath2 was analyzed@xcite , and the @xmath0 was observed in the @xmath1 mass spectrum with a statistical significance of @xmath5 . . the dots with error bars are data ; the histogram is phase space events with an arbitrary normalization . [ m6pi],scaledwidth=60.0% ] the @xmath1 invariant mass spectrum is shown in fig . [ m6pi ] , where the @xmath0 can be clearly seen . the parameters of the @xmath0 are extracted by an unbinned maximum likelihood fit . in the fit , the background is described by two contributions : the contribution from @xmath14 and the contribution from other sources . the contribution from @xmath14 is determined from mc simulation and fixed in the fit ( shown by the dash - dotted line in fig . [ m6pi_fit ] ) . the other contribution is described by a third - order polynomial . the signal is described by a breit - wigner function modified with the effects of the detection efficiency , the detector resolution , and the phase space factor . the fit result is shown in fig . [ m6pi_fit ] . the mass and width of the @xmath0 are @xmath15 mev / c@xmath7 and @xmath16 mev , respectively ; the product branching fraction of the @xmath0 is @xmath17 . in these results , the first errors are statistical and the second errors are systematic . . the dots with error bars are data ; the solid line is the fit result . the dashed line represents all the backgrounds , including the background events from @xmath18 ( dash - dotted line , fixed in the fit ) and a third - order polynomial representing other backgrounds . [ m6pi_fit],scaledwidth=60.0% ] figure [ comp_mw ] shows the comparisons of the @xmath0 with other observations at besiii@xcite . the comparisons indicate that at present one can not distinguish whether the @xmath0 is a new state or the signal of a @xmath1 decay mode of an existing state . , title="fig:",scaledwidth=67.0% ] ( -137 , 138 ) ( -137 , 123.5 ) ( -137 , 109 ) ( -137 , 94.5 ) ( -137 , 80 ) with the same data sample , the decay of @xmath6 was searched for@xcite . the mass spectrum of the @xmath1 is shown in fig . [ m6pi ] , where no events are observed in the @xmath19 mass region . with the feldman - cousins frequentist approach@xcite , the upper limit of the branching fraction is set to be @xmath20 at the 90% confidence level , where the systematic uncertainty is taken into account . with a sample of @xmath13 @xmath4 events collected at besiii , the decay of @xmath21 was analyzed@xcite . the @xmath0 was observed in the @xmath1 invariant mass spectrum . the mass , width and product branching fraction of the @xmath0 are @xmath15 mev / c@xmath7 , @xmath16 mev and @xmath17 , respectively . the decay @xmath22 was searched for . no events were observed in the @xmath19 mass region and the upper limit of the branching fraction was set to be @xmath20 at the 90% confidence level . 00 j. z. bai _ et al . _ [ bes collaboration ] , phys . lett . * 91 * , 022001 ( 2003 ) . j. p. alexander _ et al . _ [ cleo collaboration ] , phys . d * 82 * , 092002 ( 2010 ) . m. ablikim _ et al . _ [ besiii collaboration ] , phys . lett . * 108 * , 112003 ( 2012 ) . m. ablikim _ et al . _ [ bes collaboration ] , phys . lett . * 95 * , 262001 ( 2005 ) . m. ablikim _ et al . _ [ besiii collaboration ] , phys . lett . * 106 * , 072002 ( 2011 ) . m. ablikim _ et al . _ [ bes collaboration ] , phys . lett . * 96 * , 162002 ( 2006 ) . m. ablikim _ et al . _ [ besiii collaboration ] , phys . d * 87 * , 032008 ( 2013 ) . m. ablikim _ et al . _ [ besiii collaboration ] , phys . lett . * 107 * , 182001 ( 2011 ) . m. ablikim _ et al . _ [ besiii collaboration ] , chin . c * 36 * , 915 ( 2012 ) . m. ablikim _ et al . _ [ besiii collaboration ] , nucl . instrum . a * 614 * , 345 ( 2010 ) . j. z. bai _ et al . _ [ bes collaboration ] , nucl . instrum . a * 458 * , 627 ( 2001 ) . m. ablikim _ et al . _ [ besiii collaboration ] , arxiv:1305.5333 [ hep - ex ] . g. j. feldman and r. d. cousins , phys . d * 57 * , 3873 ( 1998 ) . | observation of the @xmath0 in the @xmath1 invariant mass in @xmath2 at besiii is reviewed . with a sample of @xmath3 @xmath4 events collected with the besiii detector at bepcii , the @xmath0
is observed with a statistical significance of @xmath5 .
the mass , width and product branching fraction of the @xmath0 are determined .
the decay @xmath6 is searched for , and the upper limit of the branching fraction is set at the 90% confidence level .
observation of the @xmath0 at besiii |
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weak absorbers , those with @xmath8 , constitute @xmath9% of the total absorber population @xcite at @xmath10 . they account for a fair fraction of the @xmath11 forest @xcite . unlike strong absorbers ( which almost always are associated with a @xmath1 galaxy @xcite ) , weak absorbers can usually not be associated with a @xmath1 galaxy within impact parameter @xmath12 kpc of the qso ( @xcite ; c. steidel , private communication ) . this lack of an association with bright galaxies suggests that weak absorbers may be a physically different population than strong absorbers . studying the physical conditions of weak absorbers ( eg . , metallicity , ionization conditions , total column density , and size ) is useful for two reasons : 1 ) physical conditions provide clues as to the nature of these absorbers , whose physical origin is not known ; 2 ) weak clouds provide an opportunity to study metal enriched environments over a range of redshifts . they trace metal production either in intergalactic space or in dwarf or low surface brightness galaxies . a study of weak absorbers at @xmath13 requires spectra in both the optical and in the uv in order that the several key low and high ionization transitions ( especially , , ) , as well as the lyman series , are covered . using the keck / high resolution spectrograph ( hires ) @xcite at high resolution ( @xmath14 ) and the faint object spectrograph ( fos)/_hubble space telescope _ ( hst ) at low resolution ( @xmath15 ) , @xcite applied photoionization models to @xmath16 single cloud weak absorbers . they argued that many multiple cloud weak absorbers ( which comprise @xmath17% of the weak absorbers ) are likely to be part of the same population as the strong absorbers , but that single cloud weak systems are likely to be a different population . at least half of the weak , single cloud systems of @xcite require a second phase of gas in order to reproduce the absorption and/or to fit the profile without exceeding the column density derived from the lyman limit break . while the clouds typically have doppler parameters of @xmath18@xmath19 , the second phase must have a larger effective doppler parameter ( @xmath20@xmath21 ) . with only the low resolution uv data available to @xcite , it was not clear if this implies a single broad component or multiple , blended clouds . also , in the systems for which the second phase is not required ( such as when only a limit for can be derived ) , it was not clear if the phase is just less extreme in its properties or if it is absent . by comparing the column densities to the profiles , @xcite inferred that weak , single cloud absorbers have metallicities of at least one tenth solar in the phase of gas in which the absorption arises . the sizes and densities of most of the weak , single cloud systems were unconstrained ; however , three of them had @xmath22 , which implies low ionization conditions , relatively high densities , @xmath23 , and small sizes , @xmath24 pc . those systems with limits on may be part of a `` continuum '' of single cloud weak absorbers , with some having just below the detection threshold , implying a continuous distribution of ionization conditions . alternatively , some of the systems without detected could be part of a different population , with ionization conditions , metallicities , and/or environments distinctly different from systems with detected . in may and june 2000 high resolution uv spectra ( @xmath25 ) of the @xmath26 quasar , pg @xmath0 , became public in the hst archive . we have high resolution optical ( hires / keck ) spectra of this same quasar . for the first time , weak absorbers can be studied through simultaneous high resolution coverage of numerous transitions . this allows more direct inferences of the physical properties of the gas , such as metallicity and phase structure . in particular , it should enable a determination of the nature of the second , broad phase required for some absorbers by the large relative strength of absorption in lower resolution data . is this higher ionization , broad phase centered on the cloud or is it offset ? is it produced by multiple clouds or by a single , smooth structure ? are systems for which was not detected at low resolution fundamentally different , or do they merely have a weaker second phase ? the kinematics and physical properties of the second phase , and its relationship to the cloud phase , are important diagnostics of the type of structure responsible for the weak absorption . in anticipation of the release of the high resolution stis / hst spectra , we previously pursued an in depth study of four absorbers along the pg @xmath0 line of sight @xcite using the available low resolution fos / hst spectra and the hires / keck spectra . since only the brightest quasars can , in the near future , feasibly be studied at high resolution in the uv , we aimed to compare inferences drawn on the basis of low resolution spectra to those that would be obtained once the higher resolution spectra were released . confirmation of our conclusions would lend credibility to larger statistical studies that rely on a combination of high and low resolution spectra @xcite . the present paper focuses on three single cloud weak absorbers along the pg @xmath0 line of sight . two of these absorbers , at @xmath27 and @xmath28 , were detected in the hires / keck spectra at the sensitivity of our original weak survey ( @xmath29 for @xmath30 ) @xcite . these two systems are dramatically different from each other in that the @xmath27 system has a stringent limit on ( @xmath31 at @xmath32 ) while the @xmath28 absorber has detected at @xmath33 . in the latter system , the arises not in the cloud phase , but in a second phase of gas . this provides an excellent opportunity both to study a system with very weak and to address the nature of the second phase . the metallicity could not be derived for the @xmath27 system because the fos / hst spectrum did not cover any of the lyman transitions ; however , the new stis / hst spectra can address metallicity for this system . models of the @xmath28 absorber did , however , imply a super solar metallicity and/or a depleted or @xmath34enhanced abundance pattern . this conclusion will be re assessed in this paper . also , as we will describe in [ sec : data ] , we have now identified a third single cloud weak system at @xmath35 that was just below the detection threshold of our previous study . that system was not modeled in @xcite , but will be considered in the present paper . we begin in [ sec : data ] by presenting , for the three single cloud weak systems toward pg @xmath0 , high resolution profiles of all relevant transitions covered by stis / hst or hires / keck spectra . in [ sec : techniques ] , we discuss our strategy to infer the physical conditions of these systems by applying cloudy photoionization models . we also consider the possibility that collisional ionization dominates . the results inferred for the phase structure , ionization parameters / densities , metallicities , kinematics , and abundance patterns of the three systems are presented in [ sec : results ] . in [ sec : caveats ] , the effect of relaxing assumptions of abundance pattern and the shape of the ionizing spectrum are considered . the three weak systems are compared in [ sec : discussion ] . in that discussion , we particularly focus on the nature of the second phase and compare the results to those obtained on the basis of just the lower resolution fos / hst spectrum . finally , we summarize our conclusions in [ sec : conclude ] . we briefly describe the observations with hires / keck , stis / hst , and fos / hst of pg @xmath0 , and then present the three single cloud weak absorbers along this line of sight at @xmath36 , @xmath28 , and @xmath35 . a wfpc2/hst image of the quasar field exists ( program 6740 , s. oliver , p. i. ) but the quasar is quite bright so it is not possible to perform adequate psf subtraction in order to detect galaxies close to the line of sight . also , without redshifts of candidate galaxies in the field , we could not separate out the identities of the three weak and two strong absorbers . the hires / keck observations covered @xmath37 to @xmath38 at @xmath14 ( fwhm @xmath39 ) ( @xmath40 pixels per resolution element ) . , , and transitions are covered for the three systems , but in all cases only is detected ( at @xmath41 ) . equivalent widths and limits are listed in table [ tab : ewtab ] . the spectra were obtained on 1995 july 4 and 5 , and the combined spectrum has @xmath42 over most of the wavelength range , but gradually falling toward the lowest wavelengths . the most stringent limits on come from the reddest of the transitions , at @xmath43 . the reduction of the spectrum , continuum fits , line identification , and procedure for voigt profile fitting were described in @xcite . the stis / hst observations provided useful coverage from @xmath44 to @xmath45 at @xmath25 ( fwhm @xmath24 ) ( @xmath18 pixels per resolution element ) . two sets of echelle spectra were obtained with two different tilts of the e230 m grating , using the @xmath46 @xmath47 @xmath46 slit . the first , with central wavelength @xmath48 , was obtained by burles et al . in 1999 may and june ( proposal i d 7292 ) . the total exposure time was @xmath49 s. the second , with central wavelength @xmath50 , was obtained by jannuzi et al . in 1999 june ( proposal i d 8312 ) , with a total exposure time @xmath51 s. the two spectra overlap in the region @xmath52@xmath53 . we co added the spectra , weighted by the exposure times , and also combined multiple order coverage in the same spectrum . however , we also considered the differences between the two realizations as an indication of systematic errors ( due to continuum fitting , correlated noise , and unknown factors ) when comparing model profiles to the data . reductions were done with the standard stis pipeline and continuum fits using the sfit task in iraf . based upon our original analysis of the hires / keck spectrum there were two single cloud , weak systems along the pg @xmath0 line of sight , at @xmath36 and @xmath28 . after analyzing the stis / hst spectra , another slightly weaker system was found in the hires / keck spectra , just below the threshold of our previous survey @xcite . first , the stis spectra were searched for doublets . only one doublet was found with comparable equivalent width to those associated with the two known absorption systems . the new system , at @xmath35 , also had detected , a doublet , , and in the stis spectrum . based upon the equivalent width of we expected that should be detected in the hires / keck spectrum . we then searched that location in the hires spectrum and found the doublet . the @xmath54 transition was detected at the @xmath55 level , and the @xmath56 transition at the @xmath57 level . ( our previous survey had a @xmath32 detection threshold for the @xmath56 transition . ) figure [ fig : data81 ] presents detected transitions and limits of interest in constraining the conditions in the @xmath36 system . figure [ fig : data90 ] displays the same for the @xmath28 system , and figure [ fig : data65 ] for the @xmath35 system . table [ tab : ewtab ] lists the equivalent widths for detected transitions and selected equivalent width limits ( @xmath32 ) for the three systems . we fit each transition with the minimum number of voigt profile components consistent with the errors @xcite . in all three systems , the and the other low and intermediate ionization tranistions could be fit with one component . however , the profiles required two components for the @xmath28 system , and three for the @xmath35 system . for selected transitions , column densities and doppler parameters of the voigt profile fits , performed separately for each transition , are given in table [ tab : vptab ] . for the @xmath35 system , numbers are estimated only for and because fits are ambiguous due to blending . there is no flux detected from pg @xmath0 shortward of a lyman limit break at @xmath58 , which is due to a strong absorber at @xmath59 @xcite . also , a partial lyman limit break at @xmath60 reduces the flux to about @xmath21% of the original continuum level due to a multiple cloud weak absorber at @xmath61 @xcite . although for the @xmath36 system is covered in a @xmath62 stis g230 m spectrum of pg @xmath0 , this reduced flux and crowding with lyman series lines from the @xmath59 system prevent us from deriving a useful constraint . we therefore have not used the g230 m spectrum as a constraint in the analysis . several descriptions of the basic technique used for modeling were given in previous papers @xcite . for those efforts , only , , and were covered at high resolution , and all other transitions were observed with fos / hst at only @xmath15 . even with the availability of numerous transitions at high resolution , the modeling technique used in the present paper is very similar to those previous efforts . we begin with the phase of gas that produces the dominant absorption , `` the low ionization phase '' . then we add in other phases of gas as needed to reproduce the observed absorption profiles for all transitions . for each single cloud absorber we begin with the column density , @xmath63 , and the doppler parameter , @xmath64 derived from a voigt profile fit to the doublet . cloudy photoionization models ( version 94.00 ; @xcite ) were applied to determine column densities of the various transitions that would result from the same phase of gas that produces this , assuming a slab geometry . the parameters for these models are the metallicity @xmath65 ( expressed in units of the solar value ) , the abundance pattern ( initially assumed to be solar ) , and the ionization parameter , @xmath66 . the ionization parameter is defined as the ratio of ionizing photons to the number density of hydrogen in the absorbing gas , @xmath67 . we assume that the ionizing spectrum is of the form specified by @xcite for @xmath68 . the normalization of the haardt and madau background at @xmath68 is fixed so that @xmath69 . the assumption that stellar sources do not make a substantial contribution is likely to be valid since weak absorbers typically do not have a nearby high luminosity galaxy , such as a starburst . the effects of alternative spectral shapes are discussed in [ sec : specshape ] . the doppler parameters of other elements are derived from @xmath64 , using the temperature output from cloudy to derive the thermal and turbulent contributions for each element . for each choice of parameters , the cloudy output column densities and doppler parameters are used to synthesize noiseless spectra , convolving with the instrumental profile characteristic of stis / e230 m . the synthetic model spectra are compared with the observed profiles to identify permitted regions of the parameter space . the model column densities and doppler parameters for permitted models are within @xmath70 of the values measured from the data . from the three systems studied in this paper , @xmath71 is covered , but not detected . for @xmath72 , the ratio @xmath73 is strongly dependent on the ionization parameter in the optically thin regime ( see @xcite ) and the limit can be used to place a lower limit on @xmath66 of the low ionization phase . to obtain an upper limit on the ionization parameter of this phase , and provide constraints . for optically thin gas , the constraints on @xmath66 do not depend significantly on the metallicity . the metallicity is constrained by fitting the and any higher lyman series lines that are covered for the system . low metallicities will overproduce the in the wings . high metallicities will underproduce the , but can not be excluded since the additional absorption can arise in a different phase . we assumed a solar abundance pattern , but note that there is a simple tradeoff between metallicity and the abundance of the metal line transition that is compared to the hydrogen , discussed further in [ sec : abun ] . in general , once the basic constraints on @xmath66 are derived , we consider whether a single cloud model can fully reproduce profiles of all of the observed transitions . _ in all three of the systems modeled in this paper , we will conclude that the absorption can not be fully produced in the same phase with the absorption . _ for the range of permitted values of @xmath66 for the low ionization phase , we constrain the properties of the high ionization phase that is required to fit the profile . first , we consider whether a single , relatively broad component is sufficient . if the profile is asymmetric ( as in the @xmath28 system ) or shows velocity structure ( as in the @xmath35 system ) , then it is clear that more than one component is needed . a voigt profile fit to the serves as a starting point for deriving the number of clouds needed in the high ionization phase , and their @xmath74 and @xmath75 . results from these fits are listed in table [ tab : vptab ] . the @xmath75 of these clouds is larger than for the clouds . cloudy models of this additional phase are constrained to match the observed clouds and @xmath66 is adjusted to determine what range of values are consistent with other transitions . an upper limit is set so that and ( if covered ) are not overproduced . a lower limit is set so that the broader components do not produce observable and . in some cases the intermediate ionization transitions , such as , , and especially , could not be fully produced in the cloud phase . in these cases , an intermediate value of @xmath66 is sought for the clouds in order to also account for the remainder of these transitions , without overproducing the lower ionization transitions . a lower limit on the metallicity of a cloud can be derived in order that absorption is not overproduced . more exactly , it is the combination of the and clouds that are constrained not to overproduce absorption , so there is a trade off between their metallicity constraints . if all transitions of a certain element ( such as , , and ) are underproduced relative to other elements , then simple abundance pattern variations ( @xmath34enhancement and depletion ) are considered . for the high ionization phase , we consider collisional ionization models as alternatives to cloudy photoionization solutions . in this case , an alternative source of heating ( eg . , shocks ) must be responsible for heating the gas to the assumed , higher @xmath76 . the measured doppler parameter of the , @xmath75 , is used to place an upper limit on the temperature , @xmath77 . for an assumed @xmath76 and @xmath65 , and with the measured @xmath74 , the collisional equilibrium tables of @xcite were used to determine the column densities of all other covered transitions . the doppler parameters of these other transitions , @xmath78 were calculated from @xmath79 , where the turbulent component of the doppler parameter is given by @xmath80 . in order to constrain @xmath76 , synthetic spectra were generated and superimposed upon the data to facilitate comparison . also , the model column densities of the various transitions were compared to the measured values . the profiles provided a constraint on the metallicity of any collisionally ionized phase . `` wings '' on these profiles can be produced by such a phase , which would be characterized by a relatively large @xmath81 parameter . the contribution of a collisionally ionized phase to is minimized at low @xmath76 and high metallicity . a lower limit on @xmath76 is therefore set so as not to overproduce at the highest reasonable metallicity ( usually taken to be solar ) . as for photoionization models , for the high ionization phase , the lower limit on the metallicity depends on the metallicity of the low ionization phases , which determines its contribution to . first , we describe the model results for the two weak , single cloud absorbers detected in the original hires / keck survey @xcite . then the results for the newly discovered @xmath35 system are presented . a range of parameters for satisfactory models are summarized in table [ tab : tabmod ] . model profiles for an example of an acceptable model are superimposed on the data for each of the three systems in figures [ fig : data81 ] , [ fig : data90 ] , and [ fig : data65 ] . parameters for these sample models are listed in table [ tab : tab4 ] . results in this section use the simplest set of assumptions that produce models consistent with the data , i.e. haardt and madau spectrum and a solar abundance pattern . the effects of alternate spectra and abundance patterns are discussed in [ sec : caveats ] . figure [ fig : data81 ] shows detections of , , , , , , and . there are useful limits for and . an obvious , but important , first result is that is now clearly detected in this system , despite the strong limit from the earlier low resolution data . we begin with the voigt profile fit to the doublet , and adjust the ionization parameter to fit as many of the other transitions as possible . we find that the cloud is constrained to have @xmath82 . there is no strict lower limit , because the @xmath83 vs. @xmath66 curve is flat for @xmath84 . very small values of @xmath66 ( as low as @xmath85 ) are permitted , but cloud sizes would be extremely small . for @xmath86 , is underproduced . to find an upper limit on ionization parameter , we first determined that , for @xmath87 , the model would produce minimum fluxes at the positions of and that are consistent with the observed profiles . however , the model @xmath88 and @xmath75 are small compared to the observed values , i.e. the model profiles are narrow compared to the observed profiles . we therefore conclude that the and are produced in a separate higher ionization phase , and that @xmath89 for the cloud phase . the contribution of the cloud phase to the absorption profiles ( negligible for all but the singly ionized transitions ) is denoted by a dotted line in figure [ fig : data81 ] . the metallicity of the cloud with detected is constrained to be @xmath90 dex greater than solar ( for a solar abundance pattern ) . for lower metallicities , absorption in the wings of the profile will exceed that observed . also , unless the metallicity is even higher for the cloud , the additional high ionization phase ( required to fit ) is constrained not to give rise to significant absorption . for @xmath91 and @xmath92 , the cloud size would @xmath93 pc . for @xmath94 and @xmath95 , the cloud size would be only @xmath96 au , and the observed profile would be somewhat underproduced by the model . primarily because of the breadth of the profiles , we concluded that a second phase is required to fit the , even in this system for which the absorption is relatively weak . a voigt profile fit to the profile yields an adequate fit for a single cloud with @xmath97 that is centered on the cloud . details of the fit are listed in table [ tab : vptab ] . we consider whether the can be produced by photoionization , and/or whether it can be produced by collisional ionization in gas that has been heated above the equilibrium value . for photoionization models of the high ionization phase , we optimized on the voigt profile fit values , and the ionization parameter was constrained by data for other intermediate and high ionization transitions . to produce the observed absorption in this phase , @xmath98 is the optimal range . the cloud size would be @xmath99 pc , considerably larger than the cloud . if @xmath100 , the system would have too much and absorption relative to . if @xmath101 , @xmath102 would be overproduced by the model . the metallicity of the high ionization phase is constrained , by the profile , to be solar or higher . for larger values of the ionization parameter , within the constrained range of @xmath98 , the lower limit on metallicity would be raised to a supersolar value . considering collisional ionization , for @xmath103 , pure thermal broadening gives an upper limit on the temperature of @xmath104 . below this limiting temperature , would be severely overproduced @xcite if we optimize on . however , raising the temperature to @xmath105 ( which is just consistent with the voigt profile fit @xmath75 within @xmath70 errors ) reproduces the observed @xmath106 . the metallicity of this collisionally ionized phase would have to be solar or greater in order that would not be overproduced . though it requires fine tuning of the temperature , a collisionally ionized phase with solar metallicity and with @xmath105 provides an adequate fit to the data . we conclude that the @xmath36 system has two phases , both with a metallicity solar or higher . the low ionization phase , with @xmath107 , has @xmath84 , while the high ionization phase , with @xmath108 , could be photoionized with @xmath109 or collisionally ionized with @xmath105 . the broader high ionization phase is centered on the cloud and is consistent with a single cloud producing the absorption . constraints are summarized in table [ tab : tabmod ] . an acceptable model , including two photoionized phases , is superimposed on the data in figure [ fig : data81 ] . column densities of selected transitions , produced by this model , are given in table [ tab : tab4 ] . the @xmath28 system is detected in , , , , , , , 1403 , , and . 1239 is not confirmed by a detection of 1242 , so it is viewed as a tentative detection . is also a likely detection , although 1038 is in a confused region of the spectrum that required an uncertain continuum fit . limits are available for and 2600 in the hires / keck spectrum . all of these transitions are shown in figure [ fig : data90 ] . in this system , the is quite strong , and shows an asymmetry in both members of the doublet . we first optimize on the @xmath63 and @xmath64 given by a voigt profile fit to the mg0.1emii @xmath110 profiles ( see table [ tab : vptab ] ) . comparing to the other transitions , the ionization parameter for this cloud is constrained to be @xmath111 . a higher value overproduces @xmath112 absorption at @xmath113 , which is well fit for @xmath66 at the upper end of this range and underproduced for @xmath114 . ( @xmath115 is blended with a stronger transition from another system and can not be used as a constraint . ) a value of @xmath116 would result in a model that exceeds the limit on @xmath43 . if @xmath117 , then there would have to be a significant contribution to the absorption from another phase . even for the maximum consistent @xmath118 , the absorption is not fully produced in this phase , as shown by the dotted curve in figure [ fig : data90 ] . the cloud giving rise to the , with @xmath119 , would overproduce in its blue wing unless it has solar or super solar metallicity . the metallicity constraint could be slightly relaxed if magnesium is enhanced due to an @xmath34enhanced abundance pattern . on the other hand , in the red wing , the observed is not exceeded by any model with @xmath120 . this difference between the metallicity constraints from the red and blue wings of the profile is a consequence of an asymmetry in the distribution of relative to the . the asymmetry in , along with the need to fully produce the absorption , requires an additional phase of gas . in addition to the cloud , two more clouds are needed in order to fit the profile . from a simultaneous voigt profile fit of the @xmath121 and @xmath122 transitions , the first , centered on the velocity of the absorption , would have @xmath123 and @xmath124 . a small amount of the absorption is produced in the same phase with the for @xmath118 , so the column density of the cloud would be reduced slightly in that case . the second cloud , offset by @xmath16 to the red , is fit with @xmath125 and @xmath126 . first , we consider the @xmath127 cloud centered on the profile . it is too narrow ( @xmath128 ) for the to be produced by collisional ionization . from cloudy photoionization models , to simultaneously fit the and the absorption , we derive the constraint , @xmath129 , for this high ionization phase . if so , little arises in this phase , which implies that absorption should be produced in the low ionization phase with the . revisiting the constraints for the cloud , we now find the constraint @xmath130 for the low ionization cloud , at the upper end of the previous constrained range discussed above . the metallicity of the high ionization gas that produces the absorption must be high enough not to overproduce in its blue wing . in conjunction with a solar metallicity cloud phase , this yields a solar metallicity for this @xmath129 phase as well . next , we consider the @xmath131 cloud , offset by @xmath16 from the cloud , that fills in the red wing of the profile . the @xmath132 cloud , at @xmath133 , can be fit with a simple , single cloud photoionization model with @xmath134 . this ionization parameter produces a consistent fit of the , , and , without overproducing the . the and data are very slightly underproduced by the model at this velocity , suggesting that a small ( @xmath135 dex ) abundance pattern enhancement of silicon might be needed . a simple model with the minimum number of phases would call for the asymmetry in the profile to be produced in this cloud along with the red wing of the . the metallicity is thus constrained to be at the solar value in order to fit the red wing of the line . our philosophy of fitting with the minimum number of phases argues against collisional ionization as the mechanism for producing the observed absorption at @xmath136 . to be consistent with this philosophy , both and should arise in the same phase . the limit on @xmath137 for this component with @xmath132 is consistent with production of absorption by collisional ionization . however , for the maximum permitted @xmath76 there would not be enough absorption in this model component unless the metallicity was well over supersolar . with the relatively low @xmath76 needed to produce the much absorption , absorption would be overproduced . we conclude that the @xmath28 system must have solar metallicity or higher in its low ionization phase . the @xmath138 cloud that produces absorption must be of a relatively high ionization state , with @xmath130 . the range of derived cloud sizes for this constrained range is @xmath21@xmath139 pc . two higher ionization , broader clouds are also required : one , with @xmath140 , is centered on the cloud , while the other @xmath141 cloud is offset by @xmath16 . both are consistent with photoionization with @xmath142 , and have sizes of @xmath143@xmath144 kpc . ranges of acceptable model parameters are presented in table [ tab : tabmod ] and model column densities contributed by the three clouds for an adequate model are listed in table [ tab : tab4 ] . the predictions for this same model are superimposed on the data in figure [ fig : data90 ] . the profile for this newly discovered system has a complex structure and the is extremely weak and narrow . the profile requires at least a three component fit . the is not symmetric about the , but it is approximately symmetric about the . these profiles as well as those of the other detected transitions , , , , are displayed in figure [ fig : data65 ] , along with the region of the spectrum that provides a limit on 2600 . is in a noisy region of the spectrum . if the detected feature is really , it is present only in the redward component , at @xmath145 . we begin by considering the cloud . since the 2600 spectrum is noisy , the low ionization parameter constraint on @xmath66 is not strong ; for @xmath146 , @xmath73 is relatively constant . however , assuming solar abundance pattern , @xmath147 provides a marginally better fit to the and . if @xmath148 , then the depth of the and absorption at @xmath149 can be reproduced in the cloud , but the model profile is too narrow and does not provide a good fit to the data . a separate higher ionization phase is needed to fit these higher ionization transitions . considering all transitions , and assuming solar abundance pattern , @xmath150 is favored for the cloud . the metallicity of the cloud would be @xmath151 if all of the in the red wing were to arise in this cloud . however , it could also be significantly higher if there were contribution from the clouds . for @xmath151 and @xmath152 , the cloud size ranges from @xmath19 pc to @xmath153 kpc . sizes scale with @xmath154 so that @xmath155 clouds would be a factor of @xmath156 smaller than @xmath151 clouds . consider the case of @xmath91 for the cloud . three more clouds are required to fit . the column densities and doppler parameters for these three clouds were obtained with a voigt profile fit . this fit is not unique because clouds can not be well separated due to blending , so there are no errorbars listed in table [ tab : vptab ] . the cloud centered on the was fit with @xmath157 , @xmath158 . the other two clouds , at @xmath159 and @xmath160 , were fit with @xmath161 , @xmath162 , and @xmath163 , @xmath164 , respectively . for photoionization models , the ionization parameters for these three clouds are constrained by the requirement that they produce the observed without producing significant lower ionization transitions . the @xmath165 cloud , centered on the cloud , is tightly constrained to have @xmath166 , by the profiles . the range already takes into account the fact that the cloud would make a small contribution to if its @xmath167 . the lower limit on @xmath66 of the cloud arises in order that the is not overproduced in its wings by this relatively broad component . the upper limit applies in order to produce sufficient absorption . the @xmath162 cloud at @xmath159 , has @xmath168 , in order to produce the optimal fit to , , and . for @xmath169 , is overproduced , and for @xmath170 , , , and are overproduced . for the @xmath132 cloud at @xmath160 , @xmath171 provides an optimal fit to the , and is consistent with and with the limit on . a value of @xmath172 is only marginally consistent , with the model slightly underproducing the . this inferred properties of this cloud are independent of the cloud since no is detected at this velocity . as with the cloud , the metallicities of the three additional clouds can not be extremely low . only @xmath173 or higher is consistent with the data . if the red side of the is to be produced by the @xmath160 cloud , @xmath155 would apply for this cloud . similarly , @xmath155 for the @xmath113 cloud would match the blue wing of . in the case of the blue wing , the could in principle be produced in a @xmath174 cloud and the metallicity of the cloud could be higher . constraints on the metallicities of these clouds , relative to the cloud that produced the narrower profile , are limited by the lack of coverage of higher order lyman series lines . however , the data are consistent with @xmath173 in the cloud and in the three clouds . for this metallicity , the sizes of the three clouds would be @xmath175@xmath176 kpc . we also consider whether collisionally ionized gas can be consistent with the observed profiles of , , and . for the @xmath113 cloud , @xmath177 does not overproduce , and @xmath178 is consistent with producing the blue side of the profile for @xmath173 . if the metallicity is higher , a higher ionization parameter is needed to fit the line , but in this case the model profile shape is not consistent with the data . the @xmath159 cloud is too narrow to be consistent with production of absorption through collisional ionization assuming the particular voigt profile fit that we have adopted . however , because this fit is not unique , there could possibly be a broader cloud component that could be reconciled with collisional ionization . finally , the @xmath160 cloud could be collisionally ionized with @xmath179 and @xmath173 , in which case it would match the red side of the profile . in summary , the @xmath35 system profile is quite weak and narrow . however , the profile can be fit with several components spread in velocity over @xmath180 . the three components are similar to each other , having @xmath181 and @xmath66 ranging from @xmath182 to @xmath183 if they are photoionized . collisional ionization with temperatures of @xmath184 could be consistent with the data , but requires fine tuning of the temperature and seems less likely . the bluest component has the detected at the same velocity , but in a narrower component ( @xmath185 ) produced in gas with a somewhat lower ionization parameter ( @xmath186 ) . the clouds have a metallicity of at least @xmath173 and must be at least a couple of kiloparsecs in size . abundance ratios measured in galactic stars show a clear range of @xmath187 \leq + 0.5 $ ] @xcite . in our presentation of model results ( [ sec : results ] ) solar abundance pattern was assumed unless the data require an alternative pattern . however , it should be noted that the inferred metallicity depends directly on the assumed abundance pattern . if , instead of solar , the abundance pattern is @xmath34group enhanced with @xmath188 } = + 0.5 $ ] , the inferred metallicity of a phase constrained by @xmath63 ( an @xmath34group element ) would proportionally drop by @xmath189 dex . constraints on the ionization parameter , @xmath66 , are also affected by changes in the assumed abundance pattern . for example , if the the abundance pattern is @xmath34group enhanced , the ratio @xmath190 would constrain @xmath66 to be larger than if the abundance pattern is solar . however , for @xmath188 } \le + 0.5 $ ] , the change in the constraint on @xmath66 ( and therefore on @xmath191 ) is less than @xmath192 dex . we have used the relative absorption strengths in several other transitions to determine @xmath66 , but @xmath192 dex is typical of the level of uncertainty due to variations in abundance pattern . based on results for other similar absorption line systems , there is likely to be no bright galaxy associated with these single cloud weak absorbers @xcite . therefore , we used the @xcite extragalactic background spectrum for @xmath68 in our detailed models presented above ( [ sec : results ] ) . however , because it is not strictly excluded , we do consider here the effect of changing the spectral shape . two alternative galaxy spectra , @xmath193 gyr and @xmath194 gyr instantaneous starburst models , were superimposed on the haardt and madau extragalactic background spectrum . both models , with solar metallicity and a salpeter imf were taken from @xcite . the normalizations of the starburst spectra are defined relative to the haardt and madau spectrum at @xmath144 rydberg , and in all cases the extragalactic and galactic contributions were superimposed . the largest reasonable value for the photon flux from even the most extreme starburst galaxy is @xmath195 , and of order @xmath144% of the photons escape @xcite . using these numbers , to have a flux ten times that of haardt and madau at @xmath68 , the absorber would have to be within @xmath19 kpc of the starburst . within this distance , we would expect stronger absorption than observed in these weak absorbers . therefore , we consider it most likely that the haardt and madau spectral shape is the appropriate assumption . nonetheless , in this section , we briefly examine the consequences of the alternatives . we consider the effect of spectral shape on the metallicity of the low ionization phase , on whether the absorption can arise from the same phase as the absorption , and on the ionization conditions of the high ionization phase . the largest impact on the metallicity would be from an ionizing spectrum with a large feature at the lyman edge , such as the bruzual and charlot @xmath194 gyr instantaneous burst model . in this case , the cloud would be more neutral so that not as much hydrogen would be required to fit the profile . in such a case , the metallicity would be even higher than we inferred assuming a pure haardt and madau spectrum . in general , we found that the starburst spectrum normalization needed to be at least ten times that of the haardt and madau spectrum in order to detect a difference in the models . even with a normalization of twenty five the difference in the inferred metallicity is less than @xmath196 dex . most importantly , even if the spectrum of ionizing radiation has a substantial lyman edge , the metallicity of the weak clouds would be even higher than the solar value an even more surprising result . a @xmath194 gyr instantaneous burst model , if it dominates over the haardt and madau spectrum by a factor of twenty five , can make significantly more relative to . however , only the column density can be made consistent with the observed profiles in the three systems studied here . the doppler parameters of these lines are still too large for them to be produced in the same phase with the . a spectrum with a sharp edge will have the largest effect on the inferred @xmath66 of a photoionized high ionization phase . a @xmath193 gyr starburst model is extreme in this respect , yet it takes a normalization of ten times haardt and madau to see even a small change . with a normalization of twenty five , the is significantly overproduced relative to for the same @xmath66 . however , the qualitative result of a relatively low density phase producing the bulk of the is unchanged . for example , for the @xmath36 system , @xmath66 must be increased from @xmath197 to @xmath198 in order that is not overproduced . we conclude , that at @xmath68 an extreme starburst spectrum would have to dominate in order to affect the conclusions of our models . even if such conditions prevailed , the conclusions would be qualitatively unchanged , only changing constraints by @xmath199 dex in parameter spaces ranging over a few orders of magnitude . this study of the multiple phases of gas and the physical conditions of the three weak systems along the pg @xmath0 line of sight indicates a heterogeneous population of objects selected by a weak doublet . figure [ fig : allsys ] is a comparison between the three systems showing the range of equivalent widths and kinematic structures , ranging from a profile consistent with a single cloud for the @xmath36 system , to the stronger asymmetric profile for the @xmath28 system , to the multiple component profile for the @xmath35 system . the profile strength does not systematically increase in proportion with the absorption . it could , however , be increasing with the kinematic spread of the . table [ tab : tabmod ] gives a summary of the range of `` acceptable models '' for each of the three systems , while table [ tab : tab4 ] gives more detailed information about a sample model that is within the acceptable range . that typical model was also superimposed on the data in figures [ fig : data81][fig : data65 ] . the @xmath36 system and the @xmath28 system provide contrasting examples of a weak absorber that clearly requires a separate broad phase to fit a strong profile and one with much weaker , not even detected in the previous low resolution fos spectrum @xcite . for both systems , in the present study we find that a separate broader phase ( @xmath7 and @xmath19 ) , with an ionization parameter @xmath200 , is needed to fit the observed stis profile . this phase is centered at the same velocity as the phase that gives rise to the weak absorption . however , in the @xmath28 system there is an asymmetry to the profile that indicates an additional offset component , @xmath201 redward of the cloud and the first broad phase . the profile can be fit with a two cloud model , though it is also possible that there is a more complex distribution of material , spread over a range of velocities , that gives rise to such an asymmetric profile shape . a general point is suggested by the comparison of the @xmath36 and @xmath28 systems . the strong absorption that is observed in many weak absorbers @xcite could be due to the presence of separate clouds that are of higher ionization and do not give rise to detectable absorption . the same would apply for the stronger absorption apparent in some weak absorbers . the @xmath35 system is a more extreme example in which three separate components are clearly apparent in the profile . a narrow , weak absorber is centered on one of the higher ionization clouds . the separate components at @xmath202 and @xmath203 are quite distinct from the centered on the cloud . like the higher ionization components of the other two systems , these two offset clouds have ionization parameters , @xmath200 . the profile is consistent with being centered around the three component , but not around the cloud . the fact that absorption is strongest in this system suggests that the spread in velocity of absorbing gas along the line of sight is an important factor in determining the equivalent width of absorption . the metallicities of the @xmath36 and @xmath28 systems are constrained to be at least as high as solar . the @xmath35 system is likely to have a metallicity of at least @xmath204th solar . these relatively high metallicities appear to be common for weak absorbers . rigby et al . ( 2001 ) inferred a high metallicity for several systems based upon low resolution fos data , and in no case did they find that a metallicity less than @xmath204th solar was required . for the @xmath36 system and the @xmath28 system , we can compare the results from this study , incorporating high resolution stis data , to our previous models , based upon the lower resolution fos data @xcite . for the @xmath36 system , was not covered in the earlier fos spectrum , so no constraints on metallicity were available . the was not detected in the fos spectrum , but we showed that it might be detected in the higher resolution stis spectrum either from a cloud with a high ionization parameter or from a broader , separate phase . no specific predictions could be made as to which possibility was more likely . now , with the high resolution stis spectrum coverage of we are able to place specific constraints as was outlined in [ sec : res81 ] . for the @xmath28 system the main conclusions of our previous study , based on the fos spectrum @xcite , were that the is present in a separate broader phase , and that the metallicity of the cloud is solar or higher . these conclusions are confirmed by the present study . the detailed properties of the broader phase were more difficult to determine based upon low resolution spectra . we suggested that the broad phase has @xmath205 because the doublet ratio of the was large compared to observations , if @xmath108 was assumed . the high resolution stis spectrum shows that the profile is asymmetric , so that it must be composed of at least two separate `` clouds '' , the broadest having @xmath206 . the `` wings '' of the profile in the low resolution spectrum were apparently due to fos fixed pattern noise rather than to an extremely broad phase , since these features are not apparent in the high resolution stis spectrum . from this very limited comparison we tentatively conclude that it should be possible to draw inferences about the presence of a separate phase based on a combination of high and low resolution spectra , i.e. drawing on the large existing fos database . also , our conclusion of solar metallicity for the @xmath28 system was confirmed by modeling of the high resolution stis profile of . this is important , since modeling of fos data was used to infer that many weak absorbers have close to solar metallicity @xcite . to understand the nature of these systems , we seek to infer the spatial distribution of absorbing material , and the relationship to star forming objects . all three single cloud weak absorbers have two phases that produce a narrower ( @xmath18@xmath176 ) and a broader ( @xmath19@xmath201 ) absorption component centered at the same velocity . the narrower component is of lower ionization than the broader component . for a fixed haardt madau spectrum intensity , this implies a higher density ( @xmath207@xmath194 ) for the narrower component , and also a smaller size , @xmath93@xmath139 pc . the broader component would arise in a higher ionization / lower density phase ( @xmath208 ) with a larger size ( @xmath93@xmath18 kpc ) . two simple scenarios could be consistent with the inferred properties of the narrow and broad components : 1 ) the first would have the lower ionization region embedded within the higher ionization region , and the higher ionization region would present a larger cross section . we would then expect many systems to be observed for lines of sight that pass through only the higher ionization region of such structures . these `` only systems '' might typically have lower column densities than two phase weak systems . they would be produced at large impact parameter in the structure , at which the pathlength would be shorter and the gas densities would be lower . 2 ) in the second scenario the lower ionization components could be produced by parts of a shell structure surrounding a lower density , higher ionization region . the covering factors for the low ionization shell fragments would be limited ( @xmath209 ) by the lack of observation of many two cloud weak systems . however , with such a small covering factor , we would again expect a large incidence of `` only systems '' which in this scenario would tend to have similar phases to those of two phase weak systems . we searched the pg @xmath0 spectra and found no `` only systems '' with comparable in strength even to that of the @xmath36 weak absorber . however , we have identified several weaker candidate doublets at @xmath210 , confirmed by a line detected at the expected position of . many other lines of sight need to be systematically surveyed , but there clearly will be limits on the geometry and covering factors of the two phases of gas . an alternative to the idea of embedded phases is to have separate clouds along the line of sight with different densities and sizes . this is consistent with the presence of an offset cloud in the @xmath28 system , and the spread of three clouds over @xmath180 in the @xmath35 system . these separate clouds could exist as condensations in larger structures with velocity dispersions of tens of , e.g. , dwarf galaxies . however , if the phases were all completely separate from each other then it would be hard to explain the close alignment in velocity of the lower ionization cloud with one of the higher ionization clouds . the metallicities of these absorbing structures present a clue as to their place of origin . we do not expect that they are in the vicinity of luminous galaxies ( @xmath211 galaxies are not found within impact parameters of @xmath12 kpc from the quasar ) . although no useful image is available for the pg @xmath0 field in particular , other single cloud weak absorbers are rarely found near such luminous galaxies ( c. steidel , private communication ; @xcite ) . dwarf galaxies have metallicities significantly lower than solar , which would appear inconsistent with solar metallicities for single cloud weak absorbers . however , it is possible that the weak absorbers are concentrations of higher metallicity within lower metallicity structures . most strong absorbers ( @xmath212 ) also require a phase in addition to the clouds in order to produce the observed absorption . these absorbers are associated with @xmath213 galaxies @xcite , and the high ionization phase is inferred to have an `` effective doppler parameter '' or velocity spread of @xmath145 @xcite . at high resolution some of the profiles will separate into multiple components , while others may be due to a more uniform distribution of gas @xcite . the phase of some strong absorbers is reminiscent of what would be expected for a corona such as that observed in and absorption around the milky way disk . the single cloud weak absorbers , although they do have a second phase , do not appear to be related to such a corona . the combination of stis / hst and hires / keck high resolution spectra , covering multiple chemical transitions , provided the first opportunity to collect direct information on the metallicities and phase structure of weak absorbers . there are three weak absorbers , at @xmath36 , @xmath28 , and @xmath35 along the line of sight toward the quasar pg @xmath0 . all three of these absorbers have a second , higher ionization phase , giving rise to the absorption . the broad phase in one case is consistent with a single cloud , and in the other two cases requires one or two additional clouds separated in velocity space from the one aligned with the absorption . two of the weak absorbers are constrained to have solar or greater than solar metallicity , and the other one to have a metallicity greater than @xmath214th solar . thus , in general , weak absorbers are not weak because of a low metallicity ( confirming the result of @xcite ) . as introduced in [ sec : intro ] , it is also possible that they have weak absorption because they are more highly ionized than their strong counterparts , or because their total column densities are smaller . the likely answer is that there is some combination of these two effects , perhaps leading to different populations of weak absorbers arising in different environments . the ionization parameters of the absorbers studied here are higher than inferred for many of the clouds in strong absorbers for which is also detected . [ is a tracer of low ionization conditions @xcite . ] however , there is also a population of weak absorbers with strong lines , amounting to about one third of the weak absorber population @xcite . detailed study of the phase structure of this sub group awaits spectra of additional quasar lines of sight with , and other transitions covered at high resolution . weak absorbers are potentially of general importance because they provide a sensitive probe of particular types of star forming environments . in principle , this population of objects can be used to track the chemical and ionization history of the universe in regions that are not in luminous galaxies which can be studied by other methods . for example , do they exist with solar metallicity to high redshifts , i.e. are there selected environments with extreme enrichment even at early times ? answering this question will require a large systematic study of many weak systems over a range of redshifts . such a study could address whether there is always absorption centered in velocity on a cloud . it could tabulate the distribution of clouds in velocity space , and address whether a large equivalent width is typically due to the presence of multiple clouds along the line of sight . spectral coverage of and will also provide better constraints on the ionization conditions of the high ionization phase . finally , it is highly desirable to search the quasar fields for faint galaxies that could be related to these weak absorbers . support for this work was provided by the nsf ( ast9617185 ) and by nasa ( nag 56399 and hst go08672.01a ) , the latter from the space telescope science institute , which is operated by aura , inc . , under nasa contract nas526555 . s. zonak , j. rigby , and n. bond were supported by an nsf reu supplement . we thank c. howk and k. sembach for their invaluable guidance on the analysis of high resolution stis spectra . & @xmath215 & @xmath216 & @xmath217 + & & @xmath218 & + 2853 & @xmath219 & @xmath220 & @xmath221 + 2796 & @xmath222 & @xmath223 & @xmath224 + 2803 & @xmath225 & @xmath226 & @xmath227 + 2600 & @xmath228 & @xmath229 & @xmath230 + 1193 & & @xmath231 & @xmath232 + 1260 & @xmath233 & @xmath234 & @xmath235 + 1335 & @xmath236 & @xmath237 & @xmath238 + 1084 & @xmath232 & @xmath239 & + 1207 & @xmath240 & @xmath241 & + 1394 & @xmath242 & & @xmath243 + 1403 & @xmath242 & @xmath244 & @xmath245 + 989 & & @xmath246 & + 1548 & @xmath247 & @xmath248 & @xmath249 + 1551 & @xmath250 & @xmath251 & @xmath252 + 1239 & @xmath219 & @xmath253 & @xmath254 + 1243 & & @xmath255 & @xmath256 + 1032 & & @xmath257 & + -0.05 in + & @xmath258 & @xmath259 & @xmath149 + & @xmath260 & @xmath261 & @xmath149 + & @xmath262 & @xmath263 & @xmath149 + & @xmath264 & @xmath265 & @xmath149 + & @xmath266 & @xmath267 & @xmath149 + + & @xmath268 & @xmath269 & @xmath149 + & @xmath270 & @xmath271 & @xmath149 + & @xmath272 & @xmath273 & @xmath149 + & @xmath274 & @xmath275 & @xmath149 + & @xmath276 & @xmath277 & @xmath149 + & @xmath278 & @xmath279 & @xmath16 + + & @xmath280 & @xmath281 & @xmath149 + & @xmath282 & @xmath283 & @xmath149 + & @xmath284 & @xmath285 & @xmath202 + & @xmath286 & @xmath287 & @xmath203 + -0.05 in 0.8182 & @xmath149 & @xmath288 to @xmath289 & @xmath290 & @xmath291 to @xmath292 & @xmath18 & @xmath293 to @xmath294 + & @xmath149 & @xmath295 to @xmath296 & @xmath297 & @xmath298 to @xmath299 & @xmath300 & @xmath301 to @xmath302 + 0.9056 & @xmath149 & @xmath303 to @xmath304 & @xmath297 & @xmath305 to @xmath194 & @xmath40 & @xmath306 + & @xmath149 & @xmath296 to @xmath197 & @xmath297 & @xmath143 to @xmath144 & @xmath19 & @xmath307 to @xmath301 + & @xmath16 & @xmath308 to @xmath296 & @xmath297 & @xmath143 to @xmath196 & @xmath309 & @xmath301 to @xmath310 + 0.6534 & @xmath149 & @xmath289 to @xmath303 & @xmath305 to @xmath144 & @xmath311 to @xmath144 & @xmath176 & @xmath312 to @xmath313 + & @xmath149 & @xmath182 to @xmath314 & @xmath194 to @xmath144 & @xmath18 to @xmath176 & @xmath201 & @xmath315 to @xmath316 + & @xmath202 & @xmath295 to @xmath183 & @xmath194 to @xmath144 & @xmath18 to @xmath176 & @xmath317 & @xmath301 to @xmath316 + & @xmath203 & @xmath295 to @xmath318 & @xmath194 to @xmath144 & @xmath18 to @xmath176 & @xmath309 & @xmath319 to @xmath320 + -0.05 in + @xmath321 & @xmath149 & @xmath322 & @xmath289 & @xmath323 & @xmath194 & @xmath324 & @xmath325 & @xmath326 & @xmath327 & @xmath328 & @xmath329 & @xmath317 & @xmath18 & @xmath40 + @xmath321 & @xmath149 & @xmath322 & @xmath318 & @xmath330 & @xmath331 & @xmath332 & @xmath333 & @xmath307 & @xmath334 & @xmath335 & @xmath336 & @xmath16 & @xmath300 & @xmath300 + + @xmath321 & @xmath149 & @xmath337 & @xmath304 & @xmath338 & @xmath339 & @xmath340 & @xmath341 & @xmath342 & @xmath343 & @xmath344 & @xmath345 & @xmath346 & @xmath40 & @xmath176 + @xmath321 & @xmath149 & @xmath337 & @xmath197 & @xmath311 & @xmath347 & @xmath348 & @xmath349 & @xmath350 & @xmath351 & @xmath352 & @xmath353 & @xmath354 & @xmath176 & @xmath19 + @xmath355 & @xmath16 & @xmath337 & @xmath296 & @xmath356 & @xmath357 & @xmath348 & @xmath358 & @xmath301 & @xmath359 & @xmath360 & @xmath353 & @xmath361 & @xmath309 & @xmath309 + + @xmath321 & @xmath149 & @xmath194 & @xmath289 & @xmath323 & @xmath18 & @xmath362 & @xmath363 & @xmath325 & @xmath364 & @xmath365 & @xmath366 & @xmath309 & @xmath176 & @xmath367 + @xmath321 & @xmath149 & @xmath194 & @xmath182 & @xmath368 & @xmath347 & @xmath369 & @xmath370 & @xmath371 & @xmath364 & @xmath372 & @xmath373 & @xmath374 & @xmath346 & @xmath201 + @xmath355 & @xmath202 & @xmath194 & @xmath295 & @xmath375 & @xmath376 & @xmath377 & @xmath370 & @xmath342 & @xmath378 & @xmath379 & @xmath380 & @xmath361 & @xmath381 & @xmath317 + @xmath382 & @xmath203 & @xmath194 & @xmath295 & @xmath375 & @xmath383 & @xmath377 & @xmath384 & @xmath306 & @xmath385 & @xmath343 & @xmath386 & @xmath387 & @xmath309 & @xmath309 + -0.05 in | high resolution optical ( hires / keck ) and uv ( stis / hst ) spectra , covering a large range of chemical transitions , are analyzed for three single cloud weak absorption systems along the line of sight toward the quasar pg @xmath0 .
weak absorption lines in quasar spectra trace metal
enriched environments that are rarely closely associated with the most luminous galaxies ( @xmath1 ) .
the two weak systems at @xmath2 and @xmath3 are constrained to have @xmath4 solar metallicity , while the metallicity of the @xmath5 system is not as well constrained , but is consistent with @xmath6th solar .
these weak clouds are likely to be local pockets of high metallicity in a lower metallicity environment .
all three systems have two phases of gas , a higher density region that produces narrower absorption lines for low ionization transitions , such as , and a lower density region that produces broader absorption lines for high ionization transitions , such as .
the profile for one system ( at @xmath2 ) can be fit with a single broad component ( @xmath7 ) , but those for the other two systems require one or two additional offset high ionization clouds . two possible physical pictures for the phase structure are discussed : one with a low ionization , denser phase embedded in a lower density surrounding medium , and the other with the denser clumps surrounding more highly ionized gas . |
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recent successful measurements of ortho positronium ( ps ) scattering cross sections by h@xmath2 , n@xmath2 , he , ne , ar , c@xmath3h@xmath4 , and c@xmath5h@xmath6 @xcite have spurred renewed theoretical activity in this subject @xcite . of these , the ps - he system is of special interest as it is the simplest system in which there are experimental results for total cross section @xcite and pickoff quenching rate @xcite . the experimental results for partial and differential cross sections for this system should be available soon @xcite . a complete understanding of this system is necessary before a venture to more complex targets . the pioneering calculations in this system using the static exchange approximation were performed by barker and bransden @xcite and by fraser and kraidy @xcite . there have also been r - matrix @xcite , close - coupling ( cc ) @xcite and model - potential @xcite calculations for ps - he scattering . more recently , there has been successful calculation of ps scattering by h @xcite , he @xcite , ne @xcite , ar @xcite , and h@xmath2 @xcite using a regularized model exchange potential in a coupled - channel formulation . however , there is considerable discrepancy among the different theoretical ps - he cross sections at zero energy which we discuss below . the static - exchange calculation by sarkar and ghosh @xcite , and by blackwood _ @xcite yielded @xmath7 ( at 0.068 ev ) , and @xmath8 ( at 0 ev ) , respectively , for the elastic cross section . the inclusion of more states of ps in the cc @xcite and r - matrix @xcite calculations does not change these results substantially . the pioneering static - exchange calculations by barker and bransden @xcite yielded 13.04 @xmath9 and by fraser @xcite yielded @xmath10 for zero - energy ps - he cross section . these results are in good agreement with each other . however , the model potential calculation by drachman and houston @xcite yielded @xmath11 and by this author @xcite yielded @xmath12 for the zero - energy ps - he cross section . so there is considerable discrepancy in the results of different theoretical calculation of low - energy ps - he elastic scattering . on the experimental front , there have been conflicting results for the low - energy ps - he elastic cross section by nagashima _ @xcite , who measured a cross section of @xmath13 at 0.15 ev , by coleman _ @xcite , who reported @xmath14 at 0 ev , by canter _ @xcite , who found @xmath15 at 0 ev , and by skalsey _ @xcite , who measured @xmath16 at 0.9 ev . it is unlikely that these findings could be consistent with each other . the results for the total cross section of ps scattering obtained from the coupled - channel calculation employing the model potential @xcite are in agreement with experiments of refs . @xcite at low energies . for ps - he , this model , while agrees @xcite with the experimental total cross sections @xcite in the energy range 0 to 70 ev , reproduces @xcite successfully the experimental pickoff quenching rate @xcite . all other calculations could not reproduce the general trend of cross sections of ps - he scattering in the energy range 0 to 70 ev and yielded a much too small quenching rate at thermal energies @xcite . however , the very low - energy elastic cross sections of the model - potential calculation @xcite are at variance with the experiments of refs . @xcite . pointing at the discrepancy above among different theoretical and experimental studies , blackwood _ _ @xcite called for a `` fully fledged calculation '' to resolve the situation . here we present a variational basis - set calculational scheme for low - energy ps - he scattering in s wave below the lowest ps - excitation threshold at 5.1 ev . using this method we report numerical result for the scattering length of ps - he using a one parameter uncorrelated he ground - state wave function @xcite . we present the formulation for the variational basis - set calculation in sec . ii , the numerical result for ps - he scattering length in sec . iii and a summary in sec . because of the existence of three identical fermions ( electrons ) in the ps - he system , one needs to antisymmetrize the full wave function . the position vectors of the electrons @xmath17 @xmath18 of ps , and @xmath19 and @xmath20 of he @xmath17 and positron ( @xmath21 ) measured with respect to ( w.r.t . ) the massive alpha particle at the origin are shown in fig . 1 . in this configuration the wave function for elastic scattering in the electronic doublet state of ps - he is taken as @xmath22 where @xmath23 is the incident ps momentum and @xmath24,\end{aligned}\ ] ] represents the doublet wave function of ps - he and where @xmath25 denotes spin up state and @xmath26 down and @xmath27 denotes the ps wave function of electron 1 . the he ground state wave function @xmath28 and the scattering function @xmath29 are symmetric under the exchange of electrons 2 and 3 . the spin function @xmath30 is antisymmetric under the same exchange . the full antisymmetrization operator for the three electrons is @xmath31 where @xmath32 is an operator for exchange in both space and spin of electrons @xmath33 and @xmath34 . as the scattering wave function ( [ wf ] ) above is already antisymmetrized with respect to electrons 2 and 3 , the operator @xmath35 in the antisymmetrizer is redundant and the relevant antisymmetrizer in this case is @xmath36 . hence , the fully antisymmetric state @xmath37 of ps - he scattering is given by @xmath38 the projection of the schrdinger equation on the doublet state @xmath39 is @xmath40 with @xmath41 the full ps - he hamiltonian . the incident ps energy @xmath42 ev . using the identities @xmath43 and @xmath44 @xmath45 , we see that the two terms on the right - hand side of eq . ( [ yyt ] ) give equivalent contribution which are combined in eq . ( [ xxt ] ) which is rewritten as @xmath46 hence after the spin projection to the doublet state the effective antisymmetrizer to be used on state ( [ wf ] ) is @xmath47 . we shall use this antisymmetrizer in the following and supress the spin functions . the full ps - he hamiltonian @xmath41 can be broken in the convenient form as follows : @xmath48 where @xmath49 includes the full kinetic energy and intracluster interaction of he and ps for the arrangement shown in fig . 1 and @xmath50 is the sum of the intercluster interaction between he and ps in the same configuration : @xmath51.\ ] ] we employ the position vectors @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 . the fully antisymmetric state satisfies the lippmann - schwinger equation @xcite @xmath57 where the channel green s function is given by @xmath58 and the incident wave @xmath59 satisfies @xmath60 we are using atomic units ( au ) in which @xmath61 , where @xmath62 ( @xmath63 ) is the electronic charge ( mass ) and @xmath64 the bohr radius . the properly symmetrized transition matrix for elastic scattering is defined by @xmath65 @xcite . a basis - set calculational scheme for the transition matrix can be obtained from the following expression @xcite @xmath66 using eq . ( [ 2x ] ) , it can be verified that eq . ( [ 61 ] ) is an identity if exact scattering wave functions @xmath67 are used . if approximate wave functions are used , expression ( [ 61 ] ) is stationary w.r.t . small variations of @xmath68 but not of @xmath69 . this one - sided variational property emerges because of the lack of symmetry of the formulation in the presence of explicit antisymmetrization operator @xmath70 . however , this variational property can be used to formulate a basis - set calculational scheme with the following trial functions @xcite @xmath71 where the suffix @xmath72 denotes trial and @xmath73 are the basis functions . substituting eq . ( [ 62 ] ) into eq . ( [ 61 ] ) and using this variational property w.r.t . @xmath74 we obtain @xcite @xmath75 @xmath76g_1{\cal a } _ 1 v_1 | f_n \rangle}.\ ] ] using the variational form ( [ 63 ] ) and definition @xmath77 we obtain the following basis - set calculational scheme for the transition matrix @xmath78 ) and ( [ 4 ] ) are also valid in partial - wave form . in the present s - wave calculation , the basis functions are taken in the following form @xmath79 where @xmath80 and @xmath81 are nonlinear variational parameters . the ground - state wave function of the he atom is taken to be @xmath82 with @xmath83 and @xmath84 @xcite and @xmath85 represent the ps(1s ) wave function . for elastic scattering the direct born amplitude is zero and the exchange correlation dominates scattering . to be consistent with this , the direct terms in the form factors @xmath86 and @xmath87 are zero with the above choice of correlations in the basis functions via @xmath88 and @xmath81 . this property follows as the above function is invariant w.r.t . the interchange of @xmath21 and @xmath89 whereas the remaining part of the integrand in the direct terms changes sign under this transformation . in ps - he elastic scattering the electron 2 of he is the active electron undergoing exchange with the electron 1 of ps whereas the electron 3 of he is a passive spectator . in this calculation we include in eq . ( [ 8 ] ) correlation between electrons 1 and 2 . consequently , we deal with integrals in three vector variables @xmath90 and @xmath21 . if we also include correlation involving electron 3 we shall have to deal with integration in four vector variables , which is beyond the scope of the present study . however , we believe that a meaningful calculation can be performed only with correlation between the active electrons 1 and 2 . hence , to avoid complication we ignore correlation involving electron 3 , which is expected to lead to correction over the present study . in s wave at zero energy , @xmath91 in eq . ( [ 8 ] ) ; also , @xmath92 . the useful matrix elements of the present approach are explicitly written as @xcite @xmath93 f_n({\bf r}_2,{\bf r}_3,\rho_1,{\bf s}_1 ) d{\bf r}_2 d{\bf r}_3 d{\rho_1 } d{\bf s}_1,\\ & = & - \frac{1}{2\pi}\int \varphi({\bf r}_1 ) \eta({\rho}_2)\frac{\sin(ps_2)}{ps_2 } [ { \cal v}_1 ] g_n({\bf r}_2,\rho_1,{\bf s}_1 ) d{\bf r}_2 d{\rho_1 } d{\bf s}_1,\\ \label{52 } \langle f_m|{\cal a}_1v_1|\phi^1_p \rangle & = & - \frac{1}{2\pi}\int g_m({\bf r}_1 , \rho_2,{\bf s}_2 ) [ { \cal v}_1 ] \varphi({\bf r}_2)\eta({\rho}_1)\frac{\sin(ps_1)}{ps_1 } d{\bf r}_2 d{\rho_1 } d{\bf s}_1,\\ \label{53 } \langle f_m|{\cal a}_1v_1|f_n\rangle & = & - \frac{1}{4\pi}\int g_m ( { \bf r}_1 , \rho_2,{\bf s}_2 ) [ { \cal v}_1 ] g_n({\bf r}_2 , \rho_1,{\bf s}_1 ) d{\bf r}_2 d{\rho_1 } d{\bf s}_1,\end{aligned}\ ] ] with @xmath94 , \quad \quad h(x)= \frac{1}{x}+\frac{\exp(-2\lambda x)}{x } + \lambda \exp(-2\lambda x),\ ] ] @xmath95 where the so called off - shell term @xmath96 has been neglected for numerical simplification in this calculation . this term is expected to contribute to refinement over the present calculation . in this convention the on - shell t - matrix element at zero energy is the scattering length : @xmath97 all the matrix elements above can be evaluated by a method presented in ref . . we describe it in the following for @xmath98 of eq . ( [ 7 ] ) . by a transformation of variables from @xmath99 to @xmath100 with jacobian @xmath101 and separating the radial and angular integrations , the form factor ( [ 7 ] ) is given by @xmath102 d\hat s_1 d\hat s_2 d\hat x , \label{13}\end{aligned}\ ] ] where @xmath103 , @xmath104 , @xmath105 and @xmath106 . recalling that @xmath107 , @xmath108 , @xmath109 , we employ the following expansions of the exponentials in eq . ( [ 13 ] ) @xmath110 @xmath111 @xmath112 @xmath113 @xmath114 where the @xmath115 s are the usual spherical harmonics . using eqs . ( [ 5x ] ) @xmath17 ( [ 9x ] ) in eq . ( [ 13 ] ) we get @xmath116.\end{aligned}\ ] ] where the @xmath117-sum is truncated at @xmath118 . this procedure avoids complicated angular integrations involving @xmath119 , @xmath120 and @xmath21 . the matrix element takes a simple form requiring straightforward numerical computation of certain radial integrals only . the functions @xmath121 , @xmath122 , @xmath123 etc . are easily calculated using eqs . ( [ 5x ] ) @xmath17 ( [ 9x ] ) : @xmath124 where @xmath125 is the usual legendre polynomial and @xmath126 is the cosine of the angle between @xmath127 and @xmath21 . the integrals ( [ 52 ] ) and ( [ 53 ] ) can be evaluated similarly . for example @xmath128,\end{aligned}\ ] ] where @xmath129 , @xmath130 , @xmath131 , @xmath132 and @xmath133 . we tested the convergence of the integrals by varying the number of integration points in the @xmath134 , @xmath135 and @xmath136 integrals in eqs . ( [ 8z ] ) and ( [ 8zx ] ) and the @xmath126 integral in eq . ( [ 8xz ] ) . the @xmath134 integration was relatively easy and 20 gauss - legendre quadrature points appropriately distributed between 0 and 16 were enough for convergence . in the evaluation of integrals of type ( [ 8xz ] ) 40 gauss - legendre quadrature points were sufficient for adequate convergence . the convergence in the numerical integration over @xmath135 and @xmath136 was achieved with 300 gauss - legendre quadrature points between 0 and 12 . the maximum value of @xmath117 in the sum in eqs . ( [ 8z ] ) and ( [ 8zx ] ) , @xmath137 , is taken to be 7 which is sufficient for obtaining the convergence with the partial - wave expansions ( [ 5x ] ) @xmath17 ( [ 9x ] ) . we find that a judicial choice of the parameters in eq . ( [ 8 ] ) is needed for convergence . the present method does not provide a bound on the result . consequently , the method could lead to a wrong scattering length if an inappropriate ( incomplete ) basis set is chosen . after some experimentation we find that for good convergence the nonlinear parameters @xmath138 and @xmath139 should be taken to have both positive and negative values and @xmath140 should have progressively increasing values till about 1.5 . if no care is taken in choosing the parameters a large number of functions could be necessary for obtaining convergence . the results reported in this work are obtained with the following parameters for the functions @xmath141 : @xmath142 @xmath143 @xmath144 @xmath145 @xmath146 @xmath147 @xmath148 @xmath149 @xmath150 @xmath151 @xmath152 @xmath153 @xmath154 @xmath155 by employing a suitably chosen set of the parameters we have kept the number of functions to a minimum . .4 cm table i : ps - he scattering length in au for different @xmath137 and @xmath156 . .2 cm & @xmath157 & @xmath158 & @xmath159&@xmath160 & @xmath161 & @xmath162 + 1&2.056 & 1.867&1.782 & 1.721&1.681 & 1.655 & 1.638 + 3&2.418 & @xmath163&28.016 & 4.151&3.061 & 2.839 & 2.933 + 5&@xmath164 & 1.563 & 1.234&1.128&1.135 & 1.168 & 1.190 + 6&0.712 & 0.982&0.782 & 0.637&0.650 & 0.769 & 0.878 + 7&3.372 & 0.983 & 0.792 & 0.694&0.727 & 0.824 & 0.910 + 8&@xmath165 & 1.124 & 0.944&0.877&0.907 & 0.971 & 1.023 + 9&6.283 & 0.976 & 0.832&0.801&0.841 & 0.897 & 0.943 + 10&1.332 & 1.123 & 0.941 & 0.897&0.929 & 0.981 & 1.023 + 11&1.225 & 1.112 & 0.945&0.886&0.909 & 0.963 & 1.011 + 12&0.756 & 1.468 & 0.995&0.918&0.931 & 0.964 & 0.977 + 13&1.229 & 1.197&1.008 & 0.944 & 0.970 & 1.028 & 1.022 + 14 & 1.061 & 1.249 & 1.060&0.980&0.983 & 1.026 & 1.019 + 0.4 cm in table i we show the convergence pattern of the present calculation w.r.t . the number of partial waves @xmath137 and basis functions @xmath156 used in the calculation . the convergence is satisfactory considering that we are dealing with a complicated five - body problem . however , as the present calculation does not produce a bound on the result , the convergence is not monotonic with increasing @xmath156 . the final result of the present calculation is that for @xmath166 and @xmath162 : @xmath167 au . although it is difficult to provide a quantitative measure of convergence , from the fluctuation of this result for large @xmath156 and @xmath137 we believe the error in our result to be less than 10@xmath168 , so that the final ps - he scattering length is taken as @xmath169 au . the results for large @xmath156 and @xmath137 reported in table i all lie in this domain . the maximum number of functions ( @xmath166 ) used in this calculation is also pretty small , compared to those used in different kohn - type variational calculations for electron - hydrogen ( @xmath170 ) @xcite , positron - hydrogen ( @xmath171)@xcite , and positron - helium ( @xmath172 ) @xcite scattering . because of the explicit appearance of the green s function , the present basis - set approach is similar to the schwinger variational method . using the schwinger method , convergent results for electron - hydrogen @xcite and positron - hydrogen @xcite scattering have been obtained with a relatively small basis set ( @xmath173 ) . these suggest a more rapid convergence in these problems with a schwinger - type method . to summarize , we have formulated a basis - set calculational scheme for s - wave ps - he elastic scattering below the lowest inelastic threshold using a variational expression for the transition matrix . we illustrate the method numerically by calculating the scattering length in the electronic doublet state : @xmath174 au . this corresponds to a zero - energy cross section of @xmath175 in reasonable agreement with a model calculation by this author ( @xmath176 ) @xcite and the experiment of skalsey _ et al . _ [ @xmath177 at 0.9 ev ] @xcite . this calculation as well as our previous studies of ps - he scattering using a model exchange potential @xcite possibly consolidate the experimental result of skalsey _ et al . _ however , these low - energy ps - he elastic scattering cross sections are in disagreement with other experiments by nagashima _ [ @xmath178 at 0.15 ev ] @xcite , by canter _ ( @xmath179 ) @xcite , and by coleman _ et al . _ ( @xmath180 ) @xcite , as well as with conventional static - exchange calculations of refs . @xcite ( @xmath181 ) and a model potential calculation of ref . @xcite ( @xmath182 ) . as the effective interaction for elastic scattering between ps and he is repulsive in nature , a smaller scattering length as obtained in this study and in refs . @xcite would imply a weaker effective ps - he interaction . this would allow the ps atom to come closer to he and would lead @xcite to a large pickoff quenching rate and a large @xmath183 @xmath184 in agreement with experiment @xcite . the conventional close - coupling @xcite , r - matrix @xcite and static - exchange @xcite models yielded a much too large scattering length corresponding to a stronger repulsion between ps and he . consequently , these models led to a much too small @xmath183 @xmath185 @xcite in disagreement with experiment @xcite . this is addressed in detail in ref . @xcite where we established a correlation between the different scattering lengths and the corresponding @xmath183 . this correlation suggests that a small ps - he scattering length as in this work is consistent with the large experimental @xmath183 . although we have used a simple wave function for he in this complex five - body calculation we do not believe that the use of a more refined he wave function would substantially change our findings and conclusions . however , independent calculations and accurate experiments at low energies are welcome for a satisfactory resolution of this controversy . the work is supported in part by the conselho nacional de desenvolvimento - cientfico e tecnolgico , fundao de amparo pesquisa do estado de so paulo , and financiadora de estudos e projetos of brazil . p. g. coleman , s. rayner , f. m. jacobsen , m. charlton , and r. n. west , j. phys . b * 27 * , 981 ( 1994 ) . a. j. garner , a. zen , and g. laricchia , nucl . instrum . & methods phys . b * 143 * , 155 ( 1998 ) ; p. k. biswas , _ ibid . _ * 171 * , 135 ( 2000 ) . s. k. adhikari and i. h. sloan , phys . c * 11 * , 1133 ( 1975 ) ; nucl . phys . * a241 * , 429 ( 1975 ) ; s. k. adhikari , phys . rev . c * 10 * , 1623 ( 1974 ) ; _ variational principles and the numerical solution of scattering problems _ , ( john - wiley , new york , 1998 ) . figure number 1 . different position vectors for the ps - he system w.r.t . the massive alpha particle at the origin in arrangement 1 with electrons 2 and 3 forming he and 1 forming ps . the arrows on the electrons indicate the orientations of spin @xmath17 up and down . | we present a variational basis - set calculational scheme for elastic scattering of positronium atom by helium atom in s wave and apply it to the calculation of the scattering length .
highly correlated trial functions with appropriate symmetry are used in this calculation .
we report numerical result for the scattering length in atomic unit : @xmath0 .
this corresponds to a zero - energy elastic cross section of @xmath1 .
* pacs number(s ) : 34.90.+q , 36.10.dr * = 1.5 true cm = -.6 true cm |
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the stage leading up to dynamic collapse of a magnetically subcritical cloud core to a protostar or a group of protostars is believed to be largely quasi - static , if the responsible process is ambipolar diffusion ( e.g. , mestel & spitzer 1956 , @xcite , @xcite , @xcite , @xcite ) . to describe the transition between quasi - static evolution by ambipolar diffusion and dynamical evolution by gravitational collapse , @xcite introduced the idea of a pivotal state , with the scale - free , magnetostatic , density distribution approaching @xmath8 for an isothermal equation of state ( eos ) when the mass - to - flux ratio has a spatially constant value , a condition that @xcite and @xcite termed `` isopedic '' . numerical simulations of the contraction of magnetized clouds justify the assumption of a nearly constant mass - to - flux ratio in the pivotal core .. outside the starred point , the mass - to - flux value exhibits greater variation , but this occurs only because @xcite impose starting values for the mass - to - flux in the envelope that are @xmath9 times the critical value ( see also figs . 4a and 8b in @xcite ) . such small ratios for the bulk of the mass of a molecular cloud are probably ruled out by the zeeman oh measurements summarized by crutcher ( 1998 ) . ] the small dense cores of molecular clouds that give rise to low - mass star formation are effectively isothermal ( @xcite ; @xcite ) . the situation may be different for larger regions that yield high - mass or clustered star formation . it has often been suggested that the eos relating the gas pressure @xmath10 to the mass density @xmath11 of interstellar clouds can be represented by a polytropic relation @xmath12 with negative index @xmath0 . @xcite pointed out the utility of this idealization within the context of the classic two - phase model of the diffuse interstellar medium [ @xcite ; @xcite ; @xcite ] , while @xcite published extensive tables analyzing the stability of non - magnetized , self - gravitating spheres of such gases . @xcite examined the linewidth - size and density - size relations of molecular clouds , first found by @xcite and subsequently studied by many authors [ e.g. , @xcite , @xcite , @xcite , @xcite , @xcite ] . maloney pronounced the results consistent with the properties of negative index polytropes . for a polytropic eos , the sound speed @xmath13 increases with decreasing density if @xmath14 . the latter behaviour may be compared with the empirical linewidth - density relation for molecular clouds , @xmath15 , with @xmath16 for low - mass cores ( @xcite ) and @xmath17 for high - mass cores ( @xcite ) , implying that @xmath0 lies between @xmath18 and @xmath19 , or that a static @xmath20 lies between @xmath21 and 0.7 . the case @xmath22 is of particular relevance for the equilibrium properties of molecular clouds . @xcite found that the pressure of alfvn waves propagating in a stratified medium , @xmath23 , a consequence of conservation of the wave energy flux @xmath24 . this result was later derived more rigorously by @xcite in the wkb approximation for mhd waves propagating in mildly inhomogeneous media , and , more recently by @xcite and @xcite in a specific astrophysical context . in numerical simulations of the same problem , @xcite found indication of a much shallower relation ( @xmath25 ) for a self - gravitating medium supported by nonlinear alfvn waves . on the other hand , for the adiabatic contraction of a cloud supported by linear alfvn waves , @xcite found a dynamic @xmath26 larger than 1 . @xcite confirmed a similar behaviour in numerical simulations of the gravitational collapse of clouds with an initial field of hydrodynamic rather than hydromagnetic turbulence . in the limit of @xmath27 ( or @xmath28 ) , the eos becomes `` logatropic , '' @xmath29 , a form first used by @xcite to mimic the nonthermal support in molecular clouds associated with the observed supersonic linewidths . the sound speed associated with the nonthermal contribution , @xmath30 becomes important at the low densities characteristic of molecular cloud envelopes ( as contrasted with the cloud cores ) since the thermal contribution is independent of density if the temperature @xmath31 remains constant . this nonthermal contribution decreases with increasing density and will become subsonic at high densities as recently observed in the central regions of dense cores ( @xcite ) . @xcite and @xcite have modeled the equilibrium and collapse properties of unmagnetized , self - gravitating , spheres with a pure logatropic eos and claim to find good agreement with observations . adams , lizano , & shu ( in 1987 ) independently obtained the similarity solution for the gravitational collapse of an unmagnetized singular logatropic sphere ( sls ) , but they chose not to publish their findings until they had learned how to magnetize the configuration in a nontrivial way ( see the reference to this work in fuller & myers ( 1992 ) , who considered the practical modifications to the protostellar mass - infall rate introduced by `` nonthermal '' contributions to the support against self - gravity ) . magnetization constitutes an important program to carry out if we try to justify a nonthermal eos as the result of a superposition of propagating mhd waves ( see also holliman & mckee 1993 ) . in this paper , we extend the study of li & shu ( 1996 ) to include the isopedic magnetization of pivotally self - gravitating clouds with a polytropic equation of state . as a by - product of this investigation , we obtain the unanticipated and ironic result that the only way to magnetize a singular logatropic configuration and maintain a scale - free equilibrium is to do it trivially , i.e. , by threading the sls with straight and uniform field lines ( see 6 ) . a basic consequence of treating the turbulence as a scalar pressure , coequal to the thermal pressure except for satisfying a different eos , is that we do not change the basic topology of the magnetic field . this assumption may require reassessment if mhd turbulence enables fast magnetic reconnection ( @xcite ) and allows the magnetic fields of highly flattened cloud cores ( mestel & strittmatter 1967 , @xcite ) or pseudodisks ( @xcite ) to disconnect from their background . recent mhd simulations carried out in multiple spatial dimensions ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) find turbulence in strongly magnetized media to decay almost as fast as in unmagnetized media . such decay may be responsible for accelerating molecular cloud core formation above simple ambipolar diffusion rates ( @xcite , @xcite , @xcite ) . although this result also cautions against treating turbulence on an equal footing as thermal pressure , we attempt a simplified first analysis that includes magnetization to assess the resulting configurational changes when we adopt an alternative eos for the pivotal state . in particular , different power - law dependences of the radial density profile translate immediately to different time dependences in the mass - infall rate for the subsequent inside - out collapse ( @xcite , mclaughlin & pudritz 1997 ) . the paper is organized as follows . in 2 we formulate the equations of the scale - free problem and show that each solution depends only on the polytropic exponent @xmath6 and a nondimensional parameter @xmath5 related to the cloud s morphology . in 3 we present the numerical results . in 4 , 5 , and 6 we discuss the limiting form of the solutions . finally , in 7 we give our conclusions and discuss the possible implications of our results for star formation and the structure of giant molecular clouds . to begin , we generalize the singular polytropic sphere in the same way that @xcite generalized the singular isothermal sphere ( sis ) . in the absence of an external boundary pressure , the only place the pressure @xmath10 enters in the equations of magnetostatic equilibrium is through a gradient . consider then the polytropic relation @xmath32 by integrating equation ( [ dpdrho ] ) we recover for @xmath33 the isothermal eos , @xmath34 , ( where @xmath35 is the square of the isothermal sound speed ) and for @xmath36 the logatropic eos , @xmath37 . we adopt axial symmetry in spherical coordinates and consider a poloidal magnetic field given by @xmath38 where @xmath39 is the magnetic flux . force balance along field lines requires @xmath40 where @xmath41 is the gravitational potential and @xmath42 is the bernoulli `` constant '' along the field line @xmath43 constant . poisson s equation now reads @xmath44+{1\over r^2\sin^2\theta } { \partial\over\partial\theta}\left[\sin\theta\left({dh\over d\phi } { \partial\phi\over\partial\theta}-k\rho^{-(2-\gamma ) } { \partial\rho\over\partial\theta}\right)\right]=4\pi g\rho;\ ] ] whereas force balance across field lines reads @xmath45 we look for scale - free solutions of the above equations by nondimensionalizing and separating variables : @xmath46 @xmath47 @xmath48 where @xmath5 is a dimensionless constant that measures the deviation from a force free magnetic field , and @xmath49 and @xmath50 are dimensionless functions of the polar angle @xmath51 . or @xmath52 . ] these assumptions imply that the equilibria will have spatially constant mass - to - flux ratios ( see below ) . substitution of equation ( [ nondim ] ) into equations ( [ along ] ) and ( [ across ] ) yields @xmath53\right\}=\ ] ] @xmath54,\ ] ] and @xmath55=-c_\gamma h_0r \phi^{-(2-\gamma)/(4 - 3\gamma)},\ ] ] where a prime denotes differentiation with respect to @xmath51 , and @xmath56 @xmath57 @xmath58 in particular , for @xmath59 , eq . ( [ alongn ] ) gives the dimensionless density for the non - magnetized singular polytropic sphere @xmath60^{1/(2-\gamma)},\ ] ] whereas eq . ( [ acrossn ] ) implies @xmath61 for @xmath62 , in order to satisfy the boundary conditions eq . ( [ bc ] ) . in this case , the mass - to - flux ratio @xmath1 is infinite . however , for @xmath63 , eq . ( [ acrossn ] ) admits also the analytic solution of @xmath64 corresponding to a straight and uniform field , while the density function is @xmath65 . therefore , a spherical logatropic scale free cloud can be magnetized with a uniform magnetic field of any strength , and any value of the spherical mass to flux ratio is allowed.^{-1}$ ] . ] for arbitrary values of @xmath6 and @xmath5 the ordinary differential equations ( odes ) ( [ alongn ] ) and ( [ acrossn ] ) are to be integrated subject to the two - point boundary conditions ( bcs ) : @xmath66=0,\ ] ] @xmath67 the first bc implies that there is no contribution from the polar axis to the mass inside a radius @xmath68 . the second bc comes from the definition of magnetic flux , i.e. no trapped flux at the polar axis . the last two bcs imply no kinks at the midplane . the equilibria are characterized by : ( _ a _ ) the spherical mass - to - flux ratio , is not defined for the polytropic scale free magnetized equilibria because the integral @xmath69 diverges since it can be shown that @xmath70 for @xmath71 . ] @xmath72 where @xmath73 is the mass enclosed within a radius @xmath68 ; ( _ b _ ) the factor @xmath74 by which the average density is enhanced over the non - magnetized value because of the extra support provided by magnetic fields , @xmath75^{-1/(2-\gamma ) } \int_0^{\pi/2}r(\theta)\sin \theta \ , d\theta , \ ] ] which is equal to @xmath76 if @xmath59 ( see eq . [ [ nonmagr ] ] ) ; ( _ c _ ) the sound speed , @xmath77 and ( _ d _ ) the alfvn speed @xmath78 { 1 \over r(\theta ) \sin^2\theta}.\ ] ] both the sound speed and the alfvn speed scale as @xmath79 for @xmath80 , and @xmath81 for @xmath36 ; for other values of @xmath6 , the exponent of @xmath68 lies between these two values . it is also of interest to define the ratio @xmath82 of the square of the sound speed and the square of the alfvn speed , each weighted by the density , which is a physical quantity that can be compared with observations : @xmath83 /\sin\theta d \theta}.\ ] ] if @xmath84 represents only the thermal sound speed , then the observational summary given by fuller & myers ( 1992 ) would imply that @xmath85 in the quiet low - mass cores of gmcs , whereas @xmath86 in their envelopes . if we include in @xmath84 , however , the turbulent contribution , then the turbulent speed is likely to be sub - alfvnic or marginally alfvnic , and @xmath87 everywhere is probably a better characterization of realistic clouds . to obtain an equilibrium configuration for given values of @xmath6 and @xmath5 , equations ( [ alongn ] ) and ( [ acrossn ] ) are integrated numerically . the integration is started at @xmath88 using the expansions : @xmath89 , @xmath90 , with @xmath91 , and @xmath92 a_\gamma h_0 a_0^{2(1-\gamma)/(4 - 3\gamma ) } b_0^{2-\gamma}$ ] . the values of @xmath93 and @xmath94 are varied until the two bcs at @xmath95 ( eq . [ bc ] ) , are satisfied . for flattened equilibria ( see below ) it is more convenient to start from @xmath96 , where the bcs @xmath97 and @xmath98 are imposed , and integrate toward @xmath88 . the values of @xmath99 and @xmath100 are then varied until a solution is found that satisfies the two bcs at @xmath101 . figure 1 shows the resulting flux and density functions @xmath50 and @xmath49 computed for @xmath102 and values of @xmath6 between 0.2 and 1 . we reproduce the results of @xcite for @xmath33 , which is the only case that obtains perfect toroids ( i.e. , @xmath103 = 0 $ ] ) ; models with @xmath104 have nonzero density at the polar axis . figure 2 shows the corresponding density contours and magnetic field lines . in the limit @xmath27 , independent of @xmath5 as long as it is nonzero , the pivotal configuration becomes thin disks with an ever increasing magnetic field strength . table 1 shows the spherical mass to flux ratio @xmath1 , the overdensity parameter @xmath74 , and the ratio of the square of the sound and alfvn speeds @xmath82 . this table shows that , for fixed @xmath5 , @xmath82 decreases as @xmath6 decreases because the magnetic field becomes stronger . for the same reason @xmath74 increases . in contrast , @xmath1 goes through a minimum as @xmath6 decreases . figures 1 and 2 demonstrate that for @xmath105 ( the logatropic limit ) , @xmath5 is not a measure of the strength of the magnetic fields since @xmath106 diverges as @xmath107 ( see 5 below ) . for fixed @xmath6 , a sequence from small @xmath5 to large @xmath5 progresses through configurations of increasing support by magnetic fields , as demonstrated explicitly for the isothermal case by @xcite . this behavior is illustrated here for the @xmath108 case in figure 3 , which shows the density contours and magnetic field lines corresponding to values of @xmath5 from 0.05 to 1.5 . table 2 shows the corresponding values of @xmath1 , @xmath74 , and @xmath82 . for small @xmath5 , the equilibria have nearly spherically symmetric isodensity contours and weak quasiuniform magnetic fields that provide little support against gravity . with increasing @xmath5 , the pivotal configurations flatten . the case @xmath109 is already quite disklike : the pole to equator density contrast is @xmath110 . for a thin disk , the analysis of @xcite demonstrates that magnetic tension provides virtually the sole means of horizontal support against self - gravity , with gas and magnetic pressures being important only for the vertical structure . in the limit of a completely flattened disk ( @xmath111 ) , @xmath112 independent of the detailed nature of the gas eos ( see next section ) . table 2 shows the spherical mass to flux ratio @xmath1 , the overdensity parameter @xmath74 , and the ratio of the square of the sound and alfvn speeds @xmath82 . again @xmath74 increases monotonically and @xmath82 decreases monotonically as the magnetic support increases with @xmath5 , while @xmath1 goes through a minimum and tends to 1 for large @xmath5 . since the mass - to - flux ratio @xmath1 is a fundamental quantity that will not change unless magnetic field is lost by ambipolar diffusion , in figure 4 we consider sequences where @xmath1 is held fixed , but @xmath6 is varied . this figure shows the locus of the set of equilibria with @xmath113 and @xmath114 in the ( @xmath5 , @xmath6 ) plane . equilibria with @xmath115 are highly flattened when @xmath105 even for small but fixed values of @xmath5 ( see 6 ) . in fact , to obtain incompletely flattened clouds when one takes the limit @xmath105 , one also needs simultaneously to consider the limit @xmath116 . unfortunately , because both the density and the strength of the magnetic field at the midplane diverge as the equilibria become highly flattened , we are unable to follow numerically the limit @xmath105 to verify if these sequences of constant @xmath115 will hook to a finite value in the @xmath5 axis , or will loop to @xmath117 , consistent with our demonstration in 4 that flattened disks do not exist in the logatropic limit . or large @xmath5 , it becomes necessary to determine the constants of the expansions of @xmath49 and @xmath50 near the origin with prohibitively increasing accuracy . ] we speculate that the results for @xmath115 have the following physical interpretation . according to the theorem of shu & li ( 1997 ) , only if @xmath118 itself rather than @xmath1 is less than unity , the magnetic field is strong enough overall to prevent the gravitational collapse of a highly flattened cloud . however , for moderate @xmath5 and @xmath6 when @xmath115 , even the singular equilibria are probably magnetically subcritical , since there can be little practical difference between the spherical mass - to - flux ratio @xmath1 and the `` true '' mass - to - flux ratio @xmath118 for highly flattened configurations . the latter is formally infinite when @xmath119 only because the mass column goes to zero a little slower than the field column when we perform an integration along the central field line ( see footnote 3 ) . in this interpretation , scale - free clouds with @xmath115 and intermediate values of @xmath6 can become highly flattened because magnetic tension supports them laterally against their self - gravity while the soft eos does not provide much resistance in the direction along the field lines . the squeezing of the cloud toward the midplane is compounded by the _ confining _ pressure of bent magnetic field lines that exert pinch forces in the vertical direction . both the magnetic tension and the vertical pinch of magnetic pressure disappear when the field lines unbend , as they must to maintain the scale - free equilibria in the limit @xmath105 ( see below ) . as a consequence , logatropic configurations become spherical for any value of @xmath1 . we leave as an interesting problem for future elucidation the determination whether there is still a threshold in @xmath1 below which the sls , embedded with straight and uniform field lines , will not collapse dynamically . in the limit @xmath121 , the cloud flattens to a thin disk for any @xmath122 . dominant balance arguments applied to the two odes of the problem reveal the following asymptotic behaviour : . see li & shu ( 1996 ) for the correct asymptotic expansion in this case . ] @xmath123 @xmath124 to the lowest order in @xmath5 the equation of force balance along field lines ( eq . [ alongn ] ) becomes : @xmath125= \nonumber \\ & & -2\left[{4 - 3\gamma\over ( 2-\gamma)^2}s^{-(1-\gamma ) } + \left({4 - 3\gamma\over 2-\gamma}\right)b_\gamma f^{2(1-\gamma)/(4 - 3\gamma)}\right],\end{aligned}\ ] ] valid over the interval @xmath126 , plus the the integral constraint @xmath127 obtained by integrating eq . ( [ alongn ] ) from @xmath128 to @xmath129 , and applying the first bc ( eq . [ bc ] ) on the polar axis . the constant @xmath130 is proportional to the surface density of the polytropic disks , given by @xmath131 which , for @xmath33 gives @xmath132 , as found by @xcite . ( [ acrossn ] ) expressing force balance across field lines reduces to @xmath133=-c_\gamma f^{-(2-\gamma)/(4 - 3\gamma)}r_0\delta(\theta-\pi/2),\ ] ] where the parameter @xmath134 is defined by @xmath135 this is equivalent to the equation for force free magnetic fields @xmath136 valid over the interval @xmath126 , plus the condition @xmath137 obtained integrating eq . ( [ acrossn ] ) from @xmath138 to @xmath139 , and taking the limit @xmath140 . for integer @xmath134 , solutions of eq . ( [ forcefree ] ) regular at @xmath88 are gegenbauer polynomials of order @xmath134 and index @xmath141 , @xmath142 ( see e.g. abramowitz & stegun 1965 ) . in general , it can be shown ( chandrasekhar 1955 ) that any axisymmetric force free field , separable in spherical coordinates , can be expressed in terms of fundamental solutions whose radial dependence is given by a combination of bessel functions of fractional order , and the angular dependence by gegenbauer polynomials of index @xmath141 . in our case , the choice of @xmath6 determines a particular exponent of the power - law for the radial part of the flux function , and hence the corresponding value of @xmath134 ( non - integer , except for @xmath63 and 1 ) . therefore , the magnetic field is force free everywhere except at the midplane where @xmath143 and the condition of force balance across field lines has to be satisfied . in the thin disk limit discussed here , the boundary condition @xmath144 is clearly not fullfilled : the kink of @xmath106 at the midplane provides the magnetic support against self - gravity on the midplane . currents must exist in the disk to support these kinks . with the definitions @xmath145 eq . ( [ asymp ] ) transforms into @xmath146 which has the solution @xmath147 where @xmath148 is a solution of the homogeneous equation @xmath149 therefore , @xmath150^{-1/(1-\gamma)},\ ] ] and the integral constraint eq . ( [ int1 ] ) becomes @xmath151 the problem is thus reduced to the solution of the two homogeneous equations eq . ( [ forcefree ] ) and eq . ( [ eqq ] ) for the functions @xmath152 and @xmath148 which are determined up to an arbitrary constant . however , the two integral constraints that would have determined these latter constants ( eqs . [ int2 ] , [ qconst ] ) , contain the additional unknown parameter @xmath130 . the system of equations is closed by the requirement that @xmath153 substituting eq . ( [ rh0 ] ) and eq . ( [ fh0 ] ) in eq . ( [ lambdar ] ) , one obtains @xmath154 i.e. , @xmath155 which gives the remaining condition . ( [ forcefree ] ) and ( [ eqq ] ) can be solved numerically by starting the integration at @xmath88 with the series expansions : @xmath156 $ ] , and @xmath157 \theta^4 + \ldots\right\}$ ] , where @xmath158 and @xmath159 are arbitrary constants . are of little use here : the one at @xmath88 reduces to the condition @xmath160 , trivially satisfied ; the second bc , @xmath98 can not be applied because of the @xmath161-function at @xmath162 . ] the constants @xmath163 and @xmath130 are then determined by the constraints expressed by eqs . ( [ int1 ] ) , ( [ int2]),and ( [ int3 ] ) . figure 5 shows the functions @xmath152 and @xmath164 obtained for @xmath22 and increasing values of @xmath5 from 0.4 to 1.5 compared with the asymptotic expressions computed here . already for @xmath165 , the actual @xmath152 and @xmath164 are very close to the corresponding asymptotic functions eq . ( [ rh0 ] ) and eq . ( [ fh0 ] ) . table 3 shows the value of the angle @xmath166 of the magnetic field with the plane of the disk , the flux function @xmath167 evaluated at @xmath96 ( indicative of the magnetic field stength ) , and the surface density parameter @xmath130 , as functions of @xmath6 . the angle @xmath166 ranges from @xmath168 for the isothermal case @xmath33 to @xmath169 in the logatropic case @xmath7 . correspondingly , the magnetic flux in the disk and the surface density both diverge as @xmath105 for any large but finite value of @xmath5 . for the isothermal case @xmath171 , @xcite have shown how the sis is recovered for @xmath172 from a family of toroids with zero density on the polar axis . for @xmath173 , in the limit @xmath170 , the asymptotic expansions are given by : @xmath174^{1/(2-\gamma ) } + p(\theta ) h_0^{(4 - 3\gamma)/(3 - 2\gamma ) } + \ldots\ ] ] @xmath175 to the lowest order in @xmath5 , eqs . ( [ alongn ] ) and ( [ acrossn ] ) become : @xmath176\right\}=\ ] ] @xmath177,\ ] ] and @xmath178=-c_\gamma \left[{4 - 3\gamma\over ( 2-\gamma)^2}\right]^{1/(2-\gamma ) } g^{-(2-\gamma)/(4 - 3\gamma)},\ ] ] the bc for the functions @xmath179 and @xmath180 are the same as those for @xmath49 and @xmath50 in eq . ( [ bc ] ) . figure 6 shows the convergence of the solutions of the full set of equations ( [ alongn ] ) and ( [ acrossn ] ) obtained for @xmath22 and decreasing values of @xmath5 from 0.4 to 0.05 , to the asymptotic solutions obtained by integrating the equations above . notice that @xmath181 and @xmath182 showing that the sequence of equilibria with @xmath22 originates from the corresponding unmagnetized spherical state ( eq . [ nonmagr ] ) by reducing the density on the pole and enhancing it on the equator . the same behaviour is found for any value of @xmath6 in the range @xmath183 . for @xmath171 , the function @xmath179 diverges at @xmath88 , indicating that this expansion is not appropriate in the isothermal case , as in the case @xmath121 . for the same reason , the expansion also fails for @xmath63 , since both @xmath180 and @xmath179 diverge on the equatorial plane . these flattened configurations are supported by magnetic and gas pressure against self - gravity . the intensity of the magnetic field can become very high even though @xmath5 is small , because the latter parameter measures not the field strength but the deviations from a force free field ( see eq . [ nondim ] ) . we consider in this section the logatropic limit @xmath184 . as anticipated in 2 , for @xmath63 eq . ( [ alongn ] ) and ( [ acrossn ] ) admit the analytical solution @xmath185 and @xmath186 corresponding to a sls threaded by a straight and uniform magnetic field . this solution represents the only possible scale - free isopedic configuration of equilibrium for a magnetized cloud with a logatropic eos . to show this , we use the results of 4 and 5 for @xmath121 and @xmath172 to find the limit of the equilibrium configurations for @xmath27 and fixed ( small or large ) values of @xmath5 . in the limit @xmath120 , @xmath105 , analytic solutions to equations ( [ forcefree ] ) and ( [ eqq ] ) exist . the magnetic field tends to become uniform and straight , @xmath187 , but @xmath188 diverges , as shown in table 3 , and therefore @xmath189 also diverges ( see eq . [ sol ] ) . eq . ( [ surfden ] ) shows in this limit that the surface density @xmath190 is independent of @xmath68 , therefore , no pressure gradients can be exerted in the horizontal direction . the value @xmath191 diverges as @xmath105 for any value of @xmath5 because @xmath192 ( see table 3 ) . the magnetic flux threading the disk , @xmath193 , becomes infinite in order to keep the mass to flux ratio @xmath194 equal to 1 . therefore , the limiting configuration approaches a uniform disk with infinite surface density , threaded by an infinitely strong uniform and straight magnetic field . if we now examine the case @xmath170 , in the limit @xmath195 , it is easy to show from eq . ( [ fluxsmallh ] ) that the magnetic field tends to become uniform , @xmath196 , but @xmath197 . consequently , the density function @xmath179 also diverges in @xmath198 , and the configuration again approaches a thin disk threaded by an uniform , infinitely strong , magnetic field . we conclude that scale - free logatropic clouds can not exist as magnetostatic disks except in some limiting configuration . in the absence of such limits , the equilibria are spherical and can be magnetized only by straight and uniform field lines ; i.e. , the magnetic field is force - free and therefore given by @xmath59 . the inside - out gravitational collapse of such a sls would still proceed self - similarly as in the solution of mclaughlin & pudritz ( 1997 ) , but the frozen - in magnetic fields would yield a dependence with polar angle that eventually produces a pseudodisk ( galli & shu 1993a , b ; allen & shu 1998a ) . we have solved the scale - free equations of magnetostatic equilibrium of isopedic self - gravitating polytropic clouds to find pivotal states that represent the initial state for the onset of dynamical collapse , as first proposed by @xcite for isothermal clouds . compared to unmagnetized equilibria , the magnetized configurations are flattened because of magnetic support across field lines . the degree of this support is best represented by the ratio of the square of the sound to alfvn speeds @xmath82 , or the overdensity parameter @xmath74 , since they are always monotonic functions of @xmath5 and @xmath6 . configurations with @xmath33 become highly flattened as the parameter @xmath5 increases . when @xmath104 ( softer eos ) the equilibria get flattened even faster at the same values of @xmath5 , since along field lines there is less support from a soft eos than for a stiff one . however , it seems that in the logatropic limit flattened disks do not exist : the singular scale - free equilibria can only be spherical uniformly magnetized clouds . figure 7 shows a schematic picture of the @xmath199 plane indicating the topology of the solutions for scale free magnetized isopedic singular self - gravitating clouds . in self - gravitating clouds , the joint compression of matter and field is often expressed as producing an expected relationship : @xmath200 , with different theorists expressing different preferences for the value of @xmath201 ( e.g. , mestel 1965 , fiedler & mouschovias 1993 ) . no local ( i.e. , point by point ) relationship of the form @xmath202 holds for the scale - free equilibria studied in this paper . however , if we average the magnetic field strength and mass density over ever larger spherical volumes centered on @xmath203 , we do recover such a relationship : @xmath204 , where angular brackets denote the result of such a spatial average and @xmath205 . we may think of the result @xmath206 as arising physically from a combination of two tendencies . ( a ) slow cloud contraction in the absence of magnetic fields and rotation tends to keep roughly one jeans mass inside every radius @xmath68 , which yields @xmath207 , or @xmath208 if @xmath209 . ( b ) slow cloud contraction in the absence of gas pressure tends to keep roughly one magnetic critical mass inside every radius @xmath68 , which yields @xmath210 = constant , or @xmath211 if gas pressure ( thermal or turbulent ) plays a comparable role to magnetic fields in cloud support . notice that our reasoning does not rely on arguments of cloud geometry , e.g. , whether cloud cores flatten dramatically or not as they contract ; nor does it depend sensitively on the precise reason for core contraction , e.g. , because of ambipolar diffusion or turbulent decay . crutcher ( 1998 ) claims that the observational data are consistent with @xmath212 . if we take crutcher s conclusion at face value , we would interpret the observations as referring mostly to regions where the eos is close to being isothermal @xmath213 , which is the approximation adopted by many theoretical studies that ignore the role of cloud turbulence . the result is not unexpected for low - mass cloud cores , but we would not naively have expected this relationship for high - mass cores and cloud envelopes , where the importance of turbulent motions is much greater . unfortunately , the observational data refer to different clouds rather than to different ( spatially averaged ) regions of the same cloud , so there is some ambiguity how to make the proper connection to different theoretical predictions . there may also be other mechanisms at work , e.g. , perhaps a tendency for observations to select for regions of nearly constant alfvn speed , @xmath214 constant ( bertoldi & mckee 1992 ) . thus , we would warn the reader against drawing premature conclusions about the effective eos for molecular clouds , or the related degree to which observations can at present distinguish whether molecular clouds are magnetically supercritical or subcritical . if molecular clouds are magnetically supercritical , with @xmath1 greater than 1 by order unity ( say , @xmath215 ) , then an appreciable fraction ( say , 1/2 ) of their support against self - gravity has to come from turbulent or thermal pressure ( elmegreen 1978 , mckee et al . 1993 , crutcher 1998 ) . modeled as scale - free equilibria , such clouds with @xmath82 of order unity are not highly flattened ( see tables 1 , 2 and figs . 2 , 3 ) . suppose we try gravitationally to extract a subunit from an unflattened massive molecular cloud , where the cloud as a whole is only somewhat supercritical , @xmath216 . if the subunit s linear size is smaller than the vertical dimension of the cloud by more than a factor of 2 , which will be the case if we consider subunits of stellar mass scales , then this subunit will not itself be magnetically supercritical . magnetically subcritical pieces of clouds can not contract indefinitely without flux loss , so star formation in _ unflattened _ clouds , if they are not highly supercritical , needs to invoke some degree of ambipolar diffusion in order to produce small dense cores that can gravitationally separate from their surroundings . on the other hand , if molecular clouds are magnetically critical or subcritical , with @xmath217 , then almost all scale - free equilibria are highly flattened , with @xmath82 appreciably less than unity . on a small scale , any subunit of this cloud , even subunits with vertical dimension comparable to the cloud as a whole , would also be magnetically critical or subcritical . for such a subunit to contract indefinitely , we would again need to invoke ambipolar diffusion to make a cloud core magnetically supercritical . thus , although the decay of turbulence can accelerate the formation of cloud cores , the ultimate formation of stars from such cores may still need to rely on _ some _ magnetic flux loss ( but perhaps not more than a factor of @xmath218 ) to trigger the evolution of the cores toward gravomagneto catastrophe and a pivotal state with a formally infinite central concentration . on the large scale , if gmcs are modeled as flattened isopedic sheets , @xcite proved that magnetic pressure and tension are proportional to the gas pressure and force of self - gravity . their theorems hold independently of the detailed forms of the eos or the surface density distribution . if gmcs are truly highly flattened with typical dimensions , say , of 50 pc @xmath219 50 pc @xmath219 a few pc or even less then many aspects of their magnetohydrodynamic stability and evolution become amenable to a simplified analysis through the judicious application and extension of the theorems proved by @xcite ( e.g. , see allen & shu 1998b , @xcite ) . this exciting possibility deserves further exploration . acknowledges support by cnr grant 97.00018.ct02 , asi grant ars-96 - 66 and ars-98 - 116 , and hospitality from unam , mxico . s.l . acknowledges support by j. s. guggenheim memorial foundation , grant dgapa - unam and conacyt , and hospitality from osservatorio di arcetri . f.c.a . is supported by nasa grant no . nag 5 - 2869 and by funds from the physics department at the university of michigan . the work of f.h.s . is supported in part by an nsf grant and in part by a nasa theory grant awarded to the center for star formation studies , a consortium of the university of california at berkeley , the university of california at santa cruz , and the nasa ames research center . shu , f. h. , allen , a. , shang , h. , ostriker , e. c. , & li , z. y. 1999 , in the physics of star formation and early stellar evolution ii , nato asi , ed . c. j. lada and n. d. kylafis ( dordrecht : kluwer ) , in press 1 & 1.94 & 1.50 & 0.668 0.9 & 1.80 & 1.68 & 0.632 0.8 & 1.63 & 1.76 & 0.543 0.7 & 1.45 & 1.80 & 0.434 0.6 & 1.27 & 1.84 & 0.321 0.5 & 1.09 & 1.91 & 0.218 0.4 & 0.927 & 2.14 & 0.122 0.3 & 0.947 & 5.06 & 0.0165 0.2 & 0.992 & 13.8 & 0.00181 0 & @xmath220 & 1 & @xmath220 0.05 & 4.35 & 1.18 & 6.93 0.1 & 2.83 & 1.21 & 2.81 0.2 & 1.85 & 1.31 & 1.08 0.3 & 1.45 & 1.45 & 0.577 0.4 & 1.23 & 1.64 & 0.348 0.5 & 1.09 & 1.91 & 0.218 0.6 & 1.01 & 2.34 & 0.134 0.7 & 0.980 & 3.00 & 0.0786 0.8 & 0.977 & 3.91 & 0.0448 0.9 & 0.982 & 4.95 & 0.0265 1.0 & 0.987 & 6.06 & 0.0165 1.1 & 0.991 & 7.22 & 0.0108 1.2 & 0.993 & 8.44 & 0.00733 1.3 & 0.995 & 9.71 & 0.00514 1.4 & 0.996 & 11.04 & 0.00369 1.5 & 0.997 & 12.42 & 0.00272 1 & @xmath168 & 1 & 2 0.9 & @xmath221 & 1.19 & 2.65 0.8 & @xmath222 & 1.52 & 3.61 0.7 & @xmath223 & 2.05 & 5.10 0.6 & @xmath224 & 2.93 & 7.55 0.5 & @xmath225 & 4.50 & 11.9 0.4 & @xmath226 & 7.59 & 20.5 0.3 & @xmath227 & 14.7 & 40.4 0.2 & @xmath228 & 36.8 & 102 0.1 & @xmath229 & 166 & 467 0 & @xmath169 & @xmath220 & @xmath220 | we investigate the equilibrium properties of self - gravitating magnetized clouds with polytropic equations of state with negative index @xmath0 .
in particular , we consider scale - free isopedic configurations that have constant dimensionless spherical mass - to - flux ratio @xmath1 and that may constitute `` pivotal '' states for subsequent dynamical collapse to form groups or clusters of stars . for given @xmath2 , equilibria with smaller values of @xmath1 are more flattened , ranging from spherical configurations with @xmath3 to completely flattened states for @xmath4 . for a given amount of support provided by the magnetic field as measured by the dimensionless parameter @xmath5 , equilibria with smaller values of @xmath6
are more flattened .
however , logatropic ( defined by @xmath7 ) disks do not exist .
the only possible scale - free isopedic equilibria with logatropic equation of state are spherical uniformly magnetized clouds . |
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this paper is concerned with the bianchi groups , i.e. the class of groups defined by psl@xmath0 of the integers of an imaginary quadratic number field @xmath3 $ ] where @xmath4 is a product of different primes . any such group is a group of orientation - preserving isometries of hyperbolic three space @xmath5 by means of the right - coset identification @xmath6 . we aim at computing the left hand side of the baum - connes - conjecture for these groups . our main result is theorem [ statement ] that accomplishes the calculation in the example case @xmath7 . + there is vast literature on the bianchi groups @xcite and the conjecture @xcite . the latter is concerned with the difficult problem to parametrize k - theory classes of projective modules over the reduced @xmath2-algebra of @xmath1 . this @xmath2-algebra is defined as the closure in the operator norm of the left regular representation of @xmath1 on @xmath8 . the conjecture asserts that the k - theory of @xmath9 is isomorphic to equivariant k - homology with @xmath1-compact supports . we shall use the `` official '' definition @xmath10 of equivariant k - theory with @xmath1-compact supports as in @xcite , where @xmath11 is a classifying space for proper actions of @xmath1 , and @xmath12 is equivariant bivariant k - theory as defined in @xcite . in the present context of a closed subgroup @xmath1 of a semisimple lie group @xmath13 , the symmetric space @xmath14 is a typical universal space @xmath11 @xcite , where @xmath15 is a maximal compact subgroup of @xmath13 . + the baum - connes conjecture applies equally to lie groups and has , in fact , been proved for all connected lie groups @xcite . however , most often the plancherel formula allows to understand the structure of the left regular representation of lie groups , whereas the @xmath2-algebras of arithmetic groups have turned out to be very hard to understand . one of the reasons is that for many arithmetic groups , the @xmath2-algebra is not type i , i.e. there are irreducible representations not contained in the operator ideal of compact operators . + hence , the knowledge of the k - theory of the @xmath2-algebra yields valuable information which seems not to be accessible by more elementary or more explicit approaches . furthermore , the baum - connes conjecture is , so far , the method of choice for computing the k - theory whenever the structure of the @xmath2-algebra is unknown . + however , it often turns out that the left - hand side is not immediately calculable either . the present paper is in spirit quite close to @xcite which computes the left - hand - side for the group @xmath16 ; however , it is unknown if the assembly map is surjective in this case . in contrast , julg and kasparov have verified the baum - connes conjecture for all discrete subgroups of @xmath17 and @xmath18 @xcite . since @xmath19 , this readily implies that the assembly map is an isomorphism for all bianchi groups . therefore , the bianchi groups are arguably the most interesting arithmetic groups for which complete knowledge of ( the isomorphism type of ) @xmath20 is available . + the canonical way to understand arithmetic groups is via actions on retracts of the symmetric space , as computed by reduction theory . here , we shall use ( a suitable subset of ) flge s cw - complex @xcite , a union of a 2-dimensional retract of hyperbolic 3-space with a countable subset of the satake ( spherical ) boundary of @xmath5 . these boundary points of the complex are the @xmath1-orbits of so - called singular vertices of the flge complex , and such orbits are in bijection with the non - trivial elements of the class group @xcite . the singular points are clearly visible in the visualizations of several exemplary fundamental domains in @xcite and @xcite . contractibility of the flge complex has been shown by rahm and the author in @xcite . + the equivariant @xmath15-homology of bianchi groups has been calculated by rahm in @xcite . however , he considers only the cases of trivial class number @xmath21 . the reason for this restriction is that in these cases there are no singular points . however , for higher class numbers the flge complex is not proper anymore , as the stabilizers of the singular points are parabolic . an algorithmic approach for computation of the complex has been introduced , described and exploited for the computation of the integral homology by a. rahm and the author in @xcite . furthermore , a program for the calculation of fundamental domains in the pari / gp language has been published as part of rahm s phd thesis @xcite . + since the presence of singular points in the flge complex leads to the action being non - proper , in these cases the flge complex is not a model for the universal classifying space @xmath11 of proper actions , and therefore not immediately suitable for calculation of equivariant @xmath15-homology for higher class numbers . moreover , the flge complex is not locally finite since there are infinitely many edges emanating from the singular points . so , there is no obvious locally compact topology on it , leading to severe difficulties in defining an associated commutative @xmath2-algebra . + there is a general construction for turning arbitrary , possibly non - proper , @xmath1-cw - complexes into proper actions @xcite which could in principle be applied to the flge complex ; however , the construction is not in general cofinite and therefore difficult to use for computational purposes . + in the present paper , we show how to overcome these difficulties . the approach pursued in this paper is more akin of classical @xmath2-algebraic techniques , and hence closer to the original object under study , the @xmath2-algebra @xmath9 . the four main ingredients are : reduction theory in the form of the flge complex , the existence of the borel - serre compactification , a certain amount of kasparov theory , and topology in the form of an atiyah - hirzebruch spectral sequence . + let us give a short summary of the steps and results . section [ fl ] introduces the topological objects under study . we identify the universal cover of the borel - serre compactification of the locally symmetric space associated to @xmath1 as a universal @xmath1-space ( lemma [ isup ] ) . one may note that for torsion - free subgroups of bianchi groups , borel - serre compactifications of the quotient @xmath22 ( which is then a manifold ) have already been mentioned in @xcite . there is an extension [ 55 ] of @xmath2-algebras naturally associated to the inclusion of the boundary component into this universal space . we use this to show that the equivariant @xmath15-homology of @xmath1 forms a 6-term exact sequence ( sequence [ hy2lhs ] ) with topological @xmath15-homology of a disjoint sum of tori and the kasparov k - homology of the crossed product @xmath23 . the latter , in turn , is determined ( lemma [ iszero ] ) by the k - homology of the subset @xmath24 of the flge complex @xmath25 . it is necessary to work with @xmath26 instead of @xmath25 in order to overcome the problem that @xmath25 is not locally finite . the space @xmath26 is naturally endowed with a locally compact topology , so there is no problem in associating a well - behaved commutative @xmath2-algebra to it . the goal of section [ ahs ] is to introduce the spectral sequence used to compute the k - homology of @xmath26 . in principle , the atiyah - hirzebruch spectral sequence starting from bredon homology associated to a group action on a complex is the classical tool to compute its equivariant k - homology . that spectral sequence is explained in @xcite and was used in @xcite . here , we have to consider a slightly more general form of the spectral sequence ( lemma [ bigone ] ) in order to be able to treat the space @xmath26 instead of @xmath25 . this generalization is done by identifying the atiyah - hirzebruch spectral sequence for the equivariant k - homology of a complex as the schochet spectral sequence associated to a filtration by closed ideals of a @xmath2-algebra . in this case , these ideals come from the intersections of the skeleta of @xmath25 with @xmath5 . at that point , all technical difficulties are resolved , and we can illustrate the computation by means of the concrete example @xmath7 in section [ concrete ] . this leads to the main theorem [ statement ] . + the bianchi groups , although far from being amenable , are k - amenable @xcite , as they are closed subgroups of a k - amenable group , see @xcite for the proof for @xmath27 . as a consequence , there is a kk - equivalence between the reduced and full group @xmath2-algebra . therefore , by calculating the k - theory of the former we simultaneously calculate that of the latter . hence , the bianchi groups may also be the most interesting lattices for which complete knowledge of the k - theory of the full algebra is now available . + i would like to thank prof . ralf meyer very much for several inspiring discussions on the subject , which alone made this paper possible . i would also like to thank prof . edgar wingender for the opportunity to complete the present work under his hood at the medical center gttingen . briefly , the flge complex @xmath28 is defined as follows , see @xcite for details and references . let now @xmath29 be square - free , and let @xmath30}$ ] and @xmath31 . denote the hyperbolic three - space by @xmath32 . the space @xmath5 is a homogeneous space under the lie group @xmath33 . the satake ( or spherical ) boundary @xmath34 of @xmath5 is @xmath35 , and the action of @xmath13 , and hence that of @xmath1 , extend continuously to actions on the boundary by mbius transformations @xcite*section 12 . a point @xmath36 is called a _ singular point _ if for all @xmath37 , we have @xmath38 . + we shall denote the union of all orbits of singular points by @xmath39 . note that the point @xmath40 is never in the orbit of a singular point . the set @xmath39 can either be viewed as a subset of @xmath41 , or of @xmath42 , or of @xmath43 . in the first case , @xmath39 is identified with the number field @xmath44 $ ] @xcite . + the set @xmath39 is a subset of the set of cusps of @xmath1 as defined in @xcite . serre considers the set @xmath45 of rational boundary points defined as @xmath3\cup \{\infty\}$ ] as a subset of the boundary viewed as the projective space @xmath46 . the action of @xmath1 falls into orbits which are in natural bijection with the class group , and the orbit corresponding to the trivial element is the orbit of @xmath47 , whereas the set @xmath39 is the union of all orbits corresponding to non - trivial elements of the class group . + furthermore , one considers the union of all hemispheres @xmath48 for any two @xmath49 with @xmath50 , as well as the `` space above the hemispheres '' @xmath51 let @xmath52 denote the modulus in @xmath53 . as a set , @xmath54 is the union @xmath55 ( where we omit , for simplicity , the standard identification @xmath56 . ) the topology of @xmath57 is generated by the topology of @xmath5 together with the following neighborhoods of the translates @xmath58 of singular points : @xmath59 this definition makes @xmath39 a closed subset of @xmath57 . + there is a retraction @xmath60 from @xmath57 onto the set @xmath61 of the union @xmath39 with all @xmath1-translates of @xmath62 , i. e. there is a continuous map @xmath63 such that @xmath64 for all @xmath65 . the set @xmath25 admits a natural structure as a cellular complex @xmath28 , such that @xmath1 acts cellularly on @xmath28 . we shall refer to the complex thus defined , as well as to its structure as a @xmath1-subset of @xmath57 , as to the `` flge complex '' . we shall not make use of any topology on @xmath25 , only of the structure as a subset of @xmath57 and of the combinatorial structure . since @xmath66 comprises bianchi s fundamental polyhedron ( which is called @xmath67 in @xcite ) , we have @xmath68 and hence this definition of @xmath25 coincides with flge s original definition as the @xmath1-closure of @xmath69 . note that the topology of @xmath57 is not the one inherited from the satake ( spherical ) compactification of @xmath5 . however , @xmath57 coincides on @xmath5 with the usual topology ; furthermore , it is path - connected , locally path - connected and simply connected @xcite*satz 1 , and contractible @xcite*lemma 8 , as is the cellular complex @xmath28 @xcite*corollary 7 . + since @xmath25 is not locally finite at the points of @xmath39 , we shall not work with @xmath25 directly , but only with the pruned intersection @xmath70 of @xmath25 with @xmath5 . on @xmath71 , the cellular topology coincides with the topology inherited from @xmath5 , and is locally compact ( unlike that of @xmath57 ) , whence there are no difficulties in associating a @xmath2-algebra @xmath72 . although @xmath71 is not a complex anymore , the @xmath2-algebra still possesses a filtration by closed ideals , defined by functions that vanish on the intersections of the skeleta of @xmath25 with @xmath5 . the associated schochet - spectral sequence will be one of the keys to the calculation of the equivariant k - homology of @xmath1 . in the following , the pruned cellular complex @xmath73 endowed with the subset topology of @xmath5 shall be denoted by @xmath26 , and its pruned skeleta by @xmath74.[defofcirc ] + it is important to note that any two edges adjacent to a common singular point in @xmath25 define half - open intervals that are _ disjoint _ from each other as subsets of @xmath26 . furthermore , the space @xmath26 is , as subset of @xmath5 , locally compact . this contrasts the failure of local finiteness of @xmath25 . as a warning , one should not expect to have @xmath26 the same equivariant k - homology as the subcomplex of @xmath25 that consists of all cells that do not touch a singular point although these spaces are equivariantly homotopic . the reason is that these homotopies are not proper since the singular points are at infinity , for the geometry of @xmath26 . ( recall that the `` compactly supported '' version @xmath75 of k - homology is only invariant under proper homotopies . ) besides the flge construction @xmath57 , we shall need another enlargment of @xmath5 , the borel - serre construction @xmath76 for the set @xmath77 of rational boundary points . very roughly speaking , the space @xmath76 is obtained from @xmath5 by gluing copies of @xmath78 onto each @xmath79 ( whereas @xmath57 was obtained from @xmath5 by merely gluing copies of a point onto each @xmath80 ) . let us recall the setup and notation of serre s introduction of the borel - serre boundary for linear algebraic groups of @xmath81-rank one @xcite*appendix 1 . + serre s notation translates as follows . the space @xmath5 is hyperbolic space , @xmath34 is the ordinary spherical ( satake ) boundary of @xmath5 , @xmath67 is a translate of a singular point or of the point @xmath47 , the subgroup @xmath82 is its stabilizer inside @xmath13 . we are going to view @xmath5 as a space acted upon by @xmath13 from the right ; thus , there is the natural identification @xmath83 . the group @xmath82 is a minimal parabolic subgroup , and the singular points defined by flge naturally embed into the satake boundary . for instance , consider the simplest choice where @xmath67 is the origin of the poincar plane . then @xmath84 is the set of upper triangular matrices in @xmath1 , hence isomorphic to @xmath30}$ ] . even though there are several orbits of singular points , all stabilizer groups @xmath84 are free abelian of rank two @xcite . + the group @xmath85 is conjugated inside @xmath86 to the group @xmath87 whence @xmath88 is isomorphic to @xmath89 , the multiplicative group of non - zero complex numbers . the space @xmath90 is defined as the union of hyperbolic space with a boundary component @xmath91 , defined as the space of rank one - tori of @xmath82 . in our cases , all @xmath91 are diffeomorphic to @xmath78 . more specifically , we obtain for @xmath92 , the regular cusp at infinity of the riemann sphere : @xmath93 furthermore , there is a unique fixed point @xmath94 of the cartan involution on @xmath91 defined by @xmath15 . this defines an iwasawa decomposition @xmath95 associated to the subgroup @xmath15 , where @xmath96 is the neutral connected component of @xmath94 , and @xmath97 denotes the ordinary multiplication in the group @xmath13 . the group @xmath96 identifies with @xmath98 in a canonical way ( induced by the positive root of @xmath94 with respect to the order relation coming from @xmath85 ) , and for @xmath99 , the associated element in @xmath96 is denoted by @xmath100 . these observations permit to topologize the space @xmath90 by means of the family of bijective maps @xmath101 , defined by @xmath102 where @xmath15 is a maximal compact subgroup of @xmath13 . these maps are compatible for different choices of @xmath15 and therefore define a unique structure of manifold with boundary on @xmath90 , independent of @xmath15 . + let now @xmath45 be the set of cusps of @xmath1 , i.e. the set of orbits of singular points together with the orbits of the trivial cusp at @xmath47 . in serre s terminology , the space @xmath76 is the union @xmath103 defined by gluing the spaces @xmath90 together along their common intersection @xmath5 . hence , as a set , @xmath76 is the disjoint union @xmath104 . serre shows that @xmath105 is compact , and called the borel - serre compactification of @xmath106 . the space @xmath76 is its universal cover . + of course , it is also possible to consider a slightly smaller @xmath1-space , defined by considering only the subset @xmath39 of @xmath45 , @xmath107 in order to obtain a space that surjects onto @xmath57 . in fact , one has there is a natural map @xmath108 defined by collapsing each boundary component to the corresponding singular point . this map is continuous and surjective . this is proved by checking directly that preimages of the open sets that define the topology on @xmath57 are open in @xmath109 . the relation between the spaces @xmath110 and @xmath111 can be summarized by the equivariant diagram @xmath112&\partial{\mathcal{h}}(s)\ar[l]\ar[d]\\ { \mathcal{h}}\ar[r]\ar[u ] & \widehat{{\mathcal{h}}}. } \ ] ] the following lemma uses the notion of amenable transformation group , discussed in detail in @xcite . [ biglemma ] let @xmath113 with @xmath114 . then the transformation group @xmath115 is amenable . in particular , the @xmath2-algebraic crossed product @xmath116 is nuclear and unique , i.e. the quotient map from the maximal to the reduced crossed product is an isomorphism . + the set @xmath45 can be replaced by any @xmath1-closed subset of @xmath45 , but we shall not need that fact . by @xcite*theorem 5.3 , it suffices to show amenability of the transformation group . + for @xmath117 , write the symmetric space as a coset space @xmath118 where @xmath15 is a maximal compact subgroup of @xmath13 . for two closed subgroups @xmath1 and @xmath15 of a locally compact group @xmath13 , the associated transformation group @xmath119 is an amenable transformation group @xcite*example 2.7(5 ) . + for @xmath120 , observe that the action of @xmath1 on @xmath121 is proper in the sense that the map @xmath122 defined by @xmath123 is topologically proper because the action is free and cocompact . any proper action defines an amenable transformation group . in fact , the functions @xmath124 satisfy the conditions of @xcite*propositions 2.2(2 ) where @xmath125 is a continuous non - negative function on @xmath121 such that @xmath126 for all @xmath127 . + form the associated extension of maximal crossed products associated to the invariant ideal @xmath128 in @xmath129 ( for maximal crossed products , exactness is automatic ) . we have shown that the ideal and the quotient are nuclear ( and therefore coincide with minimal crossed product ) . nuclearity for @xmath130 then follows from the fact that nuclearity is stable under extensions @xcite . moreover , since @xmath1 is discrete , it has property @xmath131 @xcite*example 4.4(3 ) , and therefore also the transformation groups @xmath132 and @xmath133 are amenable @xcite*theorem 5.8 . + for @xmath134 , write @xmath45 as the disjoint union of orbits @xmath135 ( this is a decomposition in finitely , namely @xmath4 components , where @xmath4 is the class number ) . any crossed product splits into a direct sum over these components , so it is enough to prove the statement separately for each . on such a component , we can apply @xcite*example 2.7(5 ) again , this time to the ambient group @xmath1 ( which is locally compact ) and the two closed subgroups @xmath1 itself and the stabilizer @xmath136 , since the orbit @xmath137 is then equal to the @xmath1-space @xmath138 , and @xmath139 is abelian , hence amenable . lemma [ isup ] below uses the following simple criterion , well - known in the literature , for a space to be universal , i. e. to serve as a model for the classifying space for proper actions for @xmath1 . note that whereas @xmath5 is both a universal @xmath1- and @xmath13-space , the space @xmath76 is not acted upon by @xmath13 . any free and proper @xmath1-space is universal if and only if it is @xmath140-equivariantly contractible for any finite subgroup @xmath141 . [ isup ] for any @xmath1-closed subset @xmath142 , the universal cover @xmath76 , as constructed in @xcite*appendix 1 , of the borel - serre compactification of @xmath143 is a universal proper @xmath1-space . by construction , the action of @xmath1 on @xmath144 is free . moreover , for each boundary component we can choose an isomorphism of its ( non - pointwise ) stabilizer with @xmath145 such that its action on the boundary component is equivariantly homeomorphic to the action of @xmath145 on @xmath78 , hence proper . thus , the action on @xmath121 is proper . it ensues that @xmath76 is proper . let @xmath140 be a finite subgroup of @xmath1 . the quotient @xmath146 is an orbifold with boundary which has the same homotopy type as its interior @xmath147 . the latter is contractible because @xmath5 is a universal proper @xmath1-space @xcite*section 2 . so @xmath146 is contractible . moreover , since @xmath76 is contractible , the contraction of @xmath146 thus obtained lifts to an @xmath140-equivariant contraction of @xmath76 whence the assertion . in fact , the space @xmath5 itself is the typical example of a universal proper @xmath1-space , so @xmath5 and @xmath76 can both be used as models for @xmath11 and for computation of equivariant k - homology . an isomorphism @xmath148 is induced by the inclusion map @xmath149 . however , @xmath76 has , by construction , the advantage that it is cocompact unlike @xmath5 . therefore , we can pass from @xmath150 to @xmath151 as follows . @xmath152 the last isomorphism is the dual green - julg theorem @xcite*theorem 20.2.7(b ) . since the crossed product is nuclear , every ideal is semisplit @xcite*theorem 15.8.3 . + throughout the paper , the reader should bear in mind that the notations @xmath153 and @xmath154 refer to the `` original '' kasparov kk - groups instead of the compact - support group @xmath150 . ( we shall prefer to work with the former . ) for a commutative @xmath2-algebra @xmath155 , the group @xmath156 is k - homology with locally finite support , rather than the usual group with compact support . locally finite k - homology is called k - homology with compact support in @xcite . we can now make use of 6-term sequences which are available in @xmath151 , in contrast to @xmath150 . the space @xmath26 was defined in subsection [ defofcirc ] . the sequences @xmath157 defined by the respective evaluation maps , are exact and of course equivariant . note that in the latter sequence we choose to work with @xmath76 instead of @xmath109 because only @xmath76 is cocompact , whence leading to equivariant k - homology . + note that none of these algebras have a unit . we can take reduced crossed products by @xmath1 . ( we have shown that they coincide with the maximal ones . ) it is well - known @xcite*rem . 14 that a @xmath1-invariant ideal @xmath158 in a coefficient @xmath1-algebra @xmath159 induces a short exact sequence of reduced crossed products @xmath160 when @xmath1 is discrete . therefore , we arrive at a short exact sequence of reduced crossed products with the sequences [ 55 ] . these induce exact 6-term sequences in k - homology @xmath161 for this , note that all algebras occurring in extensions throughout this paper are nuclear , and therefore all extensions are semisplit , so excision in kk - theory holds . + for a countable discrete group @xmath1 , there is an identification of @xmath162 with @xmath163 @xcite . + the k - homology of the quotient @xmath2-algebras is easy to compute . to start with the second extension , the action of @xmath1 on @xmath144 is free and proper , so there is a strong morita equivalence @xmath164 . let @xmath165 denote the class number of the underlying number field . then the number of orbits of singular points is @xmath165 . hence , the quotient space is a disjoint union of @xmath165 compact 2-tori @xcite , so we have @xmath166 hence , the second ses . of [ 55 ] allows to compute the equivariant k - homology of to compute that of @xmath76 from @xmath5 . + the former ses . of [ 55 ] , in turn , allows to compute that of @xmath5 from that of @xmath26 . as we have shown in lemma [ isup ] , the space @xmath76 is a classifying space , so this will achieve the computation . + the interior of bianchi s fundamental polyhedron has trivial stabilizer @xcite . therefore , the action on @xmath167 is free and proper so @xmath168 is morita equivalent to @xmath169 which is isomorphic to @xmath170 of an open 3-cell , i.e. @xmath171 . therefore , @xmath172 thus , we can compute the left hand side @xmath173 via successive computation of the groups @xmath174 and @xmath175 without support condition , by the exact 6-term sequences @xmath176 & { \operatorname{k}}^0(\gamma{\ltimes}\mathscr{c}_0({\mathcal{h } } ) ) \ar[r ] & 0 \ar[d]\\ { { \mathbb{z}}}\ar[u ] & { \operatorname{k}}^1(\gamma{\ltimes}\mathscr{c}_0({\mathcal{h } } ) ) \ar[l ] & { \operatorname{k}}^1(\gamma{\ltimes}\mathscr{c}_0(x_\circ ) ) \ar[l ] } \ ] ] and , using [ tori ] , @xmath177 & { r{\operatorname{k}}^\gamma_0(\underline{e}\gamma)}\ar[r ] & k^0(\gamma{\ltimes}\mathscr{c}_0({\mathcal{h } } ) ) \ar[d]\\ k^1(\gamma{\ltimes}\mathscr{c}_0({\mathcal{h}}))\ar[u ] & { r{\operatorname{k}}^\gamma_1(\underline{e}\gamma)}\ar[l ] & \mathbb{z}^{2k } \ar[l ] } \ ] ] which paves the way to reduce the computation of the equivariant k - homology of @xmath1 to that of the @xmath15-homology of the reduced crossed product and that of the boundary tori . the invertible in @xmath178 ( namely , kasparov s dual - dirac element ) leads to an isomorphism @xmath179 . therefore , rewriting the 6-term sequence [ hy2lhs ] and using the fact that the assembly map is an isomorphism , we arrive at the following exact 6-term - sequence . @xmath180 & { \operatorname{k}}_0({c^\ast_{\textit{red}}}\gamma)\ar[r ] & r_1(\gamma ) \ar[d]\\ r_0(\gamma)\ar[u ] & { \operatorname{k}}_1({c^\ast_{\textit{red}}}\gamma ) \ar[l ] & \mathbb{z}^{2k } \ar[l ] } \ ] ] where @xmath181 is the kasparov representation ring . this is remarkable since there are rarely exact sequences available that connect k - theory and k - homology of the same algebra ( in view of @xmath182 by k - amenability . ) [ iszero ] the connecting homomorphism of the 6-term - sequence [ x2hy ] is zero , so there is an isomorphism @xmath183 and a short exact sequence @xmath184 let @xmath185 denote the interior of the bianchi fundamental polyhedron which is called @xmath67 in @xcite ( in the present manuscript , the letter @xmath67 is already occupied ) ; hence , there are homeomorphisms @xmath186 and @xmath187 because the interior of @xmath188 is trivially stabilized . it is well known that the kk - equivalence coming from the strong morita - rieffel equivalence @xmath189 is induced by the inclusion of @xmath2-algebras @xmath190 where the first algebra is viewed as a crossed product with the trivial group , and a function on @xmath188 is viewed as a function on @xmath167 by setting it to zero outside @xmath188 . + consider the map of short exact sequences @xmath191 & \mathscr{c}_0(\mathcal{d})\ar[d ] \ar[r]^{= } & \mathscr{c}_0(\mathcal{d } ) \ar[r]\ar[d ] & 0 \ar[r]\ar[d ] & 0\\ 0 \ar[r ] & \gamma{\ltimes}\mathscr{c}_0({\mathcal{h}}- x_\circ ) \ar[r ] & \gamma{\ltimes}\mathscr{c}_0({\mathcal{h } } ) \ar[r ] & \gamma{\ltimes}\mathscr{c}_0(x_\circ ) \ar[r ] & 0 } \ ] ] we claim that the center vertical arrow induces a surjective map @xmath192 in k - homology ( recall that we are dealing with locally finite homology ) . the assertion then follows . + consider the following diagram of @xmath193-theory groups , @xmath194\\ { \operatorname{k}}^0(\text{pt } ) \ar[ur]\ar[r ] \ar[d ] & { \operatorname{k}}^1({\mathcal{h}})\ar[d]\\ { \operatorname{k}}_0({c^\ast_{\textit{red}}}\gamma)\ar[r ] & { \operatorname{k}}_1(\gamma{\ltimes}{\mathcal{h } } ) } \ ] ] where all vertical arrows are induced by @xmath195-inclusions , the diagonal arrow is the bott isomorphism associated to the standard orientation of @xmath196 , the upper horizontal arrow is bott isomorphism , and the lower horizontal arrow is multiplication with kasparov s dual - dirac element @xmath197 . the commutativity of the triangle is inherent in the definition of the bott element , and that of the square follows from the fact that the forgetful homomorphism @xmath198 sends @xmath199 to the bott element . + all arrows except the two lower vertical arrows are isomorphisms ( recall that @xmath199 is invertible ) . it follows that the dual map @xmath200 is injective , since the class of the unit in @xmath201 is non - zero and non - torsion . it is classical that this follows from the existence of the trace map @xmath202 which sends @xmath203 $ ] to @xmath204 . this completes the proof . bredon homology is the main tool for computation of equivariant k - homology in similar contexts @xcite , so we refer to these references for a more thorough introduction . here , we shall only fix the notation . + recall the usual terminology for bredon homology : the orbit category @xmath205 of @xmath1 has an object @xmath206 for each subgroup @xmath140 of @xmath1 , and all @xmath1-maps between any two objects as morphisms . a bredon module is a functor @xmath207 from the orbit category to abelian groups . here , we are going to consider the bredon module @xmath208 that associates to @xmath206 the k - homology group @xmath209 . this reduces to even degree where it is @xmath210 canonically . let @xmath211 be a homomorphism of discrete countable groups . the differentials @xmath212 of the atiyah - hirzebruch spectral sequence are given by left multiplication by an abstractly defined element @xmath213 . bredon homology of a proper complex @xmath214 with respect to the family @xmath215 of finite subgroups with coefficients in the bredon module @xmath216 is defined as the homology of the cellular complex with coefficients in @xmath216 @xmath217 where the sum extending over orbit representatives . in case @xmath140 and @xmath218 are finite , @xmath208 is a functor @xmath219 , and @xmath220 defines a map @xmath221 which reduces to the usual bredon differential as described in @xcite , and in even more detail in @xcite . specifically , the latter reference gives the information about the morphisms induced on the representation rings by the finite group inclusions occurring among stabilizers of flge simplices , as well as details about how to simplify the computations in the pari / gp language @xcite . in fact , the complex representation ring of a finite group as the free @xmath222-module the basis of which are the irreducible characters of the group . these irreducible characters are given by the character tables for the finite subgroups of the bianchi groups . one has to consider all possible inclusions , and identify the said morphism ; then , this information is fed into the program bianchi.gp in order to obtain the bredon chain complex , from which we shall easily deduce the information on the modified bredon chain complex , introduced below . lemma [ iszero ] motivates to calculate the k - homology of @xmath223 . in the usual setting of a proper cellular complex , there is the atiyah - hirzebruch spectral sequence computing equivariant k - homology . its @xmath224-term is the bredon complex , and its @xmath225-term is bredon homology . for the present purpose , we modify this setting in the following way . note that @xmath26 is not a cellular complex . however , there is still a natural notion of @xmath226-cell , with the difference that there are one - cells that are adjacent to only one vertex . in the cellular complex @xmath25 , there are one- and two - cells @xmath227 adjacent to singular points . these are compact subsets of @xmath25 . however , their intersection @xmath228 is , of course , not compact in @xmath26 . however , this makes no difference with regard to the combinatorial structure : they are still adjacent to the same edges . since the definition of bredon homology only takes into account the combinatorial structure , we may still write down a combinatorial bredon complex for @xmath26 , and this complex still satisfies @xmath229 . we shall refer to this complex as the _ modified bredon complex _ and accordingly to its homology as the _ modified bredon homology _ , denoted by @xmath230 with coefficients in the bredon module @xmath208 . the following lemma says , roughly speaking , that this modification of bredon homology is not only a modification of the @xmath224 term , but that in fact this modified @xmath224-term fits into an entire modified spectral sequence whose @xmath224-term is the modified @xmath224-term . furthermore , this modified spectral sequence computes `` the right thing '' , namely the equivariant k - homology of @xmath26 . + let @xmath208 denote the bredon module that associates to @xmath206 the k - homology group @xmath231 as above . [ bigone ] there is a homological spectral sequence computing equivariant k - homology , @xmath232 where the sum extends over representatives @xmath233 of orbits of @xmath226-cells in @xmath26 , and @xmath234 is the respective stabilizer isomorphism type . it is concentrated in the first and fourth quadrants , and 2-periodic in the index @xmath235 . its differential @xmath236 has bidegree @xmath237 . + the summand of the @xmath224-term belonging to the cells of @xmath26 that do not touch a singular point is equal to the usual bredon complex of these cells . the summand of the @xmath224-term belonging to the one - cells of @xmath26 that touch a singular point has only a differential to one vertex orbit . the summand of the @xmath224-term belonging to the two - cells of @xmath26 that touch a singular point is equal to the usual bredon complex associated to the combinatorial structure . therefore , the @xmath225-term is @xmath230 . + this is a slight generalization of the atiyah - hirzebruch spectral sequence . it is constructed as the spectral sequence associated as in @xcite to the three - step filtration by closed ideals @xmath238 of functions vanishing on the pruned skeleta . + the sub - quotient @xmath239 is a direct sum over the orbits of @xmath235-cells , and a summand corresponding to the orbit of a cell @xmath240 stabilized by the ( necessarily finite ) subgroup @xmath241 is isomorphic to @xmath242 which is strongly morita equivalent to the @xmath235-fold suspension of the complex group ring @xmath243 whose k - homology is isomorphic to @xmath244 . this yields the atiyah - hirzebruch spectral sequence for equivariant k - homology , upon implementing the @xmath235-fold dimension shifts pertaining to the cells dimensions . its @xmath224 computes ( and , therefore , its @xmath225-term is ) bredon homology with respect to finite subgroups @xcite , as it is considered in @xcite for computation of equivariant k - homology . the fact that we deal with pruned skeleta introduces no peculiarities since on the level of sub - quotients it only means to pay attention to the fact that for each edge @xmath245 touching a singular point , there is only one inclusion of vertex stabilizers instead of two ( recall that each edge touches at most one singular point ) . it is easy to visualize the modified atiyah - hirzebruch spectral sequence thus defined for a very simple toy case of a pruned cellular complex , for instance the case of a single edge acted upon by the trivial group . one can then compare with the schochet spectral sequence associated to a half - open interval with the filtration @xmath246 , where the endpoint is the single simplex . the latter s @xmath224-term takes the form @xmath247 in even rows , and zero in odd rows , according to a single edge and a single point . the homology of this vanishes , in accordance with @xmath248 . bearing this example in mind might help in understanding the slight generalization of the atiyah - hirzebruch spectral sequence to pruned cellular complexes , considered here . roughly speaking , `` it is enough to omit the missing points from the spectral sequence '' . let us write down how to pass , in our case where the spectral sequence is applied to the algebra @xmath223 , from @xmath249 to the k - homology using the edge homomorphisms . the odd rows of the @xmath250-term vanish since @xmath251 of a finite - dimensional @xmath2-algebra such as a finite group s ring necessarily vanishes . since @xmath252 , the spectral sequence is concentrated in the columns @xmath253 . therefore , @xmath254 , and there is an isomorphism @xmath255 and a short exact sequence @xmath256 let us go through all computations for @xmath257 . details , including a picture , of a simple fundamental domain for the action of @xmath25 are given in @xcite and @xcite*section 3.2 . we shall stick to the notations therein . + recall that we set out to calculate the spectral sequence , associated to a commutative filtrated algebra , that almost comes from the skeleton filtration of a cellular complex algebra . however , there is `` one vertex orbit missing '' in @xmath26 . so , there are only four orbits of vertices , namely those of @xmath258 , but the same numbers of edges and faces as in @xcite*section 3.2 , namely seven orbits of edges , those of @xmath259 , and three orbits of faces . stabilizers are given in explicit form in the same references . there is only one orbit of singular points , since the class number of @xmath260 $ ] is two . + all information on the representation rings @xmath261 of finite stabilizers of vertices @xmath227 is given in @xcite and used as explained in subsection [ bredon ] . + the @xmath224-term is a complex @xmath262 in even rows , and zero in odd rows . the @xmath263-matrices are displayed in tables [ d_one ] and [ d_two ] . their elementary divisors are easily determined ( for instance by the pari / gp software @xcite ) by calculating the smith normal forms , yielding @xmath264 and @xmath265 therefore , the modified bredon homology is @xmath266 where we write @xmath267 . ( a quick check of a necessary criterion is the correct rational euler characteristic @xmath268 . ) using [ k1 ] and [ extension ] , we obtain @xmath269 using lemma [ iszero ] , we have @xmath270 we can now state the main theorem . it only remains to solve the 6-term extension problem [ hy2lhs ] . recall that the class number @xmath165 is two . the strategy for solving this 6-term problem is similar to the one used in @xcite where the analogous long term sequence in homology was computed . as stated in the introduction , this simultaneously computes @xmath273 and @xmath274 . + analogous computations for other @xmath4 should present no fundamentally new difficulties . however , they are beyond the scope of the present paper . + another possibly interesting question asks for the least matrix size over @xmath9 containing an idempotent that realizes any given k - theory class associated to a list of generators of the left hand side thus computed . | we compute the equivariant k - homology of the groups psl@xmath0 of imaginary quadratic integers with trivial and non - trivial class - group .
this was done before only for cases of trivial class number .
+ we rely on reduction theory in the form of the @xmath1-cw - complex defined by flge .
we show that the difficulty arising from the non - proper action of @xmath1 on this complex can be overcome by considering a natural short exact sequence of @xmath2-algebras associated to the universal cover of the borel - serre compactification of the locally symmetric space associated to @xmath1 .
we use rather elementary @xmath2-algebraic techniques including a slightly modified atiyah - hirzebruch spectral sequence as well as several 6-term sequences . +
this computes the k - theory of the reduced and full group @xmath2-algebras of the bianchi groups . |
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polymer networks are unique soft solids which can be significantly deformed without irreversible damage to their structure . a network is formed by cross - linking a melt or a semidilute solution of polymer chains . once a homogeneous ( on length scales large compared to its `` mesh '' size ) network is formed , one can generate large - scale patterns in it by further cross - linking , followed by swelling ( and possibly stretching ) of the network , resulting in a gel inhomogeously swollen by solvent . this can be done , for example , by adding light - sensitive cross - links to a transparent network . focusing a laser beam in regions inside the gel one can write information into gel structure in the form of @xmath1 or @xmath2 patterns of cross - linking density . in this paper we show that although such information is hidden at preparation conditions , it can be recovered by swelling the gel since unobservable variations of cross - link density in the melt are transformed into observable variations of monomer density in the swollen gel . regions of a gel with increased cross - link concentration can be considered as inclusions with enhanced elastic modulus . if such inclusions deform differently from polymer matrix , as in case of any normal elastic solids , they would induce elastic stresses in the gel and initial pattern would be significantly distorted due to long range character of elastic interactions . this scenario determines , for example , the elastic properties of amorphous polycrystalline solids but it does not apply to polymer gels , because of the unusual character of gel elasticity . we show that in swollen gels that are isotropically stretched by absorption of solvent , the observed _ monomer density pattern _ is not distorted and is simply an affinely stretched variant of the _ initial cross - linking pattern_. such gels can serve as a magnifying glass that enlarges the initially written pattern without distorting its shape . the corresponding magnification factor can be very large in case of super - elasic networks . in this paper we use the simplest mean field model of a gel with free energy @xcite:@xmath3 \right ) d\mathbf{x}_{0 } \label{ai}\ ] ] here @xmath4 is monomer density as function of coordinates @xmath5 in preparation state . we assume that the gel was initially cross - linked in a polymer melt and then a pre - programmed pattern in cross - link concentration ( i.e. , a well - defined region of higher cross - link density compared to that of the surrounding network ) is created in the network using , say , a light - sensitive cross - linking technique ( the case of cross - linking in semi - dilute solution in good solvent is analyzed in si ) . here @xmath6 is the polymer contribution to the elastic modulus of the cross - linked melt ( which is proportional to the local cross - link density)@xmath7 and @xmath8 represents the variations of cross - link density introduced by the second cross - linking step ( fig . [ pattern]a ) . pattern3d @xmath9 is the osmotic ( interaction ) part of the free energy of the gel , with monomer density @xmath10 . @xmath11 is the deformation gradient tensor@xmath12 and @xmath13 are coordinates of deformed gel . it is convenient to assume that the gel is deformed with respect to preparation state in two stages:@xmath14 thus , the gel is first stretched with respect to preparation state by factors @xmath15 along axes @xmath16 . for such a deformation @xmath17 with components @xmath18 and we get @xmath19 where @xmath20 is the uniform monomer density in the undeformed state of preparation ( fig . [ pattern]b ) . notice that the coordinates @xmath13 describe a stretched network with inhomogeneous cross - link density but a _ homogeneous monomer density _ [ pattern]c and d ) . even though such a homogeneous ( in monomer density ) state does not minimize the free energy and therefore is not an equilibrium state of the deformed gel , we use it as a reference state . the true equilibrium state of the deformed network has an inhomogeneous monomer density profile and is defined by introducing a displacement field @xmath21 defined with respect to the above reference state : @xmath22 and we get gradient tensor and monomer density as function of coordinates @xmath13@xmath23 minimizing the free energy in eq . ( [ ai ] ) with respect to displacements @xmath24 at the preparation state ( all @xmath25 ) we conclude that in a melt the cross - links and the monomers will remain at their previous possitions and the elastic reference state will not change after relaxation . we conclude that information about the pattern written on network structure is hidden in preparation state and can only be revealed after swelling . in a swollen state the monomer density is small and the interaction energy can be expanded as @xmath26 , where @xmath27 is boltzmann constant , @xmath28 is temperature and @xmath29 is second virial coefficient . expanding the free energy in powers of @xmath24 and integrating over the volume of the undeformed network with measure @xmath30 we get@xmath31 \frac{d\mathbf{x}}{\prod_{i}\lambda _ { i}},\quad k_{\text{os}}=k_{b}tb\bar{\rho}^{2}. \label{a}\ ] ] the equilibrium deformation of the gel is found by minimizing this free energy . its variation is@xmath32 and therefore , the minimum condition is@xmath33 we are interested only in variations of monomer density@xmath34 where @xmath35 is average density . taking the gradient of both sides of eq . ( [ gu ] ) we obtain an equation for the variations of monomer density@xmath36 where@xmath37 the solution of this equation@xmath38 \tilde{g}\left ( \lambda^{-1}\mathbf{\cdot x}\right ) \frac{d\mathbf{y}}{\prod_{j}\gamma_{j } } \label{rg}\ ] ] is expressed through green s functions of the laplace equation in 2 and 3 dimensions , respectively:@xmath39 in case of isotropically stretched / swollen gel with all @xmath40 the equilibrium monomer density depends on local cross - link concentration,@xmath41 we conclude that under isotropic deformation such as swelling , the monomer density produces an undistorted , uniformly stretched image of the pattern of cross - link density originally written on the homogeneous network ( compare figs . [ pattern]a and e ) . equilibrium displacement is expressed through the variation of monomer density , eq . ( [ dr ] ) , as@xmath42 we conclude that although density variations in isotropically deformed gels are strictly local , there is long - range strain field decaying as power law of the distance @xmath43 . this strain induces a stress distribution in the gel , which can be observed by measuring the birefringence of transmitted light ( stress - optical law @xcite ) . in anisotropically deformed networks the pattern is strongly distorted ( compare figs . [ pattern]a and f ) and @xmath44 decays as power law of a distance @xmath45 from the localized cross - link density inhomogeneity @xmath46 . observe that variations of monomer density are largest along the direction of stretching . this effect is closely related to the well known butterfly picture in contour plots of neutron scattering from random inhomogeneities of network structure in anisotropely deformed swollen gels @xcite . in order to understand the difference between gels and normal solids we recall that the free energy of any solid is a functional of the nonlinear strain tensor @xmath47 @xcite,@xmath48 while the last term is usually neglected in the linear theory elasticity of solids because solids behave elastically only under small deformations , it can be shown that only this nonlinear term contributes to the elasticity of gels and that the elastic part of the free energy of gels ( eq . ( [ ai ] ) ) is linear in this nonlinear strain@xcite . physically , the difference between elastic energy _ of a solid , _ which is a quadratic form in the linear strain and _ of a gel , _ which is linear in the nonlinear strain tensor , stems from the fact that while in solids there is a stress - free state of equilibrium ( crystal lattice ) that minimizes the energy of interaction between the atoms , the equilibrium state of gels is not stress - free . polymer networks are made of entropic springs and , in the absence of osmotic pressure due to permeation by good solvent or due to excluded volume interactions in the melt state , such networks would collapse to the size of a single spring . the finite length of entropic springs in the swollen gel is the result of osmotic pressure which can be replaced by equivalent isotropic stretching forces that act on the outer boundaries of the gel @xcite . the difference between gels and solids becomes apparent when considering two simple toy models of heterogeneous gel and solid as two hookean springs with moduli @xmath49 and @xmath50 , connected in series as in fig . [ toy ] : toy _ a ) gel model : _ osmotic pressure is represented by a force @xmath51 applied to free ends of the connected springs . in the presence of this force the equilibrium lengths of the gaussian springs become @xmath52 and @xmath53 , and if we apply additional force @xmath54 , each of the springs will deform affinally with distance @xmath55 between the ends to which the force is applied ( boundaries of the system ) : @xmath56 _ b ) solid model : _ the springs of a solid have equilibrium lengths @xmath57 and @xmath58 in the stress - free state . during stretching due to force @xmath54 applied to the ends of the two - spring system , such a solid deforms nonaffinelly:@xmath59{c}r_{1}=r_{1}^{\text{eq}}+\left ( \lambda-1\right ) \left ( r_{1}^{\text{eq}}+r_{2}^{\text{eq}}\right ) \dfrac{k_{2}}{k_{1}+k_{2}},\\ r_{2}=r_{2}^{\text{eq}}+\left ( \lambda-1\right ) \left ( r_{1}^{\text{eq}}+r_{2}^{\text{eq}}\right ) \dfrac{k_{1}}{k_{1}+k_{2}}\end{array}\ ] ] with the soft spring ( @xmath60 ) stretched more than the rigid one . these two simple toy models illustrate why under isotropic deformations , cross - linking density patterns in gels are stretched affinally , whereas soft regions in solids would undergo larger deformation compared to more rigid regions , thus distorting the original pattern . we studied the combined effect of swelling and deformation on inhomogeneous networks , prepared by cross - linking a melt of polymer chains . it is well - known that cross - link density heterogeneities that have no effect on the monomer density in the state of preparation ( a melt or a concentrated polymer solution ) , can be revealed by swelling the gel and observing the enhancement of light , x - ray and neutron scattering from the resulting monomer density inhomogeneities@xcite . in this paper we focused on a related phenomenon , namely that when large - scale cross - link density patterns are written into the network structure , the hidden image can be revealed by swelling and stretching the gel and observing the corresponding patterns of monomer density . using the mean field theory of elasticity of polymer gels we showed that stretching / swelling in good solvent acts as a magnifying glass : while isotropic stretching reproduces an enlarged but otherwise undistorted version of the original pattern , anisotropic stretching distorts this pattern , see figure [ pattern ] . we compared these results with those obtained for ordinary elastic solids with inhomogeneous elastic moduli and found that in this case even isotropic deformations lead to distorted patterns . we showed that the fundamental difference between response of inhomogeneous gels and solids to isotropic stretching can be traced back to the fact that unlike regular springs that have an equilibrium length even in the absence of stress , the equilibrium length of entropic springs is entirely determined by the osmotic forces that isotropically stretch the polymer gel . finally , we would like to comment on the possibility of experimental verification and on possible applications of our results . in most application involving gels such as biomimetic sensors , actuators and artificial muscles @xcite , macroscopically inhomogeneous ( layered ) gels undergo shape transitions when the thermodynamic conditions are changed or in response to application of external fields @xcite . in our case , the cross - link density pattern imprinted into the gel structure by , say , activation of light - sensitive cross - links , can be microscopic ( micron size ) and therefore would have little effect on the shape of the gel . upon swelling and/or isotropic stretching in good solvent , the magnified density pattern can be imaged on a light - sensitive screen . the contrast can be significantly enhanced by stretching the entire gel in poor solvent or by focusing a laser beam on the localized pattern and heating it , resulting in local change of the quality of solvent . finally , the sensitivity of the image to quality of solvent ( the distortion under anisotropic deformation disappears in @xmath0 solvent - see si ) can be useful for sensor devices . * acknowledgments * yr s research was supported by the i - core program of the planning and budgeting committee and the israel science foundation , and by the us - israel binational science foundation . * supporting information . * in the si we show that patterns obtained by cross - linking a semi - dilute polymer solution , deform affinely ( non - affinely ) under isotropic ( anisotropic ) deformation , just like in the case of cross - linking in the melt . we then analyze how the pattern deforms under several different solvent conditions . we show that the pattern always stretches affinely in a @xmath0-solvent , even under anisotropic deformations . since the contrast between the high and the low monomer density regions can be significantly enhanced in a poor solvent we proceed to analyze the density profiles in gels that are isotropically stretched in mildly poor solvents ( at lower solubility , such stretched gels will undergo a transition into a strongly inhomogeneous state characterized by the appearance of dense filamentous structures @xcite ) . we find that when the amplitude of cross - link density variations is sufficiently low , the image stretches affinely with the isotropic deformation but that for larger density contrasts the image becomes distorted , especially near the edges and corners of the pattern . 99 onuki a. , _ adv . _ * 1993 * , _ 109 _ , 63121 . panyukov s. and rabin y. , _ macromolecules _ * 1996 * , _ 29 _ , 79607975 . doi m. and edwards s.f . , _ the theory of polymer dynamics _ , clarendon : oxford , 1986 . bastide j. , leibler l. and prost j. , _ macromolecules _ * 1990 * , _ 23 _ , 18311837 . landau l.d . and lifshitz e.m . , _ theory of elasticity _ , pergamon : oxford , 1970 . panyukov s. and rabin y. , _ physics reports _ * 1996 * , _ 269 _ , 1131 . alexander s. , _ physics reports _ * 1998 * , _ 296 _ , 65236 . flory p.j . and rehner j. , _ j. chem . phys . _ * 1943 * , _ 11 _ , 521526 . candau s. , bastide j. and delsanti m. , _ adv . * 1982 * , _ 44 _ , 2771 . horkay f. , hecht a .- m . , mallam s. , geissler e. and rennie a.r . , _ macromolecules _ * 1991 * , _ 24 _ , 28962902 . shibayama m. , tanaka t. and han c.c . , . phys _ * 1992 * , _ 97 _ , 68296841 . bastide j. and candau s. , in _ the physical properties of polymeric gels _ cohen addad j.p . , wiley , new york , 1996 , p. 143 . shahinpoor m. , bar - cohen y. , simpson j.o . and smith j , _ smart mater . struct . _ * 1998 * , _ 7 _ , r15r30 . klein y. , efrati e. and sharon e. , _ science _ * 2007 * , _ 315 _ , 11161120 . peleg o. , kroger m. , hecht i. and rabin y. , _ europhys . * 2007 * , _ 77 _ , 5800758012 . * supplementary information to : * consider a gel prepared in a good solvent at the monomer density @xmath20 that is swollen to density @xmath10 . its free energy is the sum of elastic and osmotic contributions . the osmotic pressure @xmath61 of the gel in a good solvent increases proportionally to the @xmath62 power of monomer density @xmath10 @xcite@xmath63 where @xmath27 is boltzmann constant , @xmath28 is temperature and @xmath64 is monomer size . the osmotic part @xmath65 of the free energy per polymer chain between network junctions is proportional to the free energy density ( @xmath66 ) divided by the number of chains per unit volume ( @xmath67 ) , where @xmath68 is the chain degree of polymerization:@xmath69 the dimension of the chain along the main axis @xmath70 of deformation is @xmath71 , _ _ _ _ where @xmath72 is deformation factor along this axis ( defined as eigenvalue of local deformation gradient tensor @xmath11 ) and @xmath73 is the chain size in the state at which the gel was formed . the elastic free energy per chain is@xmath74 where @xmath75 is the amplitude of fluctuations of the chain in the deformed state . in a heterogeneous network the direction of the triad of deformation axes @xmath70 depends on its position , and the sum of squares of local deformation factors in eq . ( [ si_el ] ) can be rewritten through the deformation gradient tensor @xmath11 as @xmath76 since the mean - square amplitude of chain fluctuations is proportional to the mean - square polymer size at semi - dilute good solvent conditions and scales with monomer density as@xcite @xmath77 , while the mean - square chain size in the preparation conditions scales as @xmath78 , the elastic free energy per chain is@xmath79 at the equilibrium swelling ( @xmath80 ) in the absence of additional deformations the total free energy of the gel per chain is:@xmath81 \label{si_aeq}\ ] ] and it is minimized at the density@xcite@xmath82 corresponding to maximum swelling ratio@xmath83 note that similar expression for @xmath84 is obtained in mean field model of a gel with second virial coefficient @xmath85 , see main text . this conclusion can also be extended to our solution of the image storing problem . * * * * since both elastic ( eq . ( [ si_el1 ] ) ) and osmotic ( eq . ( [ si_osm ] ) ) terms in the gel free energy are multiplied by the same scaling factor @xmath86 such scaling renormalization does not change the results obtained for the mean field model . in a @xmath0-solvent the second virial coefficient vanishes ( @xmath87 ) and equation @xmath38 \tilde{g}\left ( \lambda^{-1}\mathbf{\cdot x}\right ) \frac{d\mathbf{y}}{\prod_{j}\gamma_{j } } \label{si_rg}\ ] ] reproduces without distortion affinely stretched initial pattern @xmath88 even for anisotropically stretched gels ( small deviations from affinity are expected because of the non - vanishing third virial coefficient ) strong enhancement of monomer density contrast can be obtained by placing the gel ( with fixed boundaries otherwise it would collapse ) in a poor solvent with negative second virial coefficient @xmath89 . in case of very poor solvent with @xmath90 the gel becomes unstable with respect to formation of domains with different monomer density@xcite . below we consider the case of poor solvent close to @xmath0-conditions with small @xmath89 and positive @xmath91 . at small @xmath92 the amplitude of density variations @xmath93 can be significantly increased because of the small denominator in equation@xmath94 and we have to take into account corrections due to second order in @xmath24 term @xmath95 in expression for monomer density,@xmath96 where @xmath97 to first order in @xmath98 we find@xmath99 \label{si_corr}\ ] ] where the equilibrium displacement in @xmath100 is determined as@xmath101 we conclude that the correction term in eq . ( [ si_corr ] ) enhances the contrast between the high and the low monomer density regions of the profile ( fig . [ si_poor ] ) . poor3d this effect is the mostly pronounced near the corners of the pattern where several edges converge and it leads to distortion of the otherwise affinely stretched profile at these points . frozen - in random heterogeneities of network structure can change the image beyond recognition @xcite . the free energy of a gel with frozen - in heterogeities was derived in ref . . the only source of heterogeneities in the melt with fixed monomer density is statistical distribution of cross - links in the state of preparation that arises as the consequence of the random process of cross - linking . this frozen - in distribution is described by an additional contribution to the free energy : @xmath102 where @xmath103 is random gaussian function of coordinate @xmath13 , characterized by correlation function@xmath104 comparing the amplitude of frozen - in fluctuations on a scale @xmath105 with variations of elastic modulus @xmath106 on this scale we conclude that the contribution of frozen - in heterogeneities can be neglected if@xmath107 where @xmath108 is average cross - link concentration and thus , frozen - in heterogeities have no influence on large - scale patterns . the suppression of frozen - in heterogeneities of monomer density is due to strong overlap of network chains @xcite . | using the theory of elasticity of polymer gels we show that large - scale _ cross - link _ density patterns written into the structure of the network in the melt state , can be revealed upon swelling by monitoring the _ monomer _ density patterns .
we find that while isotropic deformations in good solvent yield magnified images of the original pattern , anisotropic deformations distort the image ( both types of deformation yield affinely stretched images in @xmath0 solvents ) . we show that in ordinary solids with spatially inhomogeneous profile of the shear modulus , isotropic stretching leads to distorted density image of this profile under isotropic deformation .
using simple physical arguments we demonstrate that the different response to isotropic stretching stems from fundamental differences between the theory of elasticity of solids and that of gels .
possible tests of our predictions and some potential applications are discussed . |
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the current version of the standard model ( sm ) consists of three generations of quarks and leptons . recently we proposed @xcite a four generation lepton model based on the non - abelian discrete symmetry @xmath0 ( icosahedron ) , in which the best features of the three family @xmath2 ( tetrahedral ) model survive . besides the new heavy degrees of freedom in the @xmath3 model , which satisfy the experimental constraints , we retain tribimaximal neutrino mixings , three light neutrino masses , and three sm charged lepton masses in the three light generation sector . in this paper , we will explore a generalization of our @xmath4 model to include four generations of both quarks and leptons . but before launching into that discussion we must first discuss the viability of models with four generations given recent experimental developments . a fourth generation is now being constrained @xcite by precision electroweak data @xcite , by flavor symmetries @xcite , and by the higgs - like particle at 125 gev recently reported at the lhc @xcite . the new data provide an important step forward in distinguishing various four generation models , and in particular eliminating some from consideration . in particular , four sequential generation models are now highly disfavored @xcite ; however , it would be premature to dismiss all four generation models . while tension between four generation models and data has developed , a fourth generation is not excluded by the electroweak precision data @xcite , so the existence of a fourth generation is still a viable phenomenological possibility which can provide an explanation of the observed anomaly of cp asymmetries in the b meson system @xcite , and the baryon asymmetry of the universe @xcite , with additional mixings and cp phases . also , there are a number of way to relieve this tension . for example two higgs doublet models ( see e.g. , @xcite and references therein ) can accommodate a fourth generation of fermions and current data . for a comprehensive review typically these two higgs doublet models are low energy effective field theories that require composite higgses similar to top quark condensate models @xcite . for some recent examples see ref . another possibility is to add electroweak doublets that are in color octets @xcite . further discussion can be found in ref . @xcite . while the model we will discuss has an extended higgs structure , a full exploration of the possible composite nature of the scalar sector is beyond the scope of our present study . to generalize our @xmath4 model to include four generations of quarks and leptons , we first recall the three family scenarios in which the binary tetrahedral group @xmath5 is capable of providing a model of both the quarks and leptons with tribimaximal mixings and a calculable cabibbo angle @xcite . the @xmath6 group is the double covering group of @xmath7 . it has four irreducible representations ( irreps ) with identical multiplication rules to those of @xmath7 , one triplet @xmath8 and three singlets @xmath9 , and @xmath10 , plus three additional doublet irreps @xmath11 , and @xmath12 . the additional doublets allow the implementation of the @xmath13 structure to the quark sector @xcite , thus the third family of quarks are treated differently and are assigned to a singlet . hence they can acquire heavy masses @xcite . one should note that @xmath7 is not a subgroup of @xmath6 , therefore , the inclusion of quarks into the model is not strictly an extension of @xmath7 , but instead replaces it @xcite . based on the same philosophy , we study the model of four families of quarks and leptons by using the binary icosahedral group @xmath14 . the relation between @xmath1 and @xmath3 is similar to that for @xmath6 and @xmath7 . the icosahedral group @xmath15 has double - valued representations that are single - valued representations of the double icosahedral group @xmath16 . hence , besides the irreps of @xmath1 that are coincident with those of @xmath3 , there are four additional spinor - like irreps @xmath17 , and @xmath18 of @xmath1 . we shall be able to assign quarks to the spinor - like representations , but to discuss model building using @xmath1 , we must first review our lepton model based on @xmath3 , which will remain essentially unchanged when generalized to @xmath1 . some useful group theory details have been relegated to the appendix . the irreps of @xmath3 are one singlet @xmath19 , two triplets @xmath8 and @xmath20 , one quartet @xmath21 , and one quintet @xmath22 . the model is required to be invariant under the flavor symmetry of @xmath23 and the particle content is given by table [ a5 ] . the most general form of the higgs potential containing the scalar fields @xmath24 , @xmath25 , @xmath26 and @xmath27 , invariant under the discrete @xmath23 symmetries is given by @xmath28 where the individual terms are written as @xmath29 , \\\end{aligned}\ ] ] @xmath30 , \\ v(h'_{4 } , \phi_{3 } ) & = & \lambda^{h'\phi}_{\beta}(h'^{\dag}_{4}h'_{4})_{\textbf{$\beta$}}(\phi^{\dag}_{3}\phi_{3})_{\textbf{$\beta$ } } \nonumber \\ & & + \lambda'^{h'\phi}_{\gamma}(h'^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$}}(\phi^{\dag}_{3}h'_{4})_{\textbf{$\gamma$ } } \nonumber \\ & & + \left [ \lambda''^{h'\phi}_{\gamma}(h'^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$}}(h'^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$ } } + \rm{h.c . } \right ] , \\ v(h_{4 } , h'_{4 } , \phi_{3 } ) & = & \lambda^{hh'\phi}_{\gamma}(h^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$}}(h'^{\dag}_{4}h'_{4})_{\textbf{$\gamma$ } } \nonumber \\ & & + \lambda'^{hh'\phi}_{\gamma}(h'^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$}}(h'^{\dag}_{4}h_{4})_{\textbf{$\gamma$ } } \nonumber \\ & & + \lambda''^{hh'\phi}_{\gamma}(h^{\dag}_{4}\phi_{3})_{\textbf{$\gamma$}}(h^{\dag}_{4}h_{4})_{\textbf{$\gamma$ } } + \rm{h.c .. }\end{aligned}\ ] ] here we have introduced the @xmath1 group representation indices @xmath31 ; @xmath32 ; and @xmath33 = @xmath34 respectively . the first stage of ssb takes us from the initial @xmath1 discrete symmetry to @xmath6 , and this is accomplished with a vev for @xmath35 . consider the pure @xmath35 sector of the higgs potential @xmath36 where @xmath37 and @xmath38 are @xmath1 group matrices . since @xmath39 in @xmath1 contains three terms there are three singlets in @xmath40 ^ 2 $ ] and hence in @xmath41 . there is also a potential cubic term that can be suppressed , either by imposing an addition @xmath42 symmetry , or by making @xmath35 a complex field . for @xmath43 and @xmath44 sufficiently small and of the proper signs , the ssb is dominated by the @xmath45 term and we have @xmath46 such that @xmath47 with the sm gauge group unaffected and with the scalars decomposing as in eq . ( 6 ) . the sector of the higgs potential of the @xmath1 model that depends only on @xmath48 , @xmath49 , @xmath50 , and @xmath51 is given in the appendix of @xcite which only involves @xmath6 invariant terms and is identical to our form of the potential up to constraints due to residual @xmath1 relations on the coupling constants . the work of @xcite demonstrates how @xmath6 is broken completely and how the light quark and lepton masses and mixings arise . our only other additional scalars are the @xmath1 triplet , sm doublet fields @xmath52 . since @xmath53 under @xmath47 , a @xmath52 breaks @xmath6 . the pure @xmath52 scalar sector can be rewritten as @xmath54 where we have included the triplet indices @xmath55 and find two quartic terms due to the fact that @xmath56 in @xmath1 contains two terms and hence @xmath57 ^ 2 $ ] contains two singlets . given the @xmath6 level vevs for @xmath50 , and @xmath51 , there is no @xmath6 symmetry remaining to rotate the @xmath52 vev . hence the choice @xmath58 is stable for @xmath59 and of the form needed in the model . a similar analysis can be applied to the investigation of the ssb in our @xmath4 model . finally we note that typically , only an @xmath60 fine tuning of scalar quartic coupling constants is needed to maintain the stability of such patterns of ssb . o. eberhardt , a. lenz and j. rohrwild , phys . d * 82 * , 095006 ( 2010 ) [ arxiv:1005.3505 [ hep - ph ] ] ; 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t. schwetz , m. tortola and j. w. f. valle , new j. phys . * 13 * , 109401 ( 2011 ) [ arxiv:1108.1376 [ hep - ph ] ] . l. l. everett and a. j. stuart , phys . d * 79 * , 085005 ( 2009 ) [ arxiv:0812.1057 [ hep - ph ] ] ; f. feruglio and a. paris , jhep * 1103 * , 101 ( 2011 ) [ arxiv:1101.0393 [ hep - ph ] ] ; g. -j . ding , l. l. everett and a. j. stuart , nucl . b * 857 * , 219 ( 2012 ) [ arxiv:1110.1688 [ hep - ph ] ] ; l. l. everett and a. j. stuart , phys . * b698 * , 131 - 139 ( 2011 ) [ arxiv:1011.4928 [ hep - ph ] ] . | to include the quark sector , the @xmath0 ( icosahedron ) four generation lepton model is extended to a binary icosahedral symmetry @xmath1 flavor model .
we find the masses of fermions , including the heavy sectors , can be accommodated . at
leading order the ckm matrix is the identity and the pmns matrix , resulting from same set of vacua , corresponds to tribimaximal mixings . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
creep is a major limitation of concrete . indeed , it has been suggested that creep deformations are logarithmic , that is , virtually infinite and without asymptotic bound , which raises safety issues @xcite . the creep of concrete is generally thought to be mainly caused by the viscoelastic and viscoplastic behavior of the cement hydrates @xcite . while secondary cementitious phases can show viscoelastic behavior @xcite , the rate and extent of viscoelastic deformations of such phases is far less significant than that calcium silicate hydrate ( c s h ) , the binding phase of the cement paste @xcite . as such , understanding the physical mechanism of the creep of c s h is of primary importance . despite the prevalence of concrete in the built environment , the molecular structure of c s h has just recently been proposed @xcite , which makes it possible to investigate its mechanical properties at the atomic scale . here , relying on the newly available model , we present a new methodology allowing us to simulate the long - term creep deformation of bulk c s h ( at zero porosity , i.e. , at the scale of the grains ) . results show an excellent agreement with nanoindentation measurements @xcite . to describe the disordered molecular structure of c s h , pellenq et al . @xcite proposed a realistic model for c s h with the stoichiometry of ( cao)@xmath0(sio@xmath1)(h@xmath1o)@xmath2 . we generated the c s h model by introducing defects in an 11 tobermorite @xcite configuration , following a combinatorial procedure . whereas the ca / si ratio in 11 tobermorite is 1 , this ratio is increased to 1.71 in the present c s h model , through randomly introducing defects in the silicate chains , which provides sites for adsorption of extra water molecules . the reaxff potential @xcite , a reactive potential , was then used to account for the reaction of the interlayer water with the defective calcium silicate sheets . more details on the preparation of the model and its experimental validation can be found in ref . @xcite and in previous works @xcite . we simulated the previously presented c s h model , made of 501 atoms , by molecular dynamics ( md ) using the lammps package @xcite . to this end , we used the reaxff potential @xcite with a time step of 0.25fs . prior to the application of any stress , the system is fully relaxed to zero pressure at 300k . shear strain and potential energy with respect to the number of loading / unloading cycles . the inset shows the shape of the applied shear stress . ] the relaxation of c s h , or of other silicate materials , takes place over long periods of time ( years ) , which prevents the use of traditional md simulations , which are limited to a few nanoseconds . to study the long - term deformations of c s h , we applied a method that has recently been introduced to study the relaxation of silicate glasses @xcite . in this method , starting from an initial atomic configuration of glass , formed by rapid cooling from the liquid state , the system is subjected to small , cyclic perturbations of shear stress @xmath3 around zero pressure . for each stress , a minimization of the energy is performed , with the system having the ability to deform ( shape and volume ) in order to reach the target stress . these small perturbations of stress deform the energy landscape of the glass , allowing the system to jump over energy barriers . note that the observed relaxation does not depend on the choice of @xmath4 , provided that this stress remains sub - yield @xcite . this method mimics the artificial aging observed in granular materials subjected to vibrations @xcite . here , in order to study creep deformation , we add to the previous method a constant shear stress @xmath5 , such that @xmath6 ( see the inset of figure [ fig : method ] ) . when subjected to shear stresses of different intensities , c s h presents a shear strain @xmath7 that : ( 1 ) increases logarithmically with the number of cycles @xmath8 ( figure [ fig : method ] ) and ( 2 ) is proportional to the applied shear stress ( see figure [ fig : strain ] ) . shear strain with respect to the number of loading / unloading cycles , under a constant shear stress of 1 , 2 , and 3 gpa . the inset shows the creep modulus @xmath9 with respect to the packing fraction @xmath10 obtained from nanoindentation @xcite , compared with the computed value at @xmath11 . ] the creep of bulk c s h can then be described by a simple logarithmic law @xmath12 , where @xmath13 is a constant analogous to a relaxation time and @xmath9 is the creep modulus . a careful look at the internal energy shows that the height of the energy barriers , through which the system transits across each cycle , remains roughly constant over successive cycles . according to transition state theory , which states that the time needed for a system to jump over an energy barrier @xmath14 is proportional to @xmath15 , we can assume that each cycle corresponds to a constant duration @xmath16 , so that a fictitious time can be defined as @xmath17 @xcite . we note that the computed creep moduli @xmath9 does not show any significant change with respect to the applied stress @xmath5 . as such , it appears to be a material property that can directly been compared to nanoindentation results extrapolated to zero porosity @xcite . as shown in the inset of figure [ fig : strain ] , we observe an excellent agreement , which suggests that the present method offers a realistic description of the creep of c s h at the atomic scale . this result also suggests that , within the linear regime ( i.e. , for sub - yield stresses , when @xmath9 remain constant ) , deformations due to cyclic creep and basic creep , with respect to the number of stress cycle or the elapsed time , respectively , should be equivalent . we reported a new methodology based on atomistic simulation , allowing us to successfully observe long - term creep deformations of c s h . creep deformations are found to be logarithmic and proportional to the applied shear stress . the computed creep modulus shows an excellent agreement with nanoindentation data , which suggests that the present methodology could be used as a predictive tool to study the creep deformations of alternative binders . mb acknowledges partial financial support for this research provisioned by the university of california , los angeles ( ucla ) . this work was also supported by schlumberger under an mit - schlumberger research collaboration and by the cshub at mit . this work has been partially carried out within the framework of the icome2 labex ( anr-11-labx-0053 ) and the a*midex projects ( anr-11-idex-0001 - 02 ) cofunded by the french program `` investissements davenir '' which is managed by the anr , the french national research agency . | understanding the physical origin of creep in calcium silicate hydrate ( c s h ) is of primary importance , both for fundamental and practical interest . here
, we present a new method , based on molecular dynamics simulation , allowing us to simulate the long - term visco - elastic deformations of c s h . under a given shear stress , c s h features a gradually increasing shear strain , which follows a logarithmic law .
the computed creep modulus is found to be independent of the shear stress applied and is in excellent agreement with nanoindentation measurements , as extrapolated to zero porosity . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the nearest molecular cloud complex to the sun ( distance @xmath0 65 pc ) consists of clouds 11 , 12 , and 13 from the catalog of magnani et al . ( 1985 ) and is located at ( l , b ) @xmath0 ( 159.4,@xmath534.3 ) . this complex of clouds ( which we will refer to as mbm12 ) was first identified by lynds ( 1962 ) and appears as objects l1453-l1454 , l1457 , l1458 in her catalog of dark nebulae . the mass of the entire complex is estimated to be @xmath0 30200 m@xmath6 based on radio maps of the region in @xmath7co , @xmath8co and c@xmath9o ( pound et al . 1990 ; zimmermann & ungerechts 1990 ) . recently , there has been much interest in understanding the origin of many isolated t tauri stars ( tts ) and isolated regions of star - formation . for example , within @xmath0 100 pc from the sun there are at least two additional regions of recent star - formation : the tw hydrae association ( distance @xmath0 50 pc ; e.g , kastner et al . 1997 ; webb et al . 1999 ) and the @xmath10 chamaeleontis region ( distance @xmath0 97 pc ; mamajek et al . both of these star - forming regions appear to be isolated in that they do not appear to be associated with any molecular gas . in addition , both are comprised mainly of `` weak - line '' tts equivalent widths , w(h@xmath11 ) @xmath4 @xmath510 and `` classical '' tts ( ctts ) to be tts with w(h@xmath11 ) @xmath12 @xmath510 where the negative sign denotes emission ] . in contrast , most of the tts in mbm12 are ctts which are still associated with their parent molecular cloud . in addition to the above isolated star - forming regions , tts have been found outside of the central cloud core regions in many nearby star - forming cloud complexes ( see references in feigelson 1996 ) . several theories exist to explain how tts can separate from their parent molecular clouds either by dynamical interactions ( sterzik & durisen 1995 ) or by high - velocity cloud impacts ( @xcite ) . feigelson ( 1996 ) also suggests that some of these tts may form in small turbulent cloudlets that dissipate after forming a few tts . since the tts in mbm12 appear to still be in the cloud in which they formed , we know they have not been ejected from some other more distant star - forming region . therefore mbm12 may be an example of one of the cloudlets proposed by feigelson ( 1996 ) . moriarity - schieven et al . ( 1997 ) argue that mbm12 has recently been compressed by a shock associated with its interface with the local bubble . this shock may also have recently triggered the star - formation currently observed in mbm12 ( cf . elmegreen 1993 ) . alternatively ballesteros - paredes et al . ( 1999 ) suggest that mbm12 may be an example of a star - forming molecular cloud that formed via large scale streams in the interstellar medium . mbm12 is different from most other high - latitude clouds at @xmath13 @xmath4 30@xmath14 in terms of its higher extinction and its star formation capability ( e.g. , hearty et al . based on co observations and star counts , the peak extinction in the cloud is @xmath3 @xmath0 5 mag ( duerr & craine 1982a ; magnani et al . 1985 ; pound et al . 1990 ; zimmermann & ungerechts 1990 ) . however , molecular clouds are clumpy and it is possible that some small dense cores with @xmath3 @xmath4 5 mag were not resolved in previous molecular line and extinction surveys . for example , zuckerman et al . ( 1992 ) estimate @xmath3 @xmath0 11.5 mag through the cloud , along the line of sight to the eclipsing cataclysmic variable h0253 + 193 located behind the cloud and we estimate @xmath3 @xmath0 8.48.9 along the line of sight to a g9 star located on the far side of the cloud ( sect . [ cafos ] ) although there is evidence for gravitationally bound cores in mbm12 , the entire cloud does not seem to be bound by gravity or pressure ( pound et al . 1990 ; zimmermann & ungerechts 1990 ) . therefore , the cloud is likely a short - lived , transient , object similar to other high latitude clouds which have estimated lifetimes of a few million years based on the sound crossing time of the clouds ( @xcite ) . if this is the case , mbm12 will dissipate in a few million years and leave behind an association similar to the tw hydrae association that does not appear to be associated with any molecular material . previous searches for tts in mbm12 have made use of h@xmath11 , infrared , and x - ray observations . the previously known tts in mbm12 are listed in table [ previous ] with their coordinates , spectral types , apparent magnitudes , and selected references . we include the star s18 in the list even though downes & keyes ( 1988 ) point out that it could be an me star rather than a t tauri star since our observations confirm that it is a ctts . the previously known and new tts stars identified in this study are plotted in fig . [ iras ] with an iras 100 @xmath15 m contour that shows the extent of the cloud . .previously known t tauri stars in mbm12 [ cols="^,^,^,^,^,^ " , ] @xmath16 herbig ( 1977 ) measured a radial velocity of @xmath17 km s@xmath18 for this object . we obtained high resolution spectra of two of the t tauri stars in mbm12 with foces at the calar alto 2.2-m telescope in august 1998 . the spectra for these stars ( rxj0255.4 + 2005 and lkh@xmath11264 , see fig . 3 . ) allow us to estimate their radial velocities and confirm the w(li ) measurements of our low resolution spectra presented in sect . [ cafos ] . determinations of radial velocity , rv , and projected rotational velocity , vsin@xmath19 , have been obtained by means of cross correlation analysis of the stellar spectra with those of radial velocity and rotational standard stars , treated in analogous way . given the large spectral range covered by the foces spectra , the cross correlation of the target and template stars was performed after rebinning the spectra to a logarithmic wavelength scale , in order to eliminate the dependence of doppler shift on the wavelength . moreover , only parts of the spectra free of emission lines and/or not affected by telluric absorption lines have been used . therefore , the nai d , and h@xmath11 lines as well as wavelengths longer than about 7000 have been excluded from the cross - correlation analysis . the result of the cross - correlation is a correlation peak which can be fitted with a gaussian curve . the parameters of the gaussian , center position and full - width at half - maximum ( fwhm ) are directly related to rv and vsin@xmath19 , respectively . the method of the correlation has been fully described by queloz ( 1994 ) , and soderblom et al . more details about the calibration procedure can be found in appendix a of covino et al . ( 1997 ) . the radial velocities we measured for the two mbm12 tts listed in table [ velocity ] are similar to that of the molecular gas ( zimmermann & ungerechts 1990 ; pound et al . radial velocity measurements have not yet been made for the fainter stars . nevertheless , the superposition of the tts on the cloud and the similar radial velocities of at least two of the stars with the gas are strong evidence to support that the tts are associated with the cloud . since both ctts and wtts are typically @xmath0 10@xmath2010@xmath21 times more luminous in the x - ray region of the spectrum than average ( i.e. , older ) low - mass stars ( damiani et al . 1995 ) , we made use of the _ rosat _ pointed and the rass observations of mbm12 to identify previously unknown tts in the cloud . the 25 ks _ rosat _ pspc pointed observation ( sequence number 900138 ) was centered at ( ra , dec ) @xmath0 ( 2:57:04.8,+19:50:24 ) . although they were originally discovered by other means , all of the previously known tts in the central region of mbm12 were also detected with _ rosat_. since the extent of the molecular gas is not known ( in particular for the mbm13 region ) and tts can sometimes be displaced several parsecs from their parent clouds , we also searched in the rass database in a @xmath0 25 deg@xmath22 region around mbm12 . details about _ rosat _ and its pspc detector can be found in trmper ( 1983 ) and briel & pfeffermann ( 1995 ) , respectively . the rass broad - band image of the region investigated around mbm12 and the _ rosat _ pointed observation are displayed in fig . [ xrayfig ] . the x - ray source search was conducted in different _ rosat _ standard bands " , defined as follows : broad " = 0.082.0 kev ; soft " = 0.080.4 kev ; hard " = 0.52.0 kev ; hard1 " = 0.50.9 kev ; hard2 " = 0.92.0 kev . we identified all of the x - ray sources above a maximum likelihood exp(@xmath5ml ) . ] threshold of 7.4 in both the rass and the _ rosat _ pspc pointed observations of mbm12 . in addition , we selected only those x - ray sources above a count rate threshold of @xmath0 0.03 cts s@xmath18 in the rass observation and a count rate threshold of 0.0013 cts s@xmath18 in the _ rosat _ pspc pointed observation . the one previously known tts candidate , s18 , near the cloud mbm13 was detected in the rass with a ml = 6.4 ( i.e. , below our threshold ) , however , since our optical spectroscopic observations confirm that it is a tts we include it in our study . we identified 49 x - ray sources in the _ rosat _ pspc pointed observation of mbm12 ( including all of the previously known tts in the central region of the cloud ) and 28 x - ray sources detected in the rass ( including s18 ) in the regions displayed in fig . [ xrayfig ] . three stars were detected both on the rass and in the pointed observation . we list the sources detected in the _ rosat _ pspc pointed observation and in the rass in table 4 . we include the _ rosat _ source name , the x - ray source coordinates , the maximum likelihood for existence for each source , the broad - band count rates , the x - ray hardness ratios @xmath23 and @xmath24 ( as defined in neuhuser et al . 1995 ) , the apparent visual magnitude taken from the guide star catalog ( magnitudes for the fainter sources indicated with a `` : '' are estimated from the digitized sky survey images ) , and the broad - band x - ray to optical flux ratio . we also list the spectral type and the h@xmath11 and lithium equivalent widths of the sources that have been observed spectroscopically and comments collected from our search through the simbad and ned databases concerning the objects . assuming a mean x - ray count - rate - to - flux conversion factor of 1.1 @xmath25 10@xmath26 erg cts@xmath18 @xmath27 , which we derive from x - ray spectral fits of the tts in sect . [ xlf ] , if the cloud is at a distance of 65 pc , the limiting luminosities of the observations are @xmath28 erg s@xmath18 and @xmath29 erg s@xmath18 for the rass and _ rosat _ pointed observations , respectively . therefore , these observations are sufficient to detect most of the wtts in the cloud since the threshold is below the x - ray faintest stars in the wtts x - ray luminosity function ( e.g. , neuhuser et al . 1995 ) . although the rass observation of mbm12 in not sensitive enough to detect all of the ctts in the cloud , the objective prism survey by stephenson ( 1986 ) identified all of the h@xmath11 emission sources in this region down to a visual magnitude threshold of @xmath0 13.5 . since this limiting magnitude corresponds to the early m spectral types in mbm12 , the current population of tts in mbm12 presented in this paper should be complete for all earlier spectral types . since mbm12 is at relatively high galactic latitude , many of the 81 x - ray sources we identified are extragalactic . therefore , we used the x - ray to optical flux ratios ( see table 4 ) to remove extragalactic sources from our list of candidates ( cf . hearty et al . 1999 ) . all sources which have log(@xmath30 ) @xmath4 0.0 are considered to be extragalactic and those with log(@xmath30 ) @xmath12 0.0 are considered to be stellar objects , some of which could be pms . we also searched the literature to remove cataloged non - pms stars from our list of candidates . finally we were left with a list of x - ray sources identified in the rass and _ rosat _ pointed observations of the cloud which have stellar optical counterparts that may be pms stars . however , many of these stars may be other types of x - ray active stars ( e.g. , rs cvn and dme stars ) and nearby main sequence stars ( which may not be intrinsically x - ray bright , but are near enough so that their x - ray flux is large ) that are more difficult to separate from pms stars by x - ray observations alone . therefore , follow - up spectral observations are necessary to identify which x - ray sources are t tauri stars . in order to complete the census of the tts population of mbm12 we require follow - up observations . since lithium is burned quickly in convective stars , a measurement of w(li ) along with a knowledge of the spectral type of a star can be a reliable indicator of youth . therefore we obtained broad - band , low - resolution , optical spectra of the x - ray emitting tts candidates to determine spectral types and measure the equivalent width of the h@xmath11 emission and 6708 absorption lines . the spectra were obtained from october 911 , 1998 with the calar alto faint object spectrograph ( cafos ) at the 2.2-m telescope at calar alto , spain . the 24@xmath15 m pixels of the site-1d 2048@xmath252048 chip with the g-100 grism provided a reciprocal dispersion of @xmath0 2.1 pixel@xmath18 . the resolving power , @xmath31 = @xmath32 @xmath0 1000 , derived from the measurement of the fwhm ( fwhm @xmath0 6.4 ) of several well isolated emission lines of the comparison spectra is sufficient to resolve the lithium absorption line in t tauri stars . the wavelength range @xmath0 4900 to 7800 was chosen to detect two indicators of possible youth ( h@xmath11 emission and @xmath336708 absorption ) and to determine spectral types . all spectra were given an initial inspection at the telescope . if a particular star showed signs of youth or the integration produced fewer than @xmath0 1000 cts pixel@xmath18 , at least one additional integration was performed . the results of the spectroscopic observations of the tts in mbm12 are summarized in table [ eqw ] . we list the name of the star ; the coordinates for the optical source ; the spectral type ; the log of the effective temperature , log@xmath34 , assuming luminosity class v and using the spectral type - effective temperature relation of @xcite ; apparent magnitude , @xmath35 , taken from the guide star catalog ; the equivalent width of h@xmath11 , w(h@xmath11 ) ; both the low and high resolution ( when available ) measurements of w(li ) ; the veiling corrected w(li ) ( cf . strom et al . 1989 ) ; and the derived lithium abundance based on the non - lte curves of growth of pavlenko & magazz ( 1996 ) assuming log@xmath36=4.5 . the estimated error for the low - resolution w(li ) measurements is @xmath0 @xmath37 90 m based on the correlation with the three stars for which we have high resolution measurements . @c@c@c@c@c@c@c@c@c@c@c@ star & ra & dec & spt & log@xmath34 & @xmath35 & w(h@xmath11)@xmath38 & w(li ) & w(li ) hi - res & w(li ) deveiled & logn(li ) + & [ 2000 ] & [ 2000 ] & & & [ mag ] & [ ] & [ m ] & [ m ] & [ m ] & + + hd 17332 & 02:47:27.3 & + 19:22:24 & g1v & 3.769 & 6.8 & 2.96 & 190 & & & 3.2 + rxj0255.3 + 1915 & 02:55:16.5 & + 19:15:02 & f9 & 3.785 & 10.4 & 3.75 & 170 & & & 3.3 + + rxj0255.4 + 2005 & 02:55:25.7 & + 20:04:53 & k6 & 3.631 & 12.2 & @xmath51.26 & 380 & @xmath39 & 462 & 2.7 + lkh@xmath11262 & 02:56:07.9 & + 20:03:25 & m0 & 3.584 & 14.6 & @xmath532.1 & 290 & & 412 & 2.0 + lkh@xmath11263 & 02:56:08.4 & + 20:03:39 & m4 & 3.517 & 14.6 & @xmath532.9 & 380 & & 543 & 1.6 + lkh@xmath11264 & 02:56:37.5 & + 20:05:38 & k5 & 3.644 & 12.5 & @xmath558.9 & 490 & @xmath40 & 836 & 3.8 + e02553 + 2018 & 02:58:11.2 & + 20:30:04 & k4 & 3.657 & 12.3 & @xmath51.6 & 620 & @xmath41 & 499 & 3.1 + rxj0258.3 + 1947 & 02:58:15.9 & + 19:47:17 & m5 & 3.501 & 15.0 : & @xmath524.5 & 580 & & 783 & 1.8 + s18 & 03:02:21.1 & + 17:10:35 & m3 & 3.532 & 13.5 & @xmath579.0 & 310 & & 552 & 1.8 + rxj0306.5 + 1921 & 03:06:33.1 & + 19:21:52 & k1 & 3.698 & 11.4 & filled & 350 & & & 3.1 + @xmath42 a negative sign denotes emission . @xmath43 the high resolution measurement of w(li ) for this star is taken from martn et al . ( 1994 ) . the optical spectra of the tts in mbm12 are displayed in fig . [ ttsspectra ] . the stars which show strong h@xmath11 emission are also scaled by an appropriate factor to display the emission line . the spectra of the two stars we classify as young main sequence stars which still show lithium are displayed in fig . [ zams ] . in addition to confirming that the star s18 is a ctts with strong h@xmath11 emission and absorption , we identified 3 previously unknown tts in mbm12 . in order to estimate the relative age of the mbm12 stars with lithium we plot them in an w(li ) vs. t@xmath44 diagram ( fig . [ lithtemp ] ) along with stars from taurus ( age @xmath0 a few myr ) , the tw hydrae association ( age @xmath0 10 myr ) , the @xmath10 chamaeleontis cluster ( age @xmath0 218 myr ) , ic 2602 ( age @xmath0 30 myr ) , and the pleiades ( age @xmath0 100 myr ) . in addition , we plot isoabundance lines for the non - lte curves of growth of pavlenko & magazz ( 1996 ) for log@xmath36=4.5 stars and the isochrones for the non - rotating lithium depletion model of pinsonneault et al . the positions of the mbm12 stars in the diagram indicates they are young objects with ages much less than that of the pleiades or ic 2602 . although the relative ages between the stars in mbm12 , the tw hydrae association , and the @xmath10 chamaeleontis cluster , can not be discerned in fig . [ lithtemp ] , since most of the tts in mbm12 are ctts which are still associated with their parent molecular cloud the tts in mbm12 must be younger than those in the tw hydrae association or the @xmath10 chamaeleontis cluster which are comprised mainly of wtts not associated with any molecular cloud ( i.e. , the tts in mbm12 have ages @xmath12 10 myr ) . although the two f and g spectral type stars in which we detected lithium ( hd 17332 and rxj0255.3 + 1915 ) are located above the pleiades in the w(li ) vs. t@xmath44 diagram , since they both show h@xmath11 absorption stronger than any similar spectral type stars in ic 2602 ( e.g. , randich et al . 1997 ) , they are probably older than 30 myr . thus , we list them as main sequence stars in table [ eqw ] . covino et al . ( 1997 ) have shown that low - resolution spectra tend to overestimate w(li ) in intermediate spectral types , therefore we probably over estimated the w(li ) for these two stars . the tts in mbm12 are clearly lithium - rich relative to the stars in the pleiades . however , current age dependent stellar population models predict that there should be a population of young stars with ages @xmath12 150 myr distributed across the sky . therefore we compare the density of young x - ray sources detected in mbm12 with the age dependent stellar population model of guillout et al . ( 1996 ) to find out if we are really seeing an excess of young x - ray sources in the direction of mbm12 . in the galactic latitude range of 40@xmath14 @xmath4 @xmath45b@xmath45 @xmath4 30@xmath14 guillout et al . ( 1996 ) predict there should be 0.61.0 stars deg@xmath46 and 0.140.20 stars deg@xmath46 above an x - ray count rate threshold of 0.0013 cts s@xmath18 and 0.03 cts s@xmath18 , respectively , which have ages @xmath12 150 myr . therefore , we expect to detect @xmath0 1.93.1 young stars in this age group in the _ rosat _ pointed observation and 3.55 stars in this age group in the rass observation . since we observed several young stars which probably have ages @xmath12 150 myr but are not associated with mbm12 ( i.e. , the two intermediate spectral type stars which have not yet depleted their lithium and the 3 me and 3 ke stars listed in table 4 which have depleted their lithium but show h@xmath11 emission ) , there are a sufficient number of x - ray active stars in this region to account for the numbers predicted by guillout et al . therefore , the tts we observe represent an excess of x - ray active young stars associated with mbm12 . in addition to the x - ray selected t tauri star candidates , we also observed the reddest star from a list of stars compiled by duerr & craine ( 1982b ) which are along the line of sight to mbm12 and have v - i colors redder than 2.5 mag . the optical spectrum of this star , which we will call dc48 , indicates it is a g9 star . since duerr & craine ( 1982b ) measured @xmath35 = 18.7 and v - i = 5.6 mag , it corresponds to a main sequence star with @xmath3 @xmath0 8.9 mag at a distance of @xmath0 63 pc or a giant star with @xmath3 @xmath0 8.4 mag at a distance of @xmath0 950 pc . the spectrum of the highly reddened ( @xmath3 @xmath0 8.48.9 ) g9 star , dc48 , is displayed in fig . we tested all of the tts for x - ray variability using the methods described in @xcite . the only t tauri star which showed x - ray variability is the newly identified star rxj0255.5 + 2005 that was detected both in the rass and in the _ rosat _ pointed observation and flared during the pointed observation ( see the light curve displayed in fig . [ lc ] ) . the peak x - ray count rate during the flare increased by more than a factor of 6 from the pre - flare count rate . although we do no have a sufficient number of counts ( @xmath0 1000 counts for the non - flare phase and @xmath0 500 counts for the flare phase ) for a detailed study of the evolution of the coronal temperature during the flare , we performed a rough spectral fit using a 2 temperature raymond - smith model ( raymond & smith 1977 ) including a photoelectric absorption term using the morrison & mccammon ( 1983 ) cross sections . we fit the data for 3 time intervals : the pre - flare phase , the flare , and the post - flare phase fig . [ xspecflare ] . both temperature components increased during the flare and remained high throughout the post - flare phase . the parameters derived from the x - ray spectral fits are listed in table [ xraylum ] ( see sect . [ xlf ] for a description of the table columns ) . since the two temperature components are not well constrained by the spectral fit during the flare , these estimates should be viewed as a lower limits . the results of the spectral fits are consistent with the type of coronal heating seen in high signal - to - noise x - ray spectra of other flaring wtts ( e.g. , tsuboi et al . although our x - ray spectra do not have sufficiently high signal - to - noise for a detailed comparison of x - ray spectral models , we performed a spectral fit using a 2 temperature raymond - smith model including a photoelectric absorption term as described in sect . [ xrayvar ] for the sources with at least 100 counts . for the sources with fewer than 100 counts we calculate the x - ray flux using an x - ray count rate to flux conversion factor of 1.1 @xmath25 10@xmath26 erg cts@xmath18 @xmath27 which is the mean conversion factor derived from the spectra for which we performed spectral fits . we list the total _ rosat _ broad band ( 0.082.0 kev ) counts and count rates for the tts in mbm12 and the derived interstellar+circumstellar absorption cross sections and plasma temperatures for the spectra in which we performed spectral fits in table [ xraylum ] . the x - ray luminosities assume a distance of 65 pc . since the binary lkh@xmath11262/263 was not spatially resolved with the pspc we fit the combined x - ray spectrum to estimate the combined x - ray luminosity but we divide that value in half to generate the x - ray luminosity function . @l@c@c@c@c@c@c@c@c@c@c@ star & & & + & counts & rate & counts & rate & n@xmath47/10@xmath48 & @xmath49 & @xmath50 & @xmath51/dof & @xmath52/10@xmath53 & log@xmath54 + & & [ cts s@xmath18 ] & & [ cts s@xmath18 ] & [ @xmath27 ] & [ kev ] & [ kev ] & & [ erg s@xmath18 @xmath27 ] & [ erg s@xmath18 ] + rxj0255.4 + 2005 + `` '' total & @xmath57 & @xmath58 & @xmath59 & @xmath60 & 0.86 & 0.89 & 0.08 & 39.7/34 & 7.81 & 29.60 + `` '' pre - flare & & & @xmath61 & @xmath62 & 0.49 & 0.88 & 0.11 & 9.2/9 & 4.12 & 29.32 + `` '' flare & & & @xmath63 & @xmath64 & 0.18 & 1.69 & 0.30 & 9.5/6 & 20.1 & 30.01 + `` '' post - flare & & & @xmath65 & @xmath66 & 0.27 & 1.14 & 0.24 & 10.7/15 & 6.96 & 29.55 + lkh@xmath11262/263 & & & @xmath67 & @xmath68 & 2.65 & 0.97 & 0.16 & 6.29/8 & 2.01 & 29.01 + lkh@xmath11264 & & & @xmath69 & @xmath70 & & & & & 0.47 & 28.39 + e02553 + 2018 & @xmath71 & @xmath72 & @xmath73 & @xmath74 & 8.21 & 1.05 & 0.10 & 16.5/16 & 5.03 & 29.41 + rxj0258.3 + 1947 & & & @xmath75 & @xmath76 & 8.42 & 1.14 & 0.13 & 2.63/2 & 0.86 & 28.64 + s18 & @xmath77 & @xmath78 & & & & & & & 3.30 & 29.22 + rxj0306.5 + 1921 & @xmath79 & @xmath78 & & & & & & & 3.30 & 29.22 + in order to compare the derived x - ray luminosity function for the tts in mbm12 with other flux limited x - ray luminosity functions we used the asurv rev . 1.2 package ( isobe & feigelson 1990 ; lavalley et al . 1992 ) , which implements the methods presented in feigelson & nelson ( 1985 ) . although the currently known tts in mbm12 are all x - ray detected , the luminosity functions of other , more distant , star forming regions include upper limits . the derived x - ray luminosity function is displayed in fig . [ lumfunc ] with the x - ray luminosity function for the tts in the l1495e cloud in taurus which ( like mbm12 ) was observed in a deep ( 33 ks ) _ rosat _ pspc pointed observation ( strom & strom 1994 ) . the _ rosat _ pointed observation of l1495e is @xmath0 20 times more sensitive than previous observations with the _ einstein _ satellite . strom & strom ( 1994 ) used this observation to show that the x - ray luminosity of tts extends to fainter luminosities than were observed with einstein . we have re - reduced the pointed observation of l1495e in a way analogous to that of mbm12 . the x - ray luminosity function we derive for l1495e ( 1 ) includes only the k and m spectral type tts , ( 2 ) includes 6 upper limits , ( 3 ) assumes an x - ray to optical flux conversion factor of 1.1 @xmath25 10@xmath26 erg cts@xmath18 @xmath27 , and ( 4 ) assumes a distance of 140 pc . the x - ray luminosity functions in mbm12 and l1495e agree well : in mbm12 the log@xmath80 = [email protected] erg s@xmath18and log@xmath81 = 28.7 erg s@xmath18 ; in l1495e log@xmath80 = [email protected] erg s@xmath18 and log@xmath81 = 28.9 erg s@xmath18 . however , we note that the mbm12 x - ray luminosity function has a lower high - luminosity limit and a higher low - luminosity limit than the l1495e x - ray luminosity function . therefore , although the pointed observation of mbm12 is more sensitive than the pointed observation l1495e ( because mbm12 is much closer ) our follow - up observations of the tts in mbm12 may be incomplete for sources fainter than @xmath35 @xmath0 15.5 mag . in addition , since we know that one of our x - ray sources , s18 , is detected but below our threshold for follow - up observations , there may be other , fainter , x - ray emitting tts in mbm12 with spectral types later than @xmath0 m2 ( i.e. , the spectral type of s18 ) that will be discovered in more sensitive follow - up observations . the discrepancy at the high luminosity end of the x - ray luminosity function may also be explained if the distance to the tts in mbm12 is larger than 65 pc . although an increased distance is allowed by the recent _ hipparcos _ results it should be confirmed with further observations . although mbm12 is not a prolific star - forming cloud when compared to nearby giant molecular clouds it is the nearest star - forming cloud to the sun and offers a unique opportunity to study the star - formation process within a molecular cloud at high sensitivity . we have presented follow - up observations of x - ray stars identified in the region of the mbm12 complex . these observations have doubled the number of confirmed tts in this region . since the _ rosat _ pspc pointed observation of the central region of the cloud was sensitive enough to detect all of the previously known tts in the cloud , we believe the list of 5 ctts and 3 wtts presented in table [ eqw ] to be a nearly complete census of the tts in mbm12 for spectral types earlier than @xmath0 m2 . assuming a mean mass of @xmath0 0.6 m@xmath6 for the 8 currently known tts in mbm12 and a cloud mass of 30200 m@xmath6 ( pound et al . 1990 ; zimmermann & ungerechts 1990 ) the star - formation efficiency of mbm12 is @xmath0 224% . since the currently known tts population in mbm12 is incomplete only for the lower mass objects , unless there are a huge number of these objects yet to be discovered in the cloud , this estimate of the star - formation efficiency will not change significantly . although there is still a large uncertainty in the mass of the cloud the estimated star - formation efficiencies are consistent with that expected from clouds with masses on the order of 100 m@xmath6 ( elmegreen & efremov 1997 ) . by comparing the strengths of the h@xmath11 emission and @xmath336708 absorption lines of the tts in mbm12 with those found in other young clusters , we place an upper limit on the age of the stars in mbm12 @xmath0 10 myr . by comparing the x - ray luminosity function of the tts in mbm12 with that of the tts in l1495e we predict that there are more young , low - mass , stars to be discovered in mbm12 and the assumed distance to the cloud may have to be increased . although this prediction agrees with the recently revised distance estimate to the cloud ( @xmath0 @xmath2 pc ) based on results of the _ hipparcos _ satellite , it should be confirmed with future observations . we have also identified a reddened g9 star behind the cloud with @xmath3 @xmath0 8.48.9 mag . therefore , there are at least two lines of sight through the cloud that show larger extinctions ( @xmath3 @xmath4 5 mag ) than previously thought for this cloud . this higher extinction explains why mbm12 is capable of star - formation while most other high - latitude cloud are not . we wish to thank patrick guillout for helpful discussions about the expected population of young x - ray active stars located at high galactic latitude and loris magnani for insightful comments concerning this paper . we also thank an anonymous referee for suggestions which enable us to put firmer constraints on the age of the tts in mbm12 . project is supported by the max - planck - gesellschaft and germany s federal government ( bmbf / dlr ) . th is grateful for a stipendium from the max - planck - gesellschaft for support of this research . rn acknowledges a grant from the deutsche forschungsgemeinschaft ( dfg schwerpunktprogramm `` physics of star formation '' ) | we present the _ rosat _ pspc pointed and _ rosat _ all - sky survey ( rass ) observations and the results of our low and high spectral resolution optical follow - up observations of the t tauri stars ( tts ) and x - ray selected t tauri star candidates in the region of the high galactic latitude dark cloud mbm12 ( l1453-l1454 , l1457 , l1458 ) .
previous observations have revealed 3 `` classical '' t tauri stars and 1 `` weak - line '' t tauri star along the line of sight to the cloud . because of the proximity of the cloud to the sun ,
all of the previously known tts along this line of sight were detected in the 25 ks _ rosat _ pspc pointed observation of the cloud .
we conducted follow - up optical spectroscopy at the 2.2-meter telescope at calar alto to look for signatures of youth in additional x - ray selected t tauri star candidates .
these observations allowed us to confirm the existence of 4 additional tts associated with the cloud and at least 2 young main sequence stars that are not associated with the cloud and place an upper limit on the age of the tts in mbm12 @xmath0 10 myr .
the distance to mbm12 has been revised from the previous estimate of @xmath1 pc to @xmath2 pc based on results of the _ hipparcos _ satellite . at this distance mbm12
is the nearest known molecular cloud to the sun with recent star formation .
we estimate a star - formation efficiency for the cloud of 224% . we have also identified a reddened g9 star behind the cloud with @xmath3 @xmath0 8.48.9 mag
. therefore , there are at least two lines of sight through the cloud that show larger extinctions ( @xmath3 @xmath4 5 mag ) than previously thought for this cloud .
this higher extinction explains why mbm12 is capable of star - formation while most other high - latitude clouds are not . |
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searching for new modes in charm decays is of great interest . it not only investigates possible decay mechanism and finds its contribution to the total decay width , but is also useful to simulate accurately cascade decays of bottom mesons and to eliminate backgrounds of charm modes in studying bottom decays . in this paper , the semileptonic and hadronic @xmath9 decays in which the final state particles contain a @xmath10 meson are studied . whenever a specific state or decay mode is mentioned in this work , the charge - conjugate state or decay mode is always implied . the besii detector upgraded from the bes @xcite is a large solid - angle magnetic spectrometer described in detail elsewhere @xcite . a 12-layer vertex chamber ( vc ) surrounding the beryllium beam pipe provides the trigger and coordinate informations . a forty - layer main drift chamber ( mdc ) , located outside the vc , yields precise measurements of charged particle trajectories with a solid angle coverage of @xmath11 of @xmath12 ; it also provides ionization energy loss ( @xmath13 ) measurements used for particle identification . momentum resolution of @xmath14 ( @xmath15 in gev / c ) and @xmath13 resolution of @xmath16 for bhabha scattering are obtained for data taken at @xmath17 = 3.773 gev . an array of 48 scintillation counters surrounding the mdc measures the time of flight ( tof ) of charged particles with a resolution of about 180 ps for electrons . outside the tof is a 12 radiation length barrel shower counter ( bsc ) comprised of gas tubes interleaved with lead sheets . the bsc measures the energies of electrons and photons over @xmath18 of the total solid angle with an energy resolution of @xmath19 ( @xmath20 in gev ) and spatial resolution of @xmath21 mrad and @xmath22 cm for electrons . a solenoidal magnet outside the bsc provides a 0.4 t magnetic field in the central tracking region of the detector . the magnet flux return is instrumented with three double layers of counters , that are used to identify muons with momentum greater than 500 mev / c and cover @xmath23 of the total solid angle . the data used for this analysis were collected around the center - of - mass energy of 3.773 gev with the besii detector operated at the beijing electron positron collider ( bepc ) . the total integrated luminosity of the data set is about 33 pb@xmath0 . at the center - of - mass energy 3.773 gev , the @xmath24 resonance is produced in electron - positron ( @xmath25 ) annihilation . the @xmath24 decays predominately into @xmath26 pairs . if one @xmath27 meson is fully reconstructed , the @xmath9 meson must exist in the system recoiling against the fully reconstructed @xmath27 meson ( called singly tagged @xmath27 ) . using the singly tagged @xmath28 sample , the semileptonic decays @xmath29 and @xmath30 are searched in the recoiling system . the hadronic candidates @xmath3 and @xmath31 are reconstructed directly from the data sample of 33 pb@xmath0 . events which contain at least three charged tracks with good helix fits are selected . to ensure good momentum resolution and reliable charged particle identification , every charged track is required to satisfy @xmath32cos@xmath33 , where @xmath34 is the polar angle . all tracks , save those from @xmath35 decays , must originate from the interaction region by requiring that the closest approach of a charged track is less than 2.0 cm in the @xmath36 plane and 20 cm in the @xmath37 direction . pions and kaons are identified by means of the combined particle confidence level which is calculated with information from the @xmath13 and tof measurements @xcite . pion identification requires a consistency with the pion hypothesis at a confidence level ( @xmath38 ) greater than @xmath39 . in order to reduce misidentification , a kaon candidate is required to have a larger confidence level ( @xmath40 ) for a kaon hypothesis than that for a pion hypothesis . for electron or muon identification , the combined particle confidence level ( @xmath41 or @xmath42 ) , calculated for the @xmath43 or @xmath44 hypothesis using the @xmath13 , tof and bsc measurements , is required to be greater than @xmath39 . the @xmath45 is reconstructed in the decay of @xmath46 . to select good photons from the @xmath45 decay , the energy deposited in the bsc is required to be greater than 0.07 gev , and the electromagnetic shower is required to start in the first 5 readout layers . in order to reduce backgrounds , the angle between the photon and the nearest charged track is required to be greater than @xmath47 , and the angle between the cluster development direction and the photon emission direction to be less than @xmath48 @xcite . the singly tagged @xmath28 sample used in the analysis was selected previously @xcite . the singly tagged @xmath28 mesons were reconstructed in the nine hadronic modes of @xmath49 , @xmath50 , @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 and @xmath57 . the distributions of the fitted invariant masses of the @xmath58 combinations are shown in fig . [ dptags_9modes ] . the number of the singly tagged @xmath28 mesons is @xmath59 @xcite , where the first error is statistical and the second systematic . , ( b ) @xmath60 , ( c ) @xmath51 , ( d ) @xmath61 , ( e ) @xmath62 , ( f ) @xmath63 , ( g ) @xmath64 , ( h ) @xmath65 , ( i ) @xmath57 combinations ; ( j ) is the fitted masses of the @xmath66 combinations for the nine modes combined together.,title="fig:",width=340,height=264 ] ( -210,175)*(a ) * ( -90,175)*(b ) * ( -210,145)*(c ) * ( -90,145)*(d ) * ( -210,115)*(e ) * ( -90,100)*(f ) * ( -210,65)*(g ) * ( -90,62)*(h ) * ( -210,35)*(i ) * ( -90,45)*(j ) * ( -150,0)invariant mass ( gev/@xmath67 ( -250,50 ) candidates for @xmath70 and @xmath71 are selected from the surviving tracks in the system recoiling against the tagged @xmath28 . to select these candidates , it is required that there are three charged tracks , one of which is identified as an electron or a muon with the charge opposite to the charge of the tagged @xmath28 meson , the other two are identified as a @xmath72 and a @xmath73 mesons . the @xmath10 meson is selected by requiring that the difference between the invariant mass of @xmath74 and the nominal @xmath10 mass should be less than 20 mev/@xmath75 . in the semileptonic decays , one neutrino is undetected . a kinematic quantity @xmath76 is used to obtain the information of the missing neutrino , where @xmath77 and @xmath78 are the total energy and the total momentum of all missing particles respectively . monte carlo study shows that the background modes for @xmath79 come primarily from @xmath80 , @xmath81 and @xmath3 . the requirement @xmath82 is imposed to reduce these backgrounds , where @xmath83 is the standard deviation of the @xmath84 distribution obtained from monte carlo simulation . in fig . [ dp_phimunu_phipi_invm_umiss ] , the solid and dashed histograms show respectively the distributions of the invariant masses of @xmath85 and @xmath86 for @xmath2 and @xmath3 from monte carlo events . the candidate mode @xmath2 has a potential hadronic background of @xmath3 due to the misidentification of a charged pion as a muon . these backgrounds can be suppressed by requiring the invariant masses of the @xmath85 combinations to be less than 1.8 gev/@xmath75 . similarly the invariant masses of the @xmath87 combinations are also required to be less than 1.8 gev/@xmath75 in order to remove the @xmath3 background for the @xmath68 decay . to suppress backgrounds from decays with neutral pion , for example @xmath81 , the number of the isolated photons is required to equal two if the singly tagged @xmath28 mode contains a @xmath45 meson , otherwise zero . the events with isolated photons those energies are larger than 0.1 gev excluded in the tagged @xmath28 are not kept . ( solid ) and @xmath86 ( dashed ) for @xmath88 and @xmath3 from monte carlo events.,title="fig:",width=302,height=226 ] ( -137,13)@xmath89 ( gev/@xmath75 ) ( -225,45 ) fig . [ dp_phienu_2d ] and fig . [ dp_phimunu_2d ] show respectively the scatter plots of the invariant masses of the @xmath74 combination versus the @xmath66 combination for @xmath29 and @xmath30 before and after applying both @xmath84 and @xmath90 cuts . no event for the @xmath29 and @xmath30 decays satisfies the selection criteria . invariant masses versus the @xmath66 invariant masses for the @xmath29 candidates : ( a ) before and ( b ) after applying both @xmath84 and @xmath91 cuts , where the rectangle represents the combined signal region of @xmath10 and @xmath92.,title="fig:",width=264,height=226 ] ( -159,130)*(a ) * ( -159,65)*(b ) * ( -130,-3)@xmath93 ( gev/@xmath67 ( -208,50 ) invariant masses versus the @xmath66 invariant masses for the @xmath30 candidates : ( a ) before and ( b ) after applying both @xmath84 and @xmath94 cuts , where the rectangle represents the combined signal region of @xmath10 and @xmath92.,title="fig:",width=264,height=226 ] ( -159,130)*(a ) * ( -159,65)*(b ) * ( -130,-3)@xmath93 ( gev/@xmath67 ( -208,50 ) the candidate for @xmath3 is reconstructed from @xmath95 . the distribution of the fitted invariant mass of the @xmath96 combination is shown in fig . [ dptags_9modes](d ) . as mentioned in previous subsection , the @xmath10 meson is selected though its decay to @xmath74 . [ dp_phipi_k2p_data](a ) shows the fitted invariant mass spectrum of the @xmath74 pairs from the @xmath92 signal region for the @xmath97 candidate events . fitting the mass spectrum with a gaussian function convoluted breit - wigner gives @xmath98 @xmath10 signal events . to select the decay @xmath3 , we require the invariant mass of the @xmath74 combination to be within 20 mev/@xmath75 of the nominal @xmath10 mass , and the invariant mass distribution of @xmath96 combination is shown in fig . [ dp_phipi_k2p_data](b ) , a clear signal of the decay @xmath3 is observed . fitting the mass spectrum of the @xmath96 combination with a gaussian function as the signal and a special function @xcite for the background yields @xmath99 signal events , where the mass resolution is fixed at 0.0022 gev/@xmath75 determined from the monte carlo simulation . there may be the @xmath74 combinatorial background in the observed @xmath86 events . the background events are estimated to be @xmath100 by the @xmath10 sideband @xmath101 0.05 gev/@xmath75 . for the @xmath102 candidates , the fitted invariant mass spectrum is shown in fig . [ dptags_9modes](a ) , and @xmath103 events are yielded from the fit . combination in the @xmath92 signal region , ( b ) @xmath96 combination in the @xmath10 signal region.,title="fig:",width=302,height=264 ] ( -115,157)*(a ) * ( -115,70)*(b ) * ( -114,95)@xmath104 ( gev/@xmath67 ( -114,6)@xmath105 ( gev/@xmath67 ( -226,55 ) in the fitted yields , there are still some background contaminations which shape a peak under the @xmath92 peak . these backgrounds are estimated by analyzing the monte carlo sample which is about 14 times larger than the data . the monte carlo events are generated as @xmath106 , and both @xmath9 and @xmath107 mesons decay to all possible modes according to the decay modes and the branching fractions quoted from pdg @xcite excluding decay modes under study . the number of events satisfying the selection criteria is then normalized to the data . for the @xmath3 decay , the dominant background modes are @xmath2 , @xmath68 and @xmath81 , and the number of the background events is @xmath108 . for the @xmath31 decay , the number of the background events is @xmath109 , after subtracting the numbers of the background events , @xmath110 and @xmath111 signal events for the @xmath3 and @xmath6 decays are remained . the reconstruction efficiencies of the semileptonic decays @xmath68 , @xmath2 and the hadronic decays @xmath3 , @xmath31 are estimated by monte carlo simulation with the geant3 based monte carlo simulation package @xcite . a detailed monte carlo study shows that the efficiencies are @xmath112 , @xmath113 , @xmath114 and @xmath115 . a @xmath92 signal is observed neither in the @xmath68 decay nor in the @xmath2 decay . the upper limit of the branching fraction can be calculated using @xmath116 where @xmath117 is the upper limit of the signal yield given with the feldman - cousins prescription @xcite , which is 2.44 for zero observed event in the absence of background for the confidence level of 90% , @xmath118 is the detection efficiency , @xmath119 is the number of the singly tagged @xmath28 mesons and @xmath120 is the systematic error . the upper limits of the branching fractions are summarized in the second column of table [ uplbr ] . comparisons with those obtained by markiii @xcite and besi @xcite experiments are also listed in the third and fourth columns of table [ uplbr ] . .upper limits for the branching fractions ( % ) are given at the 90% confidence level . [ cols="^,^,^,^",options="header " , ] in addition , the branching fraction for @xmath3 is obtained to be @xmath121 by using the branching fraction of the @xmath6 decay quoted from pdg @xcite , where the first error is statistical and the second systematic . the systematic error of the branching fraction for @xmath3 includes the systematic error of the ratio @xmath122 ( 5.3% ) and the uncertainty of the branching fraction for @xmath6 ( 6.5% ) . the total systematic error is obtained to be 8.4% by adding these uncorrelated errors in quadrature . using a data sample of integrated luminosity of 33 @xmath123 collected around 3.773 gev with the besii detector at the bepc , the semileptonic decays @xmath1 , @xmath2 and the hadronic decay @xmath3 are studied . the upper limits of the branching fractions are set to be @xmath4 2.01% and @xmath5 2.04% at the 90% confidence level . the ratio of the branching fractions for @xmath3 relative to @xmath6 is measured to be @xmath7 . in addition , the branching fraction for @xmath3 is obtained to be @xmath8 . the bes collaboration thanks the staff of bepc for their hard efforts . this work is supported in part by the national natural science foundation of china under contracts nos . 10491300 , 10225524 , 10225525 , 10425523 , the chinese academy of sciences under contract no . kj 95t-03 , the 100 talents program of cas under contract nos . u-11 , u-24 , u-25 , the knowledge innovation project of cas under contract nos . u-602 , u-34 ( ihep ) , the national natural science foundation of china under contract no . 10225522 ( tsinghua university ) . | using a data sample of integrated luminosity of about 33 pb@xmath0 collected around 3.773 gev with the besii detector at the bepc collider , the semileptonic decays @xmath1 , @xmath2 and the hadronic decay @xmath3 are studied .
the upper limits of the branching fractions are set to be @xmath4 2.01% and @xmath5 2.04% at the 90% confidence level .
the ratio of the branching fractions for @xmath3 relative to @xmath6 is measured to be @xmath7 .
in addition , the branching fraction for @xmath3 is obtained to be @xmath8 . |
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odd - electron radical complexes like ho@xmath0cl@xmath3 and ho@xmath0h@xmath2o are of tremendous importance throughout chemistry and in related fields , such as in radiation chemistry , atmospheric chemistry , environmental chemistry , and cell biology@xcite . in particular , the behavior of anions in droplets is of critical importance to understanding atmospheric chemistry . common sense suggests that anions are less perfectly screened near a water - air interface , and so have lower concentration there . recent classical molecular dynamics ( md ) simulations have shown just the opposite@xcite , creating considerable controversy on this point . since anions are strongly quantum mechanical , it is logical to check traditional md simulations , using only classical force fields , against _ ab initio _ molecular dynamics ( aimd ) calculations , which use density functional theory ( dft ) to generate the potential energy surfaces ( pes ) . unfortunately , standard dft approximations have issues for such systems@xcite . several studies show dft approximations predict two minima in the ground - state pes . one of the minima is a hydrogen - bonding structure , while the other is a two - center three - electron interacting hemi - bonding structure@xcite . many dft studies predict the hemi - bonding structure as the global minimum of [ ho@xmath0cl(h@xmath2o)@xmath4 ^ -$ ] complex@xcite . this is attributed to the infamous self - interaction error@xcite , because aimd studies with self - interaction corrected ( sic ) methods agree with pes scans with high - level ( beyond dft ) quantum - chemical methods , showing no hemi - bonding configuration in the ground state@xcite . hemi - bonding configurations have been observed in experiments@xcite , but high - level quantum - chemical studies show that these are excited - state rather than ground - state configuration@xcite . thus , high - level quantum chemical calculations reveal that the ground - state pes has only one minimum , which is the hydrogen - bonding structure . the hemi - bonding structure is relatively overstabilized in dft because three electrons are incorrectly delocalized over two atoms rather than having localized electron configurations . essentially , the aimd studies raise more questions than they answer . however , recent work@xcite produces a much simpler and more general picture of such errors , and suggests alternative solutions . in any approximate dft calculation , the density is found by self consistently solving the kohn - sham ( ks ) equations , using an approximate ks potential derived from the energy approximation . this procedure finds the density that minimizes the energy approximation for the given system . the final output energy is of the approximate energy functional evaluated at the approximate density . for most ks - dft calculations using standard approximations , the ks potential appears to be of very poor quality@xcite . in particular , the eigenvalues are usually too shallow by several ev@xcite . dft approximations are designed to yield good energies , but this does not automatically imply good potentials . functional derivatives are determined by how well the approximation performs for small variations _ away _ from the density of interest . nevertheless , usually the self - consistent density is rather good , because the approximate ks potential is rather close to a simple shift of the exact ks potential . ( highlighting this point , a recent approximation yields highly accurate energies with truly terrible potentials@xcite . ) this then raises a simple issue . one can think of two distinct sources of error in such a calculation : one of the error due to the energy approximation itself , and the other error in the corresponding approximate density . in most dft calculations , it is ( correctly ) believed that the first ( functional ) error dominates . but in many interesting cases that are specific to the system and the approximation , the error in the approximate density can be unusually large , so that it contributes more to the total energy error than would usually be the case . in such cases of density - driven errors , use of a more accurate density should reduce the final error substantially . a small homo - lumo gap ( @xmath5 ) in the original approximate dft calculation is often a strong indicator of a large density - driven error@xcite . in many cases , with standard dft approximations , a hartree - fock ( hf ) calculation provides a sufficiently accurate density to garner much of the improvement in energetics . we dub such a calculation , which is quite general , density corrected dft ( dc - dft ) , meaning any dft calculation in which the density is _ not _ found self consistently , but by some other method that substantially reduces the density - driven error . we note that a better term for self - interaction error is delocalization error , which has been recently much studied by cohen et al@xcite . our analysis complements theirs . in that language , we are able to determine if a given delocalization error is density - driven or not . if the former , a better density will improve the energy ; if the latter , as in the case of stretched h@xmath6 , it will not . we expect cases like dft calculation on stretched he@xmath6 to be normal , so the density - driven error would be small compared to the functional error@xcite . in the present work , we apply this scheme to a variety of commonly - used standard approximations in dft , applied to the ho@xmath0cl@xmath1 complex . we use the hf density as our more accurate density . we use the notation a - b , where a indicates the method for finding the density , and b the method for finding the energy . thus hf - dft implies using a hf density , but evaluating its energy using dft approximation . in fig . [ hoclpesfunc2 ] we show the near - perfect agreement of many commonly - used functionals for the potential energy surface of ho@xmath0cl@xmath1 , once they are evaluated on hf densities . we find that not only does hf - dft always produce a pes with the correct global minimum , but that the difference in pes s between different approximations becomes negligible and is , up to a constant , essentially identical to the ccsd(t ) pes . thus , with this elementary correction , the pes of such systems changes from being a disappointing failure to being a resounding success for approximate dft . there is always a price to be paid for progress , and in this case , there are two ( not so hidden ) challenges to taking advantage of this improved performance . the first is that , if the density is no longer the minimizer for the given approximate energy functional , many basic theorems , such as the hellmann - feynman theorem , no longer apply , and many of these are used in standard dft codes . but the way to calculate forces has already been detailed in a pioneering work@xcite which pointed out that hf - dft generically improves most reaction barriers over self - consistent dft , and implemented it in the aces code . the second is that aimd simulations are often performed with plane waves enforcing periodic boundary conditions , and solving the hf equations can be computationally expensive in such implementations@xcite . however , some simpler scheme available in those codes , such as self - interaction corrected local density approximation ( lda - sic)@xcite , might yield densities sufficiently accurate for the purpose while using e.g. , generalized gradient approximations ( gga ) for the energy evaluation . alternatively with the recent success of hybrid functionals such as hse06@xcite in predicting fundamental gaps , it might be straightforward to run a hf calculation under these conditions . but all these are beyond the scope of the present paper . the aim here is to show just how much of an improvement is possible with dc - dft . we begin with some theory , which also includes the historical background . since this idea is so elementary , it has been touched upon several times in the distant and recent past . we also explain how this fits in with all the present attempts to go beyond standard approximations , including using exact exchange in dft@xcite , _ ab initio _ dft@xcite , and several many - body approaches using ks orbitals@xcite . next we show a variety of results for the complex at hand , using several functionals , basis - sets , and implicit solvent models . we also demonstrate similar results for the ho@xmath0h@xmath2o radical , showing that the effect is not specific to anionic species , but is much more general . in any ks - dft calculation , only a small fraction of the total energy need be approximated as a functional of the density , namely the exchange - correlation ( xc ) energy . in reality , we always use spin densities , but for the present purposes , we suppress the spin index . the self - consistent solution of the ks equations has been designed to deliver the density that minimizes the total energy in the ks scheme@xcite : e [ ] = t+ u [ ] + v [ ] + e where the @xmath7 indicates an approximation . the functionals @xmath8 $ ] , @xmath9 $ ] , @xmath10 $ ] , @xmath11 $ ] , and @xmath12 $ ] are the total energy , kinetic energy , external potential energy , hartree - energy and xc energy functionals , respectively@xcite . the energy error in a dft calculation is defined as e = e [ ] - e [ ] where @xmath13 is the approximate self - consistent density . we may write this error as the sum of two contributions . we call the first the _ functional _ error . it is the energy error made by the functional evaluated on the exact density , and comes entirely from the xc approximation : e_f = e [ ] - e [ ] = e - e . the _ density - driven _ error is the energy difference generated by having an approximate density : e_d = e[]-e[n ] , so that the total energy error is the sum of these two : e = e_f + e_d . this separation applies to _ any _ approximate dft calculation , not just a ks calculation of electronic structure . but here we focus exclusively on the latter , since we wish to use this as a tool to analyze chemical calculations using ks - dft . note that this elementary breakdown is hardly a breakthrough . most developers and many users of dft have thought along these lines or come across this in some calculation or context of theory development . what _ is _ new is how far this elementary step can be taken in analyzing all practical dft calculations , i.e. , those using approximate functionals , and how to improve many of these . the first point to note is that for most calculations using the ks scheme and modern approximations to xc , such as a gga or a global hybrid , the densities are remarkably accurate . the above tool allows us to specifically quantify this accuracy , by measuring it in terms of its effect on the quantity we almost exclusively care about , namely the ground - state energy . in any calculation , if @xmath14 , any error in the density is irrelevant for practical purposes . for example , it has long been known that the density with standard approximations is highly inaccurate at large distances from the nuclei , due to the highly inaccurate homo in such calculations@xcite . however , in most cases , this inaccuracy produces only a very small density - driven error in the energy , so such approximations remain accurate for ground - state energies . moreover , dft approximations produce far more accurate ionization energies via total energy differences than via orbital energy differences . next , we consider some popular application , such as the calculation of a bond length with dft . since the bond length is extracted as the minimum of the total energy , the contributions to the error can be split into functional - driven and density - driven , and the two compared . thus one extracts the density - driven contribution to a given property of a given system with a given approximate functional . if this error is small or negligible compared to the actual error , we classify such a calculation as _ normal_. we believe the vast majority of dft calculations fit into this category , and we gave several examples in our previous work , including cases like stretched h@xmath6 cation@xcite . in fact , in many circumstances of method development , there is an underlying assumption that the calculation _ is _ normal . with a new approximate functional , it can often be the case that the functional derivative is demanding to calculate . thus , often a lower - level , more standard approximation is used to calculate orbitals in the ks equations , and the new approximation is tested on those orbitals . if the calculation is normal , this ( almost ) guarantees that only a small error is made by this procedure , and the change upon self - consistency will be negligible . our main interest , naturally , will be in those calculations where the density - driven error is a significant fraction of the total . we denote such calculations as _ abnormal_. in such cases , a more accurate density will reduce the error ( assuming no accidental cancellation of functional- and density - driven errors ) . we can also state just how much more accurate that density need be : enough to make the density - driven error small relative to the functional error . in such cases , correcting the density in a dft calculation greatly reduces the error ; hence our name for this method . while this separation scheme can be applied to any approximate dft calculation , we are presently focused on the infamous self - interaction error inherent in standard gga and global hybrid calculations . in earlier work , we found that many cases of such errors ( small anions , underestimated transition barriers , and incorrect dissociation limits ) were in fact density - driven@xcite . in such cases , the density was particularly poorly described , and a simple hf density was sufficient to drive out the density - driven error . note that this does not mean the hf density is especially good . in the vast majority of cases ( the normal ones ) , the hf density is worse than the self - consistent dft density@xcite , and hf - dft is worse than self - consistent dft , as we have shown in cases like stretched h@xmath15 in our previous work@xcite . in these normal cases , self - consistent dft is usually sufficient enough for getting accurate energies despite having incorrect potentials@xcite . but for abnormal systems , the self - consistent dft density is especially poor in a very systematic way , a way that is largely fixed by hf . as we showed in ref . , an unusually small ks homo - lumo gap , @xmath16 , in the dft calculation indicates a likely abnormal calculation . this means the self - consistent solution is unusually sensitive to small changes in the potential , so that an error in the xc contribution to that potential can produce an unusually large effect on the density . in the case of atomic anions , this becomes extreme : the homo is positive if the basis set is used to hold it in , and zero in the basis - set converged limit@xcite . this leads to very poor densities , missing a significant fraction of an electron , and large density - driven errors . hf densities are a major improvement in such cases . .energy decomposition of h@xmath3 and he . [ cols="^,^,^,^,^,^,^ " , ] [ 2eenergy ] [ 2edenpot ] to illustrate the idea , we consider the simplest possible case , the he atom . we begin by applying our analysis to a hf calculation of the system . for two spin - unpolarized electrons , a hf calculation is equivalent to a ks - dft calculation with e= - u[]/2 . in fact , the total error in such a calculation is simply the ( quantum - chemical ) definition@xcite of the negative of the correlation energy : e = e- e[n ] = - e. in table i , we list the different errors in hf and pbe calculations , for both he and h@xmath1 . we see that , for hf applied to he , the density - driven error is minuscule ( 0.05 mh ) , i.e. , about 0.1% of the functional error . this says that , for this problem , the hf density is extremely accurate , and essentially all the error comes from the missing correlation energy . such a calculation could be considered ultra - normal . next , we repeat the analysis with the pbe approximation@xcite . here the total error is smaller ( about 11 mh ) , and the density - driven error is -1 mh . because this is still only of order 10% , this remains a normal calculation , just like the hf calculation . but in fig . [ hedenpot ] , we show the corresponding densities and ks potentials for the pbe and hf calculation . although the densities are identical to the eye , we see that the ks potential of the pbe calculation is far too shallow . this is typical of all approximate dft calculations , and leads to ks eigenvalues that are far too shallow(-0.58 ev instead of -0.903 ev ) . nevertheless , we emphasize that these are normal calculations , and the error in the potential produces very little error in the density , and so relatively small density - driven errors . we also emphasize that normality ( or otherwise ) is a characteristic of a particular calculation ( i.e. , the system and the approximation together ) . , but for h@xmath1.,width=240 ] now we repeat this exercise for the @xmath17 ion . it is often said that h@xmath1 is ` correlation bound ' , meaning that in hf theory , h@xmath1 is not stable . in a hf calculation , the ground - state energy of h is lower than that of h@xmath1 . nonetheless , we can still converge the calculation and obtain the ground - state energy . the correlation energy remains about the same , but the density - driven error is much larger . in fig . [ h - denpot ] , we can see the error in the density by eye . nonetheless , this calculation is also normal , with a density - driven error that is only about 3% of the total . it is the functional error that makes h@xmath1 correlation bound . a very different story is seen for the pbe calculation on the same system . in fact , only by allowing a fraction of an electron ( about 0.3 ) to escape the system can the calculation be properly converged at all@xcite . here the density - driven error is more than 10 times larger than the magnitude of the functional error . the density itself is very dramatically different from the exact one , and the pbe ks potential is not only too shallow , but is actually positive , and the eigenvalue is exactly zero . this is a very abnormal calculation . evaluation of the pbe approximation on the exact density removes the density - driven error , and so drops the total energy error by an order of magnitude . even using the hf density is sufficient to produce a much highly accurate electron affinity of h@xcite . we end this section by noting that the functional errors and density - driven errors have opposite signs . thus , in a normal calculation , application of the approximate functional on the exact density will _ increase _ the energy error ( albeit only slightly ) . we have found this in all our calculations so far , which suggests this is typical behavior . thus we do _ not _ recommend universally using more accurate densities than the self - consistent density . only when a calculation is abnormal do we suggest such a procedure . in the early days of dft , it was often easier to use a hf code to find densities and evaluate xc approximations on those densities , again because such calculations were assumed to be normal@xcite . when dft began to become popular in chemical applications in the early 1990 s , this mode of testing approximations , called hf - dft , was used in calculations@xcite . but very quickly , the computational and conceptual advantages of self - consistency led to self - consistent dft calculations . moreover , the hf - dft results were not systematically compared to self - consistent dft calculations , except in some pioneering works which suggested that in difficult cases , the hf - dft may yield more accurate answers@xcite . the present paper may be considered as a fuller exploration and quantification of those early results . what does this analysis say about the many attempts to improve energetics in other ways , such as _ ab initio _ dft@xcite , random - phase approximation ( rpa ) @xcite , density - matrix functionals , etc ? our analysis explains several key features . the first is that , despite having very wrong - looking xc potentials , and hence bad ks potentials , the effect of these errors on the energy via the density is minimal ( a more detailed explanation is given in ref . ) . thus despite legions of papers reporting the very incorrect energy eigenvalues of ks potentials with approximate functionals , especially the highest occupied one which , for the exact functional , matches the negative of the ionization potential , dft with these approximations continues to be very heavily used , because the energetics are unaffected . second , we now have a tool for quantifying the energetic error due to the density error . this allows us to ask questions such as when do we need to improve the density , in order to improve the energy . for example , dft calculations using so - called exact exchange ( exx ) have much more accurate ks potentials than those of standard approximations , yet usually worse energetics . this is because their functional errors usually outweigh the reduction in the density - driven error . obviously , our present answer is a purely pragmatic one . in specific circumstances , the density becomes sufficiently poor as to be the major source of error . this shows that only in certain circumstances does this problem need to be addressed , and if ways could be found to avoid the poor self - consistent densities in such cases , a variety of apparent dft errors would be avoided . we begin with the pes of the ho@xmath0cl@xmath1 complex . the pes was scanned by changing the cl - o distance ( @xmath18 ) and cl - o - h angle ( @xmath19 ) , as indicated in fig . [ hoclpesfunc1](a ) . the o - h bond length was fixed at 1.0 . we performed single - point energy calculations with dft , hf - dft , and coupled - cluster method ( ccsd(t ) ) on each geometry . we used gga functionals ( pbe@xcite and blyp@xcite ) , hybrid functionals ( pbe0@xcite and b3lyp@xcite ) and a double - hybrid functional with empirical dispersion correction ( b2plyp - d)@xcite . we did these both in the gas phase and with implicit solvent . for implicit solvent calculations , we use the conductor - like screening model ( cosmo)@xcite within the turbomole suite@xcite . we used dunning s augmented correlation - consistent basis sets with x zeta functions ( aug - cc - pvxz , x = 2 - 3 , denoted as avxz for simplicity from now on)@xcite . contour plots of the ho@xmath0cl@xmath1 complex pes evaluated with various methods are shown in fig . [ hoclpesfunc1 ] . for this simple complex , we compare dft results with ccsd(t ) , which we take as a benchmark . we notice several drastic failures of dft approximation in these calculations . the worst qualitative failure is that the minimum of the gga pes is not at 0@xmath20 , but is closer to 30@xmath20 . this is an incorrect hemi - bonding arrangement , attributed to the strong self - interaction error of the extra electron in the literature@xcite . we also note that the contours of the pes are quite incorrect in shape everywhere in the plane we have plotted . finally , the gga pes is too negative overall ( blue everywhere ) indicating it is essentially useless for performing aimd simulations of anions . while pbe is very popular for many materials simulations and static quantum chemical calculations , in fact , most aimd simulations do not use pbe but other gga s instead@xcite . this is because some key attributes of thermal simulations of water are incorrectly described by pbe . a popular alternative is blyp , even though this is rarely used in regular quantum chemical calculations ( unlike its hybrid off - spring , b3lyp ) . but a glance at fig . [ hoclpesfunc1 ] ( c ) shows that blyp is almost identical to pbe for this purpose , and suffers all the same difficulties . on the other hand , hybrid density functionals , which add some fraction of hf to the gga form , usually improve energetics@xcite and partially correct self - interaction error . so in fig . [ hoclpesfunc1 ] ( e ) and ( f ) we plot the results with pbe0 and with b3lyp , which is the most popular functional in quantum chemistry . we see that indeed there is great improvement . the minimum is now correctly at alignment , the surfaces are not entirely blue , and the shape is roughly correct . as we have shown in fig . [ hoclpesfunc2 ] , the results by evaluating dft energies on hf densities are striking . in every case , we get essentially identical results throughout the plane of the pes . all minima are in the correct locations , no curves are too blue , and all the details are correct . the results are so consistent that we can draw several important conclusions . * such good agreement confirms the theory behind dc - dft . all these dft calculations are _ abnormal _ , and the error is greatly reduced by using a better density . it also confirms that the hf density is sufficiently more accurate than the self - consistent dft density for these calculations to produce much more accurate energies . * for this problem , we no longer need the benchmark defined by ccsd(t ) . the extreme level of consistency between so many different dft approximations imply that all such calculations are yielding a very accurate answer . * while the hybrid functionals definitely improve over the gga s , the primary effect is the improvement in the self - consistent density due to the hf component in the energy . in fact , evaluated on a sufficiently accurate density , there is no need to use a hybrid functional ( but of course , finding the hf density is relatively expensive in aimd calculations ) . , of ho@xmath0cl@xmath1 complex for several approximate functionals.,width=240 ] next we consider the gap in the approximate self - consistent ks calculations . part of the dc - dft theory is that an abnormal system should have an unusually small ks gap , suggesting that its density is unusually inaccurate@xcite . [ hoclgapfunc ] shows @xmath16 for the ho@xmath0cl@xmath1 complex . each point corresponds to the energy minimum of each @xmath19 , i.e. , using @xmath18 that gives the lowest binding energy for given @xmath19 . since we scanned @xmath18 = 2.5 @xmath7 4.5 for calculations , we excluded @xmath21 70@xmath20 from this figure where the energy minimum was located at @xmath18 = 2.5 or @xmath18 = 4.5 . the gga methods clearly have small @xmath16 ( less than 1 ev ) which is consistent with the large density - driven error in these methods . in the case of hybrid methods , the @xmath16 is a mixture of a hf gap and a ks gap rather than pure ks gap , so it may not be as good as an indicator for density - driven errors . the hybrid @xmath16 are not as small as in gga , but still less than 2 ev , which explains the moderate density - driven error compared to gga methods . as some dft calculations with @xmath16 even as high as 2.5 ev have large density - driven error@xcite , calculations with @xmath16 below 2 ev should be suspected of being abnormal . to gain more insight and quantitative understanding , we show energy curves along @xmath18 = 3.0 in fig . [ hoclfunc30r ] . the binding energy @xmath22 is defined as @xmath23 - ( e[\cdot\mbox{oh } ] + e[\mbox{cl}^{-}]$ ] ) , where @xmath24 $ ] , @xmath25[@xmath0oh ] , and @xmath25[cl@xmath3 ] is the energy of ho@xmath0cl@xmath3 complex , oh radical , and cl@xmath3 anion respectively . the dissociation energy @xmath27 is defined as @xmath28 , where @xmath29 is the energy minima on @xmath18 = 3.0 for the given calculation method . we see very clear patterns . in fig . [ hoclfunc30r](a ) , the gga s ( blue and green ) produce incorrect minima at @xmath19 = 20@xmath20 . hybrid methods ( red and orange ) are much more accurate near the minimum , but show increasing error as @xmath19 gets larger . the b2plyp - d curve is highly accurate almost everywhere . on the other hand , in fig . [ hoclfunc30r](b ) , all the gga and hybrid curves line up almost perfectly , once evaluated on hf densities . the small remaining deviation among them is near the minimum , where the pbe methods ( pbe and pbe0 ) are most accurate . in any case , all are slightly shifted above the accurate ccsd(t ) curve . finally , we decompose the energy error into density - driven and functional errors for both the ggas and hybrids in fig . [ hoclfunc30r](c ) . gga methods show large density - driven error for all regions , maximizing at @xmath19 = 130 @xmath7 140@xmath20 , while hybrid methods have less but still significant density - driven error maximizing at @xmath19 = 120 @xmath7 130@xmath20 . functional errors stay almost constant for every @xmath19 . the evaluated density changes as the geometry changes and the functional used is left unchanged , resulting in this independence of functional error to geometry . we have also calculated pes for both pbe and b3lyp , self consistently and in hf - dft , using a smaller basis , namely , avdz . these are qualitatively and quantitatively almost identical to those with avtz , showing basis set convergence , and that avdz may be sufficient for most purposes for these calculations . this is illustrated in the energy error decomposition for pbe in fig . [ hoclbas30ang2](a ) , where the shifts from one basis to the next are tiny compared to all other energy error contributions . the @xmath16 of both methods in fig . [ hoclbas30ang2](b ) are also similar . next , we show what happens when we use a more modern and more accurate approximate functional for this problem . the b2plyp - d functional is a double hybrid functional combined with empirical dispersion parameters@xcite . in conventional hybrid functionals , hf exchange is added as the non - local exchange contribution . in addition to this , b2plyp - d has the non - local perturbation correction added for the correlation part by second - order perturbation theory . this is based on _ ab initio _ kohn - sham perturbation theory ( ks - pt2 ) by grling and levy@xcite . due to the large fock exchange fraction , self - interaction error is greatly reduced , while the side effects of having large fock exchange , such as incomplete static correlation , are alleviated by the second - order perturbation in the correlation@xcite . this leads to excellent results in many cases@xcite , including two - center three - electron bonding in radical complexes@xcite , which makes the method a great choice of benchmark for this work . in fig . [ hoclpesb2plyp](a ) , this approximation is doing an excellent job of reproducing the pes everywhere , on its _ self - consistent density ! _ thus this functional is sufficiently accurate for this problem that this is a _ normal _ calculation . sure enough , when we repeat the calculation using the hf density , as shown in fig . [ hoclpesb2plyp](b ) , the pes worsens . this strongly suggests that the b2plyp - d self - consistent density is better than the hf density here . thus this functional can be used for this problem without modification , so long as the user can afford to evaluate it , and should not be density corrected . but all the cruder older approximations yield abnormal results and need correction . dauria _ et al._@xcite performed aimd simulations on the ho@xmath0cl@xmath1 complex in explicit water solvents where the minimum appeared to be a hemi - bonding structure ( @xmath19 = 80@xmath20 ) with a standard approximate functional blyp , while self - interaction corrected blyp ( blyp - sic ) gave a hydrogen - bonding minimum structure on @xmath19 = 0@xmath20 . they then scanned the gas phase ho@xmath0cl@xmath1 complex pes along @xmath19 = 0@xmath20 and @xmath19 = 80@xmath20 based on those observations . as we observed in our gas phase calculation , the true minimum of gas phase ho@xmath0cl@xmath1 complex lies somewhere between @xmath19 = 0@xmath20 and 80@xmath20 . to look in more depth at solvation effects , we show contour plots of ho@xmath0cl@xmath1 complex in implicit water solvent in fig . [ hoclpessolv ] . for both pbe and b3lyp calculations , the minimum is clearly a hemi - bonding structure with @xmath19 = 80@xmath20 , in contrast to the gas phase calculation , where the global minimum was at @xmath19 = 20@xmath20 . the fock exchange in the hybrid functional indeed has some effect , producing a second local minimum along the hydrogen - bonding region ( @xmath19 = 0@xmath20 ) , yet did not correct the overstabilization of the hemi - bonding structure , resulting in the wrong global minimum . on the other hand , the sole minimum of both dc - dft calculations is the hydrogen - bonding structure for both gas phase and implicit water calculations . unlike the self - consistent dft results , the pes are quite similar regardless of functional , which was a trait also observed in gas phase pes , and no sort of local minimum is shown in the hemi - bonding region . this shows the accuracy of dc - dft is on par with sic - dft with far less computational cost , at least for non - periodic cases , even in the presence of implicit solvent . finally , we would like to mention that there are some works that show ks - dft greatly underestimates the redox potential of oh@xmath0/oh@xmath1 and cl@xmath0/cl@xmath1 in explicit solvent while this is not the case in implicit solvent simulations@xcite . even though it is possible that there is some difference between our results and the explicit solvent results , our findings matched the explicit solvent result unlike the oh@xmath0/oh@xmath1 and cl@xmath0/cl@xmath1 case . we believe this discrepancy is due to the difference between a anion - radical complex and a lone radical , alleviating the effect of extended states of explicit solvent . h@xmath30o complex . ( b ) comparison of the pes scan for ho@xmath0h@xmath2o complex using various methods . binding energies are plotted against @xmath31 . each point is using the minimum energy geometry for given @xmath31.,width=412 ] h@xmath2o complex . each point is using the minimum energy geometry for given @xmath31.,width=288 ] to confirm that the performance of dc - dft is not restricted to anion complexes , we also look at a neutral radical complex . we evaluated the pes of the ho@xmath0h@xmath2o complex using dft , hf - dft and ccsd(t ) with avtz basis set . pbe and blyp functionals are used in dft calculations . we used the same parameters used by chipman@xcite , depicted in fig . [ hoh2opescurve](a ) . the evaluated pes is depicted in figs . [ hoh2opescurve ] and [ hoh2opes ] . the binding energy @xmath22 here is defined as @xmath32ho@xmath0h@xmath2o ] - ( e[@xmath0oh ] + e[h@xmath2o ] ) , where @xmath33ho@xmath0h@xmath2o ] , @xmath34oh ] , and @xmath33h@xmath2o ] is the energy of the ho@xmath0h@xmath2o complex , oh radical , and h@xmath2o molecule , respectively . in fig . [ hoh2opescurve ] , each point indicates the minimum energy possible for a given @xmath31 . we chose to scan between @xmath35 to @xmath36 and @xmath37 to @xmath38 . the @xmath35 to @xmath36 region is where the hemi - bonding geometry was observed in excited state calculations , while the latter is where the global minimum of the ground state was discovered@xcite . the ccsd(t ) results reproduce chipman s result , where the global minimum is in the hydrogen - bonding region of @xmath39 . on the other hand , despite having a minimum in the hydrogen region , both dft methods clearly have the global minimum in the hemi - bonding region of @xmath40 . now , as one can expect from the ho@xmath0cl@xmath1 complex results , the hf - dft curve successfully resembles the ccsd(t ) results , having the global minimum in the hydrogen - bonding region of @xmath41 . also ccsd(t ) and hf - dft results have no minimum in the hemi - bonding region , so the @xmath31 value with the lowest energy is 60@xmath42 . we scanned through @xmath18 and @xmath43 on the @xmath31 values that give minimum energy for each method and region in fig . [ hoh2opes ] . once again , the pes of hf - pbe looks like the pes of ccsd(t ) with an energy shift in both hemi - bonding and hydrogen - bonding region , while pbe has a clearly different pes in the hemi - bonding region . fig . [ hoh2oerror ] shows the error decomposition of the pbe calculation . as expected , calculations in the hemi - bonding region exert a strong density - driven error . in the hydrogen - bonding region , the density - driven error is quite small compared to the hemi - bonding region , but @xmath16 in both regions is still quite small ( @xmath16 = 1.09 ev at the hemi - bonding minimum , 0.97 ev at the hydrogen - bonding minimum ) . approximate dft typically suffers from severe self - interaction error in calculations of odd - electron radical complexes@xcite . this work shows that the density correction in dc - dft , even using simple hf densities , can often give more accurate results than dft using self - consistent densities in these types of calculations . to explain this in a systematic way , we showed _ any _ approximate calculations can be classified into one of the two types . in _ normal _ calculations , the functional error dominates , while in _ abnormal _ calculations , the density - driven error is larger than the functional error . we illustrated this using simple two - electron systems , namely h@xmath1 anion and he atom . approximate dft was abnormal in h@xmath1 showing severe density - driven errors , while all other cases were normal . in these normal cases , the density - driven error was negligible , despite having very wrong - looking ks potentials . by this analysis , we stressed that dc - dft is likely to give better results than approximate dft with self - consistent densities only for abnormal calculations , and not for normal calculations . we presented pes of an anion radical complex , e.g. , the ho@xmath0cl@xmath1 complex , using various common approximate functionals including gga functionals and hybrid functionals and even more modern functionals like b2plyp - d , with both self - consistent dft and dc - dft . both gga and hybrid functionals behaved poorly self consistently and only using the highly accurate b2plyp - d self - consistent density was sufficient for getting accurate pes . on the other hand , dc - dft gave identical pes regardless of the approximate functional used , and gave correct global minima and pes slopes , showing gga and hybrid approximate functional calculations are abnormal calculations . using the concept introduced from our previous letter@xcite , we showed a very small @xmath16 can be used as an indicator of abnormality . we checked the calculations with different basis sets where we have similar results for both avdz and avtz basis set . we also showed even calculations with implicit solvent have similar tendencies , where self - consistent gga and hybrid densities give poor results , predicting hemi - bonding structures as the global minimum . pes evaluated from dc - dft were once more identical , independent of the functional used . finally , we examined the validity of dc - dft for neutral radical complexes by evaluating pes of ho@xmath0h@xmath2o . self - consistent dft predicted the hemi - bonding structure as the global minimum , while dc - dft correctly predicted the hydrogen - bonding structure as the global minimum . these results and the @xmath16 showed the abnormality in these self - consistent dft calculations . dc - dft can be used as a simple cure of abnormality , i.e. , strong density - driven error especially driven from self - interaction , which has less computational cost and is free of empirical parameters compared to various sic methods . we must mention the hf density we used in this work may not be appropriate for all cases , including cases with strong spin - contamination@xcite , or periodic boundary conditions . nonetheless , one can use dc - dft with any other source of accurate densities in cases where the hf density is not suited . additionally , we expect this method to give promising results for various problems that are challenging for approximate dft like reaction barriers and dissociation@xcite , when the errors are density - driven . we thank eberhard engel for the use of his atomic oep code and cyrus umrigar for the exact two electron system data from qmc calculation . this work was supported by the global research network grant ( no . nrf-2010 - 220-c00017 ) and the national research foundation [ 2012r1a1a2004782 ( e. s. ) ] . k.b . acknowledges support under nsf che-1112442 . m - c thanks the fellowship of the bk 21 program , and sanghyeon lee for his effort . 65ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ ( , ) link:\doibase 10.1021/cr030453x [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1063/1.555805 [ * * , ( ) ] link:\doibase 10.1080/10643380500326564 [ * * , ( ) ] link:\doibase 10.1080/10643380601163809 [ * * , ( ) ] @noop _ _ , ed . ( , ) link:\doibase 10.1021/jp077669d [ * * , ( ) ] link:\doibase 10.1063/1.1630017 [ * * , ( ) ] link:\doibase 10.1039/b311840a [ * * , ( ) ] link:\doibase 10.1063/1.1468640 [ * * , ( ) ] link:\doibase 10.1021/jz3015293 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/jp044751p [ * * , ( ) ] link:\doibase 10.1039/b501603 g [ * * , ( ) ] link:\doibase 10.1021/jp2063386 [ * * , ( ) ] link:\doibase 10.1039/j19690000446 [ ( ) ] link:\doibase 10.1021/jp964097 g [ * * , ( ) ] link:\doibase 10.1021/jp903625k [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.073003 [ * * , ( ) ] link:\doibase 10.1103/physreva.50.3827 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.80.3 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4834075 [ * * , ( ) ] link:\doibase 10.1063/1.4770226 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.2741248 [ * * , ( ) ] link:\doibase 10.1016/j.cplett.2011.12.017 [ * * , ( ) ] link:\doibase 10.1103/physrevb.48.5058 [ * * , ( ) ] link:\doibase 10.1103/physrevb.23.5048 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.2404663 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1398093 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1453958 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4755818 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.2977789 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.3043729 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.093003 [ * * , ( ) ] link:\doibase 10.1002/qua.24259 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1063/1.3590364 [ * * , ( ) ] link:\doibase 10.1039/c3cp52547c [ * * , ( ) ] @noop * * , ( ) @noop `` , '' in link:\doibase 10.1021/bk-1996 - 0629.ch003 [ _ _ ] , , chap . , pp . @noop * * , ( ) link:\doibase 10.1103/physreva.38.3098 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1002/qua.560440828 [ * * , ( ) ] @noop * * ( ) link:\doibase 10.1063/1.463977 [ * * , ( ) ] \doibase doi:10.1063/1.477479 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.2148954 [ * * , ( ) ] link:\doibase 10.1002/jcc.20495 [ * * , ( ) ] link:\doibase 10.1039/p29930000799 [ ( ) ] @noop @noop * * , ( ) http://link.aip.org/link/?jcp/98/1358/1 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1021/jp960669l [ * * , ( ) ] link:\doibase 10.1103/physrevb.47.13105 [ * * , ( ) ] link:\doibase 10.1103/physreva.50.196 [ * * , ( ) ] link:\doibase 10.1021/ar700208h [ * * , ( ) ] link:\doibase 10.1039/b704725h [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.cplett.2010.11.062 [ * * , ( ) ] | standard density functional approximations often give questionable results for odd - electron radical complexes , with the error typically attributed to self - interaction . in density corrected density functional theory ( dc - dft ) , certain classes of density functional theory calculations are significantly improved by using densities more accurate than the self - consistent densities .
we discuss how to identify such cases , and how dc - dft applies more generally . to illustrate , we calculate potential energy surfaces of ho@xmath0cl@xmath1 and ho@xmath0h@xmath2o complexes using various common approximate functionals , with and without this density correction .
commonly used approximations yield wrongly shaped surfaces and/or incorrect minima when calculated self consistently , while yielding almost identical shapes and minima when density corrected .
this improvement is retained even in the presence of implicit solvent .
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given graphs @xmath0 and @xmath1 , say that @xmath3 if every 2-edge - coloring of @xmath1 results in a monochromatic copy of @xmath0 in @xmath1 . using this notation , the ramsey number @xmath4 of @xmath0 is the minimum @xmath5 such that @xmath6 . instead of minimizing the number of vertices , one can minimize the number of edges . define the _ size - ramsey number _ @xmath7 of @xmath0 to be the minimum number of edges in a graph @xmath1 such that @xmath3 . more formally , @xmath8 the study of size - ramsey numbers was proposed by erds , faudree , rousseau and schelp @xcite in 1978 . by definition of @xmath4 , we have @xmath9 . since the complete graph on @xmath4 vertices has @xmath10 edges , we obtain the trivial bound @xmath11 chvtal ( see , _ e.g. _ , @xcite ) showed that equality holds in ( [ clique ] ) for complete graphs . in other words , @xmath12one of the first problems in this area was to determine the size - ramsey number of the @xmath5 vertex path @xmath13 . answering a question of erds @xcite , beck @xcite showed that @xmath14 since @xmath15 for any graph , beck s result is sharp in order of magnitude . the linearity of the size - ramsey number of paths was generalized to bounded degree trees by friedman and pippenger @xcite and to cycles by haxell , kohayakawa and uczak @xcite . beck @xcite asked whether @xmath7 is always linear in the size of @xmath0 for graphs @xmath0 of bounded degree . this was settled in the negative by rdl and szemerdi @xcite , who proved that there are graphs of order @xmath5 , maximum degree 3 , and size - ramsey number @xmath16 . they also conjectured that for a fixed integer @xmath17 there is an @xmath18 such that @xmath19 where the maximum is taken over all graphs @xmath0 of order @xmath5 with maximum degree at most @xmath17 . the upper bound was recently proved by kohayakawa , rdl , schacht , and szemerdi @xcite . for further results about the size - ramsey number e.g _ , the survey paper of faudree and schelp @xcite . somewhat surprisingly the size - ramsey numbers have not been studied for hypergraphs , even though classical ramsey numbers for hypergraphs have been studied extensively since the 1950 s ( see , e.g. , @xcite ) , and more recently @xcite . in this paper we initiate this study for @xmath2-uniform hypergraphs . a _ @xmath2-uniform hypergraph _ @xmath20 ( _ @xmath2-graph _ for short ) on a vertex set @xmath21 is a family of @xmath2-element subsets ( called edges ) of @xmath21 . we write @xmath22 for its edge set . given @xmath2-graphs @xmath20 and @xmath23 , say that @xmath24 if every 2-edge - coloring of @xmath25 results in a monochromatic copy of @xmath26 in @xmath25 . define the _ size - ramsey number _ @xmath27 of a @xmath2-graph @xmath20 as @xmath28 motivated by extending the basic theory from graphs to hypergraphs , we prove results for cliques , trees , paths , and bounded degree hypergraphs . for every @xmath2-graph @xmath26 , we trivially have @xmath29 where @xmath30 is the ordinary ramsey number of @xmath20 . our first objective was to generalize ( [ eq : chvatal ] ) to 3-graphs , which shows that equality holds for graphs . it is fairly easy to obtain a lower bound for @xmath31 that is quadratic in @xmath32 , but we were only able to do slightly better . [ cliquethm ] @xmath33 . the following basic questions remain open . [ ques:1 ] is @xmath34 ? [ ques:2 ] for @xmath35 let @xmath36 . define @xmath37 to be the hypergraph obtained from @xmath38 by removing one edge . is it true that @xmath39 ? clearly , the affirmative answer to the latter gives a negative answer to question [ ques:1 ] . given integers @xmath40 and @xmath5 , a @xmath2-graph @xmath41 of order @xmath5 with edge set @xmath42 is an _ @xmath43-tree _ , if for each @xmath44 we have @xmath45 and @xmath46 for some @xmath47 . we are able to give the following general upper bound for trees . [ thm : ub_tree ] let @xmath40 be fixed integers . then @xmath48 one can easily show that this bound is tight in order of magnitude when @xmath49 ( see section [ sec : trees ] for details ) . the situation for @xmath50 is much less clear . let @xmath51 be fixed integers . is it true that for every @xmath5 there exists a @xmath2-uniform @xmath43-tree @xmath52 of order at most @xmath5 such that @xmath53 here is another related question pointed out by fox @xcite . let us weaken the restriction on the edge intersection in the definition of @xmath41 . let @xmath54 be a @xmath2-graph of order @xmath5 with edge set @xmath42 such that for each @xmath44 we have @xmath45 . let @xmath51 be fixed integers . is @xmath55 polynomial in @xmath5 ? given integers @xmath56 and @xmath57 , we define an _ @xmath43-path @xmath58 _ to be the @xmath2-uniform hypergraph with vertex set @xmath59 $ ] and edge set @xmath60 , where @xmath61 and @xmath62 . in other words , the edges are intervals of length @xmath2 in @xmath59 $ ] and consecutive edges intersect in precisely @xmath43 vertices . the two extreme cases of @xmath49 and @xmath63 are referred to as , respectively , _ loose _ and _ tight _ paths . clearly every @xmath43-path is also an @xmath43-tree . thus , by theorem [ thm : ub_tree ] we obtain the following result . @xmath64 our first result shows that determining the size - ramsey number of a path @xmath58 for @xmath65 can easily be reduced to the graph case . [ pathprop ] let @xmath66 . then , @xmath67 clearly , this result is optimal . determining the size - ramsey number of a path @xmath58 for @xmath68 seems to be a much harder problem . here we will only consider tight paths ( @xmath69 ) . by we get @xmath70 the most complicated result of this paper is the following improvement of ( [ pathbound ] ) . [ thm : tightpath ] [ thm : tight_path_k ] fix @xmath71 and let @xmath72 . then @xmath73 the gap in the exponent of @xmath5 between the upper and lower bounds for this problem remains quite large ( between 1 and @xmath74 ) . we believe that the lower bound is much closer to the truth . indeed , the following question still remains open . is @xmath75 ? if true , then since @xmath76 , this would imply the linearity of the size - ramsey number of all @xmath43-paths . our main result about bounded degree hypergraphs is that their size - ramsey numbers can be superlinear . this is proved by extending the methods of rdl and szmerdi @xcite to the hypergraph case . [ thm : bounddegree ] let @xmath35 be an integer . then there is a positive constant @xmath77 such that for every @xmath5 there is a @xmath2-graph @xmath20 of order at most @xmath5 with maximum degree @xmath78 such that @xmath79 there are several other problems to consider such as finding the asymptotic of the size - ramsey number of cycles and many other classes of hypergraphs . in general , they seem to be very difficult . therefore , this paper is the first step towards a better understanding of this concept . in the next sections we prove these result for cliques ( section [ sec : cliques ] ) , trees ( section [ sec : trees ] ) , paths ( section [ sec : paths ] ) , and hypergraphs with bounded degree ( section [ sec : bounded ] ) . * proof of theorem [ cliquethm ] . * we show that if @xmath23 is a 3-graph with @xmath80 for @xmath81 , then @xmath82 . induction on @xmath83 . if @xmath84 , then there is a 2-coloring of @xmath85 with no monochromatic @xmath86 . since @xmath87 , this coloring yields a 2-coloring of @xmath23 with no monochromatic @xmath86 . suppose that @xmath88 . since @xmath80 , there are @xmath89 and @xmath90 in @xmath91 with @xmath92 . otherwise , @xmath93 a contradiction . let @xmath89 and @xmath90 be such that @xmath94 . define @xmath95 as follows : @xmath96 and @xmath97 clearly , @xmath98 and @xmath99 . by the inductive hypothesis there is a 2-coloring @xmath100 of the edges of @xmath95 with no monochromatic @xmath101 . let @xmath102 . thus , @xmath103 and @xmath104 . if there exists @xmath105 such that @xmath106 and @xmath107 $ ] is monochromatic , then set @xmath108 . if there exists @xmath109 such that @xmath110 and @xmath111 $ ] is monochromatic , then set @xmath112 . we continue this process obtaining @xmath113 where @xmath114 $ ] is monochromatic , @xmath115 , and @xmath116 $ ] contains only monochromatic cliques of order at most @xmath117 . now we define a 2-coloring @xmath118 of @xmath23 . a. if @xmath119 , then @xmath120 . b. if @xmath121 and @xmath122 , then @xmath123 . c. if @xmath124 and @xmath125 , then @xmath126 takes the opposite color to the color of @xmath114 $ ] . d. if @xmath124 and @xmath127 , then color @xmath126 arbitrarily . now suppose that there is a monochromatic clique @xmath128 in @xmath23 . such a clique must contain @xmath90 . now there are two cases to consider . if @xmath129 , then the subgraph of @xmath95 induced by @xmath130 is also a monochromatic copy of @xmath101 , a contradiction . otherwise , @xmath131 . thus , @xmath132 and @xmath133 . observe that @xmath134 and @xmath135 . but this yields a contradiction @xmath136 for @xmath81 . first for convenience we recall the definition of a hypertree . given integers @xmath40 and @xmath5 , recall that a @xmath2-graph @xmath41 of order @xmath5 with edge set @xmath42 is an _ @xmath43-tree _ , if for each @xmath44 we have @xmath45 and @xmath46 for some @xmath47 . * proof of theorem [ thm : ub_tree ] . * fix @xmath137 . we are to show that @xmath138 . recall that a _ partial steiner system @xmath139 _ is a @xmath2-graph of order @xmath140 such that each @xmath141-tuple is contained in at most one edge . due to a result of rdl @xcite it is known that there is a constant @xmath142 such that for every @xmath143 there is an @xmath144 with the number of edges satisfying @xmath145 ( see also @xcite for similar results ) . it is easy to observe that for @xmath146 every @xmath147-tuple is contained in at most @xmath148 edges . fix @xmath40 . let @xmath149 , where the constant @xmath150 is defined as @xmath151 let @xmath23 be a @xmath152 satisfying . observe that if @xmath153 , then @xmath23 can be viewed as a complete @xmath2-graph of order @xmath140 . clearly , @xmath154 . it remains to show that for any @xmath155 tree , @xmath156 . degree _ of a set @xmath157 ( @xmath158 ) by @xmath159 and for @xmath160 a _ minimum ( non - zero ) @xmath43-degree _ by @xmath161 first observe that for any 2-coloring of the edges of @xmath23 , there is a monochromatic sub - hypergraph @xmath162 with @xmath163 . indeed , suppose that @xmath164 is colored with blue and red colors . assume by symmetry that the red hypergraph @xmath165 has at least @xmath166 edges . set @xmath167 . if there exists @xmath168 with @xmath169 , then let @xmath170 ( we remove @xmath171 and all incident to @xmath171 edges ) . now we repeat the process . if there exists @xmath172 with @xmath173 , then let @xmath174 . continue this way to obtain hypergraphs @xmath175 where either @xmath176 or @xmath177 is empty hypergraph . but the latter can not happen , since the number of removed edges from @xmath165 is less than @xmath178 now we greedily embed @xmath52 into @xmath179 . at every step we have a connected sub - tree @xmath180 . assume that we already embedded @xmath181 edges of @xmath52 obtaining @xmath182 . let @xmath183 be such that @xmath184 for some @xmath185 . observe that there is always an edge @xmath186 such that @xmath187 . indeed , if @xmath188 , then this is true since @xmath189 and @xmath190 and every @xmath191-tuple of vertices of @xmath162 is contained in at most one edge in @xmath162 . otherwise , if @xmath192 , first we find a set @xmath193 such that @xmath194 and @xmath195 is contained in an edge of @xmath162 , and next apply the previous argument to @xmath195 . thus , we can extend @xmath182 to @xmath196 , as required . as mentioned in the introduction , it would be interesting to decide whether theorem [ thm : ub_tree ] is tight up to the hidden constant . this is definitely the case for @xmath49 . indeed , let @xmath52 be a @xmath2-uniform star - like tree of order @xmath5 defined as follows . assume that @xmath197 divides @xmath198 . @xmath52 consists of @xmath199 arms @xmath200 ( each with two edges ) : @xmath201 , where @xmath202 and all @xmath203 vertices are pairwise different ( see figure [ fig : star ] ) . with @xmath204 arms each of length 2 . ] assume that @xmath156 and color @xmath205 by red if degree ( in @xmath23 ) of every vertex in @xmath126 is less than @xmath199 ; otherwise @xmath126 is blue . since @xmath206 and there is no red copy of @xmath52 , there must be a blue copy of @xmath52 . every edge in such a copy has at least one vertex of degree at least @xmath199 ( in @xmath23 ) . since @xmath52 has @xmath199 vertex disjoint edges and every edge ( in @xmath23 ) can intersect at most 3 of those disjoint edges , @xmath207 in this section we prove proposition [ pathprop ] and theorem [ thm : tightpath ] . * proof of proposition [ pathprop ] . * let @xmath1 be a graph satisfying @xmath208 and @xmath209 ( _ cf . we construct a @xmath2-graph @xmath23 as follows . replace every vertex @xmath210 by an @xmath43-tuple @xmath211 ( different for every @xmath90 ) and each @xmath212 by @xmath213 where @xmath214 are different for every edge @xmath126 , too . thus , @xmath23 is a @xmath2-graph with @xmath215 and @xmath216 . now color @xmath217 . this coloring ( uniquely ) defines a coloring of @xmath218 . since @xmath1 contains a monochromatic copy of @xmath13 , @xmath23 also contains a monochromatic copy of @xmath58 . consequently , @xmath219 and the proof is complete . we now turn to the main result of this section which we restate for convenience . * [ thm : tightpath ] * [ thm : tight_path_k ] fix @xmath71 and let @xmath72 . then @xmath73 first we prove an auxiliary result . in order to do it we state some necessary notation . set @xmath220 for a graph @xmath221 let @xmath222 be the set of all cliques of order @xmath43 and let @xmath223 . let @xmath224 and @xmath225 be a family of pairwise vertex - disjoint cliques . define @xmath226 as the number of @xmath2-cliques of @xmath0 which @xmath227 vertices form a vertex set of some @xmath228 and the remaining vertex is from @xmath229 . similarly , let @xmath230 be the number of @xmath2-cliques in @xmath0 which @xmath227 vertices form a vertex set of some @xmath228 and the remaining vertex is from @xmath231 . finally , let @xmath232 ( for @xmath233 ) be the number of @xmath2-cliques containing at least one vertex from @xmath234 . [ prop : auxg_k ] let @xmath71 be an integer and let @xmath235 . then there exists a graph @xmath221 of order @xmath5 ( for sufficiently large @xmath5 ) satisfying the following : a. for every @xmath236 , @xmath237 , and every @xmath238 , @xmath239 , vertex disjoint @xmath240-cliques such that @xmath241 we have @xmath242 [ property : ik ] b. for every @xmath233 , @xmath243 , @xmath244 [ property : iiik ] c. the total number of @xmath2-cliques satisfies @xmath245 where @xmath246 [ property : iik ] it suffices to show that the random graph @xmath247 with @xmath248 and @xmath249 satisfies _ a.a.s . _ occurs _ asymptotically almost surely _ , or _ _ for brevity , if @xmath250 . ] below we will use the following bounds on the tails of the binomial distribution @xmath251 ( for details , see , _ e.g. _ , @xcite ) : @xmath252 first we show that @xmath0 _ a.a.s . _ satisfies . fix an @xmath253 and @xmath254 with @xmath255 . observe that without loss of generality we may assume that @xmath256 . note that @xmath257 . thus , @xmath258 and ( applied with @xmath259 ) implies @xmath260 now we bound from above the number of all possible choices for @xmath261 and @xmath262 . clearly we have at most @xmath263 choices for @xmath261 . observe that the number of choices for @xmath262 can be bounded from above by the number of ways of choosing an ordered subset of vertices of size @xmath264 . indeed , suppose that @xmath265 is such a choice . then @xmath262 can be defined as @xmath266 . thus we conclude that there are at most @xmath267 ways to choose @xmath261 and @xmath262 . hence , by @xmath268 similarly , since @xmath269 , @xmath270 and since @xmath271 , @xmath272 inequality ( applied with @xmath259 ) yields @xmath273 therefore , we deduce that @xmath274 consequently , by and we get that _ a.a.s._@xmath275 for any choice of @xmath261 and @xmath262 . this finishes the proof of . for each vertex @xmath276 , let @xmath277 denote the number of @xmath2-cliques of @xmath0 which contain @xmath90 . in order to show that _ a.a.s . _ @xmath0 also satisfies , we will first estimate @xmath277 for each @xmath276 . the standard application of ( applied with @xmath278 and @xmath259 ) with the union bound imply that _ a.a.s . _ the degree of every vertex @xmath279 satisfies @xmath280 the number of @xmath2-cliques which contain @xmath90 is equal to the number of @xmath240-cliques in the neighborhood of @xmath90 . therefore , in order to show it suffices to bound the number of @xmath240-cliques in any set of size at most @xmath281 . let @xmath282 with @xmath283 . first we will decompose all @xmath240-tuples of @xmath284 into linear @xmath240-uniform hypergraphs @xmath285 with @xmath286 and @xmath287 for every @xmath288 . that means that each @xmath240-tuple of @xmath284 belongs to exactly one @xmath289 and each pair of elements of @xmath284 appears in at most one @xmath240-tuple in @xmath290 . the existence of such a decomposition follows from a more general result of pippenger and spencer @xcite ( see also @xcite ) . let @xmath291 be the random variable that counts the number of @xmath240-tuples of @xmath289 which appear as @xmath240-cliques of @xmath0 . observe that @xmath292 . therefore for each @xmath181 , @xmath293 and by ( with @xmath294 ) @xmath295 consequently , the union bound over all subsets @xmath282 of size @xmath147 and over all @xmath181 for each @xmath288 implies @xmath296 since @xmath297 grows like a polynomial in @xmath5 . therefore it follows that _ a.a.s._@xmath298 where @xmath299 in a similar way one can show that @xmath300 where @xmath301 note that equation gives the bound @xmath302 which proves part . now we finish the proof of . since each @xmath2-clique is counted exactly @xmath2 times , the number of @xmath2-cliques is _ a.a.s . _ at least @xmath303 it follows now from and that given a set @xmath304 , @xmath243 , the number of @xmath2-cliques of @xmath0 which intersect @xmath234 is _ a.a.s . _ at most @xmath305 finally observe that , together with the choice of @xmath150 yield that @xmath306 implying condition , as required . now we are ready to prove main result of this section . * proof of theorem [ thm : tightpath]*. we show that there exists a @xmath2-graph @xmath23 with@xmath307 such that any two - coloring of the edges of @xmath23 yields a monochromatic copy of @xmath308 . let @xmath0 be a graph from proposition [ prop : auxg_k ] . set @xmath309 and let @xmath217 be the set of @xmath2-cliques in @xmath0 . we prove that such @xmath23 is a ramsey @xmath2-graph for @xmath310 with @xmath311 , where @xmath235 . take an arbitrary red - blue coloring of the edges of @xmath312 and assume that there is no monochromatic @xmath310 . we will consider the following greedy _ procedure _ which at each step finds a blue tight path of length @xmath181 labeled as @xmath313 . 1 . let @xmath314 be the _ trash _ set of @xmath240-tuples and @xmath315 be the set of _ unused _ vertices and set @xmath316 . at any point in the process , if @xmath317 , then stop . [ step : start_k ] ( in this step @xmath318 . ) if possible , then choose a blue edge from @xmath319 and label its vertices by @xmath320 and then set @xmath321 . otherwise , if not possible , stop . [ step : adding_k ] ( in this step @xmath322 . ) let @xmath323 be the labels of the last @xmath227 vertices of the constructed blue path . if possible , select a vertex @xmath324 for which @xmath325 form a blue edge . label @xmath89 as @xmath326 , set @xmath327 and @xmath328 . repeat this step until no such @xmath89 can be found . ( in this step also @xmath322 . ) let @xmath323 be the labels of the last @xmath227 vertices of the constructed blue path which can not be extended in a sense described in step . remove these @xmath227 vertices from the path and set @xmath329 and @xmath330 . after this removal there are two possibilities : a. if @xmath331 , then put back @xmath332 to @xmath319 ( i.e. @xmath333 ) , set @xmath316 , and return to step ; b. otherwise , return to step . + this procedure will terminate under two circumstances : either @xmath317 or no blue edge can be found in step . first let us consider the case when @xmath317 , that means , there are @xmath334 vertex disjoint @xmath240-tuples in @xmath262 . denote by @xmath261 the vertex set of the blue path which was obtained when @xmath317 . clearly , @xmath335 , otherwise there would be a blue @xmath310 . we are going to apply proposition [ prop : auxg_k ] with sets @xmath261 and @xmath262 . notice that every edge of @xmath23 which contains a @xmath240-tuple from @xmath262 and the remaining vertex from @xmath336 must be colored red . ( this is because for a @xmath240-tuple to end up in @xmath262 , there must have been no vertex @xmath89 in step that could extend the blue path . ) it also follows from step that each @xmath240-tuple in @xmath262 is contained in at least one blue edge . thus , proposition [ prop : auxg_k ] implies that @xmath337 . that means that the number of red edges which contain a @xmath240-tuple from @xmath262 and the remaining vertex from @xmath319 is at least @xmath78 times the number of blue edges with a @xmath240-tuple from @xmath262 . now remove all the blue edges from @xmath23 which contain a @xmath240-tuple from @xmath262 and denote such @xmath2-graph by @xmath338 . perform the above procedure on @xmath338 . this will generate a new trash set @xmath339 . observe that @xmath340 , since every edge of @xmath338 which contains a @xmath240-tuple from @xmath262 must be red . again , if @xmath341 , then we use the same argument as above to find that the number of red edges in @xmath338 which contain a @xmath240-tuple from @xmath339 and the remaining vertex from @xmath319 is at least @xmath78 times the number of blue edges in @xmath338 with a @xmath240-tuple from @xmath339 . indeed , we can again apply the inequality from proposition . this is because @xmath342 is smaller than the number of all blue edges in @xmath23 containing a @xmath240-tuple from @xmath339 , while ( since we do not remove red edges ) @xmath343 remains same in both @xmath338 and @xmath23 . now remove the blue edges from @xmath338 which contain a @xmath240-tuple from @xmath339 obtaining a @xmath2-graph @xmath344 . keep repeating the procedure until it is no longer possible . at some point , we will run out of blue edges in @xmath345 for some @xmath346 , and the procedure will terminate prematurely in step . in this case @xmath347 , @xmath348 and @xmath319 has no blue edges . however , there still may be some blue edges which contain a vertex from @xmath349 . proposition [ prop : auxg_k ] ( applied for @xmath350 ) implies that the number of such edges is at most @xmath244 let @xmath351 and @xmath352 be the numbers corresponding to @xmath226 and @xmath230 obtained at the end of the procedure applied to @xmath353 . thus , @xmath354 for each @xmath355 . let @xmath356 and @xmath357 denote the number of red and blue edges in @xmath23 . observe that @xmath358 furthermore , since all sets @xmath359 are mutually disjoint , each red edge in @xmath23 containing a @xmath240-tuple from some @xmath359 can be only counted at most @xmath2 times . thus , @xmath360 consequently , by and , we get @xmath361 and so @xmath362 the conclusion is that there are more red edges than there are blue edges in @xmath23 . if we reverse the procedure and look for a red path instead of a blue one , we will conclude that there are more blue edges than red edges . since these two statements contradict each other , the only way to avoid both statements is if a monochromatic path exists . in this section we prove theorem [ thm : bounddegree ] , which states that hypergraphs with bounded degree can have nonlinear size - ramsey numbers . * proof of theorem [ thm : bounddegree ] . * we modify an idea from rdl and szemerdi @xcite . for simplicity we only present a proof for @xmath363 , which can easily be generalized to @xmath35 . the hypergraph @xmath20 will be constructed as the vertex disjoint union of graphs @xmath364 each of which is a tree with a path added on its leaves . next we will describe the details of such construction . let @xmath366 consider a binary 3-tree @xmath367 on @xmath368 vertices rooted at vertex @xmath369 ( see figure [ fig : b ] ) . denote by @xmath370 the set of all its leafs . call the edge containing @xmath369 the _ root edge_. observe that latexmath:[\[\label{eq : v(b ) } let @xmath373 by an automorphism of @xmath262 . since the root edge @xmath126 is the unique edge with exactly one vertex of degree 1 , @xmath374 . the other two vertices of @xmath126 are permuted by @xmath373 . consequently , @xmath373 permutes two vertices of every other edge . hence , it is easy to observe that the order of the automorphism group of @xmath262 satisfies @xmath375 now consider a tight path @xmath377 of length @xmath378 placed on the leaves @xmath370 in an arbitrary order . considering labeled vertices of @xmath370 there are clearly @xmath379 such paths . label them by @xmath200 for @xmath380 . let @xmath359 be vertex disjoint copies of @xmath262 and @xmath381 , where @xmath382 . let @xmath373 be an isomorphism between @xmath364 and @xmath383 . since the only vertices of degree 4 are on paths @xmath200 and @xmath384 , @xmath385 . thus , @xmath386 and so @xmath359 and @xmath387 are isomorphic . thus , the number of pairwise non - isomorphic @xmath364 s is at least clearly , @xmath392 . furthermore , by , we get @xmath393 moreover , @xmath394 and the independence number of @xmath20 satisfies @xmath395 indeed , let @xmath396 be an independent set of size @xmath397 . we estimate the number of edges @xmath398 between sets @xmath399 and @xmath400 . first observe that @xmath401 next , since each triple between @xmath399 and @xmath400 intersects one of the partition classes on 2 vertices and @xmath402 , @xmath403 this implies that @xmath404 and so . recall that @xmath20 consists of @xmath413 pairwise non - isomorphic copies of @xmath364 . we estimate the number of copies of @xmath364 s contained in a sub - hypergraph induced by @xmath414 . first fix an edge @xmath126 in @xmath415 $ ] and count the number of copies of @xmath364 s for which @xmath126 is a root edge . since @xmath416 for each @xmath417 , we get that this number is at most @xmath418 where the factor 3 counts the number of choices for the root vertex , the next factors count the number of possible @xmath359 s with @xmath126 as a root , and the last factor counts the number of paths on the set of leafs . thus , there is an @xmath419 such that @xmath420 appears in @xmath415 $ ] at most @xmath421 times . denote by @xmath162 the sub - hypergraph consisting of root edges from all copies of @xmath420 in @xmath415 $ ] . thus , @xmath422 color edges in @xmath162 together with edges incident to @xmath423 blue ; otherwise red . clearly , there is no red copy of @xmath20 , since there is no red copy of @xmath420 . moreover , there is no blue copy of @xmath20 , since every blue sub - hypergraph of order @xmath424 has an independent set of size at least @xmath425 which is strictly bigger than @xmath426 ( _ cf . _ ) . r. faudree and r. schelp , _ a survey of results on the size ramsey number _ , paul erds and his mathematics , ii ( budapest , 1999 ) , bolyai soc . 11 , jnos bolyai math . budapest , 2002 , pp . | the size - ramsey number of a graph @xmath0 is the minimum number of edges in a graph @xmath1 such that every 2-edge - coloring of @xmath1 yields a monochromatic copy of @xmath0 .
size - ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs .
we initiate the study of size - ramsey numbers for @xmath2-uniform hypergraphs .
analogous to the graph case , we consider the size - ramsey number of cliques , paths , trees , and bounded degree hypergraphs .
our results suggest that size - ramsey numbers for hypergraphs are extremely difficult to determine , and many open problems remain . |
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_ markov chain monte carlo _ ( mcmc ) _ methods _ allow samples from virtually any target distribution @xmath0 , known up to a normalizing constant , to be generated . in particular , the celebrated _ metropolis hastings algorithm _ ( introduced in @xcite and @xcite ) simulates a markov chain evolving according to a reversible markov transition kernel by first generating , using some instrumental kernel , a candidate and then accepting or rejecting the same with a probability adjusted to satisfy the detailed balance condition @xcite . when choosing between several metropolis hastings algorithms , it is desirable to be able to compare the efficiencies , in terms of the asymptotic variance of sample path averages , of different @xmath0-reversible markov chains . despite the practical importance of this question , only a few results in this direction exist the literature . peskun @xcite defined a partial ordering for finite state space markov chains , where one transition kernel has a higher order than another if the former dominates the latter on the off - diagonal ( see definition [ defipeskunordering ] ) . this ordering was extended later by tierney @xcite to general state space markov chains and another even more general ordering , the covariance ordering , was proposed in @xcite . in general , it holds that if a homogeneous @xmath8-reversible markov transition kernel is greater than another according to one of these orderings , then the asymptotic variance of sample path averages for a markov chain evolving according to the former is smaller for all square integrable ( with respect to @xmath0 ) target functions . we provide an extension of this result to inhomogeneous markov chains that evolve alternatingly according to two different @xmath0-reversible markov transition kernels . to the best of our knowledge , this is the first work dealing with systematic comparison of asymptotic variances of inhomogeneous markov chains . the approach is linked with the operator theory for markov chains but does not make use of any spectral representation . after some preliminaries ( section [ secpreliminaries ] ) , our main result , theorem [ teomainresult ] , is stated in section [ secmain ] . in section [ secappl ] , we apply theorem [ teomainresult ] in the context of mcmc algorithms by comparing the efficiency , in terms of asymptotic variance , of some existing data - augmentation - type algorithms . moreover , we propose a novel pseudo - marginal algorithm ( in the sense of @xcite ) , referred to as the _ random refreshment _ algorithm , which on the contrary to the pseudo - marginal version of the _ monte carlo within metropolis _ ( mcwm ) algorithm turns out to be exact and more efficient than the pseudo - marginal version of the _ grouped independence metropolis hastings _ ( gimh ) algorithm . here , the analysis is again driven by theorem [ teomainresult ] . the proof of theorem [ teomainresult ] is given in section [ secproofmain ] and some technical lemmas are postponed to appendix [ app ] . finally , appendix [ secappb ] relates some existing mcmc algorithms to the framework considered in this paper . we denote by @xmath9 and @xmath10 the sets of nonnegative and positive integers , respectively . in the following , all random variables are assumed to be defined on a common probability space @xmath11 . let @xmath12 be a measurable space ; then we denote by @xmath13 and @xmath14 the spaces of positive measures and measurable functions on @xmath15 , respectively . the lebesgue integral of @xmath16 over @xmath17 with respect to the measure @xmath18 is , when well - defined , denoted by @xmath19 recall that a _ markov transition kernel _ @xmath1 on @xmath15 is a mapping @xmath20 $ ] such that : * for all @xmath21 , @xmath22 is a measurable function , * for all @xmath23 , @xmath24 is a probability measure . a kernel @xmath1 induces two integral operators , one acting on @xmath13 and the other on @xmath14 ; more specifically , for @xmath25 and @xmath26 , we define the measure @xmath27 and the measurable function @xmath28 moreover , the _ composition _ ( or _ product _ ) of two kernels @xmath1 and @xmath2 on @xmath15 is the kernel defined by @xmath29 we will from now on fix a distinguished probability measure @xmath0 on @xmath15 . given @xmath0 , we denote by @xmath30:= \ { f \in\mathcal{f}(\mathcal{x } ) \dvtx\pi f^2 < \infty\}$ ] the space of square integrable functions with respect to @xmath0 and furnish the same with the scalar product @xmath31 , g \in\ltwo[\pi ] \bigr)\ ] ] and the associated norm @xmath32 \bigr).\ ] ] here , we have expunged the measure @xmath0 from the notation for brevity . if @xmath1 is a markov kernel on @xmath15 admitting @xmath0 as an invariant distribution , then the mapping @xmath33 defines an operator on @xmath30 $ ] , and by jensen s inequality it holds that @xmath34 \dvtx \ltwonorm{f } \leq1 } \ltwonorm{pf } \leq1.\ ] ] recall that a kernel @xmath1 is _ @xmath0-reversible _ if and only if the detailed balance relation @xmath35 holds . if the markov kernel @xmath1 is @xmath0-reversible , then @xmath36 defines a self - adjoint operator on @xmath30 $ ] , that is , for all @xmath37 and @xmath38 belonging to @xmath30 $ ] , @xmath39 the following off - diagonal ordering of markov transition kernels on a common state space was , in the case of markov chains in a finite state space , proposed in @xcite . the ordering was extended later in @xcite to the case of markov chains in general state space . [ defipeskunordering ] let @xmath40 and @xmath41 be markov transition kernels on @xmath12 with invariant distribution @xmath0 . we say that _ @xmath41 dominates @xmath40 on the off - diagonal _ , denoted @xmath42 , if for all @xmath21 and @xmath0-a.s . all @xmath23 , @xmath43 the previous ordering allows the asymptotic efficiencies of different reversible kernels to be compared . more specifically , the following seminal result was established in @xcite , theorem 2.1.1 , for markov chains in discrete state space and extended later in @xcite , theorem 4 , to markov chains in general state space . [ teoefficiencyordering ] let @xmath40 and @xmath41 be two @xmath0-reversible kernels on @xmath44 . if , then for a.s . all @xmath45 $ ] , @xmath46 where we have defined , for a markov chain @xmath47[\mathbb{n}]$ ] with @xmath0-reversible transition kernel @xmath1 and initial distribution @xmath0 , @xmath48 note that according to @xcite , if @xmath49 is a @xmath8-reversible markov chain and @xmath45 $ ] , then @xmath50 is guaranteed to exist ( but may be infinite ) . nevertheless , the ordering in question does not allow markov kernels lacking probability mass on the diagonal , that is , kernels @xmath1 satisfying for all @xmath23 , to be compared . this is in particular the case for gibbs samplers in general state space . to overcome this limitation , one may consider instead the following covariance ordering based on lag - one autocovariances . [ deficovarordering ] let @xmath40 and @xmath41 be markov transition kernels on @xmath12 with invariant distribution @xmath0 . we say that _ @xmath41 dominates @xmath40 in the covariance ordering _ , denoted @xmath51 p_0 $ ] , if for all @xmath45 $ ] , @xmath52 the covariance ordering , which was introduced implicitly in @xcite , page 5 , and formalized in @xcite , is an extension of the off - diagonal ordering since according to @xcite , lemma 3 , @xmath53 implies @xmath51 p_0 $ ] . moreover , it turns out that for reversible kernels , @xmath51p_0 $ ] implies @xmath54 ( see the proof of @xcite , theorem 4 ) . all these results concern homogeneous markov chains , whereas many mcmc algorithms such as the gibbs or the metropolis - within - gibbs samplers use several kernels , for example , @xmath1 and @xmath55 in the case of two kernels @xcite . a natural idea would then be to apply theorem [ teoefficiencyordering ] to the homogeneous markov chain having the block kernel @xmath56 as transition kernel ; however , even when the kernels @xmath1 and @xmath57 are both @xmath0-reversible , the product @xmath56 of the same is usually not @xmath0-reversible , except in the particular case when @xmath1 and @xmath57 commute , that is , @xmath58 . thus , theorem [ teoefficiencyordering ] can not in general be applied directly in this case . in the following , let @xmath3 and @xmath59 , @xmath5 , be markov transition kernels on @xmath15 . define @xmath60 and @xmath61 as the markov chains evolving as follows : @xmath62{i } \stackrel{p_i } { \longrightarrow } \x[1]{i } \stackrel { q_i } { \longrightarrow } \x[2]{i } \stackrel{p_i } { \longrightarrow } \x[3]{i } \stackrel { q_i } { \longrightarrow } \cdots.\ ] ] this means that for all @xmath63 , @xmath5 and @xmath64 : * @xmath65{i } \in\mathsf{a } { |}\mathcal{f}_{2 k}^{(i ) } ) = p_i(\x[2 k]{i } , \mathsf{a})$ ] , * @xmath66{i } \in\mathsf{a } { |}\mathcal{f}_{2 k+1}^{(i ) } ) = q_i(\x[2 k+1]{i},\mathsf{a})$ ] , where @xmath67{i},\ldots , \x[n]{i})$ ] , @xmath68 . we impose the following assumption : @xmath69 \hspace*{-100pt}\\[-8pt ] \hspace*{-100pt}&&\mbox{(ii ) $ p_1 \pgeq[1 ] p_0 $ and $ q_1 \pgeq[1 ] q_0$.}\nonumber\end{aligned}\ ] ] as mentioned above , @xmath53 implies @xmath70p_0 $ ] ; thus , in practice , a sufficient condition for ( ii ) is that @xmath42 and @xmath71 . [ teomainresult ] assume that @xmath3 and @xmath59 , @xmath72 , satisfy and let @xmath73 , @xmath72 , be markov chains evolving as in ( [ eqeq1markov ] ) with initial distribution @xmath0 . then for all @xmath45 $ ] such that for @xmath72 , @xmath74{i } \bigr ) } { f \bigl(\x[k]{i } \bigr)}\bigr|+\bigl|\covardu { f \bigl ( \x[1]{i } \bigr ) } { f \bigl(\x[k+1]{i } \bigr)}\bigr| \bigr ) < \infty,\ ] ] it holds that @xmath75 where @xmath76{i } \bigr ) } \qquad \bigl(i \in\{0 , 1\ } \bigr).\ ] ] at present , we have not been able to extend the arguments of our current proof of theorem [ teomainresult ] ( see section [ secproofmain ] ) to inhomogeneous markov chains evolving alternatingly according to _ more _ than two different kernels . on the other hand , we have not been able to find a counterexample rejecting the hypothesis that a similar result would hold true also in that case . we leave this as an open problem . [ remcounterexsummability ] condition ( [ eqassumpfuncthm ] ) is _ not _ a necessary condition for ( [ eqmainresult ] ) ; indeed , letting @xmath77 , @xmath78 , @xmath79 , where , as in @xcite , example 5 , @xmath80 , provides a straightforward counterexample . when verifying if a given @xmath37 satisfies the condition ( [ eqassumpfuncthm ] ) it may be convenient to consider the homogeneous markov chains @xmath81 or @xmath82 or even @xmath83 . typically , none of these chains are @xmath0-reversible . nevertheless , @xmath0-reversibility is not needed for checking conditions of type ( [ eqassumpfuncthm ] ) , which can be established using upper bounds on the _ @xmath84-norm _ between the distribution given by the @xmath85th iterate of a homogeneous kernel and its stationary distribution . this will be developed in the following section . for any measurable real - valued function @xmath37 on @xmath86 , define the _ @xmath84-norm _ of the _ function _ @xmath37 by @xmath87 moreover , let @xmath88 be a finite signed measure on @xmath89 . then by the jordan decomposition theorem there exists a unique pair of positive , finite and singular measures @xmath90 and @xmath91 on @xmath92 such that @xmath93 . the pair @xmath94 is referred to as the _ jordan decomposition _ of the signed measure @xmath88 . the finite measure @xmath95 is called the _ total variation _ of @xmath88 . let @xmath84 be a nonnegative function taking values in @xmath96 ; then the _ @xmath84-norm _ of the _ signed measure _ @xmath88 is defined by @xmath97 [ defvgeomerg ] a markov kernel @xmath1 on @xmath92 is _ @xmath84-geometrically ergodic _ if it admits a unique invariant distribution @xmath0 and there exists a measurable function @xmath98 satisfying @xmath99 and such that the following hold : there exist constants @xmath100 such that for all @xmath23 and all @xmath68 , @xmath101 there exist constants @xmath102 such that @xmath103 . @xcite , theorem 1.2 , provides sufficient conditions , in terms of drift towards a _ small set _ , for ( a ) in definition [ defvgeomerg ] to hold ; see also @xcite , fact 10 , for necessary and sufficient conditions under the assumption of aperiodicity and irreducibility . moreover , the coming developments require only the bound ( [ eqvgeom ] ) to hold @xmath0-a.s . we have now all necessary tools for giving sufficient conditions that imply the absolute summability assumption ( [ eqassumpfuncthm ] ) . let the chain @xmath49 evolve according to @xmath104 { } \stackrel{p } { \longrightarrow } \x [ 1 ] { } \stackrel{q } { \longrightarrow } \x[2 ] { } \stackrel{p } { \longrightarrow } \x[3 ] { } \stackrel{q } { \longrightarrow } \cdots\ ] ] with @xmath105 { } \sim\pi$ ] , for some markov kernels @xmath1 and @xmath2 . [ propaltcondition ] if the markov kernel @xmath56 is @xmath84-geometrically ergodic , then for all functions @xmath37 such that @xmath106 and @xmath107 , @xmath108 { } \bigr ) } { f \bigl(\x[k ] { } \bigr)}\bigr|+\bigl|\covardu{f \bigl(\x [ 1 ] { } \bigr ) } { f \bigl ( \x[k+1 ] { } \bigr)}\bigr| \bigr ) < \infty,\ ] ] where @xmath109 { } ; k \in\mathbb{n}\}$ ] evolves as in ( [ eqeq1markovbis ] ) . the proof of proposition [ propaltcondition ] is found in appendix [ appproofpropaltcondition ] . before considering some applications of theorem [ teomainresult ] , we recall the following proposition , describing how to obtain a @xmath0-reversible markov chain using some instrumental kernel @xmath110 . although this result is fundamental in the metropolis hastings literature ( see , e.g. , @xcite and the references therein ) , it is restated here as it will be used in various situations in the sequel [ especially when there is no fixed reference measure dominating all the distributions @xmath111 . [ propderiveeradon ] let @xmath110 be a markov transition kernel on @xmath112 and @xmath8 a probability measure on @xmath15 . define the probability measures @xmath113 and @xmath114 . assume that the measures @xmath115 and @xmath116 are equivalent and such that for @xmath116-a.s . all @xmath117 , @xmath118 where @xmath119 denotes the radon nikodym derivative . then the markov kernel @xmath120 , where @xmath121 is @xmath0-reversible . a natural application of theorem [ teomainresult ] consists in using the result for comparing different data - augmentation - type algorithms . in the following , we wish to target a probability distribution @xmath122 defined on @xmath123 using a sequence @xmath124[\mathbb{n}]$ ] of valued random variables . to this aim , tanner and wong @xcite suggest writing @xmath125 as the marginal of some distribution @xmath0 defined on the product space in the sense that @xmath126 , where @xmath127 is some markov transition kernel on @xmath128 . in most cases , the marginal @xmath125 is of sole interest , while the component @xmath129 is introduced for convenience as a means of coping with analytic intractability of the marginal . ( it could also be the case that the marginal @xmath125 is too computationally expensive to evaluate . ) a first solution consists in letting @xmath124[\mathbb{n}]$ ] be the first - component process @xmath130[\mathbb{n } ] $ ] of the @xmath0-reversible markov chain @xmath131{1 } , \aux[k]{1 } ) ; k \in\mathbb{n}\}$ ] defined as follows . let @xmath132 and @xmath133 be instrumental markov transition kernels on @xmath134 and @xmath135 , respectively , and define a transition of the chain @xmath131{1 } , \aux[k]{1 } ) ; k \in\mathbb { n}\}$ ] by algorithm [ algalg1 ] . @xmath136{1 } , \aux[k]{1 } ) = ( y , u)$ ] : draw @xmath137 and call the outcome @xmath138 ( abbr . @xmath139 ) , draw @xmath140 , set @xmath141{1 } , \aux[k+1]{1 } \bigr ) \nonumber\\[10pt]\\[-22pt ] & & \qquad \gets \cases{\displaystyle ( \hat{y } , \hat{u } ) , & \quad with probability $ \displaystyle\alpha ( y , u , \hat{y } , \hat{u})$ \vspace*{5pt}\cr & \qquad\quad $ \displaystyle:=1 \wedge\frac { \pi^{\ast}(\hat{y } ) r(\hat{y } , \hat{u})s(\hat{y } , \hat{u } ; y ) t(\hat{y } , \hat{u } , y ; u)}{\pi^{\ast}(y ) r(y , u ) s(y , u ; \hat{y } ) t(y , u , \hat{y } ; \hat{u})}$ , \vspace*{5pt}\cr ( y , u ) , & \quad otherwise.}\nonumber\end{aligned}\ ] ] [ remgeneralradon ] in the expression ( [ eqacceptmetropolis ] ) of @xmath142 we assume implicitly that the families @xmath143 and @xmath144 of probability measures are dominated by a fixed nonnegative measure and we denote by @xmath145 and @xmath146 the corresponding transition kernel densities , respectively . in some cases ( see , e.g. , @xcite ) it may , however , happen ( typically when some dirac mass is involved ) that these kernels are not dominated by a nonnegative measure ; nevertheless , algorithm [ algalg1 ] as well as algorithm [ algalg2 ] defined below remain valid provided that the ratio in @xmath147 is replaced by the corresponding radon nikodym derivative @xmath148 where in this case , @xmath149 by applying proposition [ propderiveeradon ] , we deduce that the output @xmath131{1 } , \aux[k]{1});\break k\in\mathbb{n}\}$ ] is a @xmath8-reversible markov chain . as a consequence , the sequence @xmath150[\mathbb{n}]$ ] targets , although it is not itself a markov chain , the marginal distribution @xmath125 . note that the method requires the product @xmath151 to be known at least up to a multiplicative constant to guarantee the computability of the acceptance probability @xmath147 in ( [ eqacceptmetropolis ] ) . [ exgimh ] the grouped independence metropolis hastings ( gimh ) algorithm ( see @xcite ) is used in situations where @xmath125 is analytically intractable . in this algorithm , the quantity @xmath152 is in the acceptance probability replaced by an importance sampling estimate @xmath153 where @xmath154 is the density of some augmented target distribution @xmath155 defined on the product space @xmath156 , known up to a normalizing constant and allowing @xmath125 as marginal distribution , and @xmath157 are i.i.d . draws from the proposal @xmath158 . denoting by @xmath159 the density used for proposing new candidates @xmath138 , one obtains the acceptance probability ratio @xmath160 where @xmath161 and @xmath162 consequently , the gimh algorithm can be perfectly cast into the framework of the freeze algorithm , with the auxiliary variable @xmath163 playing the role of the dimensional monte carlo sample and @xmath164 . in the following , we use theorem [ teomainresult ] for comparing the performance of algorithm [ algalg1 ] to that of different modifications of the same obtained in the cases where : simulating @xmath127-transitions is feasible , simulating @xmath127-transitions is infeasible . @xmath165{2 } = y$ ] : draw @xmath166 , draw @xmath167 , draw @xmath168 , set @xmath169{2 } \gets \cases { \hat{y } , & \quad with probability $ \alpha(y , u , \hat{y } , \hat{u})$ [ defined in ( \ref{eqacceptmetropolis } ) ] , \vspace*{2pt}\cr y , & \quad otherwise . } $ ] in this case , an alternative to algorithm [ algalg1 ] consists in letting @xmath170[\mathbb{n}]$ ] be the sequence @xmath171[\mathbb{n}]$ ] generated through algorithm [ algalg2 ] . note that algorithm [ algalg2 ] `` refreshes , '' in step ( i ) , systematically the second component of the markov chain , which advocates algorithm [ algalg2 ] to have better mixing properties than algorithm [ algalg1 ] . the main task of the present section is to establish rigorously this heuristics . the output @xmath172 of algorithm [ algalg2 ] is , on the contrary to @xmath173 , a markov chain . it is not a classical metropolis hastings markov chain due to the auxiliary variables @xmath174 and @xmath175 that appear explicitly in the acceptance probability . however , as established in the following proposition , whose proof is found in appendix [ appproofrevsystrefresh ] , the @xmath0-reversibility of @xmath176{1},\aux[k]{1 } ) ; k\in\mathbb{n}\}$ ] implies @xmath177-reversibility of @xmath172 . [ propinduces_rever ] the sequence @xmath172 generated in algorithm [ algalg2 ] is a @xmath125-reversible markov chain . in @xcite , the authors use the terminology _ randomized mcmc _ ( r - mcmc ) for a @xmath125-reversible metropolis hastings chain @xmath178 generated using a set of auxiliary variables@xmath179 with a particular expression of the acceptance probability . although only one of these auxiliary variables is sampled at each time step , one may actually cast this approach into the framework of algorithm [ algalg2 ] by creating artificially another auxiliary variable according to the deterministic kernel @xmath180 where @xmath37 is any continuously differentiable involution on @xmath181 . even though @xmath133 is not dominated , it is possible to verify ( [ eqcondradon ] ) using that @xmath37 is an involution . we prove in appendix [ apprmcmc ] that the r - mcmc algorithm is a special case of algorithm [ algalg2 ] with this particular choice of @xmath133 and with the general form of the acceptance probability described in remark [ remgeneralradon ] . the _ generalized multiple - try metropolis _ ( gmtm ) _ algorithm _ @xcite is an extension of the _ multiple - try metropolis hastings algorithm _ proposed in @xcite . given @xmath165{}=y$ ] , one draws @xmath85 i.i.d . possible moves @xmath182 according to @xmath183 . after this , a random index @xmath184 taking the value @xmath185 with probability proportional to @xmath186 is generated , whereupon a candidate is constructed as @xmath187 . the candidate is then accepted with some probability that is computed using @xmath85 additional random variables @xmath188 , where @xmath189 are i.i.d . draws from @xmath190 , and @xmath191 is set deterministically to @xmath192 ( see appendix [ appmtm ] for more details concerning the acceptance probability ) . in appendix [ appmtm ] , proposition [ lemmtm ] , it is shown that the gmtm algorithm is in fact a special case of algorithm [ algalg2 ] with @xmath193 and @xmath194 . when the function @xmath195 is known explicitly , one may obtain another @xmath196-reversible markov chain by means of the classical metropolis hastings ratio , that is , we use again algorithm [ algalg2 ] but replace the acceptance probability @xmath197 by @xmath198 the following proposition , which generalizes a similar result obtained in @xcite , section 2.3 , for the r - mcmc algorithm , shows , when combined with @xcite , theorem 4 , that the asymptotic variance of the classical metropolis hastings estimator is smaller than that of the estimator based on algorithm [ algalg2 ] . [ propmhalg2 ] the metropolis hastings kernel associated with the acceptance probability ( [ eqmhratio ] ) is larger , in the sense of definition [ defipeskunordering ] , than the transition kernel associated with algorithm [ algalg2 ] . set @xmath199 and note that @xmath116 is a probability measure . hence , as the mapping @xmath200 is concave , jensen s inequality implies that @xmath201 ( a similar technique was used in the proof of @xcite , lemma 1 ) . the previous computation shows that the off - diagonal transition density function of the metropolis hastings markov chain associated with the acceptance probability ( [ eqmhratio ] ) is larger than that of the chain in algorithm [ algalg2 ] . this completes the proof . however , in practice a closed - form expression of @xmath202 is rarely available , which prevents the classical metropolis hastings algorithm from being implemented . thus , if the transition density @xmath203 is known explicitly and can be sampled we have to choose between algorithms [ algalg1 ] and [ algalg2 ] for approximating @xmath177 . the classical tools ( such as the ordering in definition [ defipeskunordering ] ) for comparing and can not be applied here , since is not even a markov chain . nevertheless , theorem [ teomainresult ] allows these two algorithms to be compared theoretically by embedding and into inhomogeneous @xmath8-reversible markov chains . the construction , which will be carried through in full detail below , leads to the following result . [ teocompalg1alg2 ] let and be sequences of random variables generated by algorithms [ algalg1 ] and [ algalg2 ] , respectively , where @xmath204{1},\break \aux[0]{1 } ) \sim\pi$ ] and @xmath205{2 } \sim\pi^{\ast}$ ] . then for all @xmath206 $ ] satisfying @xmath207{i } \bigr ) } { h \bigl(\y[k]{i } \bigr ) } \bigr| < \infty\qquad \bigl(i \in\{1,2 \ } \bigr)\ ] ] it holds that @xmath208{2 } \bigr ) } \leq\lim _ { n \to\infty } \frac{1}{n } \var{\sum_{k = 0}^{n - 1 } h \bigl(\y[k]{1 } \bigr)}.\ ] ] we preface the proof of theorem [ teocompalg1alg2 ] by the following lemma , which may serve as a basis for the comparison of _ homogeneous _ markov chains evolving according to @xmath209 ( or @xmath210 ) , @xmath5 , where @xmath3 and @xmath59 , @xmath5 , are kernels satisfying on some product space . [ lemhomogeneouscomp ] let @xmath3 and @xmath59 , @xmath5 , be kernels satisfying on @xmath15 , with @xmath211 and @xmath212 . in addition , assume that for all @xmath213 , @xmath214 then for all @xmath45 $ ] depending on only the first argument [ i.e. , @xmath215 for some @xmath216 and such that @xmath217 it holds that @xmath218 [ remcounterexlemma ] assumption ( [ eqassunitmass ] ) is essential in lemma [ lemhomogeneouscomp ] . indeed , let @xmath219 and @xmath220 , and define the kernels @xmath221 , @xmath222 for some @xmath223 , @xmath224 , and @xmath225 . then the kernels @xmath3 and @xmath59 , @xmath5 , satisfy , and consequently theorem [ teomainresult ] applies to the inhomogeneous chains evolving alternatingly according to the same . however , the similar result does not hold true for chains evolving according to the product kernels @xmath226 and @xmath227 , @xmath5 , as @xmath228 with @xmath37 being the identity mapping on @xmath17 . proof of lemma [ lemhomogeneouscomp ] define markov chains @xmath73 , @xmath5 , evolving as @xmath229{i } = \pmatrix{\y[k]{i } \vspace*{3pt}\cr \aux[k]{i } } \stackrel{p_i } { \longrightarrow } \x[2k+1]{i}= \pmatrix{\cy[k]{i } \vspace*{3pt}\cr \caux[k]{i } } \stackrel { q_i } { \longrightarrow } \x[2k+2]{i}= \pmatrix{\y[k+1]{i } \vspace*{3pt}\cr \aux[k+1]{i } } \stackrel{p_i } { \longrightarrow } \cdots\ ] ] with @xmath105{i } \sim\pi$ ] . by construction , @xmath230{i } \bigr ) } { f \bigl(\x[k]{i } \bigr)}\bigr|+\bigl|\covardu { f \bigl ( \x[1]{i } \bigr ) } { f \bigl(\x[k+1]{i } \bigr)}\bigr| \bigr ) \nonumber\\[-8pt]\\[-8pt ] & & \qquad = \pi f^2 - \pi^2 f + 4 \sum _ { k = 1}^\infty\bigl|\covardu{h \bigl(\y[0]{i } \bigr ) } { h \bigl ( \y[k]{i } \bigr)}\bigr| < \infty\qquad \bigl(i \in\{0 , 1\ } \bigr ) , \nonumber\end{aligned}\ ] ] where finiteness follows from the assumption ( [ eqassbddnessproduct ] ) . moreover , for all @xmath231 and @xmath232 , @xmath233{i } \bigr ) } = \var{\sum_{k = 0}^{n - 1 } h \bigl ( \cy[k]{i } \bigr ) } = \frac{1}{4 } \var{\sum_{k = 0}^{2n - 1 } f \bigl(\x[k]{i } \bigr)},\ ] ] which implies , by ( [ eqimpliedbddness ] ) , @xmath234{i } \bigr ) } \qquad \bigl(i \in\{0 , 1\ } \bigr).\ ] ] finally , by ( [ eqimpliedbddness ] ) we may now apply theorem [ teomainresult ] to the chains @xmath235{i } ; k \!\in\!\mathbb{n}\}$ ] , @xmath236 , which establishes immediately the statement of the lemma . proof of theorem [ teocompalg1alg2 ] we introduce the kernels : * @xmath237 , * @xmath238 , * @xmath239 being - defined implicitly as the transition kernel associated with the _ freeze algorithm _ ( algorithm [ algalg1 ] ) . it can be checked readily that the two sequences @xmath173 and @xmath172 generated by algorithms [ algalg1 ] and [ algalg2 ] , respectively , have indeed the same distributions as the marginal processes ( with respect to the first component ) of homogeneous chains evolving according to the products @xmath240 and @xmath241 , respectively . in addition , all kernels @xmath3 and @xmath59 , @xmath242 , are @xmath0-reversible , as : * @xmath41 is reversible with respect to any probability measure ( in particular , it is reversible ) , * @xmath243 is @xmath0-reversible as a gibbs - sampler sub - step transition kernel , * @xmath239 is @xmath0-reversible as a classical metropolis hastings transition kernel . since @xmath41 has no off - diagonal component , it holds that @xmath244 ; moreover , trivially , @xmath245 . thus , we may complete the proof by applying lemma [ lemhomogeneouscomp ] to the function @xmath215 , for which the condition ( [ eqassbddnessproduct ] ) is satisfied [ by ( [ eqcondh ] ) ] . _ pseudo - marginal algorithms _ ( see @xcite and @xcite ) are implemented using a markov kernel @xmath246 on @xmath247 and a family @xmath248 of real - valued nonnegative functions on @xmath249 such that @xmath250 for all @xmath251 . we denote by @xmath252 the transition density of the kernel @xmath253 with respect to some dominating measure . note that @xmath254 is a markov transition kernel as well . the problem at hand is to sample the target distribution @xmath255 under the assumption that : * for all @xmath256 , @xmath257 is known up to a normalizing constant , * for all @xmath251 , @xmath183 can be sampled from . the particular case where @xmath258 for all @xmath259 was discussed in the previous section , and we now turn to the case @xmath260 ( i.e. , sampling directly from @xmath127 is infeasible ) . the solution provided by pseudo - marginal algorithms consists in replacing , in algorithm [ algalg2 ] , the operation ( i ) by the sampling @xmath261 , and the computing the acceptance probability @xmath147 [ as defined in ( [ eqacceptmetropolis ] ) ] via the formula @xmath262 the output of this algorithm , which will be referred to as the _ noisy algorithm _ in the following , is typically not on the contrary to algorithm [ algalg2]@xmath125-reversible due to the replacement of @xmath127 by @xmath253 . this justifies the denomination . however , when @xmath263 is close to unity the noisy algorithm is close to algorithm [ algalg2 ] , which is , according to theorem [ teocompalg1alg2 ] , more efficient than algorithm [ algalg1 ] in terms of asymptotic variance . the monte carlo within metropolis algorithm ( mcwm ; see @xcite ) resembles closely the gimh algorithm ( see example [ exgimh ] ) , however , with the important difference that the importance sampling estimates @xmath264 [ given by ( [ eqgimhmcest ] ) ] are _ not _ stored and propagated through the algorithm along with the @xmath265-values . instead each estimate of the marginal density is recomputed using a `` fresh '' mc sample before the calculation of the acceptance probability . thus , the mcwm algorithm can be cast into the framework of the noisy algorithm with @xmath266 and with the auxiliary variables @xmath163 and @xmath267 playing the roles of @xmath268-dimensional monte carlo samples . considering this , we now propose a novel algorithm which will be referred to as the _ random refreshment _ _ algorithm _ and which is a hybrid between algorithm [ algalg2 ] and the noisy algorithm . this novel algorithm , which is described in algorithm [ algalgrefresh ] below , targets _ exactly _ @xmath125 and turns out to be more efficient than algorithm [ algalg1 ] . @xmath136{3 } , \aux[k]{3 } ) = ( y , u)$ ] : * draw @xmath269 , * set @xmath270 draw @xmath271 , draw @xmath272 , set @xmath273{3 } , \aux[k+1]{3 } ) \gets \cases { ( \hat{y } , \hat{u } ) , & \quad with probability $ \alpha(y , \check { u } , \hat{y } , \hat{u})$ , \vspace*{3pt}\cr ( y , \check{u } ) , & \quad otherwise . } $ ] in step ( i ) in algorithm [ algalgrefresh ] , the auxiliary variable @xmath274 can be either `` refreshed , '' that is , replaced by a new candidate @xmath275 , or kept at the previous state @xmath276{3}$ ] according to an acceptance probability that turns out to be a standard metropolis hastings acceptance probability ( which will be seen in the proof of theorem [ teocompalg1alg4 ] below ) . interestingly , this allows the desired distribution @xmath0 as the target distribution of @xmath131{3 } , \aux[k]{3 } ) ; k \in\mathbb{n}\}$ ] . in comparison , the noisy algorithm described above differs only from algorithm [ algalgrefresh ] by step ( i ) , in that the new candidate is always accepted in the noisy algorithm . this `` systematic refreshment '' makes actually the noisy algorithm imprecise in the sense that @xmath0 is no longer the target distribution except when @xmath277 for all @xmath256 , in which case @xmath278 in ( [ eqdefrho ] ) becomes identically equal to unity and algorithm [ algalgrefresh ] translates into algorithm [ algalg2 ] . compared to algorithm [ algalg1 ] , step ( i ) allows the second component to be refreshed randomly according to the probability @xmath279 whereas this component remains unchanged in algorithm [ algalg1 ] . thus , in conformity with algorithm [ algalg2 ] , it is likely that algorithm [ algalgrefresh ] has better mixing properties than algorithm [ algalg1 ] . that this is indeed the case may be established by reapplying the embedding technique developed in the previous part . before formalizing this properly , we propose an example showing a typical situation where a random refreshment algorithm may be used . [ exrrabc ] in @xcite ( contributing to the discussion of @xcite ) , the authors propose a novel algorithm , _ rejuvenating gimh - abc _ @xcite , algorithm 1 , preventing the original _ gimh - abc _ @xcite , algorithm 2 ( termed _ mcmc - abc _ in the paper in question ) , from falling into possible trapping states . the gimh - abc is an instance of algorithm [ algalg1 ] targeting @xmath280 , where , in the abc context : * @xmath281 is the desired posterior of a parameter @xmath282 given some observed data summary statistics @xmath283 , * @xmath183 is the likelihood of the data ( from which sampling is assumed to be feasible ) , * @xmath284 / \int\check{r}(y , \mathrm{d}u ' ) k[(s(u ' ) - { s_{\mathrm{obs } } } ) / h ] $ ] , where @xmath110 is a kernel integrating to unity , providing the classical abc discrepancy measure between the observed data summary statistics @xmath283 and that evaluated at the simulated data @xmath129 . rejuvenating gimh - abc comprises an intermediate step in which the simulated data @xmath129 , generated under the current parameter @xmath282 , are refreshed systematically . however , since sampling from @xmath285 is typically infeasible , the auxiliary variables are refreshed through @xmath253 in the spirit of algorithm [ algalg2 ] . therefore , in accordance with algorithm [ algalgrefresh ] , a _ @xmath0-reversible _ alternative to rejuvenating gimh - abc is obtained by , instead of refreshing systematically the data , performing refreshment with probability ( [ eqdefrho ] ) note that the fact that the constant in the denominator of @xmath286 is typically not computable does not prevent computation of ( [ eqdefrho ] ) , since this constant appears in @xmath286 as well as @xmath287 . this provides a _ random refreshment gimh - abc _ , which can be compared quantitatively , via the theorem [ teocompalg1alg4 ] below , to the gimh - abc while at the same time avoiding the possible gimh - abc trapping states mentioned in @xcite . [ teocompalg1alg4 ] let and be the sequences of random variables generated by algorithms [ algalg1 ] and [ algalgrefresh ] , respectively , where @xmath204{i},\break \aux[0]{i } ) \sim \pi $ ] , @xmath288 . then the following hold true : the output of algorithm [ algalgrefresh ] is @xmath0-reversible . for all @xmath289 $ ] satisfying @xmath290{i } \bigr ) } { h \bigl(\y[k]{i } \bigr ) } \bigr| < \infty\qquad \bigl(i \in\{1,3 \ } \bigr)\ ] ] it holds that @xmath208{3 } \bigr ) } \leq\lim _ { n \to\infty } \frac{1}{n } \var{\sum_{k = 0}^{n - 1 } h \bigl(\y[k]{1 } \bigr)}.\ ] ] let the kernels @xmath41 and @xmath291 be defined as in the proof of theorem [ teocompalg1alg2 ] and introduce furthermore : * @xmath292 defined implicitly by the transition @xmath136{3 } , \aux [ k]{3 } ) \rightarrow(\y[k]{3 } , \check{u})$ ] according to step ( i ) in algorithm [ algalgrefresh ] ( note that the first component is held fixed throughout the transition ) , * @xmath293 . in conformity with the proof of theorem [ teocompalg1alg2 ] , it can be checked readily that the two sequences @xmath173 and @xmath294 generated by algorithms [ algalg1 ] and [ algalgrefresh ] , respectively , have indeed the same distributions as the marginal processes ( with respect to the first component ) of homogeneous chains evolving according to the products @xmath240 and @xmath295 , respectively . the @xmath0-reversibility of the kernels @xmath41 and @xmath296 was established in the proof of theorem [ teocompalg1alg2 ] . to verify @xmath0-reversibility of @xmath292 as well , note that @xmath292 is a metropolis hastings kernel associated with the target distribution @xmath0 , whose acceptance probability includes a radon nikodym derivative of the type given in proposition [ propderiveeradon ] ; it is therefore @xmath8-reversible . indeed , note that @xmath292 updates only the second component according to @xmath297 with the acceptance probability @xmath298 . assuming first that @xmath253 is dominated and denoting by @xmath252 its transition density , we have @xmath299 where @xmath300 in the density of the target @xmath0 . this shows that @xmath301 is indeed the acceptance probability of a metropolis hastings markov chain targeting @xmath0 , with proposal kernel @xmath302 ; the @xmath0-reversibility of @xmath292 follows . the proof can be adapted easily to the case where @xmath253 is not dominated . as a consequence , the product @xmath295 is also @xmath8-reversible , which establishes the statement ( i ) of the theorem . finally , since @xmath41 has zero mass on the off - diagonal , it holds that @xmath303 and , clearly , @xmath304 . the proof of ( ii ) is now concluded by applying lemma [ lemhomogeneouscomp ] along the lines of the proof of theorem [ teocompalg1alg2 ] . we preface the proof of theorem [ teomainresult ] with some preliminary lemmas . [ lemlem1 ] assume that @xmath305 are @xmath0-reversible markov transition kernels . then , for all @xmath306 \times\ltwo [ \pi]$ ] , @xmath307 as each @xmath308 is @xmath0-reversible , it holds that @xmath309 for all @xmath310 \times\ltwo[\pi]$ ] and @xmath311 . applying repeatedly this relation yields @xmath312 [ lemlem2 ] let @xmath313 and @xmath2 be markov transition kernels on @xmath12 such that @xmath314 and let @xmath315 be a markov chain evolving as @xmath316 { } \stackrel { p } { \longrightarrow } \x[1 ] { } \stackrel{q } { \longrightarrow } \x[2 ] { } \stackrel { p } { \longrightarrow } \x[3 ] { } \stackrel{q } { \longrightarrow } \cdots\ ] ] with initial distribution @xmath105{}\sim\pi$ ] . then , for all @xmath317 $ ] such that @xmath318 { } \bigr ) } { f \bigl(\x[k ] { } \bigr)}\bigr|+\bigl|\covardu { f \bigl(\x[1 ] { } \bigr ) } { f \bigl(\x[k+1 ] { } \bigr)}\bigr| \bigr ) < \infty,\ ] ] the limit , as @xmath85 tends to infinity , of @xmath319{})}$ ] exists , and @xmath320 { } \bigr)}\nonumber \\ & & \qquad = \pi f^2 - \pi^2 f \\ & & \quad\qquad { } + \sum_{k = 1}^{\infty } \covardu{f \bigl(\x[0 ] { } \bigr ) } { f \bigl(\x[k ] { } \bigr ) } + \sum_{k = 1}^{\infty } \covardu{f \bigl(\x[1 ] { } \bigr ) } { f \bigl(\x[k+1 ] { } \bigr)}. \nonumber\end{aligned}\ ] ] as covariances are symmetric , @xmath321 { } \bigr)}=\pi f^{2 } - \pi^{2}f + 2 n^{-1 } \sum_{0 \leq i < j \leq n - 1}\covardu{f \bigl(\x[i ] { } \bigr ) } { f \bigl(\x[j ] { } \bigr)}.\ ] ] we now consider the limit , as @xmath85 tends to infinity , of the last term on the right - hand side . let @xmath322 and @xmath323 denote the two complementary subsets of @xmath324 consisting of the even and odd numbers , respectively . for all @xmath325 such that @xmath326 , we have @xmath327 { } \bigr ) } { f \bigl(\x[j ] { } \bigr ) } = \cases { \covardu{f \bigl(\x[0 ] { } \bigr ) } { f \bigl(\x[j - i ] { } \bigr ) } , & \quad if $ i \in\mathcal{e}$ , \vspace*{3pt}\cr \covardu{f \bigl(\x[1 ] { } \bigr ) } { f \bigl(\x[j - i+1 ] { } \bigr ) } , & \quad if $ i \in \mathcal{o}$.}\ ] ] this implies that @xmath328 { } \bigr ) } { f \bigl(\x[j ] { } \bigr ) } \\ & & \qquad = \sum_{k = 1}^{n - 1 } n^{-1 } \biggl ( \biggl\lfloor\frac{n - 1 - k}{2 } \biggr\rfloor+ 1 \biggr ) \covardu { f \bigl(\x[0 ] { } \bigr ) } { f \bigl(\x[k ] { } \bigr)}\end{aligned}\ ] ] and @xmath329 { } \bigr ) } { f \bigl(\x[j ] { } \bigr ) } \\ & & \qquad = \sum_{k = 1}^{n - 2 } n^{-1 } \biggl ( \biggl\lfloor\frac{n - 2 - k}{2 } \biggr\rfloor+ 1 \biggr ) \covardu { f \bigl(\x[1 ] { } \bigr ) } { f \bigl(\x[k+1 ] { } \bigr)}.\end{aligned}\ ] ] under ( [ eqeq2asslem2 ] ) , the dominated convergence theorem applies , which provides that the limit , as @xmath85 goes to infinity , of @xmath330{})}$ ] exists and is equal to ( [ eqasvaraltexpression ] ) . [ lemlem4 ] let @xmath3 and @xmath59 , @xmath331 , be @xmath0-reversible markov kernels on @xmath92 such that @xmath332 and @xmath333 . for all @xmath334 and @xmath5 , denote by @xmath335{i}$ ] the markov kernel @xmath335{i}:=p_i \mathbh{1}_{\mathcal{e}}(n ) + q_i \mathbh{1}_{\mathcal{o}}(n)$ ] . in addition , let @xmath336 $ ] be such that for @xmath337 , @xmath338{i } \cdots\r[k-1]{i}f \bigr\rangle \bigr\vert < \infty.\ ] ] then for all @xmath339 , @xmath340{1 } \cdots\r[k-1]{1}f \bigr\rangle + \bigl\langle f , \r [ 1]{1 } \cdots\r[k]{1 } f \bigr\rangle \bigr ) \\ & & \qquad \leq\sum_{k=1}^{\infty } \lambda^k \bigl ( \bigl\langle f , \r[0]{0 } \cdots\r[k-1]{0}f \bigr\rangle + \bigl\langle f , \r [ 1]{0 } \cdots \r[k]{0}f \bigr\rangle \bigr).\end{aligned}\ ] ] for all @xmath334 and all @xmath341 , define @xmath342{\alpha}:=(1-\alpha)\r[n]{0}+\alpha\r[n]{1}$ ] . in addition , set , for @xmath339 , @xmath343{}(\alpha):= \kh [ \lambda]{\mathcal{e}}(\alpha)+\kh[\lambda]{\mathcal{o}}(\alpha ) $ ] , where @xmath344{\mathcal{e}}(\alpha ) & : = & \sum _ { k = 1}^{\infty } \lambda^k \bigl\langle f , \r[0 ] { \alpha } \cdots \r[k-1]{\alpha } f \bigr\rangle , \\ \kh[\lambda]{\mathcal{o}}(\alpha ) & : = & \sum_{k = 1}^{\infty } \lambda^k \bigl\langle f , \r[1]{\alpha } \cdots\r[k]{\alpha } f \bigr \rangle.\end{aligned}\ ] ] now , fix a distinguished @xmath339 ; we want show that for all @xmath345 $ ] , @xmath346{}}{\mathrm{d}\alpha } ( \alpha ) \leq0.\ ] ] thus , we start with differentiating @xmath343{\mathcal{e}}$ ] : @xmath347{\mathcal { e}}}{\mathrm{d}\alpha } ( \alpha ) = \frac{\mathrm{d}}{\mathrm{d } \alpha } \sum _ { k = 1}^{\infty } \lambda^k \bigl\langle f , \r[0 ] { \alpha}\cdots \r[k-1]{\alpha}f \bigr\rangle.\ ] ] to interchange @xmath348 and @xmath349 in the previous equation , we first note that @xmath350{\alpha } \cdots \r[k-1]{\alpha}f \bigr\rangle&= & \sum _ { \ell= 0}^{k - 1 } \frac{\partial}{\partial \alpha_\ell } \bigl\langle f , \r[0]{\alpha_0 } \cdots\r [ k - 1]{\alpha_{k - 1}}f \bigr \rangle \bigg\vert_{(\alpha_0,\ldots,\alpha_{k - 1})=(\alpha,\ldots,\alpha ) } \\ & = & \sum_{\ell= 0}^{k - 1 } \bigl \langle f , \r[0 { \nearrow}\ell-1]{\alpha } \bigl(\r[\ell]{1 } - \r[\ell ] { 0 } \bigr ) \r[\ell+ 1 { \nearrow}k - 1]{\alpha}f \bigr\rangle,\end{aligned}\ ] ] where @xmath351{\alpha}:=\r[s]{\alpha } \r[s+1]{\alpha } \cdots\r[t]{\alpha}$ ] for @xmath352 and @xmath351{\alpha } : = \operatorname{id}$ ] otherwise . by ( [ eqmajonormp ] ) , @xmath353{\alpha } \|\leq1 $ ] , which implies that @xmath354}|\frac{\mathrm{d } } { \mathrm{d } \alpha } \langle f , \r[0]{\alpha}\cdots\r[k-1]{\alpha}f \rangle \pi(f^2)$ ] . thus , as @xmath355 we may interchange , in ( [ eqderivh ] ) , @xmath356 and @xmath349 , yielding @xmath357{\mathcal{e}}}{\mathrm{d}\alpha } ( \alpha)=\sum_{k=1}^{\infty } \lambda^{k}\sum_{\ell=0}^{k-1 } \bigl \langle f , \r[0 { \nearrow}\ell-1]{\alpha } \bigl(\r[\ell ] { 1}-\r [ \ell]{0 } \bigr ) \r [ \ell+1 { \nearrow}k-1]{\alpha}f \bigr\rangle.\ ] ] similarly , it can be established that @xmath357{\mathcal{o}}}{\mathrm{d}\alpha } ( \alpha ) = \sum_{k=1}^{\infty } \lambda^{k}\sum_{\ell=1}^{k } \bigl \langle f , \r[1 { \nearrow}\ell-1]{\alpha } \bigl(\r[\ell ] { 1}-\r [ \ell]{0 } \bigr ) \r[\ell + 1 { \nearrow}k]{\alpha}f \bigr\rangle.\ ] ] we now apply lemma [ lemlem1 ] to the two previous sums . for this purpose , we will use the following notation : @xmath358{\alpha } : = \r[s]{\alpha } \r[s-1]{\alpha } \cdots\r[t]{\alpha}$ ] for @xmath359 and @xmath358{\alpha}:=\operatorname{id}$ ] otherwise . then @xmath360{}}{\mathrm { d}\alpha } ( \alpha ) & = & \sum_{k=1}^{\infty } \lambda^k \biggl\{\sum_{\ell=0}^{k-1 } \bigl\langle\r[\ell-1 { \searrow}0]{\alpha}f , \bigl(\r[\ell]{1}-\r[\ell]{0 } \bigr)\r [ \ell+1 { \nearrow}k-1]{\alpha}f \bigr\rangle \\[1pt ] & & \hspace*{33pt}{}+\sum_{\ell=1}^{k } \bigl\langle\r[\ell-1 { \searrow}1]{\alpha}f , \bigl(\r[\ell]{1}-\r[\ell]{0 } \bigr)\r [ \ell+1 { \nearrow}k ] { \alpha}f \bigr\rangle \biggr\ } \\[1pt ] & = & \sum_{\ell=0}^{\infty}\sum _ { m=0}^{\infty}\lambda^{\ell+m+1 } \bigl\langle\r [ \ell-1 { \searrow}0]{\alpha}f , \bigl(\r[\ell]{1}-\r[\ell]{0 } \bigr)\r [ \ell+1 { \nearrow}\ell+m]{\alpha}f \bigr\rangle \\[1pt ] & & { } { } + \sum_{\ell=1}^{\infty}\sum _ { m=1}^{\infty}\lambda^{\ell+m-1 } \bigl\langle\r [ \ell-1 { \searrow}1]{\alpha}f , \bigl(\r[\ell]{1}-\r[\ell]{0 } \bigr)\r [ \ell+1 { \nearrow}\ell+m-1]{\alpha}f \bigr\rangle.\end{aligned}\ ] ] now , note that @xmath335{\alpha}=\r[n']{\alpha}$ ] for all @xmath361 and @xmath362{\alpha}=\r[m']{\alpha}$ ] for all @xmath363 ; hence , separating , in the two previous sums , odd and even indices @xmath364 provides @xmath360{}}{\mathrm { d}\alpha } ( \alpha ) & = & \sum_{\ell\in\mathcal{e}}\sum_{m=0}^{\infty } \lambda^{\ell+m+1 } \bigl\langle\r[1 { \nearrow}\ell]{\alpha}f , \bigl(\r[0]{1}- \r[0]{0 } \bigr)\r[1 { \nearrow}m]{\alpha}f \bigr\rangle \\[1pt ] & & { } + \sum_{\ell\in\mathcal{e}\setminus\{0\}}\sum_{m=1}^{\infty } \lambda^{\ell + m-1 } \bigl\langle\r[1 { \nearrow}\ell-1]{\alpha}f , \bigl(\r [ 0]{1}- \r[0]{0 } \bigr)\r[1 { \nearrow}m-1]{\alpha}f \bigr\rangle \\[1pt ] & & { } + \sum_{\ell\in\mathcal{o}}\sum_{m=0}^{\infty } \lambda^{\ell+m+1 } \bigl\langle\r[0 { \nearrow}\ell-1]{\alpha}f , \bigl ( \r[1]{1}- \r[1]{0 } \bigr)\r[0 { \nearrow}m-1]{\alpha}f \bigr\rangle \\[1pt ] & & { } + \sum_{\ell\in\mathcal{o}}\sum_{m=1}^{\infty } \lambda^{\ell+m-1 } \bigl\langle\r[0 { \nearrow}\ell-2]{\alpha}f , \bigl ( \r[1]{1}- \r[1]{0 } \bigr)\r[0 { \nearrow}m-2]{\alpha}f \bigr\rangle.\end{aligned}\ ] ] finally , by combining the even and the odd sums , @xmath360{}}{\mathrm { d}\alpha } ( \alpha ) & = & \biggl\langle\sum_{\ell=0}^{\infty } \lambda^{\ell}\r[1 { \nearrow}\ell]{\alpha}f , \bigl(\r[0]{1}-\r[0]{0 } \bigr ) \sum_{m=0}^{\infty } \lambda^m \r[1 { \nearrow}m]{\alpha}f \biggr\rangle \\ & & { } + \biggl\langle\sum_{\ell=0}^{\infty } \lambda^\ell\r[0 { \nearrow}\ell-1]{\alpha}f , \bigl(\r[1]{1}-\r[1]{0 } \bigr ) \sum_{m=0}^{\infty}\lambda^m\r [ 0{\nearrow}m-1]{\alpha}f \biggr\rangle.\end{aligned}\ ] ] since @xmath335{1}\pgeq[1 ] \r[n]{0}$ ] , the operator @xmath335{0}-\r [ n]{1}$ ] is nonnegative on @xmath30 $ ] ( by @xcite , lemma 3 ) , and for all @xmath336 $ ] it holds that @xmath365{1}-\r [ n]{0})f \rangle\leq0 $ ] . this shows ( [ eqnegder ] ) , which implies that the function @xmath366{}(\alpha)$ ] is nonincreasing on @xmath367 . the proof is complete . proof of theorem [ teomainresult ] according to lemma [ lemlem2 ] , for all functions @xmath368 $ ] and @xmath331 , @xmath369\\[-8pt ] & & { } + \sum_{k=1}^{\infty } \bigl ( \covardu{f \bigl ( \x[0]{i } \bigr ) } { f \bigl(\x[k]{i } \bigr ) } + \covardu{f \bigl(\x [ 1]{i } \bigr ) } { f \bigl ( \x[k+1]{i } \bigr ) } \bigr ) . \nonumber\end{aligned}\ ] ] for the kernels @xmath3 and @xmath59 , @xmath331 , in the statement of the theorem , let @xmath370 , @xmath331 , be defined as in lemma [ lemlem4 ] , which then implies that for all @xmath371 , @xmath372{1 } \bigr ) } { f \bigl(\x[k]{1 } \bigr ) } + \lambda^{k}\covardu{f \bigl(\x[1]{1 } \bigr ) } { f \bigl(\x[k+1]{1 } \bigr ) } \bigr ) \nonumber\\[-8pt]\\[-8pt ] & & \qquad \leq \sum_{k=1}^{\infty } \bigl ( \lambda^{k}\covardu{f \bigl(\x[0]{0 } \bigr ) } { f \bigl(\x[k]{0 } \bigr)}+ \lambda^{k}\covardu{f \bigl(\x[1]{0 } \bigr ) } { f \bigl(\x[k+1]{0 } \bigr ) } \bigr ) . \nonumber\end{aligned}\ ] ] we conclude the proof by letting @xmath373 tend to one on each side of the previous inequality . under ( [ eqassumpfuncthm ] ) , we may , by the dominated convergence theorem , interchange limits with summation , which establishes inequality ( [ eqth4proof ] ) also in the case @xmath374 . combining this with ( [ eqth4proof1 ] ) completes the proof . in this paper , we have extended successfully the theoretical framework proposed in @xcite and @xcite as a means of comparing the asymptotic variance of sample path averages for different markov chains and , consequently , the efficiency of different mcmc algorithms to the context of inhomogeneous markov chains evolving alternatingly according to two different markov transition kernels . it turned out that this configuration covers , although not apparently , several popular mcmc algorithms such as randomized mcmc @xcite , multiple - try metropolis @xcite and its generalization @xcite , and the pseudo - marginal algorithms @xcite . it should be remarked however that our results do not take possible additional computational cost into consideration , which may be of importance in practical applications . while these algorithms are inapproachable for the standard tools provided in @xcite and @xcite , our results allow , without heavy technical developments , rigorous theoretical justifications advocating the use of these algorithms . as illustrated by our novel _ random refreshment _ algorithm in the context of pseudo - marginal algorithms , the results of the present paper can also be used for designing new algorithms and improving , in terms of asymptotic variance , existing ones . first , set @xmath375 ; then by jensen s inequality , @xmath376 and since @xmath377 , @xmath378 now , without loss of generality we may assume that @xmath379 , @xmath380 , and @xmath381 . then applying ( [ eqineq - pn - v - half ] ) yields for all @xmath23 , @xmath382 hence , for all @xmath68 , @xmath383 { } \bigr ) } { f \bigl(\x[2n ] { } \bigr)}\bigr| & = & \bigl|\mathbb{e } \bigl(f(x_0 ) ( pq)^n f(x_0 ) \bigr)\bigr| \\ & \leq & \bigl(2 c \rho^n \bigr)^{1/2 } \mathbb{e } \bigl(\bigl|f(x_0)\bigr|v^{1/2}(x_0 ) \bigr ) \leq \bigl(2 c \rho^{n } \bigr)^{1/2 } \pi v.\end{aligned}\ ] ] in the same way , for all @xmath384 , @xmath385 { } \bigr ) } { f \bigl(\x[2n+1 ] { } \bigr)}\bigr|=\bigl|\mathbb{e } \bigl(f(x_0 ) ( pq)^n pf(x_0 ) \bigr)\bigr| \leq \bigl(2 c \rho^{n } \bigr)^{1/2 } \pi v.\ ] ] by applying successively the cauchy schwarz and jensen inequalities , we obtain @xmath386^{1/2 } \leq\pi v,\ ] ] where the last inequality follows from @xmath387 and @xmath388 . this implies that for all @xmath231 , @xmath389 { } \bigr ) } { f \bigl(\x[2n ] { } \bigr)}\bigr| & = & \bigl|\mathbb{e } \bigl(f(x_1)q(pq)^{n-1 } f(x_1 ) \bigr)\bigr| \\ & \leq & \bigl(2 c \rho^{n-1 } \bigr)^{1/2 } \mathbb{e } \bigl(\bigl|f(x_1)\bigr|qv^{1/2}(x_1 ) \bigr ) \\ & \leq & \bigl(2 c \rho^{n-1 } \bigr)^{1/2 } \pi v.\end{aligned}\ ] ] in the same way , for all @xmath231 we have , using that @xmath390 , @xmath391 { } \bigr ) } { f \bigl(\x[2n+1 ] { } \bigr)}\bigr| = \bigl|\mathbb{e } \bigl(f(x_1)q(pq)^{n - 1 } pf(x_1 ) \bigr)\bigr| \leq \bigl(2 c \rho^{n - 1 } \bigr)^{1/2 } \pi v.\ ] ] the statement of the proposition follows . let @xmath110 be the transition kernel of the markov chain @xmath392 , that is , for all @xmath393 , @xmath394 where @xmath395 . thus , establishing @xmath125-reversibility of @xmath110 amounts to verifying , for all @xmath37 and @xmath38 in @xmath396 , @xmath397 indeed , by @xmath0-reversibility of @xmath131{1},\aux[k]{1 } ) ; k\in \mathbb{n } \}$ ] it holds , for all @xmath398 and @xmath399 in @xmath400 , @xmath401 which establishes ( [ eqstargrev ] ) by letting @xmath402 and @xmath403 . this completes the proof . @xmath165{2 } = y$ ] : draw @xmath404 , draw @xmath405 , set @xmath406{2 } \gets\cases { \hat{y } , & \quad w.pr . $ \displaystyle \alpha^{(\mathrm{r})}(y , u , \hat{y})$ \vspace*{5pt}\cr & \quad\qquad $ \displaystyle:=1\wedge\frac{\pi^{\ast}(\hat{y } ) \check{r}(\hat{y } , y ) \check{s}(\hat{y } , y ; f(u))}{\pi^{\ast}(y)\check{r}(y,\hat{y } ) \check{s}(y , \hat{y } ; u ) } \bigg\vert\frac{\partial f}{\partial u}(u ) \bigg\vert$ , \vspace*{3pt}\cr y , & \quad otherwise.}\ ] ] as proposed initially by @xcite , the r - mcmc algorithm generates a markov chain @xmath172 with transitions given by algorithm [ algalgrmcmc ] below . denote by @xmath407 the jacobian determinant of a vector - valued transformation @xmath37 . in this algorithm , @xmath37 is any continuously differentiable involution on @xmath408 . in addition , @xmath253 and @xmath409 are instrumental kernels on @xmath410 and @xmath411 , respectively , having transition densities @xmath252 and @xmath412 with respect to some dominating measure and lebesgue measure on @xmath413 , respectively . [ apprmcmc ] the r - mcmc algorithm is a special case of algorithm [ algalg2 ] . since @xmath414 and @xmath163 , obtained in steps ( i ) and ( ii ) of algorithm [ algalgrmcmc ] , are not drawn in the same order as in algorithm [ algalg2 ] , we first derive the expression of the corresponding kernels @xmath127 and @xmath132 , that is , @xmath415 where @xmath416 is lebesgue measure on @xmath417 . also note that @xmath418 moreover , introduce another auxiliary variable @xmath267 taking values in @xmath181 and being drawn according to @xmath419 . note that the kernel @xmath133 is not dominated by a common nonnegative measure regardless the value of @xmath129 ; still , following remark [ remgeneralradon ] , the r - mcmc algorithm may be covered by algorithm [ algalg2 ] , provided that the ratio in the acceptance probability @xmath420 corresponds to the radon nikodym derivative in proposition [ propderiveeradon ] for @xmath421 and @xmath422 the proof is completed by applying lemma [ lemrmcmclem2 ] below . [ lemrmcmclem2 ] the acceptance probability @xmath423 in ( [ eqacceptrmcmc ] ) is equal to @xmath424 where @xmath425 , @xmath426 , and @xmath427 denotes the radon nikodym derivative between the measures @xmath428 and @xmath429 defined by @xmath430 write @xmath431 , where @xmath432 to show ( [ eqapp2 ] ) , we will prove that for all bounded measurable functions @xmath433 on @xmath434 it holds that @xmath435 & = & \int g(x,\hat{x } ) \nu^{(\mathrm{r } ) } ( \mathrm{d}x \times\mathrm{d}\hat{x } ) \\ & = & \int g(x , \hat{x } ) \gamma ^{(\mathrm{r } ) } ( y , u , \hat{y } ) \mu^{(\mathrm{r})}(\mathrm{d}x \times\mathrm{d } \hat{x})\end{aligned}\ ] ] [ where @xmath436 and @xmath437 . now , using the change of variables @xmath438 , which is equivalent to @xmath439 ( since @xmath37 is an involution ) and using the relation ( [ eqtechnicos ] ) we obtain @xmath440 \\ & & \qquad = \int g^{(\mathrm{r } ) } \bigl(y , f(\hat{u } ) , \hat{y } , \hat{u } \bigr ) \pi^{\ast}(\mathrm{d}\hat{y } ) r(\hat{y } , \hat{u } ) s(\hat{y } , \hat{u } ; \mathrm{d}y ) \lambda _ d(\mathrm{d } \hat{u } ) \\ & & \qquad = \int g^{(\mathrm{r } ) } \bigl(y , u , \hat{y } , f(u ) \bigr ) \\ & & \hspace*{42pt } { } \times \pi^{\ast } ( \mathrm{d } \hat{y } ) r \bigl(\hat{y } , f(u ) \bigr ) s \bigl(\hat{y } , f(u ) ; \mathrm{d}y \bigr ) \bigl| ( \partial f/\partial u ) ( u)\bigr| \lambda_d ( \mathrm{d}u ) \\ & & \qquad = \int g^{(\mathrm{r } ) } \bigl(y , u , \hat{y } , f(u ) \bigr ) \frac{\pi ^{\ast } ( \hat{y } ) \check { r}(\hat{y } , y ) \check{s}(\hat{y } , y ; f(u ) ) } { \pi^{\ast}(y ) \check{r}(y , \hat{y})\check{s}(y , \hat{y } ; u ) } \bigg\vert\frac { \partial f}{\partial u}(u ) \bigg\vert \\ & & \hspace*{9pt}\quad\qquad { } \times\pi^{\ast}(\mathrm{d}y ) \check{r}(y,\mathrm { d } \hat{y } ) \check{s } ( y , \hat{y } ; \mathrm{d}u ) \\ & & \qquad = \int g^{(\mathrm{r})}(x , \hat{x } ) \gamma^{(\mathrm{r})}(y , u , \hat{y } ) \mu ^{(\mathrm{r})}(\mathrm{d}x \times\mathrm{d}\hat{x}),\end{aligned}\ ] ] which completes the proof . the gmtm algorithm proposed in @xcite generates a markov chain @xmath172 with transitions given by algorithm [ algalgmtm ] below . @xmath165{2 } = y$ ] : draw @xmath441 , let @xmath184 take the value @xmath442 w.pr . @xmath443 , let @xmath444 , draw @xmath445 , let @xmath446 , let @xmath447{2 } \gets\cases { \hat{y } , & \quad with probability $ \displaystyle \alpha^{(\mathrm{m})}(y , v , \hat{y } , \hat{v})$ \vspace*{5pt}\cr & \quad\qquad $ \displaystyle : = 1 \wedge\frac{\pi^{\ast}(\hat{y } ) \check{r}(\hat{y } , y ) \omega(\hat{y } , y ) \sum_{k = 1}^n \omega(y , v_k)}{\pi^{\ast } ( y ) \check{r}(y , \hat{y } ) \omega(y , \hat{y } ) \sum_{k = 1}^n \omega(\hat{y},\hat{v}_k)}$ , \vspace*{3pt}\cr y , & \quad otherwise.}\ ] ] in algorithm [ algalgmtm ] , the auxiliary variables @xmath448 are defined on @xmath249 and for all @xmath251 and @xmath449 , @xmath450 are sample weights . moreover , @xmath253 is an instrumental kernel defined on @xmath451 having the transition density @xmath252 with respect to some dominating measure on @xmath123 . [ lemmtm ] the gmtm algorithm is a special case of algorithm [ algalg2 ] . denoting by @xmath182 the random variables generated in step ( i ) in algorithm [ algalgmtm ] , the proposed candidate @xmath414 is obtained as @xmath452 , where @xmath184 is generated in step ( ii ) . let @xmath453 , where @xmath454 to obtain the joint distribution of @xmath455 conditionally on @xmath456{2}$ ] , write for any bounded measurable function @xmath433 on @xmath457 , @xmath458{2 } = y \bigr ] \\ & & \qquad = \sum_{j = 1}^n \mathbb{e } \bigl[g ( v_j , v_{-j } ) \mathbh{1}_{j = j } { |}\y[k]{2 } = y \bigr ] \\ & & \qquad = { \int\cdots\int}\check{r}(y , \mathrm{d}\hat{y } ) \prod_{k = 1}^{n - 1 } \check{r}(y , \mathrm{d}u_k ) \frac { n \omega(y , \hat{y})}{\sum_{\ell = 1 } ^{n - 1 } \omega(y , u_\ell ) + \omega(y , \hat{y})}g(\hat{y } , u ) \\ & & \qquad = { \int\cdots\int}r(y , \mathrm{d}u ) s(y , u ; \mathrm{d}\hat{y } ) g(\hat { y } , u),\end{aligned}\ ] ] where we introduced the kernels @xmath459 now , set @xmath460 where the @xmath461 s are sampled in step ( iv ) . the distribution of @xmath267 conditionally on @xmath136{2 } , u , { \hat{y } } ) = ( y , u , \hat{y})$ ] is given by @xmath462 if @xmath253 is dominated by a nonnegative measure , then ( [ eqmtmr ] ) , ( [ eqmtms ] ) and ( [ eqmtmt ] ) show that the kernels @xmath127 , @xmath132 and @xmath133 are dominated as well . denoting by @xmath203 , @xmath145 and @xmath146 the corresponding transition densities , it can be checked readily that @xmath463 so that @xmath464 defined in ( [ eqacceptmtm ] ) corresponds to the acceptance probability @xmath147 defined in ( [ eqacceptmetropolis ] ) with these particular choices of @xmath203 , @xmath145 and @xmath146 . consequently , the gmtm algorithm is a special case of algorithm [ algalg2 ] . note that in the previous proof , we have chosen the auxiliary variable @xmath163 as the vector of rejected candidates after step ( ii ) . another natural idea would consist in choosing @xmath465 , where the @xmath466s are obtained in step ( i ) ; however , since @xmath414 belongs to this set of candidates , the model would then not be dominated , which would make the proof more intricate . we thank the anonymous referees for insightful comments that improved significantly the presentation of the paper . a special thanks goes to the referee who provided the two counterexamples in remarks [ remcounterexsummability ] and [ remcounterexlemma ] , as well as the possible application of our methodology to the abc context in example [ exrrabc ] . | in this paper , we study the asymptotic variance of sample path averages for inhomogeneous markov chains that evolve alternatingly according to two different @xmath0-reversible markov transition kernels@xmath1 and @xmath2 .
more specifically , our main result allows us to compare directly the asymptotic variances of two inhomogeneous markov chains associated with different kernels @xmath3 and @xmath4 , @xmath5 , as soon as the kernels of each pair @xmath6 and @xmath7 can be ordered in the sense of lag - one autocovariance . as an important application
, we use this result for comparing different data - augmentation - type metropolis hastings algorithms .
in particular , we compare some pseudo - marginal and propose a novel exact algorithm , referred to as the _ random refreshment _ algorithm , which is more efficient , in terms of asymptotic variance , than the grouped independence metropolis
hastings algorithm and has a computational complexity that does not exceed that of the monte carlo within metropolis algorithm . , |
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supernovae ( sne ) are associated with the deaths of stars , in particular core - collapse supernovae ( ccsne ) are associated with the deaths of massive stars , which have initial masses greater than about 8 . sne are principally separated into two categories , those without hydrogen ( type i ) and those with ( type ii ) . only type ia sne are thought to be thermonuclear explosions , which arise from accreting white dwarfs in binary stellar systems . all the other sub - types are thought to be initiated by the core collapsing in massive stars . the type of sn that occurs depends on the massive star s evolutionary stage at the time of the explosion . the plateau subclass of type ii sne ( sne ii - p ) are thought to arise from the explosions of red supergiants ( rsgs ) , which have initial masses greater than 810 and have retained their hydrogen envelopes before core collapse @xcite . until the discovery of the red supergiant ( rsg ) that exploded as sn 2003gd @xcite , there had been no direct confirmation that sne ii - p did indeed arise from the explosions of rsgs . before this detection there had been only two other unambiguous detections of type ii progenitors , neither of which fitted the evolutionary scenario that is commonly accepted . these were the progenitors of the peculiar type ii - p sn 1987a , which was a blue supergiant , and the type iib sn 1993j that arose in a massive interacting binary system @xcite . the recent discovery of sn 2005cs ( ii - p ) in m51 , a galaxy with deep multi - colour pre - explosion images from the _ hubble space telescope _ ( _ hst _ ) , led to the discovery of another rsg progenitor of a ii - p @xcite . the estimated mass of the star was @xmath8 , similar to the mass ( @xmath9 ) for the progenitor of sn 2003gd @xcite . a supergiant of mass @xmath10 was found to be coincident with the type ii - p sn 2004et by @xcite , although it is likely not to have been as cool as an m - type supergiant , and the sn itself may be peculiar . there have been other extensive attempts to detect progenitors of nearby sne on ground- and space - based archival images e.g. @xcite , which have set upper mass limits mostly on ii - p events . the low mass of the progenitors discovered and upper limits set has led to the suggestion that sne ii - p come only from rsgs with masses less than about 15@xcite sn 2004a is another example of a nearby sn ii - p which has _ hst _ pre - explosion images , allowing the search for a progenitor star . sn 2004a was discovered by k. itagaki of teppo - cho , yamagata , japan on january 9.84 ut using a 0.28-m f/10 reflector . itagaki confirmed his discovery on january 10.75 ut , with a location of r. a. @xmath11 , dec . @xmath12 , around 22 arcsec west and 17 arcsec north of the centre of ngc 6207 . itagaki reported that no object was visible on his observations of 2003 december 27 , which had a limiting magnitude of 18 , or any of his observations prior to this date @xcite . itakagi s observations allow the explosion epoch to be fairly well constrained , suggesting that sn 2004a was discovered when it was quite young at less than 14d after explosion . an optical spectrum was obtained by @xcite on january 11.8 and 11.9 ut , and showed a blue continuum with p - cygni profiles of the balmer lines , consistent with a type ii sn . the emission features were somewhat weak suggesting that the sn was indeed young , in line with itagaki s observations . the expansion velocity , measured from the minima of the balmer lines , was around 12000 . in _ cycle 10 , we had a snapshot programme to enhance the _ hst _ archive with 100200 wide field planetary camera 2 ( wfpc2 ) multi - colour images of galaxies within approximately 20mpc . in the future , sne discovered in these galaxies could have pre - explosion images available to constrain the nature of the progenitor stars . this strategy is now beginning to bear fruit , ngc 6207 was one of those targets and the pre - explosion site of sn 2004a was imaged in three filters . in this paper we present photometric and spectroscopic data of sn 2004a in section [ sec : obs ] followed by an analysis of the photometry in section [ sec : epoch ] , where an explosion date is estimated . we estimate the reddening towards the sn in section [ sec : red ] and obtain the expansion velocity in section [ sec : expvel ] . the distance to ngc 6207 is not well known and only two distance estimates , which are both kinematic , exist in the literature . we estimate the distance using two further methods and compile the distances within the literature , in an attempt to improve the situation , in section [ sec : d ] . using the distance found we then calculate the amount of nickel synthesised in the explosion in section [ sec : ni ] . we present the discovery of the progenitor in section [ sec : prog ] and a discussion of the implications and conclusion in sections [ sec : diss ] and [ sec : con ] , respectively . throughout this work we have assumed the galactic reddening laws of @xcite with @xmath13 . _ bvri _ photometry was obtained shortly after discovery from the following telescopes : the 2.0-m liverpool telescope ( lt ) , la palma ; the 4.2-m william herschel telescope ( wht ) , la palma ; and the robotic palomar 60-inch telescope ( p60derekfox / p60/ ] ; cenko et al . 2006 , in prep . ) as part of the caltech core - collapse programavishay / cccp.html ] ( cccp , * ? ? ? * ; * ? ? ? * gal - yam et al . 2006 in prep . ) the lt observations were taken with the optical ccd camera , ratcam , using its bessel _ bv _ and sloan @xmath14 filters . the data were reduced using the lt data reduction pipeline . the wht observations were taken with the auxiliary port imaging camera ( aux ) , using its _ bvri _ filters , and were reduced using standard techniques within iraf . the frames were debiased and flat - fielded using dome flats from a few nights later . details of the p60 camera and data reduction can be found in @xcite . a summary of these observations can be found in table [ tab : phot ] as well as the results from the sn photometry . [ cols= " < , > , > , > , > , > , > , < " , ] [ tab : mag ] sn 2004a appears to be a ` normal ' sn ii - p with a spectrum and light curve very like the well observed sn 1999em ( figs . [ fig : s04a99em ] and [ fig:04a99em ] ) . the nickel mass found in section [ sec : ni ] was also comparable to that of sn 1999em , as we would have expected . the existence of a low luminosity , low nickel mass , sub - group of sne ii - p ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , has been suggested . one of the proposed characteristics of this group is a rapid excess in the and colours at the end of the photospheric phase @xcite . and @xcite noted this rapid excess in the ` faint ' sn 1997d ( see fig . [ fig : diffsnc ] ) , and it was also observed by @xcite in sn 1999eu , another ` faint ' sn , in the form of a sharp spike . [ fig : c04a99em ] compares the colour evolutions of sn 2004a and sn 1999em . the colour evolution of sn 2004a clearly shows such a rapid excess at the end of the plateau , which is not seen in the colour curve of sn 1999em . [ fig : diffsnc ] shows a comparison of the colour curves of the prototypical peculiar ` faint ' sn 1997d @xcite , and the ` normal ' sne 1999em and 2003gd @xcite . although the excess is present in the colour curve of sn 2004a , it does not show as great an excess as sn 1997d , but instead follows a similar curve to sn 2003gd with a slightly bluer tail . there is too large a scatter in the colour curve to be of much use , but there does not appear to be any evidence of a colour excess . the appearance of the spectrum and the light curves imply that sn 2004a is a normal sn ii - p , suggesting that this excess is not confined to the ranks of the ` faint ' sne . until we have a larger sample of sne ii - p , both ` faint ' and ` normal ' , it is difficult to confirm or rule out this as a characteristic . a comparison between the uvoir light curves of sne 2004a , 2003gd , 1999em , 1999gi and 1997d is shown in fig . [ fig : uvoir ] . all the uvoir light curves , apart from sn 2004a , are from @xcite and a full description of how they were constructed can be found there . the uvoir light curve of sn 2004a , which was constructed in the same way , also confirms the similarities between sn 2004a and the other normal sne ii - p . sn 2004a differs slightly from sn 1999em in the plateau with sn 2004a being marginally fainter , whereas the tail luminosities reflect the comparable nickel mass ejected . we compared the observed properties of sn 2004a with the progenitor mass , presented in section [ sec : prog ] , using equation ( 2 ) of @xcite . the equations of @xcite relate the explosion energy , mass of the envelope expelled ( ) and initial radius of the star just before the explosion , to the observed quantities of sne . @xcite used a sample of 14 supernovae to test these equations and found that they gave reasonable results . equation ( 2 ) of @xcite , is shown here in equation ( [ equ : m ] ) , where is the ejected envelope mass , @xmath15 is the length of the plateau and @xmath16 is the velocity of the photosphere material in the middle of the plateau . @xmath17 if we consider the uvoir light curve in fig . [ fig : uvoir ] , the length of the plateau is @xmath1880d , although it is difficult to be confident as the sn was not observed close to explosion . we therefore estimate the length of the plateau to be @xmath19d , where the positive error is very conservative . using the reddening estimated in section [ sec : red ] , the parameters discussed in section [ sec : scm ] for the scm at 50d post explosion and the distance estimated in section [ sec : d ] , we found that the ejecta mass was @xmath20 . if we assume that the compact remnant plus other mass losses from the system is @xmath21 then it suggests that @xmath22 for sn 2004a , where the upper mass limit of 23 is a hard upper limit as the error in the plateau is very conservative . the lower limit of this is consistent with the progenitor mass , @xmath5 , found from pre - explosion images in section [ sec : prog ] . this result may also suggest that equation ( [ equ : m ] ) overestimates the mass somewhat as the lower limit is only just consistent even though our lower limits are fairly well constrained . the observations of progenitors do not only have important implications for stellar evolution theory , but also for the progenitor models of ` faint ' sne ii - p . there are two very different plausible models for these , one being the low - energy explosion of massive stars ( e.g. * ? ? * ; * ? ? ? * ; * ? ? ? * ) . in this model the collapsing core forms a black hole and a significant amount of fallback of material occurs . an alternative scenario is the low - energy explosion of low - mass stars , presented by who successfully reproduced the observations of sn 1997d with an explosion of @xmath23 ergs and an ejected mass of 6 . the findings of @xcite support the high - mass progenitor scenario . the authors find a bimodal distribution of sne in the nickel mass , plane . a reproduction of the authors fig . 1 ( right ) is shown here in fig . [ fig : zamp ] . we are beginning to populate this figure with direct measurements of the main - sequence mass of the progenitors , as opposed to model dependent values . the most interesting feature of this figure , and the most controversial , is the existence of the faint branch . the sne which populate this branch are proposed to have high initial mass progenitors . the points with direct measurements , or robust limits , of progenitor mass from pre - explosion imaging are highlighted with the filled circles . this figure however should be treated with caution as the theoretical estimates of from @xcite , shown with open circles in the figure , will most probably need to be revised using the value of the opacity from @xcite . the observed nickel masses are estimated from the light curves , and are from @xcite for sn 1999br and from ( * ? ? ? * and references therein ) for sne 2003gd and 1999em . the @xcite nickel masses are however an input parameter of the semi - analytical light curve code and are therefore theoretical . for sn 2003gd is from @xcite , whereas sne 1999em and 1999br are upper mass limits only and are from @xcite and @xcite , respectively . sn 2004a comfortably sits in the ` normal ' branch of the bimodal distribution . even though the mass limit of sn 1999br rules out a high mass progenitor for this ` faint ' sn , it does not necessarily rule out a high mass progenitor scenario for other ` faint ' sne . it does however suggest that for at least some ` faint ' sne a low mass progenitor is likely . with more upper mass limits or direct detections of the progenitor stars of these sne , we should be able to determine the validity of the high mass progenitor scenario . we presented photometric and spectroscopic data of the type ii - p sn 2004a , comparing the _ bvri _ light curves with those of the well observed sn 1999em using a @xmath24-fitting algorithm . this analysis allowed us to estimate an explosion epoch of jd @xmath25 , corresponding to a date of 2004 january 6 . we estimated the extinction of the sn as @xmath26 , using _ acs photometry of the neighbouring stars and confirmed the reddening using the sn s colour evolution . the expansion velocity was measured from our only spectrum , and was found to be comparable to the velocities of similar sne ii - p . this enabled us to extrapolate the velocity evolution of sn 2004a forwards 7 days to find the velocity at a phase of 50d , to be used in the scm distance estimate . three new distances to ngc 6207 were calculated using two different methods , the standard candle method ( scm ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the brightest supergiants method . firstly , using the extrapolated velocity and interpolated _ vi _ magnitudes , we estimated a distance of @xmath27mpc with the scm . we then used _ hst _ ( acs hrc ) photometry to estimate the distances of @xmath1 and @xmath28mpc using two different bsm calibrations . using these three distances and other distances within the literature we estimated an overall distance of @xmath29mpc to ngc 6207 . this distance allowed the nickel mass synthesised in the explosion to be estimated , @xmath30 , comparable to that of sn 1999em . the probable discovery of the progenitor of sn 2004a on pre - explosion _ hst _ wfpc2 images was presented . the star that exploded was likely to have been a rsg with a mass of @xmath5 . if the 4.8@xmath6 detection is not believed then the 5@xmath6 upper limit in the f814w images implies a robust upper mass limit for a rsg progenitor of 12 . this is only the seventh progenitor star of an unambiguous core - collapse sn that has had a direct detection . the first two progenitors discovered were those of sn 1987a , which was a bsg , and sn 1993j , which arose in a massive interacting binary system . neither of these fitted in with the theory that rsgs explode to give type ii sne . however , it appears that the elusive rsg progenitors are now being found , with the discovery of the progenitor of sn 2004a being the third ( after sne 2003gd and 2005cs ) . the observations of sn 2004a were compared to those of other sne ii - p and were found to be consistent with other normal sne ii - p , although there is a small peak in the uvoir light curve at around 40d , which could just be an artifact of the photometry . the sn observations were then compared to the mass of the progenitor using theoretical relationships which relate the mass of the envelope expelled to the observed quantities of sne , and were found also to be consistent . we conclude that sn 2004a was a normal type ii - p sne that arose from the core - collapse induced explosion of a red supergiant of mass @xmath5 . based on observations made with the nasa / esa hubble space telescope , obtained from the data archive at the space telescope science institute . stsci is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . the spectroscopic data presented were obtained at the w.m . keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california and the national aeronautics and space administration . the observatory was made possible by the generous financial support of the w.m . keck foundation . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . some data presented were taken with the liverpool telescope at the observatorio del roque de los muchachos , la palma spain . the authors would like to express their thanks to the research staff at caltech and palomar observatory who made the p60 automation possible . the initial phase of the p60 automation project was funded by a grant from the caltech endowment , with additional support for this work provided by the nsf and nasa . a. gal - yam acknowledges support by nasa through hubble fellowship grant # hst - hf-01158.01-a awarded by stsci , which is operated by aura , inc . , for nasa , under contract nas 5 - 26555 . s. smartt acknowledges funding from pparc and the european science foundation in the forms of advanced and euryi fellowships . m. hendry thanks pparc and queen s university , belfast for their financial support . m. , 2004a , in measuring and modeling the universe , from the carnegie observatories centennial symposia . carnegie observatories astrophysics series . edited by w. l. freedman , 2004 . pasadena : carnegie observatories , the latest version of the standardized candle method for type ii supernovae , astro - ph/0301281 m. , 2004b , in cosmic explosions . on the 10th anniversary of sn 1993j ( iau colloquium 192 ) . springer , heidelberg , p. 535 the standard candle method for type ii supernovae and the hubble constant , astro - ph/0309122 a. , ramina m. , zampieri l. , navasardyan h. , salvo m. , fiaschi m. , 2004a , in cosmic explosions . on the 10th anniversary of sn 1993j ( iau colloquium 192 ) . springer , heidelberg , p. 195 observational properties of type ii plateau supernovae , astro - ph/0310056 l. , ramina m. , pastorello a. , 2004 , in supernovae ( 10 years of 1993j ) , marcaide j. m. , weiler k. w. , eds , proc . 192 , springer - verlag , berlin , in press , understanding type ii supernovae , astro - ph/0310057 | we present a monitoring study of sn 2004a and probable discovery of a progenitor star in pre - explosion _ hst _ images .
the photometric and spectroscopic monitoring of sn 2004a show that it was a normal type ii - p which was discovered in ngc 6207 about two weeks after explosion .
we compare sn 2004a to the similar type ii - p sn 1999em and estimate an explosion epoch of 2004 january 6 .
we also calculate three new distances to ngc 6207 of @xmath0 , @xmath1 and @xmath2mpc .
the former was calculated using the standard candle method ( scm ) for , and the latter two from the brightest supergiants method ( bsm ) .
we combine these three distances with existing kinematic distances , to derive a mean value of . using this distance
we estimate that the ejected nickel mass in the explosion is @xmath3 .
the progenitor of sn 2004a is identified in pre - explosion wfpc2 f814w images with a magnitude of @xmath4 , but is below the detection limit of the f606w images .
we show that this was likely a red supergiant ( rsg ) with a mass of @xmath5 .
the object is detected at 4.7@xmath6 above the background noise . even if this detection is spurious
, the 5@xmath6 upper limit would give a robust upper mass limit of 12 for a rsg progenitor .
these initial masses are very similar to those of two previously identified rsg progenitors of the type ii - p sne 2004gd ( @xmath7 ) and 2005cs ( @xmath5 ) .
[ firstpage ] stars : evolution - supernovae : general - supernovae : individual : sn 2004a - galaxies : individual : ngc 6207 - galaxies : distances and redshifts |
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via imaging , spectroscopy and time - series analysis , the microwave part of the spectrum provides vital information on the properties of flare - accelerated particles and the plasma and the magnetic field in which their emission is formed . although there are considerable complexities in modeling and interpreting the data , microwaves are uniquely rich in diagnostic information and are crucial for flare studies . however , flares are characterised in part by the fact that - for the few minutes of the impulsive phase at least - emission is generated across the entire electromagnetic spectrum . therefore we have the ability to set our microwave observations in context , though in practice the number of flares with excellent multi - wavelength coverage including microwaves remains small . this highlights the ongoing need for continued operation of facilities such as the nobeyama radioheliograph ( norh ) and radio polarimeters ( norp ) in the current era of multi - wavelength observations . in this short paper we will review some multi - wavelength flare observations , focusing on recent results relevant to the flare impulsive phase . these include hints of fine structuring in chromospheric footpoints , very compact footpoint sources , rapid changes in the tilt of the magnetic field during the flare impulsive phase , and hot chromospheric footpoints . in the light of these results we will speculate on what the combination of multi - wavelength and microwave flare data can potentially bring . the ` natural ' partner for microwave emission is hard x - rays , and an extensive review of the strengths of combining radio and hard x - ray data can be found in @xcite . the most directly interpretable signature of non - thermal electrons in solar flares are the non - thermal hard x - rays ( hxrs ) emitted in bremsstrahlung interactions , particularly in the dense plasma of the solar chromosphere . hxr emission from the chromosphere is usually interpreted in terms of the collisional thick target model ( cttm ) , which means that the emission is generated as electrons slow down , under the influence of collisions only , and stop within the target . to interpret a given observation in this way also requires that the slowing - down timescale ( fractions of a second ) is less than the integration time used to make an image or spectrum , which is generally the case . under the assumptions of the collisional thick target model , the total number of electrons that must be injected into the thick target to produce the observed spectrum can be inferred , in a way that does not depend on the precise density structure of the atmosphere . however , with imaging from the ramaty high energy solar spectroscopic imager ( rhessi , * ? ? ? * ) the density structure can be probed in more detail . the systematic offset of source position ( first moment of the source intensity distribution ) as a function of energy when interpreted in the cttm gives a value for the target density scale - height @xcite . the source full - width at half - maximum intensity ( 2nd moment ) as a function of energy can also be measured @xcite , and compared with the predictions of the cttm ; doing so is rather interesting as it is not possible to easily reconcile the modelled and observed behaviour of this quantity . the sources have a much larger observed fwhm than straightforward models of an electron distribution in a ` monolithic ' loop would predict @xcite . this may point to much finer sub - structure in the chromosphere composed of multiple sub - resolution structures of different density scale - heights , as shown in figure [ fig : strands ] , such that the mean source position as a function of energy as expected from the cttm is preserved , but the ` variance ' is increased @xcite . the implication is that chromospheric sub - structures on scales of a tenth of an arcsecond or less could exist . the fine - structuring would presumably have a coronal counterpart . however , the result may also point to problems with the applicability of the cttm . ( 80mm,40mm)kontaretal2010.eps another important result to emerge from rhessi is the inference , using the photosphere s albedo to hxr photons , that the electron angular distribution where the hxr footpoint emission is produced is consistent with one in which as many electrons are traveling ` upwards ' as ` downwards ' @xcite . again this presents a challenge to the cttm . the possibilities of developing such an electron distribution from a combination of scattering ( collisional and non - collisional ) and magnetic mirroring in the chromosphere are still being investigated , but already it is clear that considerable fine - tuning of the electron , field and density parameters is needed to recover the observational results . the ongoing investigations into the causes of these observationally deduced properties of the electron spatial and energy distributions have yet to be concluded , but it is clear that a model of a downward - beamed ` monolithic ' electron beam entering a simple , uniform , collisional chromosphere , is not correct . the electron distribution looks likely to be finely structured in space , and probably also time ( though some average properties can be recovered ) , and may have a complex angular distribution . co - ordinated observations with flares in the optical also make clear that electrons arrive at the chromosphere over small patches . it is not clear that optical footpoints are resolved on a scale of 1@xmath0 @xcite , so the typical optical footpoint is more like @xmath1 , rather than the @xmath2 often used as a ` canonical ' footpoint size ( see section [ sect : optical ] ) . this may have significant implications for the usual ` trap - plus - precipitation ' models used in microwave modeling . [ sect : optical ] optical , or ` white light ' flares , previously thought of primarily as accompanying large flares , are in fact common phenomena , but ill - understood . the flare optical and uv emission contains the majority of the flare s radiation output , and a wealth of spectral lines are available to probe the conditions in the flare chromosphere . for those reasons one would expect this part of the spectrum to have received more attention . however , optical flares are difficult to observe , as they have a low contrast compared to the bright photosphere , and spectroscopic observations require the good fortune to have a spectrometer slit on a flaring kernel at exactly the right time , which has in the past been rare . nonetheless broad - band optical emission is observed in flares from c - class to x - class . the key is to have stable , high - cadence imaging or photometric observations which can be used to perform reliable differencing observations to pick up the faint flare signatures against the bright photospheric background . doing this has revealed that optical footpoints are strongly correlated in space and time with hxr footpoints , and thus with the presence of large numbers of fast electrons @xcite and that optical footpoints are very compact , with areas of around @xmath3 or perhaps less . the energy contained in the white - light continuum is around 70% of the total flare energy , independent of the flare class . the emission mechanism of this broad - band optical radiation is not known . it seems unlikely that it is due to direct heating of the photosphere by electrons accelerated in the corona , as the typical electron energy required to reach the photosphere is around 2 mev , assuming a column mass to the photosphere of 1 gm @xmath4 . an interesting recent analysis by @xcite of a limb flare observed by rhessi and also by one of the stereo spacecraft uses careful triangulation to place both the flare optical emission and the hxr emission at 30 - 50 kev at only 300 km above the photosphere . this corresponds to an electron stopping energy of around 1 mev . ( this single observation has yet to be repeated ) . for the optical luminosity to be produced at or near the photosphere by electrons arriving in a beam from the corona , a large fraction of the electron energy in the injected spectrum would have to be above @xmath5 1 mev this is inconsistent with parameters for the electron distribution derived from hxr measurements . microwave and millimeter observations are informative here , as the emission is generated by electrons in the 100 kev - plus energy range , and it is interesting that these observations suggest that the spectrum may be substantially harder at energies above a few 100 kev ( e.g. * ? ? ? * ) than would be implied by the continuation of the hxr - emitting spectrum . however , this is still not adequate . for example , application of gyrosynchrotron models to microwave emission by @xcite for the large flare sol2002 - 07 - 23t00:35 ( x4.8 ) results in a non - thermal electron density distribution of @xmath6 , corresponding to an electron energy density above @xmath71 mev of @xmath8 if we assume @xmath9 g . this is far too small compared to the energy density of the photospheric plasma ( around @xmath10 ) to produce an observable optical intensity perturbation . for now the observation of @xcite remains a puzzle . another mechanism for producing optical emission is the free - bound continuum that results from the ionisation and recombination in a flare - heated chromosphere ( note , the heating can be , but does not need to be , provided by non - thermal electrons ) . the uv ( balmer and lyman ) continua may also penetrate downwards and backwarm the photosphere , effectively recycling this radiation as optical emission . by indirect means , optically thin emission has been deduced in one flare with an extended white - light ribbon @xcite . this would be consistent with free - bound emission from a hot plasma . the temperature of the emitting plasma should be high enough that the hydrogen is substantially ionised , i.e. above @xmath11k . the brightness temperature of the surrounding non - flaring chromosphere is also in this range , so associated microwave emission would not be visible unless the free - bound emitting plasma were hotter . moreover , in the free - bound model the electron density in the region emitting in the optical may be around @xmath12 @xcite , so any microwaves generated here with frequencies below the corresponding plasma frequency ( @xmath5 28ghz ) will not propagate . although we do not expect to make direct microwave observations of the plasma that radiates the optical emission , the observed optical source properties , coupled with those inferred from hard x - ray spectra , have implications for beam parameters which should be recognised in future microwave modeling . the optical footpoint area is small , and in the event presented by @xcite it is consistent with an unresolved hxr footpoint size . interpreted in the cttm the _ non - thermal _ electron beam density at the location of hxr emission in this event is at least @xmath13 electrons @xmath14 above 20 kev . this is consistent with the value inferred in another x - flare by @xcite from the microwave emission in another large flare , but this time in the corona , where the 17 ghz emission is located . if magnetic mirroring occurs , due to field convergence between the corona and chromosphere ( as one might expect given the increasing evidence for very small footpoints ) then the hxr - generating electrons in the chromosphere would represent only the component that can escape the magnetic trap , giving a lower limit to the overall non - thermal coronal electron density in the coronal loop . on the other hand if the electrons were beamed exactly along the magnetic field then there would be no trapped component and , in a field that diverges into the corona , the coronal beam density requirement would be reduced . but a highly - beamed distribution is at present inconsistent with the angular distributions inferred using photospheric hxr albedo ( section [ sect : hxr ] . ) such high beam densities challenge electron transport models , but may have some bearing on electron spectra relevant to microwave and x - ray comparisons , because beam - return current instabilities can substantially distort the electron spectrum . unless the background plasma through which the beam passes is much denser than the beam , such that the return current speed is low , the beam - return current system will be subject to plasma instabilities causing the majority of its energy to be dissipated as electron and ion heating , via wave generation . analytical considerations suggest that a beam with even a fraction of the density implied by the combination of x - ray and optical observations should , together with its return current system become unstable to the ion - acoustic instability ( when its return - current speed is greater than the ion - acoustic speed ) , with the beam energy redistributed in heating , unless the ambient density is much larger than the beam density @xcite . numerical simulations of the beam - return current system are now very elaborate , including vlasov and pic simulations , in magnetised and non - magnetised scenarios . for example , work by @xcite and indicates that the majority of the beam energy - around 70% - is redistributed as plasma heating , but that the remaining fraction may be available to re - accelerate higher energy electrons . this is interesting for the comparison between hxr and microwave radiation , since it suggests that the lower energy x - ray generating electrons and the higher energy microwave - emitting electrons need not be described by one spectral index . a number of joint studies of hxr and microwave flare conclude that there is a substantial difference in the electron spectral indices at low energy and high energy ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , where microwave emission interpreted as optically - thin gyrosynchrotron radiation implies that electron spectra at high energies ( above @xmath15kev ) are substantially flatter than at low energies . ( 80mm,80mm)petriesudolfig.eps the impulsive phase of a solar flare has detectable consequences for the low solar atmosphere , i.e. the photosphere , apart from the possible photospheric origin for optical flares . it is now well established that significant abrupt ( step - like ) , non - reversing change in the line - of - sight photospheric field occurs for major ( x - class and m - class ) flares , co - temporal with the flare impulsive phase ( e.g. * ? ? ? * ; * ? ? ? an example of such a change is shown in figure [ fig : petriesudol ] . the location of this change can be in the umbra , the penumbra or elsewhere in the active region , and is co - spatial with increases in uv footpoint intensity @xcite . the onset of the goes soft x - ray emission leads the field changes @xcite as do peaks in the uv intensity - by on average 4 minutes @xcite . as remarked on by these authors , the timing pattern is consistent with the flare causing the photospheric field changes , and not vice versa . the hxr footpoints are associated with some , but not all , locations ( e.g. * ? ? ? * ; * ? ? ? * ) , though no systematic study of this has yet been carried out . the change of the line - of - sight field is taken as a sign that the coronal magnetic field is reconfigured by the flare , as shear relaxes or the field shrinks , and that this reconfiguration propagates through the atmosphere to the field s anchor points in the photosphere . the tendency is for the field to become more horizontal @xcite . ( 80mm,80mm)tilt_angle_newcolour.eps the change in the field direction over the duration of the flare impulsive phase corresponds of course to a variation in the viewing angle . since gyrosynchrotron emission from the high - energy non - thermal electrons which are also present during the flare impulsive phase is anisotropic this variation in the viewing angle will influence the microwave emission observed . we do not know of any comparisons yet being made between field changes and variations in the microwave , and indeed it might be difficult to disentangle variations due to changes in the magnetic field direction from those due to variations in the intrinsic properties of the population of emitting electrons . however , we can anticipate the effects . in figure [ fig : tilt ] we show the variation in the microwave spectrum as the viewing angle changes while all other parameters of the source stay the same . vertical lines on this plot show the norh observing frequencies . the emission is calculated for a non - thermal electron density of @xmath16 and electron spectral index of 3 in a compact source of diameter 5 " , thickness 5000 km , and temperature 5mk in a magnetic field of 500 g and ambient density of @xmath17 . this could correspond to a gyrosynchrotron source near the footpoints of a coronal loop , or flare - heated upper chromosphere ( see section [ sect : hotfp ] ) . the three curves correspond to the same field strength , viewed at an angle to the field direction of 20@xmath18 , 40@xmath18 and 80@xmath18 . a decrease in the line - of - sight component of the field strength caused by an increasing field tilt leads to higher microwave intensity and an increase in the peak frequency . it would be interesting to search for a systematic effect such as this in the data , but in any such effort the many other parameters affecting the microwave spectrum must also be accounted for . [ sect : hotfp ] ( 90mm,80mm)dems.eps the chromosphere in solar flares is strongly heated . this is readily seen in e.g. euv images of flare ribbons which indicate plasma of at least a million degrees . however , it has been known at least since the days of _ yohkoh _ , though not widely appreciated , that more extreme plasmas exist in the chromospheric footpoints during the flare impulsive phase . impulsive soft x - ray footpoints observed via their bremsstrahlung emission by @xcite and , in a large sample by show temperatures close to 10 mk , and densities of at least a few times @xmath19 ( depending on assumptions about their size ) . using euv spectroscopy of flare footpoints from _ @xcite have determined the emission measure distribution for impulsive phase footpoints in a number of small ( b- and c - class ) events , and these typically also peak at 10 mk . an example of a footpoint emission measure distribution for a c1.1 flare is shown in figure [ fig : fpems ] , compared with the loop emission measure distribution from the same time in this event , and non - flaring active region emission measure distribution . independently , density diagnostics of this event return values of around @xmath20 at a temperature of 1.8 mk @xcite . direct density diagnostics for higher temperatures were not available . the gradient of @xmath21 for the footpoint is 1 in all cases studied ; we note that this is consistent with conductive heating balanced by radiative cooling @xcite . the consequences of these hot , dense compact footpoints for microwave footpoint emission have not been explored . in the usual microwave flare modelling , the characteristics of the _ coronal _ flaring source are carefully studied , e.g. the inhomogeneities of the coronal field or the effects of pitch - angle distribution of the electrons on the emission ( e.g. * ? ? ? but the modelling assumes that footpoints are rather cool as well as dense , and microwave emission will thus be free - free absorbed . the hot footpoint plasmas that we find are essentially at ` coronal ' temperatures but located at chromospheric altitudes and with density and magnetic field strength higher than typically found in the corona . they would be expected to produce intense compact sources , with spectral properties similar to those computed for coronal loops , and dominating any coronal emission in their optically - thin ranges due to higher densities and fields . the high plasma densities might however lead to razin suppression at low frequencies . in figure [ fig : hotfps ] we show calculations of footpoint emission for different temperatures and densities in the ranges deduced from the euv observations . of course , the observations also suggest that the hot footpoint plasma would be very inhomogeneous , with temperature varying by a factor 10 over a distance of probably 1000 km or so . so these spectra are for the moment only indicative . the parameters used in these calculations are : field strength of 100 , 500 or 1000 g ( magenta , green and blue curves , respectively ) and viewing angle of 45@xmath18 , isotropic electron distribution with flux spectral index @xmath22 , having a minimum electron energy of 200 kev and a maximum energy of 5 mev . the source angular diameter is 5@xmath0 and depth along the line of sight is 2@xmath0 , comparable to the depth of the chromosphere . the non - thermal electron density in the footpoint is around one part in @xmath23 of the background thermal density , or @xmath24 . ( 80mm,80mm)mw2_v2.eps for the temperatures @xmath25k and @xmath26k ( shown in figure [ fig : hotfps ] by continuous and dashed curves ) and densities @xmath27 and @xmath20 ( shown by thin and thick lines ) found from euv and soft x - rays , observable footpoint microwave sources in the norp frequency ranges are predicted . the emission is mostly non - thermal gyrosynchrotron , and source intensity is determined mainly by the magnetic field strength , where stronger fields shift the spectrum peak towards higher frequencies with stronger emission ( e.g. * ? ? ? * ) . the contribution of the non - thermals to the microwave spectrum as shown in figure [ fig : hotfps ] is greatly in excess of the thermal gyrosynchrotron which would be expected from these same plasma parameters . for a given field strength the intensity in the norh range , and norp above 3.8 ghz , is rather insensitive to the different values of density and temperature used here . the exception is in the weak - field case where the emission is affected by free - free absorption and razin suppression . the razin suppression ( e.g. * ? ? ? * ) is significant for microwave frequencies below the razin frequency @xmath28 , which is below 5 ghz for all cases considered , except for @xmath29 g and @xmath30 , where @xmath31 ghz . the resulting spectra are thus strongly suppressed , and show slighly different contributions from that produced by the free - free mechanism , for @xmath32k or @xmath26k . the steep emission decrease towards lower frequencies is caused by absorption below the plasma frequency ( around 1 ghz and 3 ghz for densities of @xmath27 and @xmath20 , respectively ) . imaging spectropic analysis of future observations with e - ovsa , coupled with euv , uv and optical observations , will provide an excellent diagnostic tool for deriving the magnetic field , plasma density and temperature in chromospheric flaring regions . we should also note that the recently discovered sub - thz spectral component above 100 ghz @xcite is likely to be generated in the chromosphere , when considering the proposed radiation mechanisms @xcite . in terms of gyrosynchrotron emission , it can be shown that the second spectral peak can be produced in hot and dense footpoints with strong magnetic fields and strong razin suppression , self - consistently with the typical microwave spectrum generated in the coronal source @xcite . observational understanding of the energetically dominant processes in the flare lower atmosphere during the impulsive phase , drawing on the many space- and ground - based instruments currently observing the sun , is developing rapidly and in some unexpected directions . the new results presented here on the plasma and field parameters in the chromosphere during the flare impulsive phase are important for future microwave modelling , and the multi - wavelength data that we now have at our disposal must also be confronted with ongoing microwave imaging and spectra observations , which provides unique diagnostics of both plasma and field . to this end , the continued operation of the nobeyama radioheliograph and radio polarimeters remain crucial for our exploration and understanding of flares . l.f . is very grateful to the conference organisers for the financial support which allowed her participation in this most simulating meeting . this work was supported by stfc grant st / i001808/1 and by ec - funded fp7 project hespe ( fp7 - 2010-space-1 - 263086 ) . | this short paper reviews several recent key observations of the processes occurring in the lower atmosphere ( chromosphere and photosphere ) during flares .
these are : evidence for compact and fragmentary structure in the flare chromosphere , the conditions in optical flare footpoints , step - like variations in the magnetic field during the flare impulsive phase , and hot , dense ` chromospheric ' footpoints .
the implications of these observations for microwaves are also discussed . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
clusters of galaxies are the largest gravitationally bound and collapsed systems in the universe . they are filled with x - ray emitted hot gas with the temperature of @xmath010 kev . high - resolution x - ray observations have revealed that the hot gas in many cluster cores is not smoothly distributed ( e.g. * ? ? ? * ; * ? ? ? , we define the cluster cores as the dense regions within @xmath1300 kpc from the cluster centers . the complicated structures are often attributed to activities of active galactic nuclei ( agns ) . in fact , bubbles of high energy particles have been found in some clusters ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which is the evidence that agns affect the surrounding hot gas . however , the complicated structures have also been observed in clusters in which agns are not active at the centers ( e.g. * ? ? ? * ) . in the outer regions of the cores , edge - shaped discontinuities in the gas density and temperature ( ` cold fronts ' ) are often found ( e.g. * ? ? ? they may be attributed to the ` sloshing ' of the gas in the cluster gravitational potential well , although the origin remains an open question @xcite . from the above observations , one may think of complex gas motion , that is , turbulence in the cores . actually , turbulence is expected to be prevailing in cluster cores . although it has not yet directly been observed in x - rays ( but see * ? ? ? * ) , cold gas ( @xmath2 k ) moving with the velocity of 100@xmath3 has been observed . if the ambient x - ray gas did not move with the cold gas , the latter would immediately mix with the former @xcite . moreover , the turbulence may be playing an important role in heating of the cluster cores . it had been expected that the gas in cluster cores cools by radiating away its thermal energy , which induces gas motion toward the cluster centers ( cooling flows ; * ? ? ? however , x - ray spectra have revealed that the mass of the cooling gas is far smaller than that expected from this model @xcite . the turbulence may transport thermal energy from the outside of the cluster cores , and may balance the radiative cooling of the cores @xcite . however , the actual mechanism that creates the turbulence has not been understood . the scenario we newly propose here is that the core turbulence is created by bulk gas motions , which are naturally produced in the x - ray hot gas in clusters . cosmological numerical simulations have shown that clusters are knots of larger - scale filaments in the universe @xcite . along the direction of the filaments , small clusters ( or galaxies ) successively fall into a cluster , which increases the cluster mass . the simulations have also shown that the infall velocities of the smaller clusters are more than @xmath3 @xcite . the interactions of the hot gas and dark matter between the clusters should produce a large - scale gas motion ; the velocity is typically @xmath4 and sometimes reaches @xmath5 @xcite . the cluster cores are exposed to these violent gas motions ; this would create possible turbulence and the complicated structures observed in the cores even if there are no agn activities . in order to test this scenario , we performed two - dimensional high - resolution hydrodynamic simulations to follow the long - term evolution of the core in ` a stormy cluster ' for the first time . since we are interested in the cluster core and we need resolution high enough to resolve turbulence , we limited the calculations to the central region of a cluster ( within @xmath6 kpc from the cluster center ) . these simulations were performed using a nested grid code @xcite , and the coordinates are represented by @xmath7 . while the resolution near the outer boundary is @xmath8 kpc , that at the cluster center is @xmath9 pc . free boundaries were chosen . thermal conduction , viscosity , magnetic fields , and the self - gravity of gas were ignored . radiative cooling is included . we adopted a cooling function for the metal abundance of 0.3 solar . the gas is isothermal and the temperature is 7 kev at @xmath10 . the fixed gravitational potential and the initial gas distribution are the same as those in @xcite . we approximated the bulk gas motions in a cluster by plane wave - like velocity perturbations represented by @xmath11 at @xmath12 kpc , where @xmath13 is the initial sound velocity , and @xmath14 is the wave length . this assumption comes from the idea that continuous infall of matter and small clusters along a large - scale filament should create velocity perturbations in a particular direction . the factor @xmath15 is a free parameter and @xmath16 is the cluster center . we studied the perturbations with @xmath15 and @xmath14 shown in table 1 . note that cosmological numerical simulations suggested that the velocity of the hot gas is about 0.2@xmath17 or larger even when a cluster is relatively relaxed @xcite . moreover , it is expected that the scale of bulk gas motion in a cluster is @xmath18 kpc or larger @xcite . therefore , we think that the parameters we took are reasonable . however , since the large bulk motions in a cluster do not always generate large amplitude acoustic - gravity waves , our model may be appropriate for clusters undergoing hierarchical structure formation or similar violent events . since the energy input rate through waves per unit area is given by @xmath19 , where @xmath20 is the gas density , the integrated input rate for @xmath21 kpc at @xmath12 kpc is @xmath22 for our model cluster . on the other hand , at @xmath10 , the x - ray luminosity of the gas for @xmath23 kpc and @xmath24 kpc is @xmath25 and is comparable to the wave energy input rate . in figures [ fig1 ] and [ fig2 ] , we present the temperature distributions for @xmath26 and @xmath27 kpc ; the velocity perturbations propagate upwards as acoustic - gravity waves , which were called ` tsunamis ' in @xcite . since the velocity amplitude is relatively large , the waves steepen and become weak shocks as shown in one - dimensional simulations @xcite . because of the pressure coming from the momentum of the waves , the coolest and densest gas noticeably shifts from the cluster center at @xmath28 gyr and the shift is clearly seen at @xmath29 gyr ( fig . [ fig1 ] ) . while the position of the coolest gas is at @xmath30 kpc ( fig . [ fig1 ] ) , that of the densest gas is @xmath31 kpc . the difference of the positions is made by the wave just passed ( fig . [ fig1 ] ) ; the gas slightly deviates from pressure equilibrium there . comparing the result of @xmath26 with that of @xmath32 , we found that the gravitational energy of the gas for the former is larger than that for the latter by @xmath33 ergs for @xmath21 kpc and @xmath34 kpc at @xmath29 gyr . the energy increase is smaller than the energy input by waves during the first one gyr for @xmath21 kpc ( @xmath35 ergs ) , suggesting that a part of the injected energy is used to produce the shift of dense gas . after that , radiative cooling proceeds and the core region becomes cooler and denser . since waves no longer sustain the cooling core , the core falls in the potential well of the cluster . during the fall , rayleigh - taylor ( rt ) and kelvin - helmholtz ( kh ) instabilities develop around the core . they non - linearly develop , and turbulent motion is eventually formed in and around the core as seen in figure [ fig2 ] , which shows the temperature distribution at @xmath36 gyr . the cool core oscillates around the cluster center , and temperature jumps are formed ( fig . [ fig2]a ) . associated with these temperature jumps , gas density also has discontinuities . small cool and dense blobs randomly moving in the core cause new rt and kh instabilities around them , and smaller eddies are generated ( fig . [ fig2]b ) . for the models of @xmath26 and 0.5 , the turbulence is maintained until our calculations are stopped at @xmath37 gyr . it is interesting that the turbulence presented here is naturally caused even by the regular , wave - like , linear perturbations . no irregular triggers , which are usually essential for initiating or maintaining turbulence , are necessary . in table 1 , we present the time when the gas temperature in any of the numerical grid points reaches zero ( @xmath38 ) . the gas cooling is suppressed by heat transport through turbulent mixing , especially when @xmath15 is large . in other words , in addition to the wave input energy ( see [ sec : models ] ) , the thermal energy transported from the outside of the core is used to prevent the core from cooling . we note that although there were several studies treating core heating by acoustic - gravity waves , the mechanism of the heating presented here is completely different from that proposed in the previous studies . in those previous studies , based on the analytical approach @xcite or one - dimensional numerical simulations @xcite , it was predicted that weak shocks evolved from acoustic - gravity waves directly heat the cluster core , while in this study the turbulence is the major player of the heating . since our simulations are multi - dimensional , the results should be much more realistic than those of the previous studies . in fact , since turbulence is essentially multi - dimensional , those previous studies were not able to directly treat the turbulence and complicated structures of the cores . even with multi - dimensional simulations , the turbulent heating could not be found if the resolution were not high enough . the present results show that the turbulence starts to develop after the core becomes dense through cooling . before that , waves pass the cluster center without much changing the gas structure . thus , we predict that this mechanism of turbulence generation works only for clusters with dense and cool cores , which had been called ` cooling flow clusters ' . since the turbulence is spatially limited to the cool core , it does not totally mix the hot gas in a cluster . thus , as long as violent mergers of clusters with comparable masses , which completely destroy the central gas structures of the clusters , do not happen , the metal abundance excess observed in cluster cores ( e.g. * ? ? ? * ) would not completely be erased . because of the turbulent motion , the fine structures of the core are not steady . our simulation results sometimes show fine structures similar to the peculiar structures observed in clusters such as a1795 , centaurus , and 2a @xmath39 @xcite . the temperature jumps seen in figure [ fig2]a may correspond to the ` cold fronts ' observed in some clusters . the bulk gas motion in clusters would result in the formation of acoustic - gravity waves and the weak shocks as shown above . direct observations of the waves may be difficult unless the wave fronts are almost parallel to the line of sight . in a133 , however , a weak shock just passing through the core has been observed @xcite . the maximum velocity of the turbulence in a cluster core is @xmath40 for @xmath26 . with a high spectral resolution detector like the x - ray satellite _ astro - e2 _ , the turbulence in cluster cores could be detected in the near future . if turbulence is being developed , the metal lines in the x - ray spectra would have very complicated features owing to the gas motion @xcite . on the other hand , turbulence could also be created by agn activities , especially by the buoyant motion of agn - origin bubbles . the lifetime of the eddies associated with the bubble motion is @xmath41 , where @xmath42 and @xmath43 are the size and velocity of a bubble , respectively . for the virgo cluster , for example , the size of the observed bubbles is @xmath44 kpc , and the predicted velocity of a bubble is @xmath45 @xcite . thus , the lifetime of the eddies is @xmath46 yr , which is much shorter than the lifetime of the bubble itself ( @xmath47 yr ; * ? ? ? this means that turbulence is unlikely to exist in a cluster core without agn - origin bubbles . thus , if turbulence is detected in such a core , it could be associated with the bulk gas motions outside of the core . our model predicts that cooling of a cluster core is more suppressed for larger velocity amplitude . the suppression should also work in smaller objects such as groups of galaxies and elliptical galaxies because their overall structures are similar to those of clusters of galaxies @xcite , and we expect that the bulk gas motions and turbulence are excited in them by the same mechanism presented here . this is in contrast with the suppression by thermal conduction that does not work in the smaller objects because of their low temperatures @xcite . finally , we note the limitations of our model . first , we approximated the bulk gas motions by waves with constant amplitude and length , and direction for propagation is also assumed to be fixed , but in reality all these conditions should be changed during formation and evolution of clusters . even bulk gas motions that are not represented by regular waves considered here could cause the motion of the cool cores and produce turbulence through rh and kt instabilities . moreover , it is likely that waves come from several directions corresponding to large - scale filaments . these effects should be ultimately studied by fully three - dimensional , ultra - high - resolution cosmological simulations , although the basic mechanism initiating the turbulence would not be different from the one in the two - dimensional case here . second , we did not include the heating by turbulent dissipation , agns , and thermal conduction . if they are effective as thermal energy sources , the wave energy required to suppress the cooling of a core would be smaller than that we predicted . note that the turbulence may tangle magnetic fields and reduce an effective conduction rate , however , @xcite indicated that chaotic magnetic fields do not much reduce the conduction rate . thus , we may need to follow the evolution of magnetic field lines when we include the effect of thermal conduction . third , we fixed the gravitational potential of a cluster . a change in the potential may also shift the cool gas core from the gravitational center of the cluster , which may lead to the motion of the core and the development of turbulence @xcite . fourth , we assumed a two - dimensional axi - symmetric geometry . this might affect development and structure of turbulence . it would be expected that cascade processes of eddies is different in full three - dimensional turbulence . interaction between the turbulence and plane - waves in three - dimensions should be also clarified in future simulations . we thank the anonymous referee for useful suggestions . the authors were supported in part by a grant - in - aid from the ministry of education , culture , sports , science , and technology of japan ( y. f. : 14740175 ; t. m. : 14740134 ; k. w. : 15684003 ) . all simulations were run on fujitsu vpp5000 at naoj . | based on high - resolution two - dimensional hydrodynamic simulations , we show that the bulk gas motions in a cluster of galaxies , which are naturally expected during the process of hierarchical structure formation of the universe , have a serous impact on the core .
we found that the bulk gas motions represented by acoustic - gravity waves create local but strong turbulence , which reproduces the complicated x - ray structures recently observed in cluster cores .
moreover , if the wave amplitude is large enough , they can suppress the radiative cooling of the cores .
contrary to the previous studies , the heating is operated by the turbulence , not weak shocks .
the turbulence could be detected in near - future space x - ray missions such as _ astro - e2_. |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the spontaneous appearance of cooperative motions called dynamic heterogeneity ( dh ) when liquids are supercooled @xcite has been extensively studied during the last decade , as scientists were searching for a cooperative mechanism at the origin of the glass - transition . despite large efforts in that direction of research however , the origin of the dynamic heterogeneity and the effect of these cooperative motions on the liquid dynamics still remain elusive . it is widely thought that dhs appear due to the presence of some defects inside the low temperature liquid , that could be structural ( a local packing fluctuation ) or purely dynamical ( excitations ) . the resulting excitations ( arising directly or from structural defects ) then grow due to facilitation@xcite processes , eventually disappearing at large timescales due to thermal diffusion . in 2004 , asaph widmer - cooper and coworkers@xcite showed that dhs appear preferentially in some regions of the liquid , a result that strongly suggests a connection between the local structure and the dhs . in 2007 , laura kaufman s group @xcite studied the effect of probes larger than the medium molecules on the dynamics of supercooled liquids . they found that the dynamics slows down or accelerates depending on the probe s roughness , while the dhs increase in both cases , and concluded that probes can promote dynamic heterogeneities . in 2009 , we found that dynamical defects created by probes isomerizing inside the medium can strongly promote dhs@xcite while accelerating the dynamics , results that were confirmed by experiments and simulations@xcite . following these works , results on colloids@xcite , and theoretical findings@xcite , in order to test the hypothesis of a packing fluctuation origin of the dhs , we investigate in this paper the effect of small structural defects on the dynamics of a model supercooled liquid and on the strength of dynamic heterogeneity . we focus our work on probes of sizes comparable with the size of the medium s molecules in order to model small cages fluctuations . we also focus on the possibility for some probes to destroy the dynamic heterogeneities . in agreement with the results of kaufman s group we find that large probes promote the dhs in our liquid , however we also find that small probes can destroy the heterogeneities . in this work we use molecular dynamics simulations@xcite ( md ) to simulate the dynamics of a fragile@xcite supercooled liquid when small or large probe molecules are diluted inside . this simulation method permits to gain information on the motion of each molecule of the medium provided that the interatomic potentials are known with enough accuracy , and is thus an invaluable tool to unravel condensed matter physics phenomena@xcite at the microscopic level . we model the molecules@xcite of the medium as constituted of two atoms ( @xmath0 ) that we rigidly bond fixing the interatomic distance to @xmath1@xmath2 . each atom of our linear molecule has a mass @xmath3 g/@xmath4 where @xmath4 is the avogadro number . atoms of the set of linear molecules constituting our liquid interact with the following lennard - jones potentials : @xmath5 with the parameters : @xmath6 kj / mol , @xmath7 kj / mol , @xmath8 and @xmath9@xmath2 . our mean molecule is thus @xmath10@xmath2 long and @xmath11@xmath2 wide . with these parameters the liquid does not crystallize when supercooled even during long simulation runs@xcite . this model has been described and studied in detail previously@xcite and was found to display the typical behaviors of fragile supercooled liquids . as they are modeled with lennard - jones atoms , the potentials are quite versatile . due to that property , a shift in the parameters @xmath12 will shift all the temperatures by the same amount , including the glass - transition temperature and the melting temperature of the material . we add to that medium @xmath13 smaller or larger molecules ( the probes ) intended to create small localized perturbations in the medium structure . we have @xmath14 probe molecules diluted within @xmath15 medium molecules inside the simulation box . this number of molecules lead to a simulation box size of @xmath16 , that insures the disappearance of size effects@xcite in our liquid at the temperature of study . the probes are similar diatomic molecules defined by the parameters : @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21@xmath2 , @xmath22 , @xmath23 and @xmath24 . the density is set constant in our calculations at @xmath25 . when rescaled , or in dimensionless units , that density value is larger than the density of the original model@xcite , and thus leads to a more viscous medium . we use the gear algorithm with the quaternion method@xcite to solve the equations of motions with a time step @xmath26 @xmath27 above @xmath28 and @xmath29 @xmath27 below that temperature . the temperature is controlled using a berendsen thermostat@xcite . to prevent aging processes we equilibrate the supercooled liquid during @xmath30 ns , a time much larger than the @xmath31 relaxation time at the lower temperature studied , before recording the simulation results . we use periodic boundary conditions . we expect slower motions around the large inclusions and more rapid motions around the small inclusions . this modification of the molecular motions around the defects will result in a local heterogeneity of the mobilities , that may induce or destroy dynamic heterogeneities if facilitation mechanisms are present in the system . we thus expect a modification of the liquid s dynamic heterogeneity around the inclusions . ( color online ) radial distribution function @xmath32 between the centers of masses ( com ) of the medium molecules ( dashed blue line ) and between the average molecules and the defects ( continuous red line ) . ( a ) large defects ; ( b ) small defects . the @xmath32 between the average molecules does nt change in our simulations when defects are included , due to the small proportion of defects . the temperature is t=120k . + _ due to the small percentage of probes , we do not find any modification of the mean structure of the liquid when we replace a few ( @xmath13 ) molecules of the medium by larger or smaller probe molecules . the local structure is however modified around the probe as shown in figure 1a and 1b . figure 1 suggests that the cage is smaller around the large probes and slightly larger around the small probe , as the first peak of the radial distribution function is shifted to a smaller distance for large probes than for medium molecules . the figure also shows that the local structure around the probe is only slightly modified when the small probes are used . ( color online ) radial distribution function @xmath32 between the centers of masses ( com ) of the medium molecules and of the defects ( continuous red line ) at a temperature t=120k ( continuous red line ) and t=165k ( dashed green line ) . ( a ) large defects ; ( b ) small defects . + _ figure 2 shows that the temperature modifies the structure around the large probes , increasing the probability of unfavored geometrical configurations . by contrast , for small probes the structure remains the same around the probes at low and high temperature . however figures 3 and 4 show that both probes modify the dynamics . this modification of the dynamics begins around @xmath33 and increases when the temperature drops . _ ( color online ) mean square displacement of ( inset ) the defects or ( main figure ) the medium molecules , when the defects included are : small ( blue dashed line ) , large ( continuous red line ) , or we are in a pure liquid ( dotted green line).t=120k . + _ because the cages around the large probes are smaller than the medium s cages , and these probes are larger than the medium s molecules , we expect a decrease of the free volume around large probes . the decrease of the free volume around large probes then results in a decrease of the diffusion around these probes . similarly we expect an increase in the free volume around small probes resulting in an acceleration of the diffusive motions . to verify that picture we evaluate the free volume @xmath34 around the probe from the height of the plateau of the mean square displacement of the probe ( see the inset of figure 3 ) . if @xmath35 is the plateau s height , we have @xmath36 . figure 3 shows that @xmath37 ( @xmath38@xmath39 , @xmath40@xmath39 , @xmath41@xmath39 ) leading to the same relation for the free volumes , with @xmath42@xmath43@xmath44 and @xmath45@xmath44 . the free volume for small probes is thus larger than for the mean medium molecules , leading to the increase of the diffusion for small probes that we observe in the inset of figure 3 . as a result the inclusion of small or large molecules respectively increase or decrease the mean square displacements around the perturbation . the defects thus slow down or accelerate the dynamics around them for large or small inclusions respectively . the effect of these modifications of the local dynamics around the probes results in a modification of the whole dynamics as shown in figure 4 . we see on the figure that the global diffusion coefficient follows the same trend and decreases for large probes ( or increase for small ones ) . we note that the modifications of the msds appear at the end of the plateau regime , around the characteristic time @xmath46 for which the cooperative motions are maximum , however the probes behave differently as shown in the inset of figure 3 . for the probes the msds change at the beginning of the plateau . we also note that this effect is temperature dependent and disappears at high temperature as shown in figure 4 . the fragility , seen here as the diffusion coefficient variation from the pure exponential arrhenius law @xmath47 , increases when large probes are included and decreases for small probes . as the increase of the activation energy when the temperature drops is usually explained by the increase of cooperative motions , that result suggests that the cooperativity increases for large probes and decreases for small ones . this result suggests that the cooperative motions increase when large probes are used and decreases for small probes . we will see below that the dhs follow that behavior . ( color online ) diffusion coefficient ( @xmath48 ) for the medium molecules , versus temperature when small or large probes are included , compared with the diffusion coefficient of the pure liquid in the same conditions . from top to bottom : large probes ( red solid circles ) , pure medium ( green solid circles ) , and small probes ( blue solid circles ) . the medium molecules diffuse less when the probes are large . ( color online ) maximum value of the dynamic susceptibility @xmath49 for the medium molecules , versus temperature when small or large probes are included , compared with values from the pure liquid in the same conditions . the function @xmath50 reaches its maximum for @xmath51 . from top to bottom : large probes ( red solid circles ) , pure medium ( green solid circles ) , and small probes ( blue solid circles ) . the dynamic susceptibility is normalized with its maximum value for the pure liquid at t=200k ( @xmath52 ) . + _ _ fig.6 . ( color online ) dynamic susceptibility @xmath50 for the medium molecules , versus time when small or large probes are included , compared with the dynamic susceptibility of the pure liquid in the same conditions . from top to bottom : large probes ( red continuous line ) , pure medium ( green dotted line ) , and small probes ( blue dashed line ) . t=120 k. the dynamic susceptibility is normalized with its maximum value for the pure liquid at t=200k ( @xmath52 ) . + _ the 4-point dynamic susceptibility@xcite measures the autocorrelation of the fluctuations of the mobility and as such is an efficient measure of the dynamic heterogeneity . we calculate the dynamic susceptibility @xmath53 from the equation@xcite : + @xmath54 with @xmath55 in these equations , @xmath56 denotes the volume of the simulation box , @xmath57 denotes the number of molecules in the box , and @xmath58 . also , the symbol @xmath59 stands for a discrete mobility window function , @xmath60 , taking the values @xmath61 for @xmath62 and zero otherwise . we use the value @xmath63@xmath2 in this work , which maximizes @xmath53 in our liquid at the density of the study . note that some caution must be taken if small values of @xmath64 are chosen in the calculation ( that may probe vibrational motions inside the cages instead of cage escaping motions ) while larger values of the parameter @xmath64 lead to qualitatively similar results than with the parameter we chose . 0.5 cm figures 5 and 6 show that the dynamic susceptibility is larger when we add large probes inside the medium , and smaller when we add small probes . as the dynamic susceptibility measures the dhs , these results show that the dhs increase or decrease depending on the defects we add to the medium . these results strongly suggest that the defects associated with the small molecules destroy the dhs , while the large molecules defects facilitate the dhs . when we include defects ( probes ) inside the medium , figure 5 shows that for both sort of probes the dhs modification from the pure medium values , increases when the temperature drops . adding large probes to the medium , results in an increase of the dhs while the dynamics slow down , following the same trend than the inverse of the diffusion coefficient in figure 4 . ( color online ) maximum value of the non gaussian parameter @xmath65 for the medium molecules , versus temperature when small or large probes are included , compared with values from the pure liquid in the same conditions . from top to bottom : large probes ( red solid circles ) , pure medium ( green solid circles ) , and small probes ( blue solid circles ) . the non gaussian parameter is normalized with its maximum value for the pure liquid at t=200k ( @xmath66 ) . + _ ( color online ) non gaussian parameter @xmath67 for the medium molecules , versus time when small or large probes are included , compared with the non gaussian parameter of the pure liquid in the same conditions . from top to bottom : large probes ( red continuous line ) , pure medium ( green dotted line ) , and small probes ( blue dashed line ) . t=120 k. the non gaussian parameter is normalized with its maximum value for the pure liquid at t=200k ( @xmath66 ) . + _ we display in figure 7 the non gaussian parameter ( ngp ) @xmath68 evolution with temperature , and in figure 8 its time variation at a temperature t=120k . the non - gaussian parameter @xmath67 is defined as : @xmath69 the ngp @xmath67 measures the variations from the gaussian shape of the self van hove correlation function , that is predicted by brownian motion . as one of the signature of the dynamic heterogeneity is the appearance of a tail in the van hove originating from fast cooperative motions , @xmath67 measures the dynamic heterogeneity . we see in figures 7 and 8 an evolution of @xmath68 with temperature and with the probe that is qualitatively similar than the evolution we have observed for the dynamic susceptibility . at high temperature the probes have no effect on the dh and the circles merge in figures 7 and 5 for @xmath70 . then , as the temperature decreases the different probes lead to increasingly different dhs . the small probes decrease the dh while the large probes increase them . we observe a smaller variation of the non gaussian parameter than of the dynamic susceptibility when decreasing the temperature in our liquid . note that while the characteristic time for the susceptibility depends on the choice of the parameter @xmath64 , the characteristic time of the ngp @xmath46 is an important characteristic of the supercooled liquid at the temperature of study . however we observe in both cases ( figures 6 and 8) that the maximum value of the ngp and of the susceptibility are shifted to larger times for large probes and to smaller times for small probes . _ ( color online ) inverse of the diffusion coefficient @xmath71 for the medium molecules normalized with the diffusion coefficient @xmath72 of the pure liquid in the same conditions , versus the relative size @xmath73 of the probe . t=120k . the red ( dark ) circle corresponds to the pur medium . this point was obtained with high accuracy . + _ _ fig.10 . ( color online ) maximum value of the non gaussian parameter @xmath65 for the medium molecules , normalized with the non gaussian parameter of the pure liquid @xmath66 at the same temperature t=120k , versus the relative size @xmath73 of the probe . the red ( dark ) circle corresponds to the pur medium . this point was obtained with high accuracy . note that the temperature for the normalization ( t=120 k ) is here different from that of the normalization in figures 7 and 8 ( t=200k ) . + _ we have studied the effect of two particular probes ( a small probe and a large one ) diluted inside a supercooled medium . we will now show that the effects observed are not qualitatively particular to the probes we chose previously . for that purpose we will now vary the probe size and study the resulting effects on the dynamics and heterogeneity . in the following , we use a probe that is similar to the medium molecule but with a size that has been multiplied by the factor @xmath73 ( i.e. @xmath74 and @xmath75 ) . we show in figures 9 and 10 the evolution of the diffusion coefficient @xmath48 and of the non gaussian parameter @xmath68 with the relative size of the probe @xmath73 . the diffusion coefficient decreases strongly when the size of the probe is larger than the medium molecule ( @xmath76 increases in the figure ) . larger probes result in smaller diffusion coefficients . simultaneously we observe in figure 10 an increase of the non gaussian parameter for large probes . for probes smaller than the medium molecules , the diffusion is larger than for the pure medium , and roughly constant . the non gaussian parameter is also roughly constant and smaller than the bulk value . in this work , using molecular dynamics simulations we have investigated the effect of small packing defects on the dynamic heterogeneity and diffusion processes in a supercooled liquid . our objective was to test the hypothesis of dynamic heterogeneities created by packing defects and the resulting modification of the liquid s dynamics . a secondary purpose was to better understand how diluted molecules modify the viscosity of a liquid below @xmath77 . for these purposes we have used a simple diatomic glass - former with a few ( 1@xmath78 ) probe molecules that were similar to the medium molecules but of a different size . due to the simplicity of the molecules we were able to access very long time scales ( of the order of the micro - second ) . we found that the induced defects do not modify the dynamics at high temperature but below @xmath77 the modification increases rapidly when the temperature drops . this result shows that the effects are linked to the physics of the glass - transition . we found that the relative size @xmath73 between the diluted probe and the medium molecule strongly determines the dynamics when @xmath79 , but not when @xmath80 . when @xmath80 the cooperativity decreases and the dynamics is accelerated , while when @xmath79 we observe the opposite effect . in summary , our results show that a few ( 1@xmath78 ) packing defects are able to strongly modify the dynamical heterogeneity in a supercooled liquid , in most cases increasing the heterogeneity and in some cases destroying them . these results come in support of the hypothesis of dynamic heterogeneities created by packing fluctuations in supercooled liquids . similarly , the packing defects also strongly affect the viscosity of the liquid , with larger defects leading to larger effects . | we use molecular dynamic simulations to investigate the relation between the presence of packing defects in a glass - former and the spontaneous cooperative motions called dynamic heterogeneity . for that purpose
we use a simple diatomic glass - former and add a small number of larger or smaller diatomic probes .
the diluted probes modify locally the packing , inducing structural defects in the liquid , while we find that the number of defects is small enough not to disturb the average structure .
we find that a small packing modification around a few molecules can deeply influence the dynamics of the whole liquid , when supercooled .
when we use small probe molecules , the dynamics accelerates and the dynamic heterogeneity decreases .
in contrast , for large probes the dynamics slows down and the dynamic heterogeneity increases .
the induced heterogeneities and transport coefficient modification increase when the temperature decreases and disappear around the onset temperature of the cage dynamics . |
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the process of phase separation in chemically reactive mixtures has been considered by several authors . et al _ @xcite and christensen _ et al _ @xcite used a modification of the cahn - hilliard equation to investigate the effects of a linear reaction of the type @xmath0 occurring simultaneously with phase separation following an instantaneous quench . in contrast to phase separation alone , domain coarsening was halted at a length - scale dependent on system parameters resulting in the ` freezing in ' of a spatially heterogeneous pattern . it was recognized that the steady - states resulted from competition between the demixing effects of phase separation and the equivalence of the chemical reaction term to an effective long - range repulsion @xcite . similar physics is seen in the phase ordering of block copolymers where an effective long - range interaction arises because of an interplay between interactions and steric constraints @xcite . in such systems pattern formation is a result of thermodynamic equilibrium . by contrast , in the systems we consider , the steady - states are maintained dynamically by the interplay of reaction and diffusion . a number of chemically and structurally more complicated systems have been considered , numerically and theoretically , within the same framework of a modified cahn - hilliard equation . these include ternary mixtures @xcite and systems with orientational order @xcite . here we investigate the effect of hydrodynamic interactions on phase ordering in a binary fluid mixture with chemical reactions using a lattice boltzmann method . the case of the linear reaction has been considered before by hou _ et al _ @xcite by a different numerical method . we duplicate some of their results as a means of testing our approach and then consider the quadratic reaction mechanism @xmath1 . the inclusion of hydrodynamics is known to strongly affect the way in which an unreactive fluid mixture coarsens in the aftermath of a quench @xcite . the growth exponent is found to increase from @xmath2 , for the purely diffusive case , to @xmath3 or @xmath4 for the viscous and inertial hydrodynamic regimes respectively . the new pathway for growth provided by hydrodynamics is transport of the bulk fluid down a pressure gradient established by variations in curvature @xcite . in two dimensions this minimises curvature by making domains circular , whereupon the effect vanishes and further coarsening can only occur by diffusion @xcite . in addition there is the possibility , investigated by tanaka @xcite , that the rapid decrease in interfacial area resulting from the hydrodynamic mechanism may leave the bulk phases unequilibrated and subject to a round of secondary phase separations . this suggests that coupling a modified cahn - hilliard equation to the navier - stokes equations for fluid flow may uncover behaviour different to that observed for the purely diffusive case . experimental work @xcite-@xcite has shown that a variety of mesoscopic structures can be formed when chemical reactions are photo - induced in phase separating polymer mixtures . the effects of two kinds of photo - chemistry have been considered : intermolecular photodimerisations @xcite and intramolecular photoisomerisation @xcite . both give rise to a long - range inhibition which prevents phase separation proceeding beyond a certain domain size . in the first case the inhibition is due to the formation of a network of cross - linked polymer molecules whereas in the second case it arises from the differing chemical properties of the two isomers . the similarities in the patterns formed due to phase separation arrest in simple fluids and in reactive polymer blends suggest the latter may be approached by considering first a small - molecule system . the paper is organized as follows . in section [ sec : modelsection ] we present a model of a chemically reactive binary fluid which couples the processes of reaction and diffusion to flow . we then outline the linear theory of pattern formation in the absence of hydrodynamic effects . in section [ sec : latticeboltzmethod ] we construct a lattice boltzmann scheme which solves the equations of motion of section [ sec : modelsection ] in the continuum limit . in sections [ sec : linearreact ] and [ sec : quadreact ] results are presented for the evolution of both high and low viscosity systems after a critical quench for a linear and a quadratic reaction mechanism respectively . for the reaction of type @xmath0 , comparison is made with the results of @xcite , @xcite and @xcite . in this section we summarize a model which describes the phase behavior and hydrodynamics of a two - component fluid . labeling the components @xmath5 and @xmath6 , we choose a description of the fluid in terms of the following variables : the total density , @xmath7 ; the total momentum , @xmath8 , and a compositional order - parameter , @xmath9 . the composition of the fluid evolves according to a modified version of the cahn - hilliard equation which includes the effects of chemical reaction ; advection of the order - parameter by the flow - field , * u * , and diffusion in response to gradients in chemical potential : @xmath10 here @xmath11 is a mobility constant and @xmath12 , which depends on the reaction rate constants , is the change in @xmath13 per unit time due to chemical reactions . the chemical potential of the system , @xmath14 , is given by the functional derivative of the free energy , @xmath15 , with respect to @xmath13 . we choose a free energy @xmath16(t ) = \int d{\bf x}\left(\frac{\varepsilon}{2}\phi^{2}+\frac{\gamma}{4}\phi^{4}+\frac{\kappa}{2}(\nabla\phi)^{2 } + t\rho\ln~\rho \right).\ ] ] @xmath17 is taken to be greater than zero for stability and the sign of @xmath18 determines whether the polynomial contribution to the free - energy density has one or two minima , and hence whether the fluid is above ( @xmath19 ) or below ( @xmath20 ) its critical temperature . for @xmath20 the mixture will separate into two bulk components separated by a narrow , but smooth , interface . the gradient - squared term in @xmath13 associates an energy cost with variations in composition and the parameter @xmath21 is related to the surface tension and governs the width of the interface between the two phases . the parameter @xmath22 appears in the isotropic part of the pressure tensor and is related to the degree of incompressibility of the fluid @xcite . a suitable choice is @xmath23 . we consider two types of reactive source term , @xmath12 . a linear source @xmath24 = \rho ( \gamma_{2 } - \gamma_{1 } ) - \phi ( \gamma_{1 } + \gamma_{2}),\ ] ] corresponding to the reversible chemical reaction @xmath25 . and a quadratic source @xmath26 = \frac{1}{2 } ( \gamma_{1 } + \gamma_{2 } ) ( \phi - \rho ) \left ( \phi - \frac{(\gamma_{2}-\gamma_{1})\rho}{\gamma_{1 } + \gamma_{2 } } \right),\ ] ] corresponding to the reversible chemical reaction @xmath1 . the constants @xmath27 and @xmath28 are the rates of the forward and backward reactions respectively . we note that , for a spatially homogeneous system , the linear mechanism has a single stable fixed point whereas the quadratic mechanism has a stable fixed point at @xmath29 and an unstable one at @xmath30 . here we consider only cases where @xmath31 . the velocity field obeys a navier - stokes equation , @xmath32 where @xmath33 is the pressure tensor , @xmath34 is the viscosity and @xmath35 is the viscous stress tensor . the pressure tensor is derived from the free - energy : @xmath36 this provides a further coupling between the evolution of @xmath13 and @xmath37 in addition to the advection term in ( [ eq : rda ] ) . from ( [ eq : pressurerelation ] ) and ( [ eq : freeenergy ] ) it follows that @xmath38 \delta_{\alpha\beta } + \kappa \partial_{\alpha}{\phi } \partial_{\beta}{\phi},\ ] ] where @xmath39 denotes the polynomial contribution to the free - energy density . the total mass density of the fluid @xmath40 is also conserved and obeys @xmath41 linear stability analysis of the reaction - diffusion equation ( [ eq : rda ] ) with @xmath42 and source term ( [ eq : linearsource ] ) shows that only those modes , @xmath43 , with @xmath44 are unstable , where @xmath45 and @xmath46 depend on the parameters @xmath18 , @xmath17 and @xmath11 in equation ( [ eq : rda ] ) @xcite . this is in contrast to spinodal decomposition ( the case @xmath47 ) where only short - wavelength modes are stable . the damping of long - wavelength modes in the reactive case prevents continued growth of domains . instead , phase separation is halted at some length - scale set by the reaction . in addition , there is a threshold value of @xmath48 , @xmath49 above which the reaction is strong enough to completely inhibit phase separation by rendering all linear modes stable . we also note , following @xcite and @xcite , that there is an equivalence between this behaviour and phase ordering in a system with competing short and long - range interactions . the equivalence can be demonstrated by incorporating the reactive source into an effective free - energy for the system where it appears as a non - local term . formally , we rewrite ( 1 ) as @xmath50,\ ] ] where @xmath51 and @xmath52 is the green s function of the laplace operator . the reaction is then seen to act as an effective long - range , intra - species repulsion , with strength governed by @xmath48 , in contrast to the short - range attraction of like - molecules which drives phase separation . for the case of equal forward and backward rates , the linearised behaviour of the quadratic source ( [ eq : quadsource ] ) with reaction rate @xmath48 is the same as for the linear source with reaction rate @xmath53 . this can be seen from the linearisation of ( [ eq : rda ] ) with @xmath54 and source term ( [ eq : quadsource ] ) . hence , at early times , the linear and quadratic cases segregate in the same way . however , after the formation of separate a and b - rich regions , non - linear contributions to the source term become important and this is expected to lead to growth of the a - rich phases at the expense of the b - rich ones . this follows from the asymmetry of @xmath12 between the two phases : in the a - rich phase the production of b is limited by the amount of b already present . hence it is limited by the _ minority _ component of the fluid in these regions . in the b - rich regions the production of a is limited by the amount of the majority phase present . therefore production of a in the b - rich phase is the more rapid process . the lattice boltzmann method is a well - established numerical technique for hydrodynamic problems @xcite . initially it was a kinetic - theory based method for the simulation of isothermal ideal flows which was introduced to circumvent some of the problems which rendered its predecessor , lattice - gas cellular automata , impractical . however , it has since been modified and applied to a variety of problems in the simulation of complex fluids . examples include binary fluids @xcite , liquid - gas systems @xcite , liquid - crystals @xcite and colloidal suspensions @xcite . a lattice boltzmann scheme for the simulation of two or more species undergoing reaction and diffusion in a moving , viscous solvent was formulated by dawson _ et al _ @xcite . in comparision , our model incorporates the thermodynamics of the multi - component fluid via the cahn - hilliard equation . to simulate the binary fluid model described in section [ sec : modelsection ] we utilise the free - energy lattice boltzmann method of swift _ et al _ @xcite . to this end we define two populations of dynamical variables @xmath55 and @xmath56 on the sites of a simple lattice in three - dimensions . on each site the variables @xmath57 and @xmath58 correspond to a velocity direction @xmath59 for @xmath60 . the dynamical variables are referred to as distribution functions since their moments over the velocity set define the macroscopic physical quantities : @xmath61 the distribution functions on each site are updated in discrete time with a time - step @xmath62 . the velocities are chosen so that @xmath63 and , for all @xmath64 , @xmath65 lies between two lattice sites . the choice of lattice and velocity set are subject to certain restrictions @xcite . for this work we used a face - centred cubic lattice in three dimensions with the set of fifteen velocities , @xmath66 , illustrated in figure [ fig : lattice ] . , and time - step , @xmath62 , are set to unity . , width=226 ] the distribution functions @xmath57 and @xmath58 evolve according to @xmath67 , \\ \label{eq : evolg } g_{i}({\bf x}+\delta t { \bf e}_{i};t+\delta t ) = g_{i}({\bf x};t ) + \delta_{i}^{\tau_{g}}[g ] + f_{i } , \ ] ] where @xmath68 = - \frac{\delta t}{\tau } \left ( f_{i}({\bf x};t ) - f_{i}^{eq}({\bf x};t ) \right),\ ] ] and the relaxation times @xmath69 and @xmath70 are free parameters . equations ( [ eq : evolf ] ) and ( [ eq : evolg ] ) are both lattice equivalents of the bhatnager - gross - krook ( bgk ) , or single relaxation time , approximation to the full continuum boltzmann equation @xcite . we now need to specify the local equilibria functions @xmath71 and @xmath72 and the forcing term @xmath73 . @xmath74 and @xmath75 are taken to be series expansions in the velocity @xmath76 the coefficients in ( [ eq : fequil ] ) and ( [ eq : gequil ] ) are chosen so the moments of the equilibrium distributions satisfy @xmath77 in addition , the lattice forcing term is chosen to obey @xmath78 , \;\;\ ; \sum_{i=0}^{n } e_{i\alpha } f_{i } = 0.\end{gathered}\ ] ] one possible choice of the coefficients , such that constraints ( [ eq : fmoments])-([eq : fmoments ] ) hold , is given by @xmath79 , \\ e_{1 - 6}^{\alpha\beta } = 8e_{7}^{\alpha\beta } , \\ h_{1 - 14 } = \frac{1}{10}d\mu,\;\;\ ; h_{0 } = \phi - 14h_{1 } , \\ j_{7 - 14 } = \phi/24,\;\;\ ; j_{1 - 6 } = 8j_{7 } , \\ k_{7 - 14 } = -\phi/24,\;\;\ ; k_{0 } = 16k_{7 } , \;\;\ ; k_{1 - 6 } = 2k_{7 } , \\ q_{7 - 14 } = \phi/16,\;\;\ ; q_{1 - 6 } = 8q_{7 } \\ f_{0 } = \delta t\frac{1}{2}j,\;\;\ ; f_{1 - 6 } = \delta t\frac{1}{24}j,\;\;\ ; f_{7 - 14 } = \delta t\frac{1}{32}j.\end{gathered}\ ] ] the constraints ( [ eq : fmoments])-([eq : fmoments ] ) ensure that , on length and time - scales large compared to the lattice - spacing and time - step , the evolution of the moments ( [ eq : moments ] ) satisfies the partial differential equations set out in section [ subsec : eqnmotion ] . to check that this is indeed the case the task of reducing the description of the dynamics in terms the distribution functions to one in terms of their moments must be addressed . the reduction can be performed by a chapman - enzkog expansion of equations ( [ eq : evolf ] ) and ( [ eq : evolg ] ) . since the details of this are essentially no different to those found in @xcite we present only the result here . the zeroth moment of the @xmath57 satisfies equation ( [ eq : consmass ] ) for conservation of mass . the first moment of the @xmath57 satisfies the navier - stokes equation ( [ eq : ns ] ) with @xmath80 and @xmath81 where the @xmath82 denotes unwanted error terms @xcite . the first moment of the @xmath58 satisfies the reaction - diffusion - advection equation @xmath83 + \omega_{g}\partial_{\alpha } \left ( u_{\alpha } j \right ) + j[\phi ] , \label{eq : rda2}\end{aligned}\ ] ] where @xmath84 and @xmath85 is defined in ( [ eq : secondgmoments ] ) . equation ( [ eq : rda2 ] ) corresponds to equation ( [ eq : rda ] ) with mobility @xmath86 but with two extra terms . the term in gradients of the components of the pressure tensor is present in some other free - energy lattice boltzmann methods and has been shown numerically to be small in comparison to the desired terms @xcite . the term in gradients of the reactive source can be seen , to first order in @xmath13 , to be a correction to the advecting velocity of order @xmath87 . for our choices of parameters , this contribution is small in the low viscosity regime where the advecting flow field is important . this completes the specification of our lattice boltzmann method . although the model is inherently three - dimensional , we consider only its restriction to two dimensions . as an initial condition we choose the total density of the fluid @xmath88 at each lattice lattice . the near - incompressibility of the fluid ensures that this value remains approximately the same at later times . to imitate the conditions following a rapid cooling of a fluid from above to below its critical temperature we initialise @xmath89 where @xmath90 is random noise with @xmath91 . the parameters @xmath18 and @xmath17 in the free - energy are chosen so that in the unreactive case the fluid phase separates into regions where @xmath92 . we choose @xmath93 to ensure a narrow interface and choose @xmath94 and @xmath95 which fixes the diffusion constant @xmath96 . we also choose @xmath97 and the system size @xmath98 , throughout . the viscosity of the fluid is controlled by varying the relaxation time @xmath69 . for @xmath99 , domains in the unreactive fluid grow as @xmath100 and the fluid can be taken to be in the diffusive regime where hydrodynamic flows are negligible . to ensure a hydrodynamic growth exponent of @xmath101 we choose @xmath102 . we consider first the case of the linear reaction mechanism ( [ eq : linearsource ] ) . figure [ fig : lsteadystates ] [ cols="^,^,^ " , ] the evolution of the inverse interfacial length for different values of @xmath48 and @xmath34 . it can be seen that when hydrodynamic effects are present the rate of reduction of interfacial length is faster . moreover , @xmath103 is no longer monotonically increasing : it attains a maximum due to the creation of more interface during secondary phase separation events . in conclusion , we have extended the lattice boltzmann method to study the effect of hydrodynamics on structures arising in phase - separating reactive mixtures for two simple reaction mechanisms . we have found that hydrodynamic flow significantly alters both the way in which the domain structure in these fluids evolves and the eventual steady states of the system . the results obtained for the linear reaction are in agreement with previous work on the subject . for a quadratic reaction an asymmetric domain structure was obtained , with the inclusion of hydrodynamic effects leading to secondary phase separation within majority phase . it would be interesting to extend the model to incorporate viscoelastic effects arising from constituent molecules with internal microstructure : one can envisage chemical processes which can change the local microstructural elements . the technique also makes feasible a study of the interaction of reaction and diffusion with imposed flow - fields . we would like to thank c. pooley for intersting discussions . k.f . acknowledges funding from eprsc grant no . gr / r83712/01 . glotzer , e.a . di marzio and m. muthukumar , phys . lett . * 74 * 2034 ( 1995 ) . christensen , k. elder and h.c . fogedby , phys . e * 54 * r2212 ( 1996 ) . s.c . glotzer and a. coniglio , phys . e * 50 * 4241 ( 1994 ) . l. leiber , marcomolecules * 13 * 1602 ( 1980 ) . t. ohta and k. kawasaki , marcomolecules * 19 * 2621 ( 1986 ) . t. okuzono and t. ohta , phys . e * 67 * 056211 ( 2003 ) . b. liu , c. tong and y. yang , j. phys . b * 105 * 10091 ( 2001 ) . c. tong , h. zhang and y. yang , j. phys . b * 106 * 7869 ( 2002 ) . travasso , g.a . buxton , o. kuksenok , k. good and a.c . balazs , j. chem . phys . * 122 * 194906 ( 2005 ) . r. reigada , f. sagues and a.s . mikhailov , phys . lett . * 89 * 038301 ( 2002 ) ; r. reigada , a.s . mikhailov and f. sagues , e * 69 * 041103 ( 2004 ) . a.j . wagner and j.m . yeomans , phys . * 80 * 1429 ( 1998 ) . y. huo , x. jiang and y. yang , j. chem . phys . * 118 * 9830 ( 2003 ) ; macromol . theory simul . * 13 * 280 ( 2004 ) . q. tran - cong and a. harada , phys . lett . * 76 * 1162 ( 1996 ) ; q. tran - cong , a. harada , k. kataoka , t. ohta and o. urakawa , phys . e * 55 * r6340 ( 1997 ) ; qui tran - cong , katsunari kataoka and osamu urakawa , phys . e * 57 * r1243 ( 1998 ) . qui tran - cong , takashi ohta and osamu urakawa , phys . e * 56 * r59 ( 1997 ) ; t. ohta , o. urakawa and q. tran - cong , macromolecules * 31 * 6845 ( 1998 ) . q. tran - cong , j. kawai and k. endoh , chaos * 9 * 298 ( 1991 ) . kendon , m.e . cates , i. pagonanarraga , j - c . desplat and p. bladon , j. fluid . mech . * 440 * 147 ( 2001 ) . bray , adv . phys . * 43 * 357 ( 1994 ) . siggia , phys . a * 20 * 595 ( 1979 ) . h. tanaka , phys . e * 51 * 1313 ( 1995 ) . cahn , j. chem . phys . * 42 * 93 ( 1964 ) . s. succi , _ the lattice boltzmann equation for fluid dynamics and beyond _ ( clarendon press , oxford , 2001 ) . xiaowen shan and hudong chen , phys . e * 47 * 1815 ( 1993 ) . swift , e. orlandini , w.r . osborn and j.m . yeomans , phys . e * 54 * 5041 ( 1996 ) . luo and s.s . girimaji , phys . e * 67 * 036302 ( 2003 ) . xiaowen shan and hudong chen , phys . e * 49 * 2941 ( 1994 ) . li - shi luo , phys . lett . * 81 * 1618 ( 1998 ) . s. pounce dawson , s. chen and g.d . doolen , j. chem . phys . * 98 * 1514 ( 1992 ) . c. denniston , e. orlandini and j.m . yeomans , europhys . lett . * 52 * 481 ( 2000 ) . ladd and r. verberg , j. stat . phys . * 104 * 1191 ( 2001 ) . | we use a lattice boltzmann method to study pattern formation in chemically reactive binary fluids in the regime where hydrodynamic effects are important .
the coupled equations solved by the method are a cahn - hilliard equation , modified by the inclusion of a reactive source term , and the navier - stokes equations for conservation of mass and momentum .
the coupling is two - fold , resulting from the advection of the order - parameter by the velocity field and the effect of fluid composition on pressure .
we study the the evolution of the system following a critical quench for a linear and for a quadratic reaction source term .
comparison is made between the high and low viscosity regimes to identify the influence of hydrodynamic flows . in both cases
hydrodynamics is found to influence the pathways available for domain growth and the eventual steady - states . |
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the 1.68-ev photoluminescence ( pl ) center was reported many decades ago in diamond@xcite and where later it was assumed @xcite that silicon impurities were involved in this center . this was confirmed by pl measurements at cryogenic temperature on si - doped chemical vapor deposition ( cvd ) polycrystalline diamond samples where the fine structure of the 1.68-ev pl center could be detected.@xcite a 12-line fine structure is observed close to 1.68 ev , and this can be divided into three similar groups each containing four components . the relative strengths of the optical absorption for the three groups of lines are found to be the same as the ratio of the abundancies of the natural isotopes of silicon , @xmath0si , @xmath1si , and @xmath2si@xcite . the 4-line fine structure for an individual si - isotope is assigned to doublet levels both in the ground and excited states which split by about 48 and 242 ghz , respectively.@xcite it has been assumed that this small splitting might be explained by dynamic jahn - teller effect@xcite and/or by spin - orbit effect.@xcite recently , single photon - emission from 1.68-ev pl center has been demonstrated @xcite . its zero - phonon - line ( zpl ) with 5 nm width even at room temperature and the near infrared emission makes this pl center very attractive candidate for quantum optics@xcite and biomarker@xcite applications . spin - polarized local density approximation@xcite ( lda ) within density functional theory ( dft ) calculations in a very small nanodiamond model ( with @xmath370 carbon atoms ) concluded that the negatively charged split - vacancy form ( see fig . [ fig : struct ] ) of siv defect ( siv@xmath4 ) is responsible for the 1.68-ev pl center.@xcite they exclude the neutral siv defect ( siv@xmath5 ) as a good candidate as its ground state is orbitally singlet.@xcite this model was later disputed based on a semiempirical restricted open - shell hartree - fock cyclic cluster model calculation where they claimed that tunneling of si - atom along the symmetry axis may occur for siv@xmath5 defect that can explain the doublet line in the ground state.@xcite the fingerprint of siv@xmath5 was found by electron paramagnetic resonance ( epr ) studies.@xcite the kul1 center with @xmath6 high spin ground state and d@xmath7 symmetry@xcite was recently associated with siv@xmath8 defect where 216-atom lda supercell calculations produces relatively good agreement with the measured @xmath9c and @xmath0si hyperfine couplings ( see tab . [ tab : hf ] ) . very recently , thorough epr and pl studies have been carried out to correlate the kul1 epr center with an 1.31-ev pl center , and its relation to the 1.68-ev pl center.@xcite the final conclusion was that siv@xmath8 has a zpl at 1.31-ev whereas siv@xmath4 has a zpl at 1.68-ev . photo - conductivity measurements@xcite and photo - ionization measurements@xcite indicate that the adiabatic ( thermal ) charge transition level of @xmath10 level of siv defect is at @xmath11 + 1.5 ev , where vbm is the valence band edge . while the recent measurements@xcite are very plausible still no _ ab initio _ theory was able to conclusively support the assignment of 1.68-ev center with the negatively charged siv defect . lda or any semilocal generalized gradient approximation ( gga ) functionals suffer from the band gap error@xcite which inhibits to directly compare the calculated and experimental zpl energies . recent advances in dft functionals made possible to accurately calculate zpl energies and charge transition level of defects in diamond and other semiconductors.@xcite in this paper , we apply this theory to study the charge transition levels and the zpl energies of siv defect in diamond . these calculations yield the position and nature of defect states in host diamond and are able to reveal the nature of the shelving state in 1.68-ev pl center . our paper is organized as follows . in section [ sec : method ] we describe briefly the _ ab initio _ method that we applied to study the electronic structure and excitations of the siv defect . in section [ sec : results ] we describe the structure and the basic defect level scheme of siv defect by group theory . here , we combine the results from _ ab initio _ calculations with group theory considerations in order to identify the order of important defect states and the charge state relevant for the most important 1.68-ev pl center . we discuss the results then we conclude and summarize the results in section [ sec : summary ] . ( color online ) the split - vacancy structure of the siv defect and the calculated spin density in neutral charge state . the isosurface of spin density is 0.05 . the defect has @xmath12 symmetry with the symmetry axis of [ 111 ] direction . the lattice sites of the missing carbon atoms are depicted by the smallest ( pink ) balls . ] the present calculations have been carried out in the framework of the generalized kohn - sham theory,@xcite by using the screened hybrid functional hse06 of heyd , ernzerhof and scuseria with the original parameters ( 0.2 @xmath13 for screening and 25% mixing).@xcite hse06 in diamond happens to be nearly free of the electron self - interaction error , and is capable of providing defect - levels and defect - related electronic transitions within @xmath30.1 ev to experiment.@xcite we have used the vienna ab initio simulation package vasp 5.3.2 with the projector augmented wave@xcite ( paw ) method ( applying projectors originally supplied to the 5.2 version).@xcite to avoid size effects as much as possible , a 512-atom supercell was used in the @xmath14-approximation for defect studies . parameters for the supercell calculations were established first by using the gga exchange of perdew , burke and ernzerhof ( pbe)@xcite in bulk calculations on the primitive cell with a 8@xmath158@xmath158 monkhorst - pack ( mp ) set for brillouin - zone sampling.@xcite ( increasing the mp set to 12@xmath1512@xmath1512 has changed the total energy by @xmath160.002 ev . ) constant volume relaxations using a cutoff of 370(740 ) ev in the plane - wave expansion for the wave function ( charge density ) resulted in an equilibrium lattice parameter of @xmath17= 3.570 . increasing the cutoff to 420 ( 840 ) ev has changed the lattice constant by only 0.003 . therefore , considering the demands of the supercell calculations , the lower cut - off was selected . the hse06 calculation with the 8@xmath158@xmath158 mp set and 370(740 ) ev cutoff resulted in the lattice constant @xmath18=3.545 , indirect band gap of @xmath19=5.34 ev , in good agreement with the experimental values of @xmath20=3.567 and @xmath19=5.48 ev ( see , e.g. , ref . ) . due to the different choice of the basis , the hse06 values presented here differ somewhat from those in refs . , but tests on the negatively charge nitrogen - vacancy center ( nv ) have shown that the higher cutoff would cause only negligible difference in the equilibrium geometry of that defect too . defects in the supercell were allowed to relax in constant volume till the forces were below 0.01 ev / . we calculated the hyperfine tensors of @xmath9c and @xmath0si isotopes within paw formalism@xcite as implemented in vasp 5.3.2 package . here , we applied a larger 500(1000 ) ev cutoff for the plane wave ( charge density ) expansion . the hyperfine tensor ( @xmath21 ) between the electron spin density @xmath22 of an electron spin @xmath23 , and the non - zero nuclear spin @xmath24 of nucleus j may be written as @xmath25 \label{eq : hyperfine}\ ] ] where the first term within the square brackets is the so - called ( isotropic ) fermi - contact term , and @xmath26 represents the ( anisotropic ) magnetic dipole - dipole contribution to the hyperfine tensor . @xmath27 is the nuclear bohr - magneton of nucleus @xmath28 and @xmath29 the electron bohr - magneton . the fermi - contact term is proportional to the magnitude of the electron spin density at the center of the nucleus which is equal to one third of the trace of the hyperfine tensor , @xmath30 . the fermi contact term arises from the spin density of unpaired electrons with @xmath31-character , and can be quite sizable . the spin density built up from unpaired electrons of @xmath32-character yields the dipole - dipole hyperfine coupling . the fraction of fermi - contact and dipole - dipole term implicitly provides information about the character of the wave function of the unpaired electron as well as the corresponding nuclei ( via @xmath33 ) . according to our recent study@xcite the contribution of the spin polarization of core electrons to the fermi - contact hyperfine interaction@xcite is significant for @xmath9c isotopes , thus , we include this term in the calculation of hyperfine tensors . the excitation energies were calculated within constrained dft method that was successfully applied to nv center in diamond.@xcite in this method one can calculate the relaxation energy of the nuclei due to optical excitation . the formation energy of the defect with defect charge state @xmath34 is defined as @xmath35 by ignoring the entropy contributions , where @xmath36 is the total energy of the defect in the supercell , @xmath37 is the calculated energy of vbm in the perfect supercell , @xmath38 and @xmath39 are the chemical potentials of c and si atoms in diamond with @xmath40 number of c atoms in the supercell , @xmath41 is the chemical potential of the electron , i.e. , the fermi - level , and @xmath42 is the correction needed for charged supercells . the thermal charge transition level between the defect charge states of @xmath34 and @xmath43 is the position of the fermi - level ( @xmath41 ) in the fundamental band gap of diamond where the formation energies are equal in these charge states . this condition simplifies to a difference in the total energies in their respective charge states as follows , @xmath44 where @xmath41 is referenced to the calculated @xmath37 . for comparison of different defect configurations and charge states , the electrostatic potential alignment and the charge correction scheme of lany and zunger was applied for @xmath42.@xcite recently , this scheme was found to work best for defects with medium localization.@xcite we first study the structure and the obtained defect levels by hse06 calculation . then , we apply group theory in order to explain the symmetry of the defect states . we use the calculated thermal ionization energies , excitation energies and hyperfine couplings to identify the 1.31-ev and 1.68-ev pl centers . we also discuss the results with comparing them to the experiments . we first calculated the neutral defect siv@xmath8 by substituting the c - atom by a si - atom adjacent to a nearby vacancy . the si automatically left the substitutional site creating a split - vacancy configuration which may be described as a si - atom placed in a divacancy where the position of the si - atom is equidistant from the two vacant sites ( see fig . [ fig : struct ] ) . the position of si - atom is a bond center position which is an inversion center of perfect diamond lattice . in our special coordinate frame the two vacant sites reside along the [ 111 ] direction of the lattice which has a @xmath45 rotation axis . the symmetry of the defect may be described as @xmath46 or @xmath12 , where @xmath47 is the inversion . we note that nv center has @xmath48 symmetry with no inversion . the defect has @xmath6 high - spin ground state . this finding agrees with the lda calculations.@xcite after establishing the symmetry of the defect one can apply group theory analysis for this defect.@xcite . one can build the defect states of this defect as an interaction between the divacancy orbitals and the si - impurity states . the si impurity has 6 immediate neighbor c - atoms in divacancy . the calculated distance between si- and c - atoms is about 1.97 in the neutral charge state , which is longer than the usual si - c covalent bond ( 1.88 ) . since c - atoms are more electronegative than si - atom the charge transfer from the si - atom toward the c - atoms can be relatively large leaving positively charged si ion behind . the divacancy has @xmath12 symmetry with six c dangling bonds . these dangling bonds form @xmath49 , @xmath50 , @xmath51 and @xmath52 orbitals while the si - related four @xmath53 states should form @xmath49 , @xmath51 , and @xmath50 orbitals in @xmath12 crystal field ( the explicit form of these orbitals as a function of @xmath53 states can be seen in ref . ) . please , note that the @xmath49 , @xmath50 and @xmath51 states may be combined but @xmath52 orbitals should be pure c dangling bonds state ( see fig . [ fig : leveldiag ] ) . the bonding and anti - bonding combinations of these states form the defect states of siv defect . ( color online ) the defect - molecule diagram of the neutral siv defect in diamond . the irreducible representation of the orbitals under @xmath12 symmetry is shown . the orbitals of the si - atom and the six carbon dangling bonds of divacancy recombine in diamond where the conduction band ( cb ) and valence band ( vb ) are schematically depicted . the @xmath52 orbital is very close to the valence band maximum with forming a strong resonance state at the valence band edge shown as brown ( lighter gray ) lines . the @xmath51 level falls in the valence band that also forms a strong resonance even deeper in the valence band . for the sake of the simplicity , the position of the spin - up ( majority spin channel ) levels are depicted in the spin - polarized density functional theory calculation . ] according to hse06 calculation the occupation of the defect states may be described as @xmath54@xmath55@xmath56@xmath57 which agrees again with previous lda calculations.@xcite here , the ten electrons are coming from the six electrons of c dangling bonds and the 4 @xmath53 electrons of the si impurity . as two electrons occupy the double degenerate @xmath52 state , the high - spin @xmath6 ground state naturally forms by following hund s rules . it is important to determine the position of the defect levels . interestingly , hse06 predicts that only @xmath52 appears in the band gap . in the spin - polarized calculation the occupied @xmath52 state in the spin - up channel is at @xmath37 + 0.3 ev . the @xmath51 state is resonant with the valence band and can be found just 0.64 ev below vbm . the occupied @xmath49 state lies very deep in the valence band and may play no important role in the excitation or ionization processes . our estimations indicate that the @xmath50 defect level is too deep in the vb to be excited by red excitation . however , higher energetic lasers ( in the green and blue spectrum ) can excite this state and other states within the vb with @xmath50 symmetry . the empty anti - bonding orbitals fall in the conduction band , and will not be considered any more . the most important @xmath51 and @xmath52 states are depicted in fig . [ fig : states ] . despite the @xmath51 state lies in the valence band it is still localized around the defect site . the valence band states are strongly perturbed by the presence of the defect . the vbm is triple degenerate in perfect lattice that splits to an upper @xmath49 level and a lower @xmath52 level in the presence of the defect . since this @xmath58 state has the same symmetry as the low - lying @xmath52 defect state , that @xmath58 state becomes a defect resonance state . similar phenomena occurs for the deep @xmath51 defect state as well . the shallow @xmath58 state may play an important role in the excitation / de - excitation process of the defect . we conclude that the neutral siv defect ( i ) has @xmath59 ground state , ( ii ) can be theoretically ionized as ( 2 + ) , ( 1 + ) , as well as ( 1- ) and ( 2- ) by emptying or filling the double degenerate @xmath52 state in the gap . in order to establish the relevant charge states , one has to calculate the adiabatic charge transition levels of siv defect . we found that the neutral , negatively charged and double negatively charged states can be found in diamond ( see fig . [ fig : occlev ] ) . the single positive charged state might be only found in very highly p - type doped diamond samples . the calculated formation energy of siv defect as a function of the fermi - level in the gap . the crossing points represent the charge transition levels . the chemical potential of si is taken from cubic silicon carbide in the carbon - rich limit . ] the calculated @xmath10 level at @xmath37 + 1.43 ev is very close to the level that is associated with the acceptor ionization energy of the defect from photo - conductivity measurements at @xmath3@xmath37 + 1.5 ev.@xcite interestingly , the calculated @xmath60 level at @xmath3@xmath37 + 2.14 ev is well below the midgap . since the @xmath52 level is fully occupied at @xmath37 + 1.5 ev in @xmath61 charge state intra - level optical transition can not take place , and ultra - violet excitation ( @xmath34.0 ev ) is needed to excite or ionize the defect optically to the conduction band edge . another important note that the calculated acceptor level of nv lies at @xmath3@xmath37 + 2.6 ev.@xcite this means that if both nv and siv defects are present in the diamond sample then most of the nv defects should be neutral in order to detect siv@xmath62 defect . another important point that the luminescence from siv@xmath62 defect can be more stable than that of nv@xmath62 defect in nanodiamonds as a function of surface termination because the corresponding @xmath60 charge transition level lies deeper in the band gap than the @xmath10 level of nv defect . all in all , the neutral and negatively charged siv defects are relevant for intra - level optical transitions . in the negatively charge siv defect the ground state electron configuration is @xmath56@xmath63 . this is principally a jahn - teller unstable system as the double degenerate @xmath52 level is partially filled ( in the spin - down channel in our calculation ) . this has @xmath64 symmetry in @xmath12 symmetry . in hse06 geometry optimization we allowed the systems to relax to lower symmetries . indeed , hse06 showed a @xmath65 distortion where two c - atoms have 0.03 longer distance from si - atom than the other two atoms . in this particular case @xmath66 state was formed in @xmath65 symmetry . however , we have to note that the dynamic coupling between vibrations and electronic states can not be taken into account in our calculation . thus , dynamic jahn - teller system can not be directly described by our method . for instance , static jahn - teller effect occurs for nv@xmath8 in ref . while it is known from experiments that it is a dynamic jahn - teller system . since the distortion from @xmath12 symmetry obtained by hse06 calculation is small the defect may well have @xmath12 symmetry with dynamic jahn - teller effect . we note that no such an electron paramagnetic resonance center with @xmath67 spin was found that could be associated with siv - defect . this also hints that the ground state of siv@xmath4 is a dynamic jahn - teller system which prohibits the electron spin resonance signal similar to nv@xmath8 defect . now , we discuss the possible excited states of this system . again , the fully occupied @xmath51 state is resonant with the valence band . still , one can promote one electron from this level to the @xmath52 level in the band gap . the resulted @xmath68 excited state is again jahn - teller unstable . interestingly , when the electron from the minority spin - down @xmath51 level in the valence band was promoted to the @xmath52 state in the gap then the resulted hole state pops up clearly above the valence band edge at about @xmath37 + 0.12 ev [ see fig . [ fig : sivexc](a ) ] . the @xmath69 optical transition is allowed . in hse06 calculation the excited state has also @xmath65 distortion but the effect is again small , and can be a dynamic jahn - teller system . the calculated zpl energy is about 1.72 ev which agrees well with that of 1.68-ev pl center . the calculated relaxation energy due to optical excitation is about 0.03 ev which is much smaller than that of nv center ( about 0.21 ev ) , and explains the small contribution of the vibration sideband in the emission spectrum . thus , the assignment of 1.68-ev pl center with siv@xmath62 defect is well - supported by our calculation . we argue that the sharp zero - phonon line of siv@xmath4 as well as the relative strength of the fine - structure splittings in the ground and excited states can be well - understood by our findings . the @xmath64 ground and @xmath68 excited states have very similar electron charge densities , where mostly just the phase differs between the two states . as a result , the ions will be subject to similar potentials in their ground and excited states leading to only small change in the geometry due to optical excitation . this is in stark contrast to the case of nv@xmath4 defect where the electron charge density strongly redistributes upon optical excitation leading to a large stokes - shift.@xcite another observation is that @xmath64 state virtually is not localized at all on si - atom due to symmetry reasons , however , our projected density of states analysis shows that there is a small contribution from the orbitals of the si - atom in the @xmath68 excited state , allowed by symmetry . this may explain the larger splitting in the fine - structure of the excited state than that in the ground state:@xcite the @xmath68 state has expected to have larger spin - orbit splitting due to the small contribution of the orbitals of si - atom than that in @xmath64 state where this contribution is missing , as the spin - orbit strength increases with the atomic number to the fourth power . ( color online ) schematic diagram about the electronic structure of the negatively charged siv defect in diamond . the valence band ( vb ) resonant states play a crucial role in the excitation . for the sake of the simplicity , the spin - polarization of the single particle levels are not shown . ( a ) ground and the bright ( optically allowed ) excited state . ( b ) ground and shelving states when the hole left on the split valence band edge states . ( c ) schematic energy level diagram based on hse06 constraint density functional theory calculations . thick red lines indicate strong absorption / emission while the thin red line indicates a weak radiative recombination in the case of the presence of strain which breaks the inversion symmetry of the defect . orange wavy arrows represent non - radiative recombination down to the ground state . , title="fig : " ] + ( color online ) schematic diagram about the electronic structure of the negatively charged siv defect in diamond . the valence band ( vb ) resonant states play a crucial role in the excitation . for the sake of the simplicity , the spin - polarization of the single particle levels are not shown . ( a ) ground and the bright ( optically allowed ) excited state . ( b ) ground and shelving states when the hole left on the split valence band edge states . ( c ) schematic energy level diagram based on hse06 constraint density functional theory calculations . thick red lines indicate strong absorption / emission while the thin red line indicates a weak radiative recombination in the case of the presence of strain which breaks the inversion symmetry of the defect . orange wavy arrows represent non - radiative recombination down to the ground state . , title="fig : " ] + furthermore , our hse06 calculations reveal that the heavily perturbed valence band edges can play important role in understanding the defect properties as those states lie closer to the @xmath52 state in the band gap than the @xmath51 resonant defect state . therefore , we calculate the excitation from the split vbm states . a @xmath70 and a @xmath71 excited states are built from the hole left on the @xmath58 and @xmath72 states , respectively [ c.f . , fig . [ fig : sivexc](b ) ] . we find that the @xmath70 excited state is @xmath30.1 ev lower in energy than the bright @xmath68 excited state where the calculated stokes - shift of this transition is again small . the @xmath71 excited state is @xmath31.59 ev above the @xmath70 ground state . these results can explain the nature of the shelving state in 1.68-ev pl center . unlike nv - center the siv defect has inversion symmetry that plays a crucial role in the de - excitation process . due to the inversion symmetry the @xmath70 and @xmath71 excited states are shelving states because they are optically forbidden due to the same parity of their wave function with that of the ground state . thus , the shelving states have the same spin state as the ground and the bright excited states . since @xmath70 shelving state has a level very close to that of the bright @xmath73 state the non - radiative coupling between these states can be efficient . we further note that a new and weak near - infrared ( nir ) transition at @xmath3823 nm ( 1.52 ev ) has been found associated with the negatively charged siv - defect@xcite . the measurements implied that this transition belongs to the same charge as the 1.68-ev transition . this nir transition was particularly found in ensemble measurements of nanodiamond samples when strain was present in the sample@xcite . our calculations imply [ fig . [ fig : sivexc](c ) ] that this weak radiative transition can be explained by the slightly distorted @xmath71 excited state . any distortion of the diamond lattice will break the inversion symmetry of the siv defect , so the parity of the corresponding wave functions . therefore , transitions between @xmath74 and @xmath75 ( originally @xmath76 and @xmath64 ) will be allowed . the calculated transition energy ( 1.59 ev ) is very close to the detected one which further supports our assignment . our calculations highlight the importance of the valence band edge states in understanding the optical properties of 1.68-ev center in diamond . we note again that the @xmath51 state is resonant with the valence band . thus , unlike the case of nv center with well - separated atomic - like states in the gap , it is probable that ionization of the defect can occur during optical excitation . in this process , a hole is created that might ( temporarily ) leave the defect with creating an optically inactive ( @xmath77 ) charge state , particularly , in the presence of external perturbations that create an effective electric field . this ( @xmath77 ) charge state can be optically converted back to ( @xmath78 ) state only by ultra - violet excitation . the ( @xmath77 ) charge state is a closed - shell singlet while the neutral and negatively charged siv defects have @xmath6 and @xmath67 spin states , respectively . the siv@xmath8 defect was assigned to kul1 epr center with @xmath6 state and @xmath12 symmetry.@xcite . our calculations supports this assignment ( see table [ tab : hf ] ) . the spin density is localized on four c dangling bonds in the @xmath52 orbital of divacancy . while the @xmath52 state is not localized on si impurity but the spin density from these c dangling bonds can overlap with the si atom which promotes a well measurable fermi - contact term on the @xmath1si nuclei . the hyperfine interaction with @xmath1si is almost isotropic unlike the case of @xmath9c isotopes which show typical anisotropic signal due to @xmath53 dangling bonds . .[tab : hf ] calculated and measured hyperfine constants ( @xmath79 ) for kul1 epr center and neutral silicon - vacancy defect in diamond . the kul1 epr center data was taken from ref . . previous theoretical results are taken from ref . carried out by lda 216-atom supercell calculation without taken into account the core polarization ( lda - nocp ) . the present theoretical values are obtained by hse06 functional in 512-atom supercell where the contribution of the spin - polarization of core electrons to the fermi - contact term in @xmath9c is 26% of the total ( hse06-cp ) . @xmath80 is the angle between the symmetry axis ( [ 111 ] direction ) and the parallel component of the hyperfine constant @xmath81 . [ cols="<,^,^,^",options="header " , ] @xmath82 ref . , @xmath83 ref . the calculated and measured @xmath9c hyperfine tensor agree very well . we note that the contribution of the spin - polarization of core electrons to the fermi - contact term is very significant which _ compensates _ the hyperfine interaction due to valence electrons . we found this effect also for nv and other related defects.@xcite next , we discuss the optical transitions for siv@xmath8 . intra - level transition may occur between the fully occupied @xmath51 state and the empty @xmath52 state in the spin - down channel . while the fully occupied @xmath51 state lies in the valence band a strong resonant excitation may occur from this state . in hole picture the excited state may be described as @xmath84@xmath85 while the ground state is @xmath57 . the electron configuration of @xmath57 results in @xmath86 triplet state and @xmath87 , @xmath88 singlet states . the ground state is @xmath86 . the electron configurations of @xmath84@xmath85 results in @xmath89 , @xmath90 , @xmath91 , @xmath92 , @xmath93 , and @xmath94 multiplets . among these states @xmath89 and @xmath93 states are optically allowed from the @xmath86 ground state . polarization studies of the 1.31-ev center indicates that @xmath95 transition occurs in the pl process@xcite . the @xmath89 state can be described as linear combination of slater - determinants that can not be treated within constraint dft . however , the @xmath96=1 substate of @xmath93 multiplet can be described by a single slater - determinant . the calculated excitation energy is about 1.63 ev which is significantly larger than 1.31 ev . thus , we may conclude that @xmath97 transition is not responsible for the 1.31-ev pl line . as the only other feasible transition is @xmath95 , our result indirectly confirms the experimental finding . we also study the role of the valence band resonant states similar to the case of the negatively charged defect . when the electron is promoted from @xmath72 state to the @xmath52 state in the gap then @xmath98 state is created . the calculated excitation energy of this shelving state is @xmath31.27 ev . this coincides with the small peak found in ref . , simultaneously with the peak of 1.31-ev this transition is optically allowed only when the inversion symmetry of the defect is broken . this might happen due to strain in the diamond sample . nevertheless , our calculation implies that this @xmath98 state can play an important role in the de - excitation process as a shelving or metastable state . we note that the experiments indicate@xcite that the ground state of the neutral siv defect resides at @xmath30.2 ev above the valence band edge . according to hse06 calculations the occupied single particle @xmath52 level lies at @xmath37 + 0.3 ev that agrees nicely with the implications from the experiments . in addition , we found a metastable state of the defect which may explain the temperature dependence of the 1.31-ev pl spectrum@xcite . we also propose that strain may induce a weak pl transition from this shelving state because this shelving state has the same spin as that of the ground state , and the parity selection rule may be relaxed in slightly distorted geometry of the siv defect . we further note that a very similar conclusion can be drawn for the optical excitation of the neutral siv defect as was hinted for the negatively charged defect : a hole is created in the valence band in the excitation process that may leave defect temporarily or permanently which leads to a charge conversion from neutral to negatively charged siv defect . we carried out _ ab initio _ supercell calculations on siv defect in diamond . the calculations could positively confirm the assignment of 1.68-ev and 1.31-ev pl centers to the negatively charged and neutral siv defect , respectively . the calculations reveal the high importance of the inversion symmetry of the center as well as the role of resonant valence band states in understanding the optical properties of the defect . we show that the shelving states of 1.68-ev pl center are from valence band excitations where the lowest - energy shelving state may have nir emission to the ground state in strain diamond samples . we show that holes are created in the excitation of the negatively charged and neutral siv defects that may lead to charge conversion of these centers . in addition , we show that the acceptor level of siv defect lies very deep in the band gap . as a consequence , the 1.68-ev pl center can be photo - stable when it is close to the surface of hydrogenated diamond surface . on the other hand , siv defect is double negatively charged in such diamond samples where the nv - defect is negatively charged . thus , the fermi - level should be set lower than the midgap of diamond , in order to conserve its negative charge state . discussions with christoph becher are highly appreciated . support from eu fp7 project diamant ( grant no . 270197 ) is acknowledged . j.r.m . acknowledges support from conicyt pia program act1108 . 39ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) , @noop * * , ( ) link:\doibase 10.1103/physrevb.51.16681 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.77.3041 [ * * , ( ) ] http://stacks.iop.org/0953-4075/39/i=1/a=005 [ * * , ( ) ] http://stacks.iop.org/1367-2630/13/i=2/a=025012 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.205211 [ * * , ( ) ] http://stacks.iop.org/1367-2630/15/i=4/a=043005 [ * * , ( ) ] link:\doibase 10.1039/b813515k [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1134/1.1626778 [ * * , ( ) ] link:\doibase 10.1103/physrevb.62.16587 [ * * , ( ) ] link:\doibase 10.1103/physrevb.66.195207 [ * * , ( ) ] link:\doibase 10.1103/physrevb.77.245205 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.245208 [ * * , ( ) ] link:\doibase 10.1063/1.358566 [ * * , ( ) ] link:\doibase 10.1002/pssb.201046254 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.103.186404 [ * * , ( ) ] link:\doibase 10.1103/physrevb.81.153203 [ * * , ( ) ] @noop * * ( ) link:\doibase 10.1103/physrevb.76.115109 [ * * , ( ) ] link:\doibase 10.1063/1.1564060 [ * * , ( ) ] link:\doibase 10.1063/1.2404663 [ * * , ( ) ] link:\doibase 10.1103/physrevb.50.17953 [ * * , ( ) ] link:\doibase 10.1103/physrevb.54.11169 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.77.3865 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1002/pssb.201046210 [ * * , ( ) ] link:\doibase 10.1103/physrevb.62.6158 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.71.115110 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.235104 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.045112 [ * * , ( ) ] link:\doibase 10.1103/physrevb.76.075204 [ * * , ( ) ] http://arxiv.org/abs/1310.3106 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.79.235210 [ * * , ( ) ] link:\doibase 10.1103/physrevb.85.245207 [ * * , ( ) ] | the split silicon - vacancy defect ( siv ) in diamond is an electrically and optically active color center .
recently , it has been shown that this color center is bright and can be detected at the single defect level .
in addition , the siv defect shows a non - zero electronic spin ground state that potentially makes this defect an alternative candidate for quantum optics and metrology applications beside the well - known nitrogen - vacancy color center in diamond .
however , the electronic structure of the defect , the nature of optical excitations and other related properties are not well - understood . here
we present advanced _
ab initio
_ study on siv defect in diamond .
we determine the formation energies , charge transition levels and the nature of excitations of the defect .
our study unravel the origin of the dark or shelving state for the negatively charged siv defect associated with the 1.68-ev photoluminescence center . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the question of coexistence of singlet superconductivity and magnetism has been adressed for many years . it was found that the superconducting order parameter is destroyed by a magnetic field both via the orbital effect @xcite and the paramagnetic effect.@xcite in the usual case of an isotropic three - dimensional ( 3d ) superconductor under an external magnetic field , the orbital effect prevails and leads to the well - known temperature - field phase diagram of conventional type i or ii superconductors.@xcite in contrast , superconductivity is essentially suppressed by the paramagnetic effect in presence of a ferromagnetic exchange interaction . this is also true for quasi - two - dimensional ( 2d ) superconductors under in - plane magnetic field and for heavy fermions materials wherein the orbital effect is partially quenched . in the whole paper , the magnetism is characterized by an internal exchange field @xmath1 ( given in energy units ) which may arise either from an externally applied magnetic field or from ferromagnetic ordering . note that ferromagnetism must be weak in order to avoid complete suppression of superconductivity . this is realized in rare - earth metals or actinides in which the indirect exchange interaction leads to curie temperatures of a few degrees . superconductors with internal homogeneous exchange field @xmath1 exhibit a very special behaviour . according to chandrasekhar@xcite and clogston,@xcite at zero temperature uniform superconductivity should be destroyed when the polarization energy of the free electron gas exceeds the energy gain due to cooper pairing in the bcs ground state . this criterion gives the exchange field @xmath2 where the superconductor should undergo a first - order transition to the normal state , @xmath3 being the zero temperature superconducting gap . larkin and ovchinnikov @xcite and fulde and ferrell @xcite ( fflo ) predicted the existence of a nonuniform superconducting state with higher critical exchange field @xmath4 and second - order transition to the normal state . this prediction was made for 3d superconductors . in quasi-2d superconductors the critical exchange field of the fflo state is even higher , namely @xmath5,@xcite while in quasi - one - dimensional systems there is no paramagnetic limit at all.@xcite the appearance of the modulated fflo state is related to the pairing of electrons with opposite spins which do not have the opposite momenta anymore due to the zeeman splitting . from now on we focus on the 2d case for which a generic temperature - exchange field phase diagram has been established.@xcite at low field and temperature , the ground state is characterized by a uniform superconducting order parameter . a tricritical point , located at @xmath6 and @xmath7 , is the meeting point of three transition lines separating the normal metal , the uniform and the nonuniform superconductors . at @xmath8 , the ( low - field ) uniform superconductor is separated from the ( high - field ) normal metal by a narrow fflo nonuniform superconducting phase . in contrast , at @xmath9 , the system undergoes merely a second - order phase transition from the uniform superconductor to the normal metal when increasing the exchange field . the nonuniform fflo state is settled in a small region of the phase diagram and is very sensitive to impurities,@xcite making it difficult to observe experimentally . nevertheless , several evidences of the fflo state have been obtained recently in organic superconductors @xcite and in heavy fermions compounds , see martin _ _ et al.__@xcite and references therein . in the context of organic and high-@xmath10 superconductors , layered systems made of conducting atomic planes have been extensively studied.@xcite in order to investigate the interplay of superconductivity and magnetism in such anisotropic systems,@xcite andreev _ et al . _ considered a periodic array of alternating ferromagnetic and superconducting 2d planes.@xcite solving the corresponding gorkov equations , these authors established the existence of a @xmath0 state wherein each f layer separates superconducting planes with opposite order parameter . this is relevant for the ruthenocuprate compound rusr@xmath11gdcu@xmath11o@xmath12 which comprises cuo@xmath11 superconducting planes and ruo@xmath11 magnetic planes.@xcite a related system is an isolated f / s / f trilayer which exhibits the so - called superconducting spin - valve effect . namely , its critical temperature is higher in the antiparallel ( ap ) orientation of the layers magnetizations than in the parallel ( p ) orientation both for thick layers@xcite and atomic size layers.@xcite surprisingly , in the atomic thickness limit , the superconducting gap at zero temperature is higher for p orientation of the magnetizations.@xcite hence one expects a transition from ap to p orientation by cooling the system below a finite crossing temperature . the recent progress in molecular beam epitaxy@xcite enables to fabricate such f / s / f trilayer with atomic thicknesses . in this paper , we consider a periodic array of sf bilayers . each bilayer is made of two atomic planes coupled by single electron tunneling . both exchange fields and bcs superconducting pairing are present in each sf plane . the possibility of @xmath13 phase difference between the planes inside each bilayer is also taken into account . in the whole paper , we assume that the coupling @xmath14 between successive bilayers is considerably weaker than the intra - bilayer coupling @xmath15 . our study is performed within the framework of the bcs theory of s - wave superconductivity . solving exactly the gorkov equations in the limit @xmath16 , we first derive the critical temperature and the superconducting gap both for parallel ( p ) and antiparallel ( ap ) orientation of the magnetizations . we show that the critical temperature is higher for the ap orientation than for the p orientation whereas it is the opposite for the zero temperature gap . we also investigate the interlayer josephson current in the small coupling limit : the current increases as a function of the exchange field for ap orientation whereas it is field - independent for the p orientation . furthermore , we find that for low exchange fields and high temperatures , the ground state corresponds to identical superconducting order parameters on adjacent layers . for high enough fields and/or low enough temperatures , the @xmath0 phase ground state is favoured and compete with the fflo state . for the p orientation , the full temperature - exchange field phase diagram is constructed in the two limits of extremely low and high coupling between the planes . as expected , for perturbative coupling between two sf planes , the phase diagram is very close to the quasi-2d superconductor s phase diagram . nevertheless an important change arises . indeed a new @xmath17phase is inserted inside the usual fflo phase close to the tricritical point . for higher tunneling coupling @xmath18 , this @xmath17 phase is pushed to low temperatures @xmath19 and high fields @xmath20 . in this unusual superconducting phase , the zeeman splitting is compensated by the bonding / antibonding energy splitting due to single - electron tunneling between the planes.@xcite as a result , field - induced superconductivity and enhanced paramagnetic limit are realized in this simple model . these new phenomena are encountered due to the introduction of an additional discrete degree of freedom , here the layer index @xmath21 . the layer index acts as a pseudo - spin and thus enlarges the usual spin - space for singlet pairing . this idea was introduced by kulic and hofmann@xcite in the context of two - bands superconductivity for which the pseudo - spin was the band index . nevertheless , these authors did not investigate the presently studied @xmath0 state . the outline of the paper is the following . in sec.ii , we present the model , derive the corresponding gorkov equations and give their exact solutions . in sec.iii , we investigate the critical temperature , the gap and the interlayer josephson current in the small exchange field regime for which there are only uniform superconducting phases . in the last two sections the temperature - exchange field phase diagram of the bilayer is studied thoroughly . in sec.iv , we first construct a ginzburg - landau functional to determine the transitions between the different phases in the low interlayer coupling limit . sec.v is devoted to the opposite limit of strong interlayer coupling . in conclusion , we discuss the conditions for the observation of field - induced superconductivity . we consider a superconducting ferromagnetic bilayer ( see fig.1 ) constituted of two superconducting atomic layers , labeled as @xmath22 and @xmath23 . in the whole article , we assume @xmath24 where @xmath25 is the interlayer coupling energy and @xmath26 the fermi energy . as a consequence , cooper pairs are localized within each plane.@xcite each layer @xmath21 supports a superconducting singlet bcs coupling with the energy gap @xmath27 and an internal exchange field @xmath28 . the hamiltonian of the system can be written as @xmath29 + h_{t}\text { , } \label{hamiltonien}%\ ] ] where @xmath30 is the attractive bcs interaction constant and @xmath31 is the two - dimensional coordinate within each layer . for the layer @xmath21 the kinetic and zeeman parts of the hamiltonian are written together as @xmath32 in which summation over repeated spin indexes @xmath33 and @xmath34 is implied . creation ( resp . annihilation ) operator of an electron with spin @xmath33 and two - dimensional momentum @xmath35 in the layer @xmath21 is denoted @xmath36 ( resp . @xmath37 ) . the exchange fields @xmath28 are assumed to be either equal ( @xmath38 ) or opposite ( @xmath39 ) . as a consequence the matrix @xmath40 is spin - diagonal , and the zeeman effect manifests itself in breaking the spin degeneracy of the electronic energy levels according to @xmath41 \text{,}%\ ] ] where @xmath42 . the s - wave singlet superconductivity is represented by the standard mean - field hamiltonian @xmath43 \text{,}%\ ] ] and the layers are coupled together by the hopping hamiltonian @xmath44 \text { .}%\ ] ] in order to investigate the occurence of modulated superconducting phases ( fflo ) , we choose the following spatial dependence for the superconducting order parameter @xmath45 where @xmath46 is the fflo modulation wave vector and @xmath47 the superconducting phase difference between the layers . the above model can be solved exactly using the green functions @xmath48 where @xmath21 and @xmath49 are the layer s indexes . the brackets mean statistical averaging over grand - canonical distribution.@xcite we obtain the following gorkov equations in the fourier representation:@xmath50 where @xmath51 are the fermionic matsubara frequencies . in quasi-2d superconductors @xcite the maximal fflo modulation amplitude is of the order of @xmath52 , @xmath53 being the typical superconducting coherence length . this means that with a good approximation we can consider @xmath54 , @xmath55 being the fermi velocity vector in the plane . solving the gorkov equations ( [ equationgorkov ] ) yields the anomalous gorkov green function for the @xmath22 sf layer @xmath56 where @xmath57 with @xmath58 and @xmath59 similar equation holds for @xmath60 note that in the case where a lattice made of such sf / sf bilayers is considered , the generalized anomalous green function is obtained by replacing @xmath61 by @xmath62 in eq.([fcroix ] ) , @xmath63 being the projection of the momentum @xmath35 along the @xmath64 axis and @xmath65 the period of the lattice . as a consequence , a finite inter - bilayer coupling @xmath66 introduces an anisotropy in the dispersion relation which leads to a broadening of the electronic excitation levels . in the absence of tunneling @xmath67 we retrieve from eq.([fcroix ] ) the anomalous green function of a quasi-2d superconductor with the exchange field @xmath68 @xmath69 although the dependence on momentum has been removed for simplicity , notice that @xmath70 and @xmath71 the set of basic equations ( [ equationgorkov ] ) must be completed by the self - consistency equation @xmath72 close to the critical temperature @xmath10 of the second - order phase transition , the order parameters @xmath27 are small and eq.([self ] ) can be written as @xmath73 where @xmath74 is the critical temperature for the 2d superconducting single layer in the absence of exchange field , namely for @xmath75 . at zero temperature , it is convenient to write eq.([self ] ) as @xmath76 where @xmath77 is the superconducting order parameter at @xmath78 in the absence of exchange field and interlayer coupling . in this section , we investigate phases with uniform superconductivity within each layer . we obtain the critical temperature of the second - order superconducting ( @xmath79 ) to normal metal ( @xmath80 ) phase transition and the order parameter @xmath81 as a function of the temperature , the exchange field @xmath1 and the interlayer coupling @xmath25 . we consider both parallel ( p ) and antiparallel ( ap ) orientations of the magnetizations , the superconducting phase difference being either @xmath82 or @xmath13 . we also calculate the josephson interlayer current when the bilayer is connected to external superconducting leads . most of these results are obtained in the perturbative limit of small coupling between the layers @xmath83 . the field is also assumed to be sufficiently small to prevent the occurence of a spatial modulation of the superconductivity within the planes . study of nonuniform phases and strong coupling @xmath84 are respectively postponed to sec . iv and sec . v. we consider the second - order phase transition between the normal metal and the uniform bcs superconductor . thus the order parameters @xmath85 and @xmath86 are small and the anomalous gorkov green function ( [ fcroix ] ) can be linearized in the following form @xmath87 where @xmath88 . similar equation may be found for @xmath89 . we first consider _ the parallel ( p ) orientation of the magnetizations , _ namely _ @xmath90_. the first possibility is @xmath82 wherein the layers have the same superconducting order parameters @xmath91 . in this situation the anomalous green function obtained from eq.([fcroixlinearisee ] ) is denoted @xmath92 . the identity @xmath93 and eq.([selftnonnul ] ) yield the following implicit equation for the critical temperature @xmath94 @xmath95 where @xmath96 denotes the euler digamma function . therefore the interlayer coupling disappears from the self - consistency equation and eq.([tcp0 ] ) is identical to that for the 2d monolayer in a uniform exchange field : the bilayer is equivalent to a single layer in the neighborhood of superconducting to normal state transition.@xcite the critical temperature of the bilayer decreases when the exchange field @xmath1 increases . the equation ( [ tcp0 ] ) describes the second - order phase transition between the normal metal and the uniform superconductor which is realised only for fields smaller than the tricritical one @xmath6 . for larger fields , superconductivity becomes nonuniform . a second possibility is _ the p orientation with @xmath13 phase difference between the layers_. now the anomalous gorkov green function is denoted @xmath97 . then @xmath98 , \ ] ] and the self - consistency relation eq.([selftnonnul ] ) yield a critical temperature @xmath99 given by @xmath100 from this expression one may notice that superconductivity in the @xmath0 state is destroyed by a combination of two effective exchange fields @xmath101 . in the small interlayer coupling limit @xmath102 , eq.([tcppi ] ) becomes @xmath103 where the function @xmath104 is defined and represented in appendix a. in the regime of low magnetic fields , namely for @xmath105 , the factor @xmath106 is positive and thus the critical temperature is smaller in the @xmath0 superconducting state than in the @xmath107 state . however the situation may be inverted if @xmath108 . moreover , along the critical line , the value @xmath109 corresponds to the tricritical point , @xmath110 and @xmath111 , where fflo nonuniform states appear . as a consequence one expects competition between the @xmath0 superconducting phase and fflo phases in the neighborhood of the tricritical point . this competition will be detailed in sec . iv . let us focus on the case of _ ap orientation @xmath112_. following the same procedure as previously , the equations for the critical temperatures @xmath113 are obtained . in the limit @xmath83 , it reads @xmath114 for @xmath82 , and @xmath115 for @xmath13 . from eqs.([tcap0petitt],[tcappi ] ) the critical temperature is clearly higher in the @xmath107 phase than in the @xmath0 phase . therefore the @xmath107 phase is the more stable in this region of the @xmath116 phase diagram , _ i.e. _ in the vicinity of the critical temperature and for low fields @xmath117 in conclusion , the bilayer is always in the @xmath107 superconducting state for temperatures close to the critical temperature , whatever the relative orientation of magnetizations is . a spin - valve effect is also present : the critical temperature is higher for the ap orientation than for the p orientation of the magnetizations . for @xmath67 and low fields @xmath118 it is well - known that the zero temperature gap @xmath119 is field - independent.@xcite for small interlayer coupling @xmath120 , the anomalous green function ( [ fcroix ] ) may be expanded to the second order in @xmath25 as @xmath121 where the full nonlinear dependence on @xmath122 is kept in @xmath123,@xmath124 and @xmath125 then self - consistency relation ( [ selft=0 ] ) becomes @xmath126 where @xmath127 and @xmath82 or @xmath13 . using the preceding equation in _ the p orientation _ we obtain @xmath128 , either for @xmath107 or @xmath0 phase difference . as a result , the superconducting gap @xmath129 is not affected by a small interlayer coupling , at least at the order of @xmath61 . the superconducting condensation energy gain has also been calculated and the zero state found to be more stable than @xmath0 state . _ for the ap orientation _ , the superconducting gap @xmath130 is given by @xmath131 \text { , } \label{gapapzero}%\ ] ] for _ zero phase difference . _ the unphysical divergence at @xmath132 is removed by terms of higher order in @xmath25 . expression ( [ gapapzero ] ) is the main result of this paragraph and reduces to @xmath133 in the small field regime @xmath134__. _ _ therefore in the @xmath82 state and for ap orientation , the order parameter is suppressed by the exchange field in the small coupling limit . this is surprising because ap orientation was expected to weaken the effective exchange field and thus enhance superconducting properties . nevertheless such a decrease of the superconducting order parameter has already been found in a ballistic atomic - scaled f / s / f trilayer.@xcite _ for the ap orientation and _ @xmath135 _ phase difference , _ the gap @xmath136 is field and coupling independent . moreover the energy of the @xmath13 state does not depend on the relative orientation of the magnetizations . to summarize , the lowest energy corresponds to the @xmath137 phase . the @xmath138 and @xmath139 phases are degenerate with a somewhat higher energy than the @xmath137 phase . although we have not performed the energy calculation in the case where the magnetizations are antiparallel and the phase difference is @xmath107 , we believe that the highest energy corresponds to the @xmath140 phase since its order parameter is the smallest one . we now extend our investigation of the superconducting gap to finite temperatures . in order to determine the gap @xmath81 as a function of temperature @xmath141 , exchange field @xmath1 and coupling @xmath25 , we analyse numerically the self - consistency relation ( [ selft=0 ] ) using the exact anomalous gorkov green function ( [ fcroix ] ) . the result is shown schematically in fig.2 . _ for the p orientation and @xmath82 _ , the superconducting gap @xmath142 is the same as the gap @xmath143 of a single layer whereas _ for @xmath13 _ the superconducting gap @xmath144 is lowered by finite interlayer coupling . _ for the ap orientation and @xmath82 _ , the gap is smaller than @xmath143 for @xmath145 and larger for @xmath146 where the inversion temperature @xmath147 depends only on the exchange field in the small interlayer coupling limit ( see fig.2 inset ) . this phenomenon has been called inversion of the proximity effect.@xcite moreover , the gap @xmath148 is larger than @xmath149 for all temperatures . according to these results , one may suggest several experiments . first we consider a bilayer with magnetizations pinned in the ap mutual orientation . by lowering the temperature , a @xmath107-@xmath0 transition is expected at some temperature @xmath150 . in the small interlayer coupling limit , this temperature @xmath151 is a function of the exchange field only ( see fig.2 inset ) . in contrast , the @xmath107 state is more favorable energetically for all temperatures in the case of magnetizations pinned in the p orientation . as another illustration we consider samples where the relative orientation of magnetizations is free . then the orientation is chosen by the system to minimize its energy . cooling such a bilayer will result in a switching from the ap orientation to the p orientation at the inversion temperature @xmath152 . the same prediction was made recently in a ballistic f / s / f trilayer.@xcite here we consider that the sf / sf bilayer is connected to superconducting electrodes . in this set - up , one may impose an arbitrary superconducting phase difference @xmath47 between the sf layers , and thus a non dissipative josephson current flows through the bilayer in the direction perpendicular to the planes . this interlayer josephson current is evaluated here in the tunneling limit @xmath153 and at zero temperature . within the green functions formalism , the general formula for the interlayer josephson current is @xmath154 where @xmath155 is the two - dimensional density of state per spin direction and unit surface . solving exactly the gorkov equations ( [ equationgorkov ] ) leads to @xmath156 the function @xmath157 is obtained from eq.([g21 ] ) by permuting the layer indexes @xmath158 the corresponding anharmonic current - phase relationship is given by@xmath159 in the tunneling limit @xmath160 the interlayer josephson current becomes sinusoidal as a function of the phase difference , @xmath161 where @xmath162 . the second harmonic @xmath163 has also been evaluated and is smaller than the first one by a factor @xmath164 . the preceding equation ( [ courant1 ] ) yields the current - phase relation both for parallel ( p ) and antiparallel ( ap ) orientation of magnetizations . _ in the parallel case _ the critical current does not depend on the field as already reported in other systems since @xmath165.@xcite _ for the antiparallel orientation _ and to the lowest order in @xmath25 , the current - phase relation reads @xmath166 where @xmath167 @xmath168 . therefore the critical current increases with the exchange field @xmath1 and even diverges for @xmath169 of course this divergence is unphysical and should disappear if all orders in @xmath25 were taken into account . in fig.3 , the critical current is shown as a function of the exchange field both for p and ap orientations . recently , the issue of the josephson coupling between two clean sf layers through an insulating layer was considered using eilenberger equations @xcite or bogoliubov - de gennes formalism.@xcite similar results as ours were obtained : the critical current increases with @xmath1 only if three conditions are met : low temperature , very weak coupling between the sf layers and ap orientation . otherwise the presence of an exchange interaction suppresses the josephson current . using usadel equations , krivoruchko demonstrated that this statement holds in the diffusive regime for which the divergence for @xmath170 is remplaced by a regular peak.@xcite from now on , we consider the sf / sf bilayer only for the parallel ( p ) orientation . hence the results obtained in the next sections may be also applied to a superconducting bilayer in an external in - plane magnetic field . the present section is devoted to the weak coupling regime @xmath171 in contrast to the low field restriction of sec . iii , regions of the phase diagram with @xmath172 are also investigated here . then competition between the fflo and @xmath0 phases is expected to take place . particular attention is paid to the vicinity of the tricritical point given by @xmath173 and @xmath174@xcite in order to examine this narrow region of the @xmath175 plane , we construct a ginzburg - landau ( gl ) functional from the gorkov equations used in the previous sections . in the past buzdin and kachkachi @xcite derived a generalized ginzburg - landau ( gl ) functional for a single sf layer that describes the fflo superconducting state near the tricritical point . here we extend this functional to a sf / sf bilayer for which it is possible to have not only fflo modulation within the planes but also @xmath13 superconducting phase difference between the planes . for @xmath82 , the physics of the bilayer is independent of the coupling and thus the buzdin - kachkachi gl functional is retrieved . in contrast for @xmath13 , we obtain a free energy functional which depends on the interlayer coupling @xmath25 and leads to the presence of a superconducting @xmath0 phase . the free energy of the sf / sf bilayer in a uniform superconducting state with @xmath82 ( @xmath176 state ) is given by ( see details in appendix b ) @xmath177 with @xmath178 where @xmath179 and @xmath180 are respectively the reduced exchange field and the reduced order parameter . here we retrieve the well - known case of a single sf layer . for reduced exchange fields lower than the tricritical one @xmath181 , the transition between the superconducting and the normal states is a second - order one since @xmath182 . the critical line is given by the equation @xmath183 for higher fields , the transition becomes a first - order one because @xmath184 and @xmath185 . as for any first - order transition , two conditions must be fullfilled . on one hand , the free energy ( [ freezero ] ) is minimized , ( @xmath186 , and on the other hand the free energies of the superconducting and normal phases are equal , @xmath187 hence in the @xmath175 plane the equation for this first - order line is @xmath188 @xmath189 , and the jump of the superconducting gap at the transition is given by @xmath190 . it is well - known that this scenario is not realized because it is replaced by a transition between normal metal and nonuniform superconductivity.@xcite nevertheless , this first - order transition provides a useful energy scale @xmath191 and a reference line in the @xmath175 plane that will be used to construct a universal phase diagram , namely a @xmath25-independent phase diagram valid for any weakly coupled sf / sf bilayers , see sec . the sf / sf bilayer may support opposite order parameters on the layers , superconductivity being still uniform within each sf plane . in this so - called @xmath192 state , the free energy of the bilayer depends on the reduced interlayer coupling @xmath193 according to @xmath194 for low reduced fields @xmath195 @xmath196 . hence the uniform superconducting phase with @xmath82 is more stable than the @xmath0 phase , as already found in the sec . interestingly for higher reduced fields @xmath197 this @xmath0 phase is in competition with fflo nonuniform superconducting phases having either @xmath82 ( @xmath198 ) or @xmath13 ( @xmath199 ) . according to buzdin and kachkachi the order parameter @xmath200 leads to the lowest energy.@xcite for the @xmath198 phase , the corresponding free energy reads @xmath201 whereas for the @xmath199 phase , the free energy depends on the interlayer coupling @xmath25 in the following manner @xmath202 the notation @xmath203 is introduced in appendix b. now we proceed to analyse the above free energies in order to determine the critical line between the normal and the superconducting states . we will also describe the nature of the various superconducting states and what kinds of transitions are encountered . we focus on the vicinity of the tricritical point . then @xmath204 is a linear function of the exchange field with @xmath205 , whereas @xmath185 is nearly field and temperature independent . for high reduced fields @xmath206 , it appears that the @xmath176 and the @xmath199 never lead to the highest critical temperature . then we emphasize the competition between the two remaining phases , namely @xmath192 and @xmath207 the minimization of the gl functional ( [ fflozero1 ] ) leads to a modulation wave vector given by @xmath208 in the limit @xmath209 , _ i.e. _ near the critical line . the @xmath192 phase is more stable than the @xmath198 phase under the energy condition @xmath210 or equivalently for @xmath211 therefore the uniform @xmath0 phase is `` inserted '' within the usual fflo superconducting state . the upper and lower values of @xmath212 between which this new @xmath0 phase is stable depend on the particular value of the coupling @xmath25 . it is convenient to define a dimensionless generalized coordinate @xmath213 @xmath214 that quantifies the `` distance '' from the tricritical point along the s / n transition line . indeed @xmath215 at the tricritical point and the @xmath0 phase is settled in the region @xmath216 as shown on fig . 4 , going along the critical line from low to high fields , one expects the sequence of superconducting states : uniform in the planes with @xmath82 for @xmath217 , fflo modulation along the planes with @xmath82 for @xmath218 then uniform @xmath0 state for @xmath219 and finally fflo modulation along the planes with @xmath82 for @xmath220 . in all cases , the sign of the @xmath221 coefficient in the gl free energy is always positive and thus transitions between these superconducting states and the normal metal are second - order ones . we now construct the phase diagram around the tricritical point . because each value of the coupling leads to different transition lines , we introduce the following mapping of the thermodynamic variables @xmath223 in order to obtain a universal phase diagram valid in the small coupling regime . this mapping makes use of the energy scale @xmath191 and of the function @xmath224 related to the first - order transition between the normal state and the uniform superconducting state , see sec . iv.a . then the free energies for the uniform superconducting phases eqs.([freezero],[freepi ] ) become @xmath225 and @xmath226 where @xmath227 first it is straightforward to minimize @xmath228 and @xmath229 with respect to @xmath230 . then replacing the reduced gap @xmath230 by its equilibrium value , one obtains the equilibrium energies @xmath231 and @xmath232 of each superconducting phase . these energies are functions of both field and temperature via the dimensionless thermodynamical variables @xmath233 and @xmath234 . using the same scaling eq.([scaling ] ) the free energies of the @xmath198 phase , @xmath235 and of the @xmath199 phase , @xmath236 are also obtained for the order parameter @xmath237 . the equilibrium energies @xmath238 and @xmath239 of the modulated phases are obtained after minimization of eqs.([energyfflozero],[energyfflopi ] ) with respect to @xmath230 and @xmath240 .then eq.([energyfflozero ] ) enables to study the second - order phase transition between the normal metallic state and the nonuniform fflo state@xmath241 under the assumption of second - order phase transition , it is sufficient to consider the gl free energy up to the @xmath242 order . then the free energy ( [ energyfflozero ] ) is minimal for @xmath208 . for this particular modulation , the critical fflo / n line is given by @xmath243 we now consider the transition lines between the various superconducting states obtained in the previous paragraph , in particular the @xmath198/@xmath192 , the @xmath198/@xmath176 and the @xmath244 transitions . let us focus on the transition between the uniform phases @xmath176 and @xmath192 . solving @xmath245 , we obtain the critical line @xmath246 which corresponds to a first order @xmath247 phase transition . however , this transition is not realized ( see fig.4 ) because the transition to the nonuniform superconducting state occurs before . the @xmath248 transition line is obtained in a similar way . the equation @xmath249 has the solution @xmath250 . at this value of @xmath233 , the system undergoes a first order phase transition from the uniform state to the modulated fflo state . adding higher harmonics to the order parameter @xmath251 gives a more accurate evaluation , namely @xmath252 . finally , the @xmath253 transition line is obtained from @xmath254 and shown in fig.4 in the @xmath255 plane . this transition is a first - order one . using the mapping @xmath256 of the thermodynamical variables , we have obtained a universal phase diagram fig.4 of all weakly coupled sf / sf bilayers in the vicinity of the tricritical point . an important feature of this phase diagram is the presence of a superconducting @xmath257-phase . as an example , the phase diagram has been redrawn in the @xmath175 plane on fig.5 for a particular value of the coupling @xmath25 . now we consider the ballistic sf / sf bilayer in the regime of strong interlayer coupling limit @xmath84 and low temperature . a very unusual @xmath257-superconducting state is found between a lower @xmath258 and an upper @xmath259 critical exchange field , and below a maximal temperature of the order of @xmath260 . therefore field - induced superconductivity is obtained above @xmath261 within the bcs theory of superconductivity . the underlying physical mechanism is the compensation of the zeeman splitting by the energy splitting between bonding and antibonding electronic states of the bilayer , see fig.6.@xcite thus the new zero temperature paramagnetic limit @xmath259 may be tuned far above the usual one @xcite @xmath262 merely by increasing the interlayer coupling . this compensation also occurs for small coupling , but the @xmath257-superconducting state is then less energetically favorable than the usual @xmath107-superconducting phase as demonstrated in sec.iv . therefore the @xmath175 phase diagrams are topologically distinct in the opposite limits of small ( sec.iv ) and strong ( sec.v ) coupling . we first analyse the second - order superconducting / normal phase transition in sec.v.a . then the first - order transition between uniform superconductivity and the normal state is discussed in sec.v.b . here we study the second - order phase transition between the @xmath0 superconducting state and the normal metal state , as a function of the field . we start from the the linearized anomalous green function ( [ fcroix ] ) for arbitrary coupling @xmath25 and @xmath0 superconducting phase difference , @xmath263 .\left [ t^{2}% -(i\omega+h+\xi)^{2}\right ] } \text{. } \label{fcroixgrandt}%\ ] ] from this equation and the self - consistency relation ( [ self ] ) , the critical exchange field @xmath1 is shown to satisfy @xmath264 where @xmath265 , and @xmath266 is the critical exchange field for the second - order superconducting phase transition in a two - dimensional monolayer . one must then find the value of @xmath267 which maximizes the critical field @xmath268 . if the @xmath0 phase is assumed to be uniform inside each plane , namely if @xmath269 , eq.([hcritfflo1 ] ) merely reduces to @xmath270 the lower and upper critical fields are respectively given by @xmath271 , in the limit @xmath272 . thus at zero temperature and strong enough coupling , the superconductivity destruction follows a very special scenario . at low fields , superconductivity is first suppressed as usual at the paramagnetic limit @xmath273 leading to the normal metal phase . then further increase of the field leads to a normal to superconducting phase transition at the lower critical field . this superconducting @xmath0 phase is finally suppressed at the upper critical field . this is a new paramagnetic limit which may be tuned far above the usual one merely by choosing the coupling @xmath25 greater than @xmath274 . thorough analysis of eq.([hcritfflo1 ] ) shows that the upper critical field is even increased by an in - plane modulation in analogy with the case of the two - dimensional fflo phase.@xcite the upper critical field is maximal for the choice @xmath275 , and then eq.([hcritfflo1 ] ) reduces to @xmath276 that gives the upper and lower fields @xmath277 in the @xmath278 limit . note that the period of the modulated order parameter @xmath279 is larger than the corresponding period in the two - dimensional fflo phase which coincides with the ballistic coherence length @xmath280 @xcite furthermore one may derive the full temperature - field phase diagram using eqs.([selftnonnul],[fcroixgrandt ] ) and the result is shown in fig.7 . when the temperature is increased , the lower critical field increases whereas the upper one decreases . along the upper ( resp . lower ) critical line the fflo modulation is lost at some temperature @xmath281 ( resp . @xmath282 ) . for higher temperatures a uniform @xmath0 phase ( @xmath192 ) is recovered and the temperature dependence of the critical field is given by @xmath283 , \label{hcriticaltemperature}%\ ] ] where @xmath284 is the digamma function and @xmath285 , @xmath286 being the euler constant . finally the lower and upper critical lines merge at field @xmath287 and temperature @xmath288 in the limit @xmath289 . therefore the field - induced @xmath0 superconductivity is confined to temperatures lower than @xmath290 . the structure of these @xmath192 and the @xmath199 phases is reminiscent of the corresponding @xmath176 and the @xmath198 phases although the former are shifted to higher fields and lower temperatures than the later . above results were obtained for relatively strong coupling . for lower coupling @xmath291 , the @xmath192 and the @xmath199 phases merge continuously into the usual @xmath82 phases as shown in fig.8 , and finally disappear for @xmath25 slightly smaller than @xmath274 . from an experimental point of view , one might choose a system with intermediate coupling @xmath25 small enough to settle the @xmath0 phase island in an available range of temperatures but also large enough to separate the @xmath0 phase island from the usual superconducting phases with @xmath82 . in the general sf multilayer case the inter - bilayer coupling constant @xmath66 needs to be sufficiently high to prevent from superconductivity destruction by 2d fluctuations but also sufficiently low to preserve the effect of field - induced superconductivity.@xcite in the following we investigate the first - order @xmath292 transition to determine whether it is more or less favorable than the above studied second - order transition . the zero - temperature superconducting order parameter @xmath293 is calculated from the self - consistency equation ( [ selft=0 ] ) for p orientation of magnetizations and @xmath13 phase difference . at zero temperature , the difference between the energy @xmath294 of the superconducting state and the energy @xmath295 of the normal metal state is given by @xcite@xmath296 d\delta\label{energieself}%\ ] ] in the limit @xmath297 we retrieve the well - known case of the single sf layer.@xcite then the self - consistency relation ( [ selft=0 ] ) admits two branches of solutions . the lower branch @xmath298 , labelled ( 2 ) in the inset of fig.9 , corresponds to a positive energy cost @xmath299 . thus this superconducting solution is never realized . the actual superconducting gap is given by the upper horizontal branch @xmath300 , ( 1 ) in the inset of fig.9 , which corresponds to the energy difference @xmath301 hence the superconducting phase is settled for low fields @xmath302 with a field - independent order parameter @xmath303 . for higher fields @xmath304 , the system is in the normal phase @xmath305 . finally the zero temperature gap exbibits a jump at @xmath306 which reveals the first - order transition from the uniform superconducting phase to the normal phase . in the opposite limit of strong interlayer coupling , we have obtained in sec.v.a . that field - induced superconductivity with @xmath13 phase difference occurs for fields close to @xmath25 and at low temperatures . from the self - consistency equation ( [ selft=0 ] ) one obtains several possible solutions for the zero - temperature superconducting gap @xmath307 as a function of the exchange field @xmath1 , see fig.9 . for relatively low fields @xmath308 and for high fields @xmath309 , the bilayer is in the normal phase @xmath310 the limiting fields @xmath311 are solutions of @xmath312 for intermediate fields ranging between @xmath313 and @xmath314 there are three superconducting branches . two of them , ( 2 ) and ( 2 ) are never realized owing to their energy cost @xmath315 the third branch ( 1 ) requires more detailed analysis . namely , it is given by the equation @xmath316 and the corresponding energy cost is @xmath317 \nonumber\\ & + \frac{\pi(h - t)^{2}}{2}. \label{energiemoyen}%\end{aligned}\ ] ] analysis of these equations reveals that @xmath299 is negative for @xmath318 where @xmath319 and @xmath320 hence the sf bilayer undergoes first - order transition at @xmath321 and @xmath322 . this scenario is quite similar than the one for @xmath67 , but with a smaller order parameter jump at the transition . moreover there are two first - order transitions , respectively at @xmath323 and @xmath324 instead of one at @xmath325 . in order to generalize the above gap calculations to finite temperatures and determine the first - order s / n transition line , we have solved numerically together the self - consistency equation ( [ selftnonnul ] ) and the condition @xmath326 . the result is given in the inset of fig.7 . collecting results from sec.v.a and b. we obtain the full @xmath175 phase diagram for the field - induced @xmath0 superconductivity . note that this @xmath0 superconductivity reproduces the structure of the phase diagram in quasi-2d superconductors @xcite although it is shifted to higher fields and lower temperatures . in this paper we have studied a periodic array of sf / sf bilayers in the limit of small coupling between the different bilayers . the corresponding gorkov equations have been solved exactly , taking into account both in - plane fflo modulation and arbitrary superconducting phase difference between sf layers . the superconducting state with zero phase difference is always settled in the low field regime , @xmath327 for parallel ( p ) orientation of the magnetizations . for antiparallel ( ap ) orientation , the @xmath0 state predominates at low temperatures over the @xmath107 state which is settled in the neighborhood of the critical line . consequently if the system is pinned in the antiparallel orientation , we predict a transition from the usual @xmath82 superconducting state to the @xmath0 state by cooling . while the critical temperature is higher for the ap orientation , the zero temperature order parameter is larger for the p orientation . this results in a crossing temperature @xmath152 below which the p orientation is more suitable for superconductivity . this temperature has been calculated as a function of the exchange field . in an experiment where the magnetizations might be easily reversed , one therefore expects a transition from the ap to the p orientation by cooling the system below this crossing temperature . in the low interlayer coupling limit , a ginzburg - landau functional has been derived from the exact expression of the anomalous gorkov green function . as a main result , we have obtained a @xmath0 superconducting state located in the vicinity of the tricritical point @xmath328 . details of the bilayer phase diagram are obtained in this framework , including the first - order transition lines between superconducting phases . since increasing the interlayer coupling enlarges the @xmath0 phase region , experimental observation of such details of the phase diagram requires the use of sf layers with large enough interlayer coupling , namely @xmath329 . finally the case of even stronger interlayer coupling , namely @xmath330 has been also investigated . it appears that at low temperatures the @xmath0 superconducting state is settled for exchange fields of the order of @xmath25 , which are well above the chandrasekhar - clogston paramagnetic limit . thus this new paramagnetic limit may be tuned by varying the interlayer coupling . in the present article we have reported the detailed structure of the phase diagram in this regime of high magnetic field . the first - order @xmath292 transition line is also derived . we expect that our results may be applicable to compounds like bi@xmath11sr@xmath11cacu@xmath11o@xmath12 under a magnetic field . indeed such perovskite superconductors comprise tightly coupled superconducting cuo planes separated by bio layers . however observing the field - induced superconductivity in a reasonable range of magnetic field requires relatively low critical temperatures which are realized in the heavily doped or underdoped regimes . finally the latter effect is solely related to the compensation of the energy shift in the two layers systems by the zeeman splitting . so it should be quite general and might appear also in two band superconductors or in weakly coupled superconducting grains . note that the inhomogeneous superconductivity has been obtained in the absence of magnetic field in two - bands superconductors.@xcite however since the @xmath0 state was not considered in this latter work no field - induced superconductivity had been noticed . bulaevskii @xcite studied thoroughly josephson coupling in periodic layered structures with one sf plane as unit cell . here we have demonstrated that systems with several sf planes as unit cell exhibit qualitatively new phenomena like field - induced superconductivity . the simplest case , two planes per unit cell , has been studied here . it may be regarded as a basic approach to understand the properties of more complex ferromagnetic superconducting compounds or artificial heterojunctions . we thank m. daumens , m. faure , m. houzet and m. kulic for useful discussions and comments . this work was supported , in part , by esf pi - shift program . we define the function @xmath332 by @xmath333 , \ ] ] and for any integrer @xmath334 the function @xmath331 is given by@xmath335 the variations of @xmath336 , @xmath337 , @xmath338 , with @xmath212 are represented in fig.10 . one can notice that in the vicinity of the tricritical point , _ i.e. _ @xmath339 , the functions @xmath332 and @xmath340 are negative and of the order of unity . @xmath341 cancels exactly at @xmath342 and becomes negative in the domain @xmath343 , which is studied sec.iv . this part of the appendix refers to sec . iv of the paper . in the ginzburg - landau theory , the free energy is expanded in terms of the gap @xmath122 , _ i.e. _ the order parameter , assuming the temperature close to @xmath10 . originaly it was introduced as a phenomenological theory for superconductivity before the bcs theory . here we derive the ginzburg - landau free energy from the full microscopic knowledge of our model in order to analyze the vicinity of the tricritical point . to do this , we consider the simplest case where the fflo gap modulation is exponential , namely @xmath344 , @xmath345 being the in - plane modulation wave vector . it is known that this modulation structure is not realized to the benefit of the cosine modulation discussed in the article s body . however , in sec.ii , the gorkov green functions of the sf / sf bilayer were derived for a modulated order parameter @xmath344 and @xmath82 or @xmath346 this modulation structure is then convenient to calculate the coefficients of the generalized gl functional because the exact expression of the anomalous green function ( see eq.([fcroix ] ) ) is valid for this gap modulation structure , whereas it is unknown with the cosine structure . we first expand the exact anomalous green function ( [ fcroix ] ) and the self - consistency relation in powers of the gap @xmath122 and the fflo wave vector @xmath347 then this self - consistency relation is interpreted as the stationarity condition for the ginzburg - landau free energy , which allows ( by identification ) to determine the coefficient of every term of the gl functional . in the @xmath82 case , the expansion of the anomalous green function reads @xmath348 where @xmath349 . we first consider the case of uniform superconductivity , _ i.e. _ @xmath350 . after integration over @xmath351 , we obtain:@xmath352 with @xmath353 . note that the interlayer coupling @xmath25 has disappeared in eq.([gna ] ) . we are now able to write down the self - consistency equation ( [ selftnonnul ] ) as an expansion in powers of @xmath122 @xmath354 where the functions @xmath355 are those defined in appendix a , and @xmath180 . this self - consistency relation may be interpreted as the stationnary condition @xmath356 for the ginzburg - landau free energy with uniform order parameter within each superconducting plane and @xmath82 phase difference between the planes . close to the tricritical point , @xmath357 is small and it is enough to retain only the first term in this infinite expansion as @xmath358 where @xmath179 . by identification with eq.([minim ] ) we obtain the gl free energy for the @xmath176 phase as a function of the variational parameter @xmath357 and the thermodynamical variable @xmath359 : @xmath360 \left\vert \tilde{\delta}\right\vert ^{2}+b_{1}% k_{3}(\tilde{h})\frac{\left\vert \tilde{\delta}\right\vert ^{4}}{2}-b_{2}% k_{5}(\tilde{h})\frac{\left\vert \tilde{\delta}\right\vert ^{6}}{3}\nonumber\\ & = \left [ \ln\frac{t}{t_{c0}}-k_{1}(\tilde{h})\right ] \left\vert \tilde{\delta}\right\vert ^{2}+\frac{k_{3}(\tilde{h})}{4}\left\vert \tilde{\delta}\right\vert ^{4}-\frac{k_{5}(\tilde{h})}{8}\left\vert \tilde{\delta}\right\vert ^{6 } \label{energyzero}%\end{aligned}\ ] ] which corresponds to eq.([freezero ] ) . note that in this usual @xmath107 state the same coefficients have been already reported in ref.@xcite the same procedure may be followed when the phase difference is @xmath0 . the anomalous green function is then @xmath361 and leads to the self - consistency relation which contains explicitely the coupling @xmath25 , via the normalized coupling @xmath193:@xmath362 \left\vert \tilde{\delta } \right\vert ^{2n}=0\text{.}%\ ] ] from the latter expression we deduce that the coefficients of the gl free energy for the @xmath13 state can be directly obtained using the coefficient of @xmath363 in which we replace @xmath364 by @xmath365 . finally the free energy of the @xmath192 state is @xmath366 \left\vert \tilde{\delta}\right\vert ^{2}\nonumber\\ & + \frac{k_{3}(\tilde{h}+\tilde{t})+k_{3}(\tilde{h}-\tilde{t})}{8}\left\vert \tilde{\delta}\right\vert ^{4}\nonumber\\ & -\frac{k_{5}(\tilde{h}+\tilde{t})+k_{5}(\tilde{h}-\tilde{t})}{16}\left\vert \tilde{\delta}\right\vert ^{6 } \label{energypi}%\end{aligned}\ ] ] which yields eq.([freepi ] ) in the small interlayer coupling limit @xmath367 . we have developped in a similar way the gl free energy in the case where the order parameter is modulated within each superconducting plane . using the expressions ( [ fzeroannexe ] ) and ( [ fpiannexe ] ) for the anomalous green function of the bilayer with @xmath344 fflo modulation respectively in the @xmath82 and @xmath13 cases , one obtains the expansion of the self - consistency equation in powers of @xmath122 and of the fflo wave vector @xmath46 . finally , after averaging over all possible orientations of the fflo modulation vector , the self - consistency equation reads:@xmath368 for @xmath82 . the coefficients @xmath369 are symmetric with respect to the expansion indexes @xmath370 and @xmath371@xmath372 and related to the coefficients @xmath373 by @xmath374 . from eq.([yoyo ] ) the gl free energy can be constructed using the method described in the previous paragraph for uniform phases . we retrieve all the coefficients already obtained by buzdin and kachkachi,@xcite including the coefficients of the gradient terms of the generalized functional . to derive the free energy of the @xmath198 phase , we have therefore used the bk functional with the cosine modulation which is effectively realized in each superconducting layer . as a result , it reads @xmath375 where @xmath203 . in the @xmath13 state the free energy has been derived from the bk functional in which the replacement @xmath376 has been done in order to obtain the modified coefficients . finally the free energy of the @xmath199 phase can be written as@xmath377 in the article body more convenient forms of eqs.([energyfflozeroannexe],[energyfflopiannexe ] ) involving the reduced quantities @xmath233 , @xmath230 and @xmath234 are used in order to derive the universal phase diagram . 99 v. l. ginzburg , zh . eksp . . fiz . * 31 * , 202 ( 1956 ) [ sov . jetp * 4 * , 153 ( 1957 ) ] . d. saint - 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dotted ( resp . dashed ) line . fig.5 : phase diagram in @xmath175 coordinates , for @xmath384 . only the neighborhood of the tricritical point is represented . the lines have the same meaning than in fig.4 . note that the @xmath0 phase is settled in a very narrow region of the phase diagram . fig.6 : excitation spectrum . usual singlet pairing ( thin line circles ) between opposite - spin electrons occupying the same orbital is affected by zeeman effect . in contrast , @xmath0 coupling ( thick line ) between two electrons occupying a bonding and an antibonding orbitals may lead to the cancellation of the zeeman splitting . fig.7 : phase diagram for @xmath385 . thick ( resp . thin ) solid lines represents second - order transition between @xmath386 ( resp . @xmath387 ) and normal metal phase ( @xmath80 ) for @xmath82 and @xmath0 . we expect the @xmath388 transition lines ( not calculated ) to be in the vicinity of the ( virtual ) first order @xmath389 lines ( dash - dotted ) . | we present the detailed theoretical study of a heterostructure comprising of two coupled ferromagnetic superconducting layers . our model may be also applicable to the layered superconductors with alternating interlayer coupling in a parallel magnetic field .
it is demonstrated that such systems exhibit a competition between the nonuniform larkin - ovchinnikov - fulde - ferrel ( fflo ) state and the @xmath0 superconducting state where the sign of the superconducting order parameter is opposite in adjacent layers .
we determine the complete temperature - field phase diagram . in the case of low interlayer coupling
we obtain a new @xmath0 phase inserted within the fflo phase and located close to the usual tricritical point , whereas for strong interlayer coupling the bilayer in the @xmath0 state reveals a very high paramagnetic limit and the phenomenon of field - induced superconductivity . |
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the absolute magnitude of the rr lyrae variables , @xmath9 , is integral to determining distances to old stellar systems in our galaxy and to other nearby galaxies . for example , rr lyraes are widely used to measure the distances to galactic globular clusters , to the galactic center , and to many members of the local group . in addition , the distances to individual field rr lyrae stars in the thick disk and halo of our galaxy enable us to determine the kinematic and spatial distributions of these populations . precise distances to globular clusters are also necessary to determine their ages ( e.g. , buonanno _ et al . the variation of @xmath9 with abundance , [ fe / h ] , strongly affects the derived age spread and age - metallicity relation of the galactic globular cluster system . these quantities in turn place strong constraints on scenarios describing the formation of the galaxy , specifying the rate of halo formation and whether the chemical enrichment of the halo proceeded in a uniform , global fashion , ( eggen _ et al . _ 1962 ) or within autonomous star - forming fragments ( searle & zinn 1978 ) . the zero - point of the @xmath9[fe / h ] relation sets the mean absolute age of the globular cluster system , thus placing a critical lower limit on the age of the universe . the rr lyraes in a given globular cluster are observed to have a very narrow range of intensity - mean magnitudes , typically @xmath14 = 0.060.15 mag ( sandage 1990a ) . though the @xmath9 variation from cluster to cluster over a broad range in [ fe / h ] is not so well understood , it is clear that rr lyrae stars have the potential to be excellent standard candles . historically , there has been discussion over the slope of the @xmath9[fe / h ] relation , with values ranging between @xmath15[fe / h ] = 0 to 0.4 mag dex@xmath5 ( corresponding to a globular cluster age range of @xmath165 gyr to approximately zero ) . recently , a consensus has begun to form that the slope has a value of 0.150.20 ( e.g. , carney _ et al . _ 1992 and chaboyer 1995 , though see sandage 1993 and mazzitelli _ et al . _ 1995 for dissenting opinions ) . however , the zero - point of the relationship continues to defy a consensus . at the characteristic abundance of the halo , [ fe / h ] = 1.6 , @xmath9 values range from 0.45 to 0.75 mag , and usually fall either toward the brighter or the fainter end of this range . chaboyer ( 1995 ) showed that this `` two value '' effect translates to a @xmath1722% difference in the derived ages of the galactic globular clusters , and in fact represents the dominant uncertainty in the determination of cluster ages . a number of methods have been used to estimate @xmath9 , including baade - wesselink ( surface brightness ) analyses ( jones _ et al . _ 1992 ) , main sequence fitting of globular clusters ( buonanno _ et al . _ 1989 ) , application of stellar pulsation theory to field stars ( sandage 1990b ) , horizontal branch evolution theory ( lee _ et al . _ 1990 ) , calibrating the lmc rr lyraes using other lmc distance estimates ( walker 1992 , gould 1995 ) , and the statistical parallax method ( hawley _ et al . _ 1986 , strugnell _ et al . _ 1986 ) . the essence of the statistical parallax ( `` stat-@xmath18 '' ) method is to balance the radial - velocity - derived kinematics of a homogeneous stellar sample with its kinematics as derived from proper motions . the former are independent of distance , while the latter are distance - dependent . they are balanced through a simultaneous solution for a distance scale factor . ( 1986 ) discussed the stat-@xmath18 solutions performed prior to 1985 , and described the shortcomings in the methods that were employed . they argued that only a complete treatment using a maximum - likelihood formulation , together with a minimization technique which is tolerant of inter - dependent variables , can produce accurate solutions . two modern studies have employed these techniques ( hawley _ et al . _ 1986 , and strugnell _ et al . _ 1986 , hereafter referred to as hjbw and srm , respectively ) . hjbw compiled proper motion , radial velocity , apparent magnitude , and abundance data from the literature to produce a sample of @xmath17140 stars . srm employed the virtually same set of data . the two groups obtained similar values for @xmath9 , the only difference being in the details of the adopted reddenings and determination of the apparent magnitudes . both groups concluded that the sample was too small to constrain the slope of the @xmath9[fe / h ] relation . since then , considerable new data have become available , indicating that it is an opportune time for a new stat-@xmath18 analysis . the lick northern proper motion ( npm ) program ( klemola , jones & hanson 1987 ) has measured absolute proper motions for over 1000 rr lyrae stars . these data are more uniform and have a more accurate zero - point than the proper motions employed in previous analyses , mainly because the plates , obtained from a single telescope and covering most of the northern hemisphere , were measured and reduced onto a single inertial frame tied to external galaxies . meanwhile , layden ( 1994 ) determined abundances and radial velocities for over 300 rr lyraes , including most of those in the hjbw and srm studies . blanco ( 1992 ) showed that the abundances used in those studies were of variable accuracy and zero - point . layden s [ fe / h ] measures are on a self - consistent system and are typically accurate to 0.150.20 dex . thus , a new stat-@xmath18 solution will reap the benefits of a 50% larger sample size ( 213 stars ) and higher quality data . furthermore , layden ( 1995 ) showed that the local rr lyrae sample breaks fairly cleanly into thick disk and halo populations at [ fe / h ] = 1 . the existence of two kinematically distinct populations had not been recognized in previous stat-@xmath18 studies , in part because the existing @xmath19 abundances were of insufficient accuracy to provide the the required abundance resolution . as a result , the two populations had been mixed . one worries that such mixing might have resulted in the @xmath9[fe / h ] slope being under - estimated . in the worst case , population mixing might lead to errant solutions , since the stat-@xmath18 method is a simultaneous solution for kinematics and @xmath9 . we consider it safest to treat the disk and halo separately in our solutions . this paper reports the findings of our new stat-@xmath18 solutions , based on the improvements described above . throughout the paper , we employ the stat-@xmath18 code used by hjbw . in sec . 2 , we describe in detail the data used in our stat-@xmath18 solutions . in sec . 3 , we begin our analysis with an inverted approach ; we assume an @xmath9[fe / h ] relation , and compute the distance and three space velocity components of each rr lyrae star in our sample . this gives us insight into how best to divide the sample during the stat-@xmath18 solutions . in sec . 4 , we present monte carlo simulations which enable us to investigate the accuracy of our solutions , and to search for any inherent biases produced by the stat-@xmath18 technique . in sec . 5 , we present the absolute magnitude and kinematic results of the stat-@xmath18 solutions for the observed stars . in sec . 6 , we compare our @xmath9 results with those obtained by other authors using other techniques . in sec . 7 , we discuss the implications of our results for some basic properties of the galaxy and the universe . we close with a short summary of our findings . our primary source of proper motions for this study is the lick northern proper motion ( npm ) program ( klemola , jones & hanson 1987 ) . the npm program is a photographic survey measuring precise absolute proper motions , on an inertial system defined by 50,000 faint galaxies ( @xmath20 ) , for over 300,000 stars with @xmath21 , covering the northern two - thirds of the sky ( @xmath22 ) , based on plates taken between 1947 and 1988 with the lick 51 cm carnegie double astrograph . details of the npm observing , plate measurement , and reduction procedures are given by klemola _ et al . _ ( 1987 ) . part i of the npm program , covering the 72% of the northern sky lying outside the heavily obscured regions of the milky way , was completed in 1993 , with the release of the lick npm1 catalog ( klemola , hanson & jones 1993 ; hanson 1993 ) , containing 149,000 stars . the stellar content of the npm1 catalog is detailed in the npm1 cross - identifications ( klemola , hanson & jones 1994a ; hanson & klemola 1994 ) . comprehensive error analyses ( klemola , hanson & jones 1994b ) have determined the rms error of the npm absolute proper motions to be @xmath23 in each coordinate , corresponding to a transverse velocity error @xmath24 . the npm1 catalog contains over 1000 rr lyrae variables ( klemola _ et al . _ 1994a , appendix 2 ) . some 300 of these have @xmath25 ( corresponding to @xmath26 kpc ) and may be individually useful for stat-@xmath18 studies . the particular value of the npm proper motions for this work is that they are on an absolute reference frame . by contrast , previous rr lyrae stat-@xmath18 solutions have relied on relative proper motions , measured with respect to field stars ( generally with @xmath27 ) , and corrected to absolute by assuming the net motions of the reference stars from models of galactic kinematics and rotation . using relative proper motions inevitably raises the possibility that the resulting kinematics ( and luminosities ) may depend to some extent on the input assumptions . specifically , the hjbw and srm stat-@xmath18 solutions used the list of proper motions for 168 rr lyraes compiled by wan , mao & ji ( 1980 ; hereafter referred to as wmj ) . wmj added new relative proper motions from the shanghai observatory ( wan , he , zhu & li 1979 ) and other sources to the previous such compilation by hemenway ( 1975 ) . because of their absolute character , we adopted the npm proper motions as the primary source for our stat-@xmath18 database . our search of the npm1 catalog , using the npm1 cross - identifications , found 171 rr lyrae stars from layden s ( 1994 ) list . however , the npm1 catalog does not cover low galactic latitudes ( @xmath28 ) , nor the southern sky below @xmath29 declination . to attain the complete sky coverage needed for reliable stat-@xmath18 solutions , we used the wmj compilation as a secondary source where npm proper motions were not available , adding another 42 stars . using the wmj data raises two practical problems : first , should the wmj motions be corrected to the npm absolute system ? the heterogeneous nature of the wmj compilation might make any single correction doubtful . however , clube & dawe ( 1980b ) suggested that the existing rr lyrae proper motions needed a large correction `` in the direction of galactic rotation '' @xmath30 due to incorrect motions of the reference stars , making @xmath31 to 0.1 mag fainter . so large a systematic error should be easily detectable by comparing the npm and wmj proper motions . second , how should the wmj data be weighted relative to the npm motions ? wmj estimated the errors for each star from the repeatability of the existing proper motions for that star , but the number of determinations is generally small , so these estimates may not be individually reliable . for example , the range of @xmath32 is very large , some values are incredibly small ( one is zero ! ) , and the correlation between each star s right ascension and declination proper motion errors is very poor . one - fourth of the wmj stars have @xmath33 in one coordinate more than three times the value in the other coordinate . because all the sources cited in wmj used methods which should produce errors of equal size in either coordinate , this is plainly an artifact of the wmj error analysis . however , the overall mean and rms errors ( @xmath34 and @xmath35 in @xmath36 ; @xmath37 and @xmath38 in @xmath39 ) are quite comparable to values given in the literature ( wan _ et al . _ 1979 ; hemenway 1975 ) , and may in fact be reliable error estimates . to answer these questions , we compared data for the 109 stars in common between the npm1 catalog and the wmj compilation . the comparison was done twice ; using @xmath40 in equatorial coordinates ( @xmath41 ) and in galactic coordinates ( @xmath42 ) . normal probability plots ( lutz & hanson 1992 ) were used for robust estimates of the mean differences and rms dispersions . plots of @xmath43 vs. @xmath44 and magnitude were examined for any dependence on these observational variables . finally , we assessed the significance of the wmj individual proper motion errors @xmath33(wmj ) . two principal results were found , bearing on each of the questions posed above . first , the wmj proper motions in galactic longitude have a small but significant mean difference with npm1 . we found @xmath45 @xmath46 as clube & dawe ( 1980b ) suggest , such a systematic difference may reflect erroneous reference star motions , but our result is six times smaller than theirs . consequently , any effects on @xmath9 would be at the 0.01 mag level . so , there is no need to correct the wmj data for our stat-@xmath18 solutions . second , the wmj proper motion errors are only useful as an overall average , not on a star - by - star basis . figure 1 shows the npm wmj proper motion differences versus the wmj listed errors . in each coordinate , the stars with the smaller errors ( @xmath47 ) scatter just as much as the stars with the larger errors ( @xmath48 ) . for the very smallest errors ( @xmath49 ) the scatter is smaller , but the offsets from zero are large . a constant error describes the data much better than does the large variation in @xmath32 . figure 2 quantifies this , testing whether the rms error actually increases with @xmath33 . in each coordinate , the 109 stars were sorted by the size of @xmath32 . figure 2 plots the running values of the rms nominal error and the actual rms dispersion of @xmath43 as a function of this rank . the nominal error @xmath50 is the quadratic sum of @xmath32 and @xmath51 . the smooth , slowly rising curve labeled `` tot '' in figure 2 is the running rms value of @xmath50 . if the wmj errors are correct , then @xmath52 , the rms dispersion of the proper motion differences , should follow this curve . the jagged line labeled `` @xmath43 '' traces the running value of the actual rms dispersion . in each coordinate , @xmath52 quickly rises to a large value ( @xmath53 in @xmath36 , @xmath54 in @xmath39 ) by rank @xmath55 , and remains at that level out to rank @xmath56 , rising somewhat at the end . this proves that the real error of the wmj proper motions is roughly constant , independent of the listed error @xmath32 . quadratically subtracting @xmath57 from these values gives @xmath58 in ( @xmath59 ) . almost identical error estimates were obtained from normal probability plots of @xmath60 . these estimates agree almost perfectly with the rms values of the wmj errors cited above . in view of this result , we gave all wmj proper motions equal weights in our stat-@xmath18 solutions . test solutions with @xmath61 and @xmath62 , corresponding to the mean and rms wmj errors , respectively , gave very similar results . we decided to adopt the smaller value , @xmath63 , in order to weight all areas of the sky equally in the stat-@xmath18 solutions ( section 5 ) . our primary source for rr lyrae metal abundances is the work of layden ( 1994 , hereafter referred to as l94 ) . these abundances are based on the relative strengths of the k line and the balmer lines h@xmath64 through h@xmath65 , analogous to the @xmath66@xmath67 abundance technique of preston ( 1959 ) . the abundance scale is tied to the [ fe / h ] abundance scale for globular clusters developed by zinn & west ( 1984 ) , and the individual [ fe / h ] values are typically accurate to 0.150.20 dex . lambert _ et al_. ( 1995 ) have measured new high - dispersion abundances for a number of the stars in l94 , and find excellent agreement with those results . jurcsik & kovacs ( 1996 ) also discuss the high quality of the l94 abundances . previous statistical parallax solutions have used @xmath66@xmath67 values as the metallicity indicator . since then , blanco ( 1992 ) has shown that @xmath66@xmath67 values available to those authors were of variable accuracy and zero - point . the [ fe / h ] sample of l94 is both self - consistent and contains many more stars than a sample of @xmath66@xmath67 values collected from the literature . however , there are a few bright rr lyraes in the literature which are not included in the list of l94 . for these , we adopt the literature @xmath66@xmath67 values , converted to [ fe / h ] using eqn . we note that some of these values , those taken from hemenway ( 1975 ) , are actually _ inferred _ from photoelectric indices or a period - amplitude-@xmath66@xmath67 relation . we discuss these stars further in sec . l94 also measured radial velocities for the stars in his sample , and combined them with literature velocities to produce a catalog of radial velocities , the most accurate currently available , for over 300 nearby rr lyrae stars . l94 is our primary source of radial velocities . velocities for a few stars not observed by l94 were taken from the literature compilation shown in table 1 of l94 . we include these , adopting for their errors the typical errors for each source derived in sec . 2 of l94 . the existing photometry on field rr lyrae stars is a surprisingly heterogeneous data set . there are three principal works in the johnson @xmath68-band . the work of sturch ( 1966 ) is comprised mainly of observations at minimum light ; that of bookmyer _ et al_. ( 1977 ) , which contains as a subset the better - known work of fitch , wisniewski & johnson ( 1966 ) , and is purported to be on the same photometric system ; and that of clube & dawe ( 1980b , hereafter referred to as cd80 ) . barnes & hawley ( 1986 , hereafter referred to as bh86 ) show that the photometric system of sturch is in good agreement with that of cd80 , and that both are offset from the work of fitch _ et al._. we therefore adopt the cd80 photometry as the standard to which we will compare other photometric works . many of the existing rr lyrae light curves have incomplete phase coverage , so it is difficult to obtain their intensity - mean apparent magnitudes with accuracy . however , various relations exist in the literature which allow this quantity to be calculated from light curve extrema , rise times , etc . ( e.g. , fitch _ _ 1966 , cd80 ) . bh86 recomputed the coefficients for two of these methods using modern , self - consistent data , and find that the method of cd80 gives the tighter relation . we adopt this parameterization along with their coefficients , @xmath69 where @xmath70 is the intensity - mean magnitude , @xmath71 is the magnitude at minimum light , and @xmath72 is the light curve amplitude . by using the cd80 photometry as the basis of our photometric system , and by employing intensity - mean magnitudes computed from the preceding equation , our photometry system is equivalent to that of bh86 . we computed @xmath70 for the cd80 and bookmyer _ et al_. data sets , and performed a linear regression between them to obtain a transformation between the two photometric data sets ( see table 1 , line 1 ) . we then adopted data values of cd80 ( 57 stars ) as the primary data , and supplemented it with the values from bookmyer _ et al_. that had been transformed onto the cd80 system ( 81 additional stars ) . the resulting data set was used to transform the walraven photometry of lub ( 1977 ) onto the cd80 system . the transformation , given in line 2 of table 1 , is in close agreement with the relation of pel ( 1976 ) . this resulted in 7 additional stars being added to the data set . this data set was then used to transform the ccd photometry of schmidt _ et al _ ( 1991 , 1995 ) onto the cd80 system ( see line 3 of table 1 ) , adding 36 stars to the database . preliminary photometry from layden ( 1996 ; 8 stars ) was also included , though no transformations were possible since there were no stars in common with the database . data for 24 additional stars were adopted from the photometric compilation of l94 ( his table 9 ) , after converting from the @xmath70 definition of fitch _ et al . _ 1966 to that of cd80 . clearly , this approach is not ideal , since it relies on the statistical transformations between photometric systems , and assumes that the cd80 system is equivalent to the modern systems used by observers ( e.g. , landolt 1992 ) and theorists ( e.g. , lee _ et al . the approach has the advantage of reducing systematic errors by placing all the stars on the same photometric system . ultimately , the transformations result in changes , typically 0.07 mag , which are small compared to the 0.20.3 mag disagreements which arise between different methods of measuring @xmath9 . furthermore , the sense of the transformation is to brighten the literature values ; had we used the fainter photometry , the @xmath9 values we derived would have been fainter as well . obtaining self - consistent photometry at the level of several hundredths of a magnitude is a problem shared , but seldom mentioned , by all observers attempting to measure @xmath9 . srm argued that reddenings derived from the burstein & heiles ( 1982 ) reddening maps provide the most accurate and consistent estimates of rr lyrae reddenings , and hence absorption ( we assume @xmath73 = 3.1 ) . we therefore use burstein & heiles values when they are available , i.e. , for stars more than 10@xmath74 from the galactic plane . we reduce their tabulated reddenings by an amount consistent with a uniform dust distribution with an exponential scale height of 100 pc ( see l94 ) . in most cases this is a small or negligible correction . for stars less than 10@xmath74 from the plane , we adopt the reddening values given by blanco ( 1992 ) , which are derived from the stars colors at minimum light . when neither are available , we interpolate between the burstein & heiles reddening at @xmath7510@xmath74 , and that of fitzgerald ( 1968 , 1987 ) at @xmath750.5@xmath74 , at the longitude of the star . the latter is clearly a poor solution , but it is the best available until accurate minimum - light colors can be obtained for these stars . fortunately , it was used for only 9 stars . using the data sources described above , we find that 213 stars have values for all five of the fundamental data types : proper motion , abundance , radial velocity , apparent magnitude , and reddening . this is substantially more than were used in the recent studies of hjbw ( 142 stars ) and srm ( 139 stars ) . note that we do not consider bailey type-@xmath76 rr lyraes in this study , only type-@xmath77 stars . this database is presented in table 2 . the first column gives the variable star name , and the second column gives the npm1 catalog number . following this are the galactic longitude and latitude ( in degrees ) and the adopted proper motions in right ascension and declination ( in arcsec cen@xmath5 ) . the seventh and eighth columns give the adopted radial velocity and its error ( in km s@xmath5 ) . next is the adopted abundance , [ fe / h ] . the tenth column gives the adopted intensity - mean apparent @xmath68 magnitude , and the eleventh column gives the adopted interstellar absorption . the twelfth column gives references for the sources of the proper motion , abundance , photometry , and interstellar absorption , as listed at the end of the table . the final column indicates whether a star was treated as a disk ( 1 ) or halo ( 0 ) star under the three disk / halo definitions discussed in sec . 3 . npm proper motions are used for 171 of the 213 stars in our sample . abundances from l94 are used for 187 of the stars , and apparent magnitudes computed directly from @xmath68band photometry are used for 182 of the stars . given the dominance of npm proper motions in our catalog , one wonders if the distribution of stars on the sky is skewed , and whether this would introduce a bias into our stat-@xmath18 solutions ( e.g. , croswell , latham & carney 1987 ) . regarding the former , we find that 63% of our sample lies above the celestial equator ; we are weighted to the north celestial hemisphere , but not overwhelmingly so . similarly , 67% of our stars lie north of the galactic plane . however , these asymmetries can not produce the kind of biases discussed by croswell _ , since our sample contains no proper motion bias . as klemola _ ( 1987 ) describe , samples of `` astrophysically interesting '' stars , such as rr lyraes , were selected for the npm program in advance of the plate measurements , and no star was omitted from the npm1 catalog because its measured proper motion proved to be small . this type of pre - selection is true of our secondary proper motion source as well . as confirmation , we note that our data show no sign of the `` croswell effect . '' the number of stars moving away from the galactic plane is almost identical to the number moving toward it : 106 @xmath78 108 , respectively . finally , we note specifics of interest concerning several stars . ( 1 ) bx dra was shown by schmidt _ ( 1995 ) to be an eclipsing binary rather than an rr lyrae ( kholopov 1985 ) ; it was removed from our database . ( 2 ) l94 found sv boo to have [ fe / h ] = 0.43 from a single low - quality spectrum , whereas hemenway ( 1975 ) quoted @xmath66@xmath67 = 7 ( [ fe / h]@xmath79 = 1.55 ) . since the kinematics of sv boo suggest it belongs to the halo , we adopt the hemenway abundance . ( 3 ) the radial velocity of bb pup was revised to 98 @xmath75 9 km s@xmath5 from that in l94 by eliminating the outlier velocity 255 @xmath75 16 km s@xmath5 from the list in table 2 of l94 . ( 4 ) for three stars , ae dra , bd dra , and bk eri , the proper motion was improved by removing one discordant measurement from the average value quoted in the npm1 catalog . ( 5 ) the seven stars listed with the abundance source `` 3 '' in table 2 all had photometrically - determined [ fe / h ] values from hemenway ( 1975 ; see sec . 2.2 ) which suggested that they were thick disk stars . however , their kinematics suggested that all seven stars are halo members . we therefore set [ fe / h ] = 1.5 for these stars . ( 6 ) we found that rx cvn historically has been misidentified . the proper motion of wmj and the radial velocity of joy ( 1950 ) give a space velocity greater than the escape velocity of the galaxy . the npm proper motion and the radial velocity measured by l94 ( see our table 2 ) give a reasonable space velocity for a halo star . inspection of the npm plates shows that the measured star has a nearby companion , probably a foreground dwarf , which was probably mistakenly observed by wmj and joy . the rr lyrae is the eastern - most of the pair . layden ( 1995 , hereafter referred to as l95 ) showed that the rr lyraes separate into a halo and a ( primarily thick ) disk population at [ fe / h ] = 1.0 . we wish to see if this separation persists using our improved database . to do this , we computed provisional distances to the stars in our sample using the @xmath9[fe / h ] relationship of carney , storm & jones ( 1992 , hereafter referred to as csj ) , @xmath9 = 0.15[fe / h ] + 1.01 mag . we then computed the stars @xmath80 , @xmath68 , @xmath81 space velocities , following johnson & soderblom ( 1987 ) with one exception : we take @xmath80 as positive outward . at distances of 12 kpc , typical of the distances in our sample , the sun - oriented @xmath80 , @xmath68 , @xmath81 frame can be misaligned by a small angle in the @xmath80 , @xmath68 plane ( @xmath82 deg ) from the cylindrical galactic directions ( @xmath18 , @xmath83 , @xmath84 ) at each star . so , we rotated the @xmath80 , @xmath68 , @xmath81 velocities into the @xmath18 , @xmath83 , @xmath84 frame , after adding to @xmath68 the iau standard rotational velocity @xmath85 km s@xmath5 ( kerr & lynden - bell 1986 ) , and after adding the `` dynamical '' solar motion ( 9 , + 12 , + 7 km s@xmath5 ; mihalas & binney 1981 , p.400 ) to correct @xmath80 , @xmath68 , @xmath81 to the local standard of rest ( lsr ) . the results are the velocity components @xmath86 , @xmath87 , and @xmath88 , where @xmath86 increases outwards from the axis of galactic rotation , @xmath87 increases in the direction of galactic rotation , and @xmath88 increases toward the north galactic pole . figure 3 shows @xmath86 and @xmath88 plotted against @xmath87 . clearly , the stars with [ fe / h ] @xmath89 1.0 have large velocity dispersions and little net galactic rotation , typical of the halo . meanwhile , the stars with [ fe / h ] @xmath90 are clustered around @xmath87 @xmath91 km s@xmath5 . figure 4 shows @xmath87 as a function of abundance . while the distribution of stars in figures 3 and 4 is consistent with the first - order view of a disk / halo separation at [ fe / h ] = 1.0 , there are four stars with @xmath87 @xmath92 km s@xmath5 at [ fe / h ] @xmath93 , whose extreme kinematics clearly mark them as members of the halo . similarly , there may be an excess of stars with @xmath87 @xmath94 km s@xmath5 at [ fe / h ] @xmath95 , which may belong to the `` metal - weak thick disk '' ( mwtd ; morrison , flynn & freeman 1990 ) . l95 discussed the presence of such stars in his sample of rr lyraes . it is not possible to assign these stars individually to the disk or halo populations without ambiguity . we therefore separate the disk from the halo using three distinct definitions , and perform the stat-@xmath18 solutions for each set of definitions , in order to test the effects of the different definitions on the derived kinematics and absolute magnitudes . the three disk / halo separation definitions are summarized in table 3 . all three definitions assign the four low-@xmath87 stars with [ fe / h ] @xmath90 to the halo . the first definition , similar to that of nissen & schuster ( 1991 ) , admits a small number of mwtd rr lyraes , primarily with @xmath96 [ fe / h ] @xmath95 , in agreement with l95 . the second assumes that no mwtd ( [ fe / h ] @xmath95 ) rr lyraes exist . the third definition admits a larger population of mwtd rr lyraes , which reaches to [ fe / h ] @xmath97 , more along the lines of the population of red giants described by morrison _ et al_. ( 1990 ) . two stars , ao peg and fu vir , fit one or more of the disk definitions , yet clearly belong to the halo based on their extreme kinematics : ( @xmath98 ) = ( 212 , + 236 , 207 ) and ( 178 , + 249 , 93 ) , respectively . we have moved these stars into the corresponding halo definitions , as noted in table 3 . we note that the choice of @xmath9[fe / h ] relations used to compute the distances does not significantly affect the separation . only one star crosses the sloping disk / halo line in fig . 4 when we change from the csj @xmath9[fe / h ] relation to that advocated by sandage ( 1993 ) . in sec . 5 we will present the stat-@xmath18 analysis of our data . first , however , we will test the hjbw stat-@xmath18 algorithm itself , using synthetic data with known properties ( positions , velocities , @xmath9 , etc . ) . this step seems vital to ensure reliable results . specifically , we test : ( 1 ) how accurately the stat-@xmath18 solutions reproduce the kinematics and luminosities of the input data ; ( 2 ) how reliable the error estimates are ; ( 3 ) the sensitivity to the number of stars and their distribution on the sky ; ( 4 ) whether all the free parameters in the hjbw stat-@xmath18 algorithm are necessary ; ( 5 ) whether the small misalignment ( sec . 3 ) between @xmath99 and @xmath100 has any significant effects ; ( 6 ) whether the results are biased by any input assumptions ; and ( 7 ) whether any corrections are necessary for bias in the results . the hjbw algorithm uses a simplex optimization technique to maximize the likelihood in murray s ( 1983 ) kinematic model ( also used by srm ) . there are 11 free parameters : the solar motion ( @xmath101 ) , the velocity ellipsoid ( @xmath102 ) with three covariances ( @xmath103 ) to allow an arbitrary orientation , the distance scale parameter @xmath104 and its dispersion @xmath105 . the solution also returns an approximate standard error estimate @xmath106 for each parameter . the solution returns an absolute magnitude @xmath107 by differential correction to a starting value @xmath108 , which for convenience we fix at + 1.0 . we refer the reader to hjbw for full details of the stat-@xmath18 algorithm . we made two modifications to the hjbw procedure : ( 1 ) we fixed an error which caused the large velocity dispersion errors listed in table 2 of hjbw ; and ( 2 ) when necessary , we inspected the data to reject extreme outliers and repeated the solutions . to perform monte carlo tests we generated an ensemble of simulated data sets as outlined in table 4 . these were designed to realistically simulate the halo and disk subsamples of our real data ( sec . 3 ) . for the halo and disk simulations h1 and d1 , @xmath109 were randomly assigned spatial positions ( @xmath110 ) and [ fe / h ] values from uniform distributions with appropriate limits . @xmath107 values were then computed from the @xmath9[fe / h ] relation of csj , with a gaussian cosmic dispersion @xmath111 mag . space velocities ( @xmath86 , @xmath87 , @xmath112 ) were generated from a gaussian velocity ellipsoid whose parameters are given in table 4 . each star s right ascension , declination , proper motion , radial velocity , and apparent magnitude were then calculated , with gaussian observational errors 0.5 arcsec cen@xmath5 and 20 km s@xmath5 added to the proper motion components and the radial velocity , respectively . for each simulation ( h1 , d1 ) we created @xmath113 different data sets ; @xmath114 was set larger for the disk simulations because @xmath115 is proportionately smaller . as discussed in sec . 2 , the actual distribution of our rr lyraes on the sky is far from uniform . to test whether this affects the stat-@xmath18 results , we prepared alternate data sets ( h2 , d2 ) using the observed sky positions , apparent magnitudes , and metallicities from table 2 , with the disk-1/halo-1 separation of table 3 . then , @xmath114 sets of synthetic proper motions and radial velocities were randomly generated as above . for each data set ( h1 , h2 , d1 , d2 ) , stat-@xmath18 solutions were performed as outlined in column 6 of table 4 . multiple solution sets tested particular parameters in the hjbw model . to test the effect of the hjbw distance scale dispersion parameter @xmath105 we ran two solutions for each halo data set , with @xmath116 as indicated in column 7 of table 4 . these solutions will be discussed in sec . 4.6 . for the data sets h2 and d2 we ran solutions with and without the velocity covariance parameters ( @xmath117 ) , as indicated in column 8 of table 4 . the reasons for this will be discussed in sec . fifteen additional h2 data sets ( row 5 of table 4 ) were created in order to test with higher precision the important solution sets which excluded the velocity covariance parameters . the significance of these solutions will be discussed in sec . finally , we note that for a few of the disk data sets , the solutions failed to converge ( @xmath118 ) . this will be discussed in sec . before discussing the stat-@xmath18 solutions , we need to deal with an apparent inconsistency between the data simulations and the hjbw solution method . we generated our simulated velocities using velocity ellipsoids oriented to the cylindrical coordinate frame ( @xmath86 , @xmath87 , @xmath112 ) which is physically the most appropriate frame for galactic velocities . however , the hjbw algorithm uses the ( @xmath101 ) rectangular coordinate frame to compute the solar motion and stellar velocity dispersions . in sec . 3 , we noted that the ( @xmath119 ) coordinates can be misaligned with ( @xmath86 , @xmath87 , @xmath112 ) , by a small angle ( rms @xmath120 deg ) . it is important to show that any effects of this misalignment are too small to significantly affect our results . we can do this directly with the simulated data , because for every star we know the space velocity in each coordinate frame . this lets us measure , for every data set , any differences ( 1 ) between @xmath121 and @xmath122 and ( 2 ) between @xmath123 and @xmath124 . in the remainder of this paper , we refer to the values computed directly from the simulated data as the `` true '' values for the data set . for the halo ( h1 , h2 ) , the principal effect is the projection of the long axis of the velocity ellipsoid ( @xmath125 ) partly onto the @xmath68 axis , increasing @xmath14 at the expense of @xmath126 . quantitatively , @xmath127 km s@xmath5 , and @xmath128 km s@xmath5 . clearly , these effects are far too small to be of any concern here . for the disk ( d1 , d2 ) , the principal effect of coordinate misalignment is the partial projection of the rotation vector @xmath129 km s@xmath5 onto the @xmath80 axis . distant stars in the direction of galactic rotation @xmath130 get a negative contribution to their @xmath80 velocity ; stars toward ( 270,0 ) get a positive contribution . the net effect is that @xmath131 km s@xmath5 . also , @xmath132 is decreased by 1 km s@xmath5 . again , these effects are considerably smaller than the observational errors , and can be neglected in practice . stat-@xmath18 solutions for each of the 25 halo and 40 disk data sets were performed as outlined in table 4 . the results for each set of solutions are summarized in table 5 . the left side of table 5 gives results for the @xmath68 component of the solar motion , the three velocity dispersions ( @xmath133 ) , and @xmath107 . the @xmath66 values on the right side of table 5 represent the differences ( solution @xmath134 `` true '' ) for each quantity . table 5 omits the @xmath80 and @xmath81 solar motion components , as these were always equal to the true values to @xmath135 km s@xmath5 . each solution set in table 5 lists two rows of results . the first row gives the mean of each quantity . the second row gives two different estimates of the uncertainty in the quantity : @xmath136 ( left side of table 5 ) gives the mean of the internal error estimates returned by the hjbw code for each parameter , while sd ( right side ) gives the rms dispersion about each mean difference . the latter external error estimates reflect how precisely the stat-@xmath18 program returns the `` true '' values . the values in row 5 of table 5 are more precise than those quoted for the other halo simulations , since they are based on 20 rather than 5 data sets . the major result of the monte carlo simulations is that the hjbw algorithm does an excellent job of returning the `` true '' input parameters , to within a few km s@xmath5 for the velocities and dispersions , and to within @xmath137 mag for @xmath107 . this is true for both the halo and the disk simulations , and for both the random and real space distributions . furthermore , in nearly all cases the external errors ( sd s of the @xmath66 s ) are no larger than the hjbw program s internal error estimates @xmath138 , and in some cases they are considerably smaller . detailed examination of table 5 shows some small systematic effects which are worth considering further . note that 7 out of 8 @xmath139 values are positive , all 24 @xmath140 values are negative , and 7 out of 8 @xmath141 values are positive . examination of the individual solutions shows that these same effects occur on a solution - by - solution basis , with @xmath68 , @xmath123 , and @xmath107 all varying in tandem , linked together by the hjbw distance scale parameter @xmath104 . the average values are slightly biased toward a `` short '' distance scale ( @xmath142 ) . this bias is larger for the real space distribution data sets ( h2 , d2 ) than for the random sets ( h1 , d1 ) . whether we can or should correct for this small bias will be discussed in sec . 4.8 . for the halo ( h ) solutions , the internal error estimates @xmath143 returned by the hjbw program tend to be larger than the external errors for example , @xmath138 for @xmath107 is 0.12 mag , while the sd of @xmath144 averages 0.08 mag . for the kinematic parameters , @xmath145/sd @xmath146 . these results indicate that the real accuracy of the halo solutions may be better than the internal errors claim . however , given the simplified nature of the simulations , we will conservatively adopt the internal error estimates in discussing our real data solutions ( sec . 5 ) . for the disk ( d ) solutions , the internal and external errors for @xmath107 generally agree , though there is a small discrepancy for the kinematic parameters , @xmath145/sd @xmath147 . in addition , the standard error of @xmath107 is @xmath170.3 mag , 34 times as large as for the halo . the failure of the @xmath107 errors to follow an @xmath148 relation may mean that @xmath149 is near the lower limit for successful solutions ( see sec . 4.5 ) , but it must also reflect the fact that the stat-@xmath18 method inherently works better for a population with a larger velocity dispersion . the test solutions ( h1.0 , h2.0 ) with the hjbw distance scale dispersion parameter @xmath105 set to zero will be discussed in section 4.6 . the test solutions ( h2d , d2d ) without the velocity ellipsoid covariances ( @xmath103 ) will be discussed in section 4.7 . since the number of stars in our disk samples ( both real and synthetic data ) is near or below the lower limit ( 50 stars ) that hjbw found necessary for successful solutions , we need to ask how reliable our disk solutions can be with so few stars . the symptom of having too few stars is that the hjbw likelihood function becomes ill - conditioned , and the iterative solution fails to converge to finite output parameters ( hence @xmath118 in table 4 ) . surprisingly , given the hjbw result , over 90% of our monte carlo disk solutions , and all 3 real - data disk solutions ( sec . 5 ) did in fact converge . moreover , table 5 shows that the disk output parameters are well - determined , i.e. , they have little bias and accurate error estimates . the convergence of our monte carlo solutions for @xmath150 may reflect the gaussian nature of our simulations , in contrast to the vagaries of the real data that hjbw used . in our experience , the hjbw algorithm is not resistant to large outliers ; for real data , this makes the disk / halo separation quite critical . the success of our real - data disk solutions for @xmath151 is most likely due to the better separation we achieved using [ fe / h ] and @xmath152 ( sec . 4 ) instead of @xmath19 and period ( hjbw ) . without doing many more simulations , it is not possible for us to state what the true lower limit on @xmath115 may be . nor is this necessary , since the clear result of our monte carlo simulations is that solutions that do converge give reliable results . both hjbw and srm found a strong correlation in their stat-@xmath18 solutions between @xmath9 and @xmath153 , the cosmic dispersion in @xmath9 . in the murray ( 1983 ) model , the cosmic dispersion is parameterized by @xmath105 , the dispersion in the distance scale parameter @xmath104 . this correlation effectively prevents solving for @xmath105 ; instead this parameter must be fixed at a value chosen to represent a reasonable value of @xmath153 . equation 8 of hjbw relates @xmath153 , @xmath105 , and @xmath104 . for @xmath154 , @xmath155 . for @xmath156 and @xmath157 from our solutions , @xmath158 mag . observations show this to be a reasonable range of @xmath153 . sandage ( 1990a ) found the intrinsic dispersion of rr lyrae magnitudes within a globular cluster ( i.e. , at a single metallicity ) to be @xmath14 = 0.060.15 mag . for our field rr lyraes there will be an additional dispersion @xmath159 proportional to the slope of the @xmath9[fe / h ] relation . we estimated @xmath159 numerically by populating various @xmath9[fe / h ] relations with `` stars '' having the [ fe / h ] distribution of our halo sample . the total @xmath153 is then the quadratic sum of @xmath14 and @xmath159 . for @xmath160 mag , using the csj @xmath9[fe / h ] relation ( slope = + 0.15 mag dex@xmath5 ) we obtain @xmath161 mag . using the steeper slope ( + 0.39 ) advocated by sandage ( 1990b ) gives @xmath162 mag . to reach @xmath163 mag , we must adopt an extreme value of @xmath164 mag along with the sandage ( 1990b ) slope . consequently , a reasonable range of @xmath105 to test in our monte carlo solutions is @xmath165 . solution sets ( h1.0 , h1 ) and ( h2.0 , h2 ) with @xmath166 respectively ( table 4 ) let us evaluate the effects on @xmath9 . the results in table 5 indicate that @xmath9 comes out @xmath170.03 mag brighter for @xmath156 than for @xmath167 . similar results were found by hjbw and srm . table 5 shows that the assumed value of @xmath105 does not affect the derived kinematics . clearly , our choice of @xmath105 will only have a small effect on our stat-@xmath18 results . to be conservative , we will adopt @xmath156 to analyze our real data ( sec . 5 ) . for a given solution the true value of @xmath9 thus may be a few hundredths of a magnitude fainter than the value we derive . three of the 11 parameters in the murray ( 1983 ) model are the covariances ( @xmath168 which allow the velocity ellipsoid to have an arbitrary orientation with respect to the principal galactic directions . the results of hjbw strongly suggest that this may not be necessary in practice . given the apparent benefits of eliminating unneeded parameters from the model ( especially for the disk , where @xmath169 does not greatly exceed the number of free parameters ) , it seems wise to use our monte carlo simulations to test whether the covariances ( @xmath103 ) are needed in our real - data stat-@xmath18 solutions . because our simulated data were generated with no correlations among the ( @xmath101 ) velocities , the covariances ( @xmath117 ) returned by the stat-@xmath18 solutions simply reflect random scatter in the data . thus the covariances for the real - data solutions can be tested for statistical significance by comparison with the monte carlo simulations . to do this , for each stat-@xmath18 solution we calculated the correlation coefficients ( cc s ) @xmath170 , etc . for each of the sets of simulated data listed in table 4 , we computed the mean , sd , and range of the cc s of the individual trials . for all the simulations , the mean correlations were near zero , as expected . for the halo , ( both h1 and h2 ) each of the three sd s was @xmath171 . by comparison , the rms value of the real - data halo cc s was 0.10 , with none of the cc s exceeding the range of the simulated values . for the disk , the sd s were @xmath172 for set d1 ( random space distribution ) , and @xmath173 for set d2 ( real space distribution ) . the rms value of the real - data disk cc s was 0.13 ; again none of the cc s exceeded the range of the simulated values . from these tests we conclude that there is no evidence , for either the halo or the disk rr lyraes , that the velocity ellipsoid deviates from the principal directions ( @xmath101 ) . consequently , we can run our stat-@xmath18 solutions with the covariance parameters ( @xmath174 ) removed from the model . the entries in table 5 for solution sets h2d and d2d show the results of these solutions on the data sets h2 and d2 . there is little difference in the kinematics or absolute magnitudes produced by the new solutions , save for a slight tendency for the errors to be reduced . analyzing our real data ( sec . 5 ) without the unneeded covariances reduced the disk @xmath107 errors by 10% . in sec . 4.4 we found that the results in table 5 suggested that the stat-@xmath18 solutions may be slightly biased toward a `` short '' distance scale . the solution velocities are consistently several per cent too small , and @xmath175 mag too faint . the bias was largest ( @xmath176 mag ) for the data set d2 , the disk simulation with the space distribution of the real rr lyraes . although this bias is much less than the random error ( @xmath177 mag ) of a single disk solution , it may still be worth applying corrections to our real - data results ( sec . the above - mentioned bias toward a `` short '' distance scale should not be confused with a bias toward the _ fainter _ @xmath107 regime discussed in secs . 1 , 6 , and 7 ( i.e. , toward @xmath178 mag at [ fe / h ] @xmath97 , rather than + 0.4 mag ) . it can easily be demonstrated that our solutions have no intrinsic bias toward any particular @xmath107 value ; nor are the solutions biased by the choice of the starting value @xmath108 . tests using simulated data sets based on @xmath9 0.5 mag brighter than the csj relation adopted in sec . 4.2 correctly returned @xmath107 values 0.5 mag brighter , with unchanged kinematics . these results were recovered exactly when a brighter starting value @xmath179 was used , and to within @xmath180 mag in @xmath107 and @xmath181% in the velocities even when the usual @xmath182 was used . in the spirit of exploratory data analysis , we plotted @xmath144 versus the `` solution '' and `` true '' values of the velocities ( @xmath183 and dispersions @xmath123 for all 65 simulations . this led to the discovery that , for the real space distribution sets ( h2 , d2 ) , the bias was smallest when the velocity ellipsoid was `` long '' , i.e. when @xmath184 , and largest when the velocity ellipsoid was `` round '' , i.e. when @xmath185 . thus the disk is chiefly affected , while the halo is not . the 15 additional h2d solutions outlined in table 4 were performed to illuminate this situation . figure 5 plots @xmath144 as a function of @xmath186 for solution sets h2d and d2d . because of their markedly different distributions on the sky , we consider the halo and disk separately in fig . 5 . for the halo , the small bias ( @xmath187 mag ) is well - determined because of the increased number of h2d solutions , but the slope is not statistically significant . for the disk , both the mean bias ( + 0.13 mag ) and the slope ( 0.69 mag ) are 2-@xmath188 significant . we suggest that these results may be used to apply a bias correction ( subtracted from @xmath107 ) to our real - data results ( sec . 5 ) for the disk , as a function of @xmath189 . ( we choose not to apply a @xmath144 correction to the halo results for two reasons . the halo correction would be small compared to the other sources of error in @xmath9 discussed in sec . 6 ; moreover the bias is compensated by the roughly equal , but opposite , @xmath105 effect discussed in sec . 4.6 . ) for consistency , when we apply the disk @xmath144 correction in sec . 5 , we will also correct the disk kinematics for the `` short '' distance scale by enlarging the velocities and dispersions by a factor of @xmath190 . two objections may be raised to these corrections : first , that the reason for the bias is not understood , and second , that the `` true '' value of @xmath189 is not known for the real data . the latter problem can be overcome by computing @xmath126 and @xmath14 directly from the data , as in sec . 3 . because we only need the ratio @xmath189 , any distance dependence from the assumed @xmath9[fe / h ] relation cancels out . it remains mysterious to us why the disk stat-@xmath18 solutions are slightly biased when the velocity ellipsoid is `` round '' . since the effect occurs for the real ( but not a random ) stellar distribution , it must be caused by the the uneven distribution of the disk rr lyraes on the sky . it might be possible to solve this puzzle with a much larger set of simulations , but that is clearly beyond the scope of this paper . we note that maximum - likelihood methods in general are not unbiased ; clube & dawe ( 1980a ) found an equal @xmath107 bias ( @xmath191 mag ) in the opposite direction ! we conclude simply that since our disk @xmath107 bias is relatively large and can be calibrated as a function of the observational variables , we should apply it to our real - data solutions . applying the lessons learned from our monte carlo simulations ( sec . 4 ) , we analyzed our real data ( each of the disk / halo subsamples in table 3 ) by running the stat-@xmath18 program with @xmath156 and performing two sets of solutions , with and without the velocity ellipsoid covariances ( @xmath103 ) . as in sec . 4.7 , the differences between the two sets of solutions were small . the kinematics generally changed by @xmath89 1 km s@xmath5 ; @xmath107 averaged @xmath192 mag brighter in the solutions without the covariance terms . most important , the errors returned by the stat-@xmath18 program for the disk data sets were typically 10% smaller without the covariances , presumably owing to the larger number of degrees of freedom attained by removing three free parameters from the solutions . since the correlation coefficients ( @xmath193 ) did not prove to be significant ( sec . 4.7 ) , we therefore adopt these solutions , with the velocity ellipsoid aligned to the principal galactic directions ( @xmath101 ) , as our final results . table 6 presents , for each solution as discussed below , the solar motion @xmath194 , the velocity ellipsoid @xmath123 , and the absolute magnitudes . below each of these entries is presented the standard error for that term , computed by the stat-@xmath18 program . recall that the monte carlo simulations suggested that the true uncertainties may be as much as 2 times _ smaller _ than those quoted in the table . the final column of table 6 shows the @xmath9 values after correction for the @xmath195 bias discussed in sec . examination of table 6 shows that the kinematics of the rr lyraes do not depend significantly on which disk / halo definition is employed . because definition 3 of table 3 gives the purest halo sample and the largest , best - determined disk sample , we adopt halo-3 and disk-3 as our best solutions in table 6 . thus , our best estimates of the net rotation and velocity ellipsoid of the halo rr lyraes are @xmath196 @xmath197 for the disk rr lyraes , after correcting ( by a factor of 1.07 ) for the distance scale bias ( @xmath198 mag , sec . 5.2 ) , we obtain @xmath199 @xmath200 we note ( see sec . 4.4 ) that the hjbw stat-@xmath18 program implicitly accounts for the effects of observational errors in the determination of the velocity dispersion parameters , so these results are unbiased estimates of the true velocity dispersions of the halo and disk rr lyrae populations . these kinematic values are in excellent agreement with the rr lyrae kinematics derived by l95 . they also correspond quite well with the kinematics of the thick disk and halo based on other tracer populations ( _ cf . _ casertano _ et al . _ 1990 ; l95 table 8) , with two possible exceptions . first , the vertical velocity dispersion of the disk , @xmath201 km s@xmath5 , is somewhat smaller than the typically quoted value of 3545 km s@xmath5 . l95 suggested that the rr lyrae `` disk '' subsample contains stars from both the thick disk and the old thin disk populations , such that the net kinematics are intermediate between the two . unfortunately , the disk sample contains too few stars for us to subdivide it and perform meaningful stat-@xmath18 solutions for separate thin and thick disk components . second , @xmath126 for the halo rr lyraes is large compared to many other estimates , 168 km s@xmath5 @xmath78 120155 km s@xmath5 . the cause of this effect is less clear . it may be due to our removing interloper thick disk stars more completely than other studies ( l95 ) , or it may be related to a subtle selection bias experienced by rr lyraes . for example , if the halo is composed of an accreted component and a dissipatively - formed component ( zinn 1993 , majewski 1993 ) , if the components have different kinematics ( e.g. , beers 1996 ) , and if rr lyraes are more easily formed in one component than the other , then using rr lyraes as kinematic tracers would bias the kinematic results to favor one or the other halo components , relative to their representation in samples using other stellar tracers . at present , it seems that the halo rr lyraes may be preferentially tracing the accreted halo component , though a detailed analysis outside the scope of this paper is required to further address this problem . in table 6 , @xmath107 is virtually the same for each of the three halo definitions . as above , we adopt halo-3 as our purest definition of the halo rr lyraes . this gives @xmath202 } \rangle = -1.61.\ ] ] as discussed in secs . 4.8 and 6 , no bias corrections have been applied to the halo solutions . for the disk , @xmath107 is more sensitive to which set of stars is used in the stat-@xmath18 solution . as in the monte carlo simulations ( sec 4.4 ) , the errors in the derived @xmath203 s for the disk are @xmath204 times larger than for the halo . both effects are largely due to the relatively small number of stars in the disk solutions . again we adopt disk-3 as the best solution . since the disk velocity ellipsoid is quite `` round '' ( @xmath205 ) , a bias correction of @xmath206 mag ( figure 5 ) was subtracted from the solution value , giving @xmath207 } \rangle = -0.76.\ ] ] to see what would have happened had we been unable to separate the disk and halo rr lyraes , we performed a stat-@xmath18 solution ( last line of table 6 ) using all 213 stars . this solution is comparable to `` group rr @xmath77 '' of hjbw ( 142 stars ) and to `` sample c@xmath208 '' ( 139 stars ) of srm . interestingly , the absolute magnitude ( @xmath209 ) for our `` all stars '' solution is almost exactly the same as for the three halo solutions , but the kinematics are rather different . for this mixture of disk and halo , @xmath132 , @xmath126 , and @xmath210 are smaller than for the pure halo , but @xmath14 is larger . the velocity ellipsoid is much `` rounder '' ; @xmath211 , vs. 1.65 for halo-3 . hjbw and srm found @xmath212 1.25 and 1.29 , respectively , for the comparable groups . these results point out that a good disk / halo separation is necessary to get reliable kinematic results for the rr lyraes . the stat-@xmath18 method is robust enough to produce solutions for mixed populations with distinctly different kinematics , successfully determining @xmath107 , but it derives kinematics not accurately representing either population . the fact that we have separate disk and halo solutions over a range of almost 1 dex in [ fe / h ] gives hope that we might obtain the slope of the @xmath9[fe / h ] relationship , a parameter of considerable astrophysical importance ( sec . 1 ) which previous stat-@xmath18 studies ( e.g. , hjbw , srm ) found difficult to determine . unfortunately , our disk solutions are not sufficiently precise to meaningfully constrain this slope . figure 6 depicts this fact graphically ; the error bars on the disk solutions easily admit slopes between 0 and + 0.4 mag dex@xmath5 , the extreme values currently under debate . calculating the slope using the bias - corrected @xmath9 estimates for halo-3 and disk-3 from table 6 , we obtain @xmath213 = + 0.09 \pm 0.38\ mag\ dex}^{-1}.$ ] to pursue the @xmath9[fe / h ] slope further , we divided the halo-1 sample at [ fe / h ] = 1.55 , giving equal - sized metal - rich and metal - poor sub - groups . we performed stat-@xmath18 solutions on each group ( halo-1r and halo-1p in table 6 , and the crosses in figure 6 ) . these solutions show no indication of any slope within the halo . again , we are unable to constrain the slope of the @xmath9[fe / h ] relation within meaningful limits . in a final effort to determine the @xmath9[fe / h ] slope , we attempted to incorporate this dependence directly into the stat-@xmath18 program by parameterizing the absolute magnitude term @xmath108 in eqn . 5 of hjbw with with two coefficients @xmath214 and @xmath215 , where @xmath216 - \langle [ fe / h ] } \rangle)\ + \ b.\ ] ] however , preliminary solutions indicate that this makes the maximum - likelihood solution ill - conditioned . it may be possible to avoid this problem by incorporating the [ fe / h ] dependence into the distance scale correction @xmath104 rather than @xmath108 ; we are continuing to work on this problem , and any results will be reported in a future paper . in sec . 1 , we mentioned the long history of efforts to determine @xmath9 . csj provide an extensive review of much of this work , which we shall not repeat here . table 7 presents the results of some of these efforts . values produced by the stat-@xmath18 method are for `` typical '' rr lyraes in the sample . these stars tend to be somewhat evolved off the zero age horizontal branch ( zahb ) . some @xmath9 estimates using different methods refer to @xmath9 at the zahb . we note these cases in table 7 , where we have corrected @xmath217 to @xmath9 using the eqn . 4 of csj . ] they are plotted as a function of [ fe / h ] in figure 7 to facilitate comparison with the results of our stat-@xmath18 solutions . we refer the interested reader to the individual references in table 7 for more detailed discussions on these methods . we begin by noting that our stat-@xmath18 solutions are in good agreement with the results of the two recent applications of the method ( @xmath218 , bh86 ; @xmath219 , srm ) . this is not a complete surprise , given that the stars in those studies are all included in the present study , albeit with improved data . as shown in table 7 and figure 7 , the agreement between the various stat-@xmath18 solutions becomes even better when small corrections are made to bring the previous results onto the system of reddenings and magnitudes used in this paper . like us , neither of the previous groups was able to detect a meaningful trend in @xmath9 with [ fe / h ] due to the small sample sizes . at the characteristic abundance of the halo , [ fe / h ] @xmath220 1.6 , the various results shown in figure 7 cover a range of 0.20.3 mag , and appear to separate into a brighter and a fainter group . interestingly , whether a particular result is bright or faint does not seem to be a function of the method employed . for example , buonanno _ et al . _ ( 1990 ) found a bright zero - point by fitting 19 globulars to 5 subdwarfs , while csj found a faint zero - point by fitting a single cluster to the subdwarf ( of identical abundance ) with the best trigonometric parallax . similarly , using the sandage period - shift effect , sandage ( 1990b ) obtained a bright zero - point and a steep slope . using data for a different set of field stars , adopting a different effective temperature relation , and employing a different mass - metallicity relation , csj obtained a moderate slope . fernley ( 1993 ) used infra - red rather than optical observations of field and cluster rr lyraes in his period shift analysis , and found a bright zero - point but a moderate slope . apparently , the current uncertainty in @xmath9 is dominated by differences in the details of the methods a particular author follows , and his or her choice of a particular data set or reddening correction . it is very difficult to determine which of these assumptions are correct or incorrect at this level of detail . in deciding which methods shown in figure 7 should be given the most weight , it is worth noting several strengths of the stat-@xmath18 method . first , stat-@xmath18 is independent of other distance determinations . by contrast , cluster main sequence fitting requires precise trigonometric parallaxes to the nearby subdwarfs . calibrations of @xmath9 based on lmc distances determined by other methods are similarly complicated . for example , the cepheid calibration used by walker ( 1992 ) to obtain the lmc distance is based on main sequence fits of cepheid - bearing galactic open clusters to the pleiades , @xmath221 main sequence fits of the pleiades to local dwarfs with trigonometric parallaxes . a second strength of the stat-@xmath18 method is that it relies on a simple , extremely well - tested model . the kinematics of the galaxy are described by three mean velocities , three velocity dispersions , and the orientation of this velocity ellipsoid relative to the cardinal directions of the galaxy . countless kinematic studies over the past half century have shown this to be a very complete description of local galactic kinematics . by comparison , the models or basic assumptions on which many of the other techniques rely are far more complex , and tend not to be so well - tested . for example , the baade - wesselink method depends on model atmospheres to determine both the surface brightness constant , @xmath222 , and the observed - to - pulsation velocity correction , @xmath223 ( jones _ et al . both the baade - wesselink and period shift methods rely on a color - index @xmath78 effective temperature relation , and different authors advocate different relations ( sandage 1990b , csj , fernley 1993 , sandage 1993 ) . the period shift method also relies heavily on the assumed rr lyrae mass - metallicity relation , yet the mass estimates from double - mode rr lyraes are in serious conflict with those derived from the baade - wesselink method , and to a lesser extent with masses derived from hb theory ( fernley 1993 , yi _ meanwhile , @xmath9 estimates from hb theory are dependent on the color temperature relation , the evolutionary models ( including the treatment of convection ) , and especially on the assumed main sequence helium abundance , @xmath224 ( lee 1990 ) . @xmath9 values derived from main sequence fits currently rely on theoretical isochrones to correct the colors and/or magnitudes of the clusters and/or field subdwarfs to a common metallicity ( buonanno _ et al . _ 1990 , bolte & hogan 1995 ) . the reader is referred to the papers noted above and in table 7 for detailed discussion of these topics . our @xmath9 result for the halo is 0.06 mag brighter than the csj value at [ fe / h ] = 1.61 , and agrees better with this and other `` faint '' @xmath9 values than the `` bright '' results shown in fig . 7 , which are @xmath170.2 mag brighter . given this dichotomy , it is valid to ask whether there are any parameters we could change that would push our result to a brighter value . the results of srm suggest that if we adopted the sturch ( 1966 ) reddening scale , based on the blanketing - corrected colors of rr lyraes at minimum light , we would obtain a result @xmath170.11 mag _ brighter _ than our current result . however , srm argue that the burstein & heiles ( 1982 ) based redding scale is preferred . in sec . 2.4 , we noted that the various sources of rr lyrae photometry suffer small inconsistencies in photometric standardization at the 0.07 mag level . had we used the un - transformed literature values , we would have obtained an @xmath9 for the halo about 0.07 mag _ fainter_. several other effects could alter our @xmath9 result by very small amounts . in sec . 4.6 , we showed that observations constrain @xmath105 to be between 0.0 and 0.1 . by adopting @xmath156 , we obtained the brightest @xmath9 consistent with this constraint ; adopting a smaller value of @xmath105 results in values of @xmath9 up to 0.03 mag _ fainter_. had we corrected for the small bias uncovered by our halo simulations ( sec . 4.8 ) , our result would have been @xmath170.04 mag _ brighter_. had we retained the 3 velocity dispersion covariance parameters in our model ( sec . 4.7 ) , our result would have been 0.02 mag _ brighter_. had we adopted the sample halo-2 rather than halo-3 , our result would have been @xmath170.01 mag _ fainter_. a final note in favor of our @xmath9 zero - point comes from the derived kinematics ( e.g. , sec . if the brighter sandage ( 1993 ) @xmath9 is used , the velocity dispersions for the local thick disk and halo grow by 5 - 10% . the dispersions derived from our stat-@xmath18 solutions and presented in table 6 are in good agreement with estimates of the velocity ellipsoids of these components as measured by other tracers ( in fact , the @xmath126 value for the halo is already larger than many estimates ) . enlarging them to match the brighter @xmath9 degrades this agreement . as discussed in sec . 1 , the @xmath9[fe / h ] relation , in particular its zero - point , is important in determining a number of quantities of interest to both galactic and extra - galactic astronomy . in this section , we present the implications of our stat-@xmath18derived @xmath9 zero - point . in sec . 5.4 , we were unable to obtain a meaningful value for the slope of the @xmath9[fe / h ] relation . however , our zero - point for the halo rr lyraes is quite well determined . in the following discussion , we adopt our zero - point along with a slope @xmath225[fe / h ] = 0.15 mag dex@xmath5 , in agreement with hb theory ( lee 1990 ) , rgb theory ( fusi pecci _ et al . _ 1990 ) , baade - wesselink observations ( jones _ et al . _ 1992 ) , and some analyses of the sandage period shift effect ( csj , fernley 1993 ) . specifically , we adopt the relation @xmath9 = 0.15[fe / h ] + 0.95 mag . walker & mack ( 1986 ) obtained the distance to the galactic center , @xmath226 , by finding the peak in the space density of rr lyraes as a function of distance along the line of sight through baade s window ( bw ) . they recalibrated the photographic photometry of blanco ( 1984 ) to the johnson @xmath227 band using several ccd standard fields , corrected it for interstellar absorption , and converted it to the @xmath68 band using a @xmath228 period relation obtained from ngc 6171 . they found @xmath229 if @xmath9 = + 0.60 mag . we repeat their analysis here , comparing the results obtained using both their preferred value of @xmath9 = + 0.60 mag , and our preferred value of @xmath9 = 0.15[fe / h ] + 0.95 mag . we employ the newer reddening estimates of @xmath230 = 0.50 for bw ( walker & terndrup 1991 ) and @xmath230 = 0.33 for ngc 6171 ( harris 1994 ) . we also employ the walker & terndrup ( 1991 ) @xmath66@xmath67 metallicities , rather than the photometric ones used by walker & mack ( 1986 ) . like those authors , we found that a small shift ( + 0.04 mag ) should be applied to make the fitted line in the @xmath231 period plane coincide with the bw rr lyraes listed in their table 7 , presumably to correct for slight inconsistencies in the adopted reddenings and/or metallicities of the cluster rr lyraes relative to those in bw . figure 8 shows the space density of stars ( in arbitrary units ) as a function of distance through bw . using a variety of methods , we find that the curve based on @xmath9 = 0.15[fe / h ] + 0.95 peaks at @xmath232 = 7.4 kpc ( squares ) , while that based on @xmath9 = + 0.60 peaks at @xmath232 = 8.1 kpc ( circles ) . after applying the small geometric correction of 1.03 discussed by walker & mack ( 1986 ) , we obtain @xmath233 and @xmath234 , respectively . we note that the relation @xmath9 = 0.15[fe / h ] + 0.725 mag , based on the walker ( 1992 ) zero - point ( see table 7 ) produces results almost identical with those of the @xmath235 relation . we also note that the widths of the density curves produced using these three @xmath9[fe / h ] relations are nearly identical , after they are corrected for the distance scale effects produced by the different distance zero - points . this supports the statement by walker & terndrup ( 1991 ) that the abundance dispersion in bw is too small to meaningfully constrain the slope of the @xmath9[fe / h ] relation using this method . the short distance to the galactic center based on our stat-@xmath18 zero - point is favored by the `` primary '' distance source , h@xmath236o maser proper motions ( @xmath237 kpc ) , quoted by reid ( 1993 ) . reid lists a number of other @xmath226 estimates , and from them derives a `` best value '' of @xmath238 kpc . as noted by carney _ ( 1995 ) , this value is in part determined using a `` bright '' rr lyrae calibration . if we recompute the `` best value '' excluding the optical rr lyrae - dependent methods , we obtain @xmath239 kpc . this value is further supported by carney _ ( 1995 ) , who recently found @xmath240 kpc from the @xmath241-band photometry of 58 rr lyrae stars in bw , as calibrated using the @xmath242log(@xmath243 ) relation of csj . the age of a globular cluster can be determined by comparing the absolute magnitude of its main - sequence turnoff ( msto ) with the mstos of theoretical isochrones . the difference in apparent magnitude between a cluster s rr lyraes and its msto , together with an adopted value for @xmath9 , can be used to find the absolute magnitude of the cluster s msto . it is instructive to see the difference between the ages derived using our @xmath9 zero - point and those using the brighter zero - point of walker ( 1992 ) . for both cases , we adopt a metallicity dependence of 0.15 mag dex@xmath5 , so the relations are @xmath244 [ fe / h ] + 0.95 , and @xmath244 [ fe / h ] + 0.725 , respectively . the zero - point of the @xmath9[fe / h ] relation primarily determines the mean age of the globular cluster system , while the slope of the relation determines the age distribution . thus our comparison focuses on the mean ages of the halo globular cluster system under the two @xmath9 zero - points . brian chaboyer has kindly computed ages for the 39 `` older '' clusters listed in table 3 of chaboyer , demarque & sarajedini ( 1996b ) using both of these @xmath9[fe / h ] relations . the ages were derived using the opal equation of state isochrones of chaboyer & kim ( 1995 ) . these isochrones include the effects of diffusion and the latest available equation of state ( rogers 1994 ) , and employ modern helium and @xmath245-element abundances . the weighted mean age of the 39 clusters is @xmath246 gyr using our zero - point , and @xmath247 gyr using the walker ( 1992 ) zero - point ( standard errors of the mean ) . we account for the uncertainties in the @xmath9 zero - points by computing the mean ages from @xmath9[fe / h ] relations based on zero - points of @xmath248 ( again , thanks to b. chaboyer ) . for our stat-@xmath18 @xmath9 zero - point , the mean age including formal errors is @xmath249 gyr , compared with @xmath250 gyr using the walker ( 1992 ) zero - point . these values are @xmath1714% smaller than the ages computed without the improved treatments of diffusion and the equation of state ( chaboyer _ et al . _ 1996b ) . clearly , the @xmath9 zero - point derived from the present stat-@xmath18 analysis supports an older age for the globular cluster system . the ages of the _ oldest _ globular clusters place a lower limit on the age of the universe . ( 1996a ) define a group of 17 clusters which they suspect represents the oldest galactic globulars . using the method described above , their weighted mean age is @xmath251 gyr using our stat-@xmath18 @xmath9 zero - point , and @xmath252 gyr using walker s zero - point . a word of caution is warranted here . the median [ fe / h ] of this group is @xmath170.2 dex lower than that of the field rr lyraes used to determine the @xmath9 zero - point , so if the @xmath9[fe / h ] slope is 0.30 ( 0.0 ) mag dex@xmath5 , the mean derived ages are @xmath253 gyr smaller ( larger ) than those quoted . walker ( 1992 ) lists the de - reddened @xmath70 magnitudes of the rr lyrae stars in seven large magellanic cloud ( lmc ) globular clusters . using an lmc distance modulus of 18.50 @xmath75 0.10 , based on lmc cepheid observations and an abundance - corrected galactic cepheid calibration , he obtained @xmath254 at [ fe / h ] = 1.9 , consistent with the `` brighter '' @xmath9 values shown in fig . 7 . this distance modulus implies an lmc distance of @xmath255 kpc . using walker s @xmath70 magnitudes , and adopting @xmath256[fe / h ] + 0.95 , based on our new stat-@xmath18 zero - point , we find the lmc distance modulus to be 18.28 @xmath75 0.13 , equivalent to a distance of 45 @xmath75 3 kpc . given the complexities of deriving the cepheid period - luminosity relation ( see sec . 6 , also walker ( 1992 ) and references therein ) , it seems worthwhile to explore the consequences of applying a zero - point offset to the cepheid period - luminosity relation to make it match the shorter lmc distance based on our stat-@xmath18 results . a particularly interesting consequence involves the measurement of the hubble constant , @xmath13 . many extra - galactic distance indicators are calibrated to , or are consistent with , an lmc distance modulus of 18.50 mag . if the distances determined using these indicators are recalibrated to agree with our smaller lmc distance modulus , the value of @xmath13 derived from them increases by 10% . for example , the recent cepheid - based distance of @xmath257 mpc to m100 yielded @xmath258 km s@xmath5 mpc@xmath5 ( ferrarese _ et al . if we reduce the m100 distance modulus by 0.22 mag to bring the cepheid period - luminosity relation into agreement with our lmc distance , we obtain @xmath259 mpc and @xmath260 km s@xmath5 mpc@xmath5 . freedman _ et al . _ ( 1994 ; their fig . 3 ) have already shown that the expansion age of the universe implied by @xmath261 km s@xmath5 mpc@xmath5 in the framework of the einstein de sitter cosmological model is in conflict with the observed ages of globular clusters . our stat-@xmath18 @xmath9 zero - point indicates a larger value of @xmath13 ( shorter expansion time ) and older globular clusters , thus increasing the disagreement between these two important observables . the disagreement persists even at lower values of the density parameter ( @xmath262 ) . however , other recent measurements of @xmath13 obtain lower values which are in better agreement with our cluster ages . for example , branch _ et al . _ ( 1996 ) obtained @xmath263 km s@xmath5 mpc@xmath5 from type ia supernovae . additional possible routes to reconciling observations of @xmath13 and globular cluster ages include accepting a non - zero cosmological constant ( carroll & press 1992 ) and further refinements to stellar evolution theory which result in younger cluster ages ( e.g. , mazzitelli _ et al . we have assembled high - quality data on 213 nearby rr lyrae variables . these data include new absolute proper motions from the lick northern proper motion program ( klemola _ et al . _ 1993 ) and abundances and radial velocities from layden ( 1994 ) . based on an _ a priori _ kinematic study , we defined three ways to separate the stars into thick disk and halo sub - populations . statistical parallax solutions for these sub - samples yielded the absolute magnitude and kinematics of the rr lyraes in the samples . we note that our @xmath9 values correspond to the absolute magnitude of typically - evolved rr lyrae stars , not to that of the zero age horizontal branch . for the halo population , the solutions produced a well determined absolute magnitude , @xmath0 at @xmath1[fe / h]@xmath2 = 1.61 . the derived kinematics , @xmath264 and @xmath265 , are in good agreement with previous estimates of the halo rr lyrae kinematics , and with the kinematics of other stellar tracers of the halo . for the thick disk population , the results of the three definitions scatter somewhat , and the uncertainties are larger . our best estimate for the thick disk was @xmath3 at @xmath1[fe / h]@xmath2 = 0.76 , after correction for the bias mentioned below . the derived kinematics , @xmath266 and @xmath267 , again are in good agreement with previous estimates of the thick disk kinematics based on rr lyraes and on other tracers . the large uncertainty in the disk solution prevented us from deriving a meaningful slope for the @xmath9[fe / h ] relation . an attempt to measure the slope by sub - dividing the halo sample into two metallicity bins also failed to meaningfully determine the slope . monte carlo tests using simulated data showed that our stat-@xmath18 code accurately returns the true kinematic and @xmath107 values of the input data . they also revealed the possibility that the internal errors returned by the hjbw stat-@xmath18 algorithm may be overestimates . for the halo , the true error of @xmath9 may be @xmath170.08 mag rather than 0.12 mag . the kinematic results for both the disk and halo may also be more precise than the errors cited above . however , it is outside the scope of this paper to determine which set of error values is correct . so , for the present , we have conservatively adopted the larger values . the simulations also enabled us to evaluate the effects of other factors on the solutions . all were negligible , except for a small bias towards `` short '' distance scales that is observed when the @xmath80 and @xmath68 velocity dispersions are of comparable size . we determined corrections based on the simulations : 0.15 mag for the disk and 0.04 mag for the halo solutions . the former correction was applied to the real - data solutions , while the latter was deemed too small to be of practical value . we discussed the effects of systematic errors . the main systematic uncertainty is in the adopted reddening scale . our scale ( burstein & heiles 1982 ) makes our @xmath9 value @xmath170.11 mag fainter than it would have been using the sturch ( 1966 ) reddenings , but srm argue that the former scale is preferred . our adopted photometric scale makes our quoted @xmath9 value as bright as possible ( @xmath170.07 mag ) . several other minor systematics , including the halo bias correction noted above , were also considered . it is unlikely that these systematic errors alone can account for the difference between our @xmath9 value and the brighter ( @xmath170.2 mag ) values found by some authors ( e.g. , buonanno _ et al . _ 1990 ; sandage 1990b , 1993 ; walker 1992 ) . unlike the methods used by many authors , the stat-@xmath18 method is independent of other distance calibrations . it also depends on a relatively simple and well - tested model ( galactic kinematics ) in comparison to the other methods , which employ model atmospheres , stellar evolution theory , empirical color temperature relations , rr lyrae mass determinations , etc . we investigated the implications of our halo @xmath9 by using an @xmath9[fe / h ] relation based on an adopted slope of @xmath15[fe / h ] = 0.15 mag dex@xmath5 in combination with our halo zero - point . we found the distance to the galactic center to be @xmath268 kpc based on observations of the rr lyraes in baade s window , in good agreement with many other estimates of @xmath226 . using the `` brighter '' @xmath9 values , @xmath226 is 10% larger . following chaboyer _ ( 1996b ) , we found the mean age of the 17 oldest galactic globular clusters to be @xmath269 gyr , 3.4 gyr older than the mean age obtained using the brighter rr lyrae zero - point ; this places an important lower limit on the age of the universe . we found the distance modulus of the large magellanic cloud to be @xmath12 mag . any estimates of the hubble constant , @xmath13 , which are based on an lmc distance modulus of 18.50 mag ( e.g. , the cepheid study of ferrarese _ et al . _ 1996 ) increase by 10% if their distance scales are recalibrated to match our lmc distance . this increase implies a younger age for the universe , in conflict with the older globular cluster ages derived from our @xmath9 value . the conflict is lessened or eliminated if the true value of @xmath13 is low , if the cosmological constant is non - zero , or if further refinements to stellar evolution theory result in younger cluster ages . the authors acknowledge valuable discussions with drs . brian chaboyer , jan lub , robert schommer , and alistair walker . the comments of an anonymous referee improved several sections of the paper . acl acknowledges financial support from the natural sciences and engineering research council of canada , through research grants to w.e . harris and d.l . welch , and from cerro tololo inter - american observatory . the lick northern proper motion program is supported by national science foundation grant ast 92 - 18084 . slh acknowledges the support of nsf young investigator ( nyi ) grant number ast94 - 57455 . cjh performed part of this research as an reu student at michigan state university . ajhar , e.a . , grillmair , c.j . , lauer , t.r . , baum , w.a . , faber , s.m . , holtzman , j.a . , lynds , c.r . & oneil , e.j.jr . 1996 , , 111 , 1110 barnes iii , t.g . , & hawley , s.l . 1986 , , 307 , l9 ( bh86 ) beers , t.c . 1996 , formation of the galactic halo ... inside and out , asp conf . morrison & a. sarajedini , ( asp : san francisco ) , 130 blanco , b.m . 1984 , , 89 , 1836 blanco , v.m . 1992 , , 104 , 734 bolte , m. & hogan 1995 , nature , 376 , 399 bookmyer , b.b . , fitch , w.s . , lee , t.a . , wisniewski , w.z . & johnson , h.l . 1977 , rev . astrofis . , 2 , 235 branch , d. , romanishin , w. , & baron , e. 1996 , , 465 , 73 buonanno , r. , corsi , c.e . & fusi pecci , f. 1989 , , 216 , 80 buonanno , r. , cacciari , c. , corsi , c.e . & fusi pecci , f. 1990 , , 230 , 315 burstein , d. , & heiles , c. 1982 , , 87 , 1165 carney , b.w . , storm , j. & jones , r.v . 1992 , , 386 , 663 ( csj ) carney , b.w . , fulbright , j.p . , terndrup , d.m . , suntzeff , n.b . , & walker , a.r . 1995 , , 110 , 1674 carrol , s.m . & press , w.h . 1992 , , 30 , 499 casertano , s. , ratnatunga , k.u . , & bahcall , j.n . 1990 , , 357 , 435 chaboyer , b. 1995 , , 444 , l9 chaboyer , b. & kim , y .- c . 1995 , , 454 , 767 chaboyer , b. , demarque , p. , kerman , p.j . & krauss , l.m . 1996a , science , 271 , 957 chaboyer , b. , demarque , p. & sarajedini , a. 1996b , , 459 , 558 clube , s.v.m . & dawe , j.a . 1980a , , 190 , 575 clube , s.v.m . & dawe , j.a . 1980b , , 190 , 591 ( cd80 ) croswell , k. , latham , d.w . & carney , b.w . 1987 , , 93 , 1445 eggen , o.j . , lynden - 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( 1966 ) , , 143 , 774 walker , a.r . 1992 , , 390 , l81 walker , a.r . & mack , p. 1986 , , 220 , 69 walker , a.r . & terndrup , d.m . 1991 , , 378 , 119 wan , l. , he , m .- f . , zhu , g .- l . , & li , z .- y . 1979 , chinese astronomy , 3 , 296 wan , l. , mao , y .- q . & ji , d .- s . 1980 , ann . shanghai obs . , no . 2 , 1 ( wmj ) yi , s. , lee , y .- w . & demarque , p. 1993 , , 411 , l25 zinn , r.j . 1993 , the globular cluster galaxy connection , asp conf . smith & j.p . brodie , ( asp : san francisco ) , 38 zinn , r. , & west , m.j . 1984 , , 55 , 45 sw and & + 29.0017 & 115.72 & 33.09 & 0.53 & 2.49 & 21 & 2 & 0.38 & 9.68 & 0.14 & 1121 & 111 xx and & + 38.0072 & 128.45 & 23.64 & 3.53 & 3.56 & 0 & 5 & 2.01 & 10.63 & 0.13 & 1111 & 000 xy and & + 33.0068 & 131.22 & 28.23 & 1.34 & 0.28 & 64 & 53 & 0.92 & 13.63 & 0.15 & 1161 & 000 zz and & + 26.0036 & 122.42 & 35.85 & 2.50 & 1.80 & 13 & 53 & 1.58 & 13.01 & 0.12 & 1141 & 000 ci and & + 43.0105 & 134.93 & 17.62 & 0.15 & 0.38 & 99 & 30 & 0.83 & 12.15 & 0.26 & 1141 & 111 dr and & + 33.0052 & 126.16 & 28.57 & 3.00 & 1.38 & 81 & 30 & 1.48 & 12.34 & 0.09 & 1141 & 000 wy ant & & 266.93 & 22.08 & 2.56 & 4.81 & 211 & 24 & 1.66 & 10.83 & 0.18 & 2111 & 000 sw aqr & 00.1628 & 51.31 & 31.47 & 4.67 & 6.08 & 42 & 8 & 1.24 & 11.14 & 0.22 & 1111 & 000 sx aqr & + 03.1345 & 57.90 & 34.00 & 3.87 & 5.14 & 166 & 7 & 1.83 & 11.70 & 0.11 & 1111 & 000 tz aqr & 05.1879 & 53.25 & 44.33 & 0.96 & 1.77 & 35 & 12 & 1.24 & 12.01 & 0.10 & 1121 & 000 yz aqr & 11.1939 & 48.93 & 49.76 & 1.25 & 0.21 & 150 & 14 & 1.55 & 12.65 & 0.08 & 1161 & 000 aa aqr & 10.2266 & 54.63 & 53.83 & 2.11 & 0.30 & 20 & 14 & 2.09 & 12.37 & 0.15 & 1161 & 000 bn aqr & 07.2815 & 56.22 & 50.73 & 0.96 & 3.26 & 182 & 30 & 1.33 & 12.52 & 0.10 & 1161 & 000 bo aqr & 12.2527 & 55.40 & 58.83 & 0.82 & 1.21 & 24 & 13 & 1.80 & 12.11 & 0.07 & 1111 & 000 br aqr & 09.2377 & 75.48 & 65.24 & 0.50 & 0.02 & 29 & 10 & 0.84 & 11.40 & 0.04 & 1121 & 111 bt aqr & 05.1748 & 42.94 & 30.60 & 0.30 & 1.06 & 52 & 11 & 0.29 & 12.34 & 0.12 & 1121 & 111 cp aqr & 01.1098 & 48.74 & 31.34 & 1.45 & 1.73 & 29 & 21 & 0.90 & 11.71 & 0.11 & 1131 & 111 dn aqr & & 35.76 & 69.06 & 4.60 & 1.60 & 214 & 8 & 1.63 & 11.18 & 0.02 & 2131 & 000 aa aql & 03.1592 & 43.08 & 24.99 & 1.80 & 1.54 & 32 & 4 & 0.58 & 11.74 & 0.21 & 1121 & 111 v341 aql & & 45.62 & 22.04 & 3.48 & 2.40 & 81 & 4 & 1.37 & 10.81 & 0.31 & 2121 & 000 x ari & + 10.0299 & 169.08 & 39.84 & 5.97 & 9.24 & 35 & 3 & 2.40 & 9.51 & 0.50 & 1111 & 000 tz aur & + 40.0228 & 176.79 & 20.92 & 0.30 & 1.35 & 46 & 6 & 0.80 & 11.86 & 0.18 & 1121 & 111 rs boo & + 31.0704 & 50.84 & 67.35 & 0.22 & 1.17 & 9 & 2 & 0.32 & 10.36 & 0.00 & 1121 & 111 st boo & + 35.0720 & 57.39 & 55.22 & 1.54 & 1.67 & 13 & 4 & 1.86 & 10.98 & 0.04 & 1121 & 000 sv boo & + 39.0683 & 68.75 & 65.51 & 0.17 & 2.28 & 131 & 22 & 1.55 & 13.12 & 0.00 & 1121 & 000 sw boo & + 36.0643 & 62.52 & 67.74 & 4.58 & 0.13 & 18 & 18 & 1.12 & 12.34 & 0.00 & 1111 & 000 sz boo & + 28.0840 & 41.93 & 65.50 & 0.77 & 0.85 & 38 & 21 & 1.68 & 12.60 & 0.01 & 1111 & 000 tw boo & + 41.0778 & 71.06 & 62.85 & 0.34 & 5.86 & 99 & 4 & 1.41 & 11.20 & 0.01 & 1121 & 000 uu boo & + 35.0710 & 56.50 & 58.01 & 0.86 & 3.38 & 10 & 28 & 1.92 & 12.22 & 0.01 & 1121 & 000 uy boo & + 13.0991 & 354.24 & 68.81 & 0.28 & 4.51 & 144 & 3 & 2.49 & 10.80 & 0.00 & 1121 & 000 rz cam & + 67.0052 & 147.98 & 23.17 & 0.75 & 0.57 & 266 & 26 & 1.01 & 12.73 & 0.18 & 1121 & 000 rw cnc & + 29.0370 & 197.49 & 43.53 & 0.79 & 4.17 & 85 & 7 & 1.52 & 11.85 & 0.03 & 1161 & 000 ss cnc & + 23.0300 & 198.94 & 26.28 & 0.77 & 1.72 & 27 & 16 & 0.07 & 12.16 & 0.09 & 1121 & 111 tt cnc & + 13.0361 & 212.10 & 28.38 & 4.60 & 4.00 & 49 & 5 & 1.58 & 11.24 & 0.12 & 1111 & 000 an cnc & + 15.0560 & 212.10 & 35.03 & 0.51 & 2.53 & 16 & 14 & 1.45 & 13.16 & 0.06 & 1141 & 000 aq cnc & + 12.0610 & 218.04 & 38.10 & 1.15 & 3.39 & 390 & 20 & 1.53 & 12.00 & 0.04 & 1141 & 000 as cnc & + 25.0305 & 197.89 & 31.23 & 2.79 & 0.83 & 258 & 26 & 1.89 & 12.50 & 0.05 & 1141 & 000 w cvn & + 38.0637 & 71.82 & 70.96 & 1.71 & 0.89 & 18 & 21 & 1.21 & 10.52 & 0.00 & 1111 & 001 z cvn & + 44.0884 & 124.00 & 73.35 & 0.79 & 3.97 & 14 & 10 & 1.98 & 11.93 & 0.00 & 1121 & 000 rr cvn & + 34.0596 & 154.05 & 81.09 & 1.58 & 3.17 & 5 & 21 & 1.08 & 12.55 & 0.01 & 1111 & 000 ru cvn & + 31.0668 & 53.96 & 74.51 & 2.97 & 0.22 & 27 & 21 & 1.37 & 11.96 & 0.00 & 1121 & 000 rx cvn & + 41.0711 & 87.08 & 71.53 & 0.20 & 0.10 & 158 & 28 & 1.31 & 12.57 & 0.00 & 1121 & 000 rz cvn & + 32.1089 & 61.59 & 77.15 & 5.84 & 0.36 & 12 & 7 & 1.92 & 11.42 & 0.00 & 1121 & 000 ss cvn & + 40.0608 & 83.85 & 72.63 & 1.24 & 5.10 & 15 & 22 & 1.52 & 11.89 & 0.00 & 1121 & 000 sv cvn & + 37.0978 & 139.98 & 79.40 & 0.12 & 2.55 & 29 & 28 & 2.20 & 12.59 & 0.00 & 1121 & 000 sw cvn & + 37.0987 & 134.84 & 79.80 & 0.96 & 1.98 & 18 & 21 & 1.53 & 12.74 & 0.00 & 1121 & 000 uz cvn & + 40.0532 & 139.53 & 75.93 & 0.06 & 2.32 & 38 & 26 & 2.34 & 12.02 & 0.00 & 1141 & 000 al cmi & + 05.0339 & 214.43 & 15.35 & 1.21 & 0.52 & 46 & 21 & 0.85 & 12.01 & 0.09 & 1151 & 111 rv cap & 15.2350 & 33.13 & 35.54 & 2.42 & 10.57 & 106 & 7 & 1.72 & 10.92 & 0.11 & 1111 & 000 iu car & & 269.59 & 22.95 & 1.30 & 0.60 & 328 & 18 & 1.85 & 11.91 & 0.39 & 2111 & 000 v499 cen & & 315.10 & 18.12 & 2.10 & 0.25 & 323 & 24 & 1.56 & 11.05 & 0.18 & 2111 & 000 dx cep & + 83.0238 & 119.50 & 21.95 & 1.86 & 0.91 & 6 & 30 & 1.83 & 12.67 & 0.35 & 1161 & 000 rr cet & + 01.0121 & 143.54 & 59.89 & 0.05 & 3.88 & 75 & 1 & 1.52 & 9.75 & 0.02 & 1111 & 001 ru cet & 16.0129 & 134.27 & 78.63 & 2.15 & 1.00 & 57 & 8 & 1.60 & 11.60 & 0.01 & 1111 & 000 rv cet & 11.0305 & 177.32 & 64.40 & 2.58 & 1.83 & 93 & 7 & 1.32 & 10.76 & 0.02 & 1111 & 000 rx cet & 15.0060 & 102.48 & 77.64 & 2.89 & 6.25 & 58 & 7 & 1.46 & 11.36 & 0.03 & 1111 & 000 rz cet & 08.0299 & 178.21 & 60.34 & 2.59 & 1.88 & 10 & 6 & 1.50 & 11.84 & 0.03 & 1121 & 000 xz cet & 16.0267 & 182.40 & 70.75 & 4.14 & 0.13 & 167 & 10 & 2.27 & 9.49 & 0.00 & 1161 & 000 ry col & & 246.47 & 35.05 & 3.60 & 1.80 & 482 & 15 & 1.11 & 10.86 & 0.01 & 2131 & 000 s com & + 27.0979 & 213.16 & 85.84 & 2.22 & 2.01 & 55 & 4 & 2.00 & 11.55 & 0.02 & 1111 & 000 v com & + 27.0931 & 208.67 & 80.85 & 0.84 & 0.47 & 23 & 28 & 1.75 & 13.16 & 0.02 & 1121 & 000 ry com & + 23.0619 & 342.56 & 85.06 & 0.61 & 1.78 & 31 & 8 & 1.65 & 12.30 & 0.04 & 1121 & 000 tv crb & + 27.1359 & 41.33 & 56.51 & 0.31 & 1.01 & 157 & 48 & 2.33 & 11.61 & 0.07 & 1161 & 000 w crt & 17.1378 & 276.00 & 40.47 & 2.38 & 1.48 & 65 & 13 & 0.50 & 11.51 & 0.09 & 1121 & 111 x crt & 10.1333 & 278.87 & 49.49 & 0.13 & 3.55 & 79 & 4 & 1.75 & 11.48 & 0.00 & 1111 & 000 xz cyg & & 88.21 & 16.98 & 8.43 & 2.15 & 119 & 14 & 1.52 & 9.72 & 0.33 & 2111 & 000 dm cyg & + 31.0966 & 79.46 & 12.41 & 1.34 & 0.72 & 12 & 23 & 0.14 & 11.49 & 0.69 & 1121 & 111 dx del & + 12.1761 & 58.47 & 18.84 & 1.41 & 1.17 & 45 & 3 & 0.56 & 9.92 & 0.32 & 1121 & 111 rw dra & + 57.0740 & 87.39 & 40.60 & 0.23 & 0.82 & 108 & 22 & 1.40 & 11.57 & 0.00 & 1121 & 001 su dra & + 67.0274 & 133.44 & 48.27 & 4.95 & 7.26 & 167 & 1 & 1.74 & 9.78 & 0.00 & 1111 & 000 sw dra & + 69.0237 & 127.27 & 47.33 & 3.22 & 0.93 & 30 & 1 & 1.24 & 10.49 & 0.04 & 1121 & 001 wy dra & + 80.0305 & 113.08 & 25.14 & 1.11 & 0.88 & 6 & 30 & 1.66 & 12.67 & 0.24 & 1161 & 000 xz dra & + 64.0579 & 95.65 & 22.50 & 0.56 & 0.71 & 30 & 2 & 0.87 & 10.18 & 0.22 & 1121 & 111 ae dra & + 55.0697 & 84.35 & 25.41 & 0.83 & 0.24 & 243 & 30 & 1.54 & 12.65 & 0.14 & 1161 & 000 bc dra & + 76.0341 & 107.95 & 28.48 & 2.18 & 3.30 & 161 & 26 & 2.00 & 11.60 & 0.18 & 1161 & 000 bd dra & + 77.0411 & 108.63 & 28.25 & 2.88 & 0.07 & 253 & 30 & 1.74 & 12.69 & 0.10 & 1161 & 000 bt dra & + 60.0453 & 99.41 & 51.21 & 0.13 & 3.60 & 156 & 30 & 1.55 & 11.94 & 0.00 & 1161 & 000 rx eri & 15.0672 & 214.26 & 33.88 & 1.56 & 0.70 & 66 & 1 & 1.30 & 9.68 & 0.08 & 1111 & 000 sv eri & 11.0414 & 194.26 & 53.47 & 1.45 & 3.95 & 12 & 9 & 2.04 & 9.95 & 0.19 & 1111 & 000 xy eri & 13.0580 & 207.42 & 41.69 & 1.13 & 0.68 & 221 & 11 & 2.08 & 13.02 & 0.08 & 1161 & 000 bb eri & 19.0632 & 218.81 & 34.36 & 3.45 & 0.91 & 235 & 11 & 1.51 & 11.46 & 0.03 & 1121 & 000 bk eri & 01.0185 & 175.80 & 51.70 & 3.24 & 1.98 & 141 & 10 & 1.64 & 12.67 & 0.10 & 1161 & 000 ss for & & 216.42 & 72.99 & 4.30 & 7.25 & 112 & 1 & 1.35 & 10.10 & 0.00 & 2121 & 000 sw for & & 243.27 & 60.75 & 1.25 & 0.10 & 174 & 18 & 1.95 & 12.34 & 0.00 & 2111 & 000 rr gem & & 187.44 & 19.52 & 0.35 & 0.24 & 64 & 1 & 0.35 & 11.34 & 0.21 & 2121 & 111 sz gem & + 19.0314 & 201.85 & 22.08 & 1.17 & 3.40 & 307 & 11 & 1.81 & 11.66 & 0.08 & 1121 & 000 tw her & + 30.0973 & 55.87 & 24.80 & 0.32 & 0.74 & 4 & 16 & 0.67 & 11.23 & 0.17 & 1121 & 111 vz her & + 36.0784 & 59.59 & 34.59 & 2.44 & 1.95 & 115 & 4 & 1.03 & 11.44 & 0.12 & 1121 & 000 af her & + 41.0868 & 65.15 & 41.64 & 1.62 & 0.72 & 268 & 9 & 1.94 & 12.82 & 0.00 & 1121 & 000 ag her & + 40.0809 & 64.49 & 41.46 & 2.15 & 1.91 & 103 & 21 & 2.01 & 12.66 & 0.00 & 1121 & 000 cw her & + 35.0792 & 58.01 & 38.99 & 1.65 & 1.31 & 285 & 30 & 2.09 & 12.47 & 0.05 & 1141 & 000 dl her & + 14.1435 & 36.28 & 26.60 & 1.13 & 0.13 & 61 & 14 & 1.32 & 12.37 & 0.35 & 1141 & 101 gy her & + 37.1381 & 60.70 & 41.71 & 0.29 & 1.13 & 157 & 53 & 1.92 & 12.57 & 0.01 & 1141 & 000 v394 her & + 17.1675 & 40.01 & 27.39 & 0.67 & 0.67 & 74 & 10 & 1.48 & 12.87 & 0.21 & 1161 & 000 sv hya & & 297.08 & 36.59 & 5.51 & 0.81 & 100 & 8 & 1.70 & 10.51 & 0.34 & 2111 & 000 sz hya & 09.0958 & 239.77 & 25.93 & 0.66 & 3.85 & 140 & 9 & 1.75 & 11.23 & 0.05 & 1121 & 000 uu hya & + 04.0500 & 230.41 & 38.18 & 2.51 & 1.12 & 295 & 14 & 1.65 & 12.27 & 0.05 & 1121 & 000 wz hya & 12.1166 & 254.26 & 34.41 & 0.14 & 1.49 & 304 & 8 & 1.30 & 10.82 & 0.26 & 1121 & 000 xx hya & 15.1036 & 244.59 & 21.35 & 1.86 & 2.92 & 32 & 20 & 1.33 & 11.89 & 0.15 & 1121 & 000 dd hya & + 02.0597 & 219.85 & 19.30 & 0.70 & 1.19 & 153 & 25 & 1.00 & 12.18 & 0.06 & 1151 & 011 dg hya & 05.0893 & 233.78 & 24.95 & 1.08 & 2.52 & 164 & 18 & 1.42 & 12.14 & 0.06 & 1121 & 000 dh hya & 09.0936 & 238.03 & 22.96 & 2.37 & 0.67 & 355 & 8 & 1.55 & 12.13 & 0.05 & 1111 & 000 et hya & 08.0736 & 233.50 & 18.31 & 1.66 & 0.73 & 320 & 20 & 1.69 & 12.06 & 0.08 & 1151 & 000 fy hya & & 318.76 & 31.37 & 4.12 & 0.04 & 82 & 24 & 2.33 & 12.46 & 0.15 & 2111 & 000 gl hya & + 02.0649 & 223.71 & 25.46 & 1.34 & 0.90 & 223 & 21 & 1.45 & 12.95 & 0.07 & 1161 & 000 go hya & + 06.0328 & 221.77 & 30.32 & 0.23 & 0.98 & 25 & 23 & 0.83 & 12.34 & 0.09 & 1141 & 111 v ind & & 355.33 & 43.12 & 7.00 & 9.00 & 202 & 3 & 1.50 & 9.92 & 0.05 & 2111 & 000 cq lac & + 39.0934 & 93.95 & 14.55 & 0.34 & 0.15 & 20 & 30 & 2.04 & 12.43 & 0.45 & 1161 & 000 rr leo & + 24.0416 & 208.42 & 53.10 & 1.69 & 1.26 & 88 & 1 & 1.57 & 10.68 & 0.09 & 1111 & 001 rx leo & + 26.0471 & 209.43 & 70.51 & 0.38 & 2.66 & 121 & 6 & 1.38 & 11.90 & 0.00 & 1111 & 000 ss leo & + 00.0760 & 265.32 & 57.06 & 2.38 & 2.79 & 163 & 3 & 1.83 & 11.03 & 0.04 & 1111 & 000 st leo & + 10.0701 & 253.44 & 66.15 & 0.56 & 3.37 & 153 & 4 & 1.29 & 11.46 & 0.09 & 1121 & 000 su leo & + 08.0618 & 228.92 & 43.82 & 0.60 & 0.86 & 81 & 30 & 1.41 & 13.55 & 0.02 & 1161 & 000 sw leo & 02.1164 & 255.63 & 48.98 & 0.74 & 0.68 & 46 & 11 & 1.45 & 13.08 & 0.07 & 1161 & 000 sz leo & + 08.0731 & 243.93 & 57.83 & 1.60 & 2.55 & 185 & 4 & 1.86 & 12.35 & 0.04 & 1121 & 000 tv leo & 05.1129 & 262.99 & 49.06 & 1.07 & 0.28 & 96 & 5 & 1.97 & 12.10 & 0.07 & 1111 & 000 ww leo & + 07.0715 & 226.04 & 38.45 & 0.03 & 2.63 & 66 & 20 & 1.48 & 12.47 & 0.09 & 1121 & 000 aa leo & + 10.0702 & 254.14 & 66.09 & 0.26 & 3.35 & 32 & 24 & 1.47 & 12.27 & 0.09 & 1121 & 000 ae leo & + 17.0930 & 234.20 & 68.19 & 2.38 & 1.25 & 53 & 10 & 1.71 & 12.52 & 0.00 & 1151 & 000 an leo & + 06.0502 & 253.35 & 60.72 & 0.29 & 3.06 & 68 & 17 & 1.14 & 12.45 & 0.13 & 1141 & 000 ax leo & + 12.0892 & 248.29 & 66.30 & 1.85 & 2.08 & 182 & 10 & 2.28 & 12.18 & 0.07 & 1141 & 000 bt leo & + 18.0551 & 228.38 & 65.59 & 0.90 & 0.34 & 119 & 14 & 0.81 & 13.11 & 0.00 & 1141 & 111 v lmi & + 29.0467 & 201.30 & 57.84 & 2.16 & 3.02 & 110 & 7 & 1.15 & 11.71 & 0.02 & 1121 & 000 x lmi & + 39.0396 & 182.53 & 53.70 & 1.63 & 2.01 & 82 & 18 & 1.68 & 12.31 & 0.00 & 1121 & 000 u lep & 21.0669 & 221.10 & 34.37 & 4.33 & 5.86 & 128 & 11 & 1.93 & 10.60 & 0.02 & 1111 & 000 ry lib & 21.1628 & 330.87 & 36.08 & 1.68 & 0.45 & 33 & 11 & 1.48 & 13.15 & 0.25 & 1161 & 000 tv lib & 08.1548 & 353.16 & 39.67 & 0.05 & 1.04 & 61 & 10 & 0.27 & 11.94 & 0.25 & 1121 & 111 vy lib & 15.2271 & 353.86 & 28.84 & 0.22 & 6.46 & 142 & 10 & 1.32 & 11.72 & 0.45 & 1111 & 000 tw lyn & + 43.0251 & 176.15 & 27.54 & 0.69 & 0.86 & 40 & 26 & 1.23 & 11.90 & 0.14 & 1141 & 101 y lyr & + 43.0988 & 72.67 & 20.87 & 0.08 & 0.66 & 65 & 23 & 1.03 & 13.28 & 0.18 & 1121 & 101 rr lyr & & 74.96 & 12.30 & 10.95 & 19.42 & 63 & 8 & 1.37 & 7.74 & 0.13 & 2111 & 000 rz lyr & + 32.1588 & 62.11 & 15.82 & 1.02 & 1.99 & 233 & 23 & 2.13 & 11.51 & 0.32 & 1111 & 000 cn lyr & + 28.1070 & 58.01 & 14.70 & 0.12 & 1.61 & 67 & 30 & 0.26 & 11.49 & 0.62 & 1161 & 111 cx lyr & + 28.1083 & 58.99 & 12.72 & 0.49 & 1.30 & 203 & 30 & 1.79 & 12.83 & 0.82 & 1141 & 000 io lyr & + 32.1543 & 60.59 & 19.98 & 1.22 & 2.19 & 157 & 30 & 1.52 & 11.86 & 0.15 & 1161 & 000 uv oct & & 308.40 & 23.55 & 6.93 & 12.30 & 126 & 12 & 1.61 & 9.42 & 0.21 & 2111 & 000 st oph & & 22.83 & 16.64 & 0.09 & 0.08 & 12 & 7 & 1.30 & 12.05 & 0.60 & 2111 & 101 v413 oph & 10.1983 & 4.39 & 25.97 & 1.10 & 1.62 & 39 & 30 & 1.00 & 12.08 & 0.63 & 1161 & 111 v445 oph & & 7.91 & 28.45 & 0.47 & 1.24 & 22 & 5 & 0.23 & 10.99 & 0.59 & 2121 & 111 v452 oph & + 11.1284 & 32.52 & 25.72 & 0.36 & 0.10 & 375 & 30 & 1.72 & 12.18 & 0.41 & 1121 & 000 v964 ori & 02.0731 & 202.50 & 23.91 & 0.70 & 1.21 & 178 & 11 & 1.89 & 12.95 & 0.25 & 1151 & 000 ty pav & & 330.55 & 17.10 & 2.10 & 2.20 & 245 & 9 & 2.31 & 12.58 & 0.26 & 2111 & 000 dn pav & & 332.82 & 30.80 & 0.90 & 3.00 & 69 & 12 & 1.54 & 12.42 & 0.14 & 2111 & 000 vv peg & + 18.1195 & 78.42 & 30.42 & 0.06 & 1.22 & 13 & 8 & 1.88 & 11.79 & 0.13 & 1111 & 000 av peg & + 22.1796 & 77.44 & 24.05 & 1.35 & 1.34 & 58 & 1 & 0.14 & 10.44 & 0.14 & 1121 & 111 bh peg & + 15.1616 & 85.62 & 38.36 & 2.01 & 6.71 & 278 & 2 & 1.38 & 10.44 & 0.20 & 1111 & 000 cg peg & + 24.0966 & 77.18 & 20.75 & 0.11 & 0.49 & 4 & 4 & 0.48 & 11.11 & 0.20 & 1121 & 111 dz peg & + 15.1715 & 93.09 & 41.46 & 1.70 & 2.49 & 294 & 11 & 1.52 & 12.00 & 0.05 & 1161 & 000 gv peg & + 26.1124 & 109.07 & 34.83 & 0.69 & 3.07 & 335 & 30 & 1.99 & 13.36 & 0.10 & 1161 & 000 ar per & & 154.93 & 2.27 & 0.15 & 0.50 & 5 & 1 & 0.43 & 10.43 & 1.08 & 2122 & 111 u pic & & 257.67 & 39.61 & 0.10 & 1.70 & 30 & 12 & 0.73 & 11.32 & 0.00 & 2131 & 111 ry psc & 02.0022 & 100.68 & 62.89 & 3.99 & 0.77 & 1 & 9 & 1.39 & 12.28 & 0.08 & 1121 & 000 bb pup & 19.0790 & 241.28 & 10.27 & 1.58 & 1.05 & 98 & 9 & 0.57 & 12.17 & 0.45 & 1111 & 111 v440 sgr & & 15.31 & 19.20 & 0.20 & 5.00 & 62 & 1 & 1.47 & 10.24 & 0.36 & 2121 & 000 ru scl & & 41.53 & 78.86 & 5.63 & 2.04 & 38 & 8 & 1.25 & 10.21 & 0.03 & 2111 & 000 vy ser & + 01.1004 & 6.16 & 44.09 & 9.84 & 0.51 & 145 & 1 & 1.82 & 10.13 & 0.06 & 1111 & 000 an ser & + 13.1114 & 23.80 & 45.24 & 0.27 & 1.11 & 47 & 4 & 0.04 & 10.97 & 0.09 & 1121 & 111 ar ser & + 02.1454 & 7.89 & 44.25 & 3.66 & 0.60 & 132 & 4 & 1.78 & 11.85 & 0.07 & 1131 & 000 at ser & + 08.1225 & 18.03 & 42.45 & 0.14 & 1.48 & 58 & 11 & 2.05 & 11.45 & 0.08 & 1111 & 000 av ser & + 00.1096 & 11.28 & 36.83 & 0.80 & 0.65 & 45 & 13 & 1.20 & 11.40 & 0.26 & 1111 & 101 aw ser & + 15.1229 & 28.67 & 43.35 & 0.97 & 1.63 & 126 & 15 & 1.67 & 12.79 & 0.04 & 1141 & 000 bh ser & + 19.0930 & 27.56 & 56.27 & 0.78 & 1.51 & 113 & 11 & 1.59 & 12.85 & 0.08 & 1161 & 000 cs ser & + 03.1110 & 7.22 & 45.43 & 2.37 & 2.79 & 2 & 14 & 1.57 & 12.39 & 0.08 & 1141 & 000 df ser & + 18.0911 & 26.27 & 55.92 & 0.46 & 0.55 & 10 & 14 & 0.74 & 12.69 & 0.07 & 1141 & 111 rv sex & 08.0980 & 258.12 & 43.38 & 0.98 & 0.85 & 120 & 20 & 1.10 & 12.30 & 0.02 & 1161 & 000 ss tau & + 05.0288 & 180.09 & 38.53 & 0.77 & 0.28 & 11 & 10 & 0.28 & 12.50 & 0.49 & 1121 & 111 u tri & + 33.0093 & 137.89 & 27.24 & 0.89 & 1.30 & 6 & 23 & 0.79 & 12.60 & 0.10 & 1121 & 111 w tuc & & 301.66 & 53.72 & 0.30 & 0.20 & 63 & 3 & 1.64 & 11.43 & 0.00 & 2111 & 000 yy tuc & & 325.32 & 54.21 & 0.14 & 0.34 & 56 & 9 & 1.82 & 11.98 & 0.00 & 2111 & 000 rv uma & + 54.0419 & 109.75 & 62.06 & 2.76 & 4.69 & 183 & 9 & 1.19 & 10.78 & 0.01 & 1111 & 000 tu uma & + 30.0521 & 198.80 & 71.87 & 7.64 & 4.97 & 88 & 1 & 1.44 & 9.81 & 0.00 & 1111 & 000 ab uma & + 48.0617 & 141.04 & 67.86 & 1.30 & 2.00 & 56 & 26 & 0.72 & 10.80 & 0.00 & 1141 & 111 st vir & 00.1211 & 346.37 & 53.65 & 0.06 & 2.65 & 22 & 13 & 0.88 & 11.52 & 0.07 & 1121 & 111 uu vir & & 280.73 & 60.52 & 2.96 & 0.45 & 8 & 1 & 0.82 & 10.56 & 0.01 & 2121 & 111 uv vir & + 00.0808 & 286.55 & 62.28 & 2.63 & 1.80 & 99 & 11 & 1.19 & 11.83 & 0.02 & 1121 & 000 wy vir & 06.1416 & 321.78 & 54.28 & 1.74 & 1.17 & 181 & 10 & 2.84 & 13.38 & 0.03 & 1161 & 000 ad vir & 07.2115 & 333.18 & 51.21 & 1.97 & 0.60 & 134 & 14 & 1.15 & 13.04 & 0.04 & 1151 & 000 ae vir & + 04.0956 & 351.61 & 57.27 & 0.03 & 1.77 & 208 & 10 & 1.16 & 13.26 & 0.02 & 1141 & 000 af vir & + 06.0757 & 355.48 & 59.16 & 6.16 & 0.05 & 35 & 14 & 1.46 & 11.52 & 0.01 & 1121 & 000 am vir & 16.1465 & 313.94 & 45.52 & 0.16 & 5.15 & 99 & 24 & 1.45 & 11.49 & 0.14 & 1121 & 000 as vir & 09.1409 & 303.47 & 52.61 & 1.14 & 3.69 & 70 & 23 & 1.49 & 11.90 & 0.08 & 1121 & 000 at vir & 05.1349 & 304.66 & 57.40 & 5.42 & 1.76 & 346 & 8 & 1.91 & 11.27 & 0.04 & 1121 & 000 av vir & + 09.0882 & 325.01 & 70.82 & 0.64 & 3.38 & 152 & 4 & 1.32 & 11.78 & 0.00 & 1121 & 000 bq vir & 02.1373 & 295.38 & 60.23 & 0.27 & 1.39 & 129 & 9 & 1.32 & 12.48 & 0.03 & 1161 & 000 do vir & 05.1546 & 345.60 & 48.45 & 2.72 & 0.69 & 24 & 36 & 0.80 & 14.14 & 0.10 & 1151 & 000 fu vir & + 13.0858 & 290.13 & 75.56 & 1.33 & 0.77 & 90 & 8 & 1.17 & 12.63 & 0.07 & 1161 & 000 fk vul & + 22.1711 & 67.60 & 13.92 & 0.05 & 1.51 & 76 & 30 & 0.95 & 12.87 & 0.35 & 1161 & 111 at and & & 109.76 & 18.09 & 0.20 & 4.60 & 241 & 11 & 0.97 & 10.66 & 0.38 & 2221 & 000 s ara & & 343.38 & 12.45 & 2.34 & 1.53 & 172 & 13 & 1.43 & 10.67 & 0.36 & 2231 & 000 ru boo & + 23.0728 & 30.94 & 63.87 & 1.35 & 0.32 & 60 & 35 & 1.50 & 13.60 & 0.04 & 1321 & 000 bi cen & & 294.66 & 2.44 & 0.76 & 0.15 & 210 & 30 & 0.83 & 11.86 & 0.59 & 2262 & 000 uu cet & 17.0006 & 73.26 & 75.09 & 2.68 & 0.65 & 114 & 3 & 1.32 & 11.95 & 0.01 & 1221 & 000 z com & + 18.0747 & 328.12 & 80.58 & 0.77 & 1.85 & 50 & 35 & 1.50 & 13.73 & 0.03 & 1321 & 000 st com & & 347.87 & 81.25 & 3.61 & 3.57 & 68 & 7 & 1.26 & 11.38 & 0.04 & 2221 & 000 sw cru & & 296.49 & 1.91 & 1.07 & 0.19 & 23 & 30 & 0.54 & 12.33 & 1.18 & 2262 & 111 uy cyg & & 74.54 & 9.63 & 0.13 & 0.76 & 2 & 6 & 1.03 & 11.05 & 0.22 & 2222 & 101 sw her & + 21.1016 & 41.68 & 34.00 & 1.11 & 0.04 & 130 & 35 & 1.50 & 14.14 & 0.21 & 1321 & 000 vx her & + 18.0988 & 35.22 & 39.08 & 4.70 & 1.66 & 377 & 3 & 1.52 & 10.62 & 0.18 & 1211 & 000 ar her & + 47.1123 & 74.10 & 48.20 & 6.53 & 1.24 & 349 & 8 & 1.40 & 11.18 & 0.04 & 1211 & 000 rv leo & & 232.37 & 51.14 & 0.50 & 1.30 & 0 & 35 & 1.50 & 13.85 & 0.08 & 2321 & 000 tt lyn & + 44.0496 & 176.07 & 41.65 & 8.41 & 4.01 & 67 & 1 & 1.76 & 9.87 & 0.03 & 1261 & 000 ez lyr & & 65.52 & 16.25 & 1.32 & 0.20 & 60 & 23 & 1.56 & 11.60 & 0.21 & 2261 & 001 ao peg & + 18.1149 & 69.90 & 22.60 & 0.31 & 2.94 & 115 & 35 & 0.92 & 12.83 & 0.19 & 1261 & 000 tu per & & 142.78 & 4.29 & 1.51 & 0.61 & 377 & 11 & 1.50 & 12.53 & 1.38 & 2324 & 000 rv phe & & 336.01 & 64.00 & 4.15 & 1.85 & 99 & 2 & 1.60 & 11.75 & 0.06 & 2211 & 000 xx pup & & 236.65 & 8.72 & 3.13 & 0.14 & 386 & 7 & 1.50 & 11.20 & 0.38 & 2324 & 000 v675 sgr & & 358.26 & 7.83 & 0.00 & 1.20 & 105 & 30 & 2.01 & 10.36 & 0.19 & 2212 & 000 v1640 sgr & & 0.47 & 13.64 & 0.40 & 0.90 & 41 & 10 & 0.54 & 12.68 & 0.31 & 2251 & 111 v494 sco & & 357.23 & 0.49 & 0.34 & 0.62 & 26 & 30 & 1.01 & 11.27 & 1.16 & 2233 & 101 af vel & & 284.16 & 8.60 & 5.90 & 1.75 & 236 & 16 & 1.64 & 11.34 & 0.61 & 2214 & 000 bb vir & + 06.0723 & 340.31 & 64.84 & 3.71 & 1.02 & 38 & 13 & 1.61 & 11.07 & 0.00 & 1221 & 000 bc vir & + 06.0660 & 323.42 & 67.52 & 1.55 & 2.81 & 4 & 13 & 1.50 & 12.21 & 0.00 & 1361 & 000 bn vul & & 58.63 & 3.41 & 4.85 & 3.80 & 267 & 4 & 1.52 & 11.08 & 1.36 & 2262 & 000 cl disk-1 & all stars lying above / rightward of @xmath273[fe / h]@xmath274 ( see fig . halo-1 & all stars lying below / leftward of this line . & disk-2 & all stars having [ fe / h ] @xmath275 and @xmath276 km s@xmath5 . halo-2 & all stars excluded from disk-2 . & disk-3 & all stars in disk-1 @xmath277 all stars having @xmath278 km s@xmath5 , @xmath276 km s@xmath5 , & @xmath279 km s@xmath5 , @xmath280 @xmath281 kpc , @xmath221 [ fe / h ] @xmath282 . halo-3 & all stars excluded from disk-3 . & halo-1r & all stars in halo-1 with [ fe / h ] @xmath283 . halo-1p & all stars in halo-1 with [ fe / h ] @xmath284 . h1 & halo & random & 165 & 5 & h1.0 & 0.0 & yes & 5 & & & & & h1 & 0.1 & yes & 5 h2 & halo & real & 169 & 5 & h2.0 & 0.0 & yes & 5 & & & & & h2 & 0.1 & yes & 5 & & & & ( + 15 ) & h2d & 0.1 & no & 20 & & & & & & & & d1 & disk & random & 50 & 20 & d1 & 0.1 & yes & 19 d2 & disk & real & 45 & 20 & d2 & 0.1 & yes & 18 & & & & & d2d & 0.1 & no & 18 crrrrrrrrrrrr h1.0 & mean&203 & 158 & 100 & 89 & + 0.82 & mean & + 3 & 2 & 1 & 3 & + 0.05 & @xmath138 & 11 & 11 & 8 & 7 & 0.12 & sd & 5 & 2 & 7 & 3 & 0.07 & & & & & & & & & & & & h1 & mean&204 & 158 & 99 & 89 & + 0.79 & mean & + 2 & 2 & 3 & 3 & + 0.02 & @xmath138 & 12 & 11 & 8 & 7 & 0.12 & sd & 5 & 2 & 6 & 3 & 0.07 & & & & & & & & & & & & h2.0 & mean&198 & 155 & 96 & 86 & + 0.84 & mean & + 7 & 7 & 3 & 2 & + 0.07 & @xmath138 & 11 & 12 & 7 & 6 & 0.12 & sd & 2 & 5 & 3 & 4 & 0.08 & & & & & & & & & & & & h2 & mean&199 & 155 & 94 & 86 & + 0.82 & mean & + 6 & 6 & 4 & 2 & + 0.05 & @xmath138 & 12 & 12 & 7 & 6 & 0.12 & sd & 2 & 5 & 3 & 3 & 0.08 & & & & & & & & & & & & h2d & mean&205 & 155 & 93 & 87 & + 0.81 & mean & + 4 & 4 & 4 & 2 & + 0.04 & @xmath138 & 12 & 12 & 7 & 6 & 0.12 & sd & 5 & 6 & 3 & 3 & 0.08 & & & & & & & & & & & & & & & & & & & & & & & & d1 & mean & 32 & 51 & 48 & 29 & + 0.88 & mean & 0 & 2 & 1 & 2 & 0.02 & @xmath138 & 8 & 8 & 8 & 6 & 0.34 & sd & 5 & 7 & 7 & 7 & 0.44 & & & & & & & & & & & & d2 & mean & 33 & 51 & 48 & 27 & + 1.04 & mean & + 1 & 4 & 4 & 2 & + 0.13 & @xmath138 & 9 & 9 & 8 & 6 & 0.35 & sd & 5 & 6 & 6 & 5 & 0.29 & & & & & & & & & & & & d2d & mean & 32 & 51 & 48 & 26 & + 1.04 & mean & + 1 & 4 & 4 & 3 & + 0.13 & @xmath138 & 9 & 8 & 8 & 6 & 0.34 & sd & 5 & 5 & 7 & 4 & 0.29 lrcrrrrrrrr halo-1 & 169 & 1.60 & + 9 & 205 & 11 & 165 & 102 & 95 & + 0.71 & + 0.71 & & & @xmath75 13 & 12 & 8 & 12 & 7 & 7 & 0.12 & 0.12 & & & & & & & & & & halo-2 & 175 & 1.58 & + 8 & 196 & 11 & 161 & 108 & 93 & + 0.72 & + 0.72 & & & @xmath75 13 & 12 & 7 & 12 & 8 & 7 & 0.12 & 0.12 & & & & & & & & & & halo-3 & 162 & 1.61 & + 9 & 210 & 12 & 168 & 102 & 97 & + 0.71 & + 0.71 & & & @xmath75 14 & 12 & 8 & 13 & 8 & 7 & 0.12 & 0.12 & & & & & & & & & & & & & & & & & & & & halo-1r & 86 & 1.34 & 13 & 216 & 13 & 172 & 92 & 89 & + 0.69 & + 0.69 & & & @xmath75 19 & 16 & 10 & 17 & 9 & 9 & 0.16 & 0.16 & & & & & & & & & & halo-1p & 83 & 1.86 & + 31 & 195 & 10 & 154 & 111 & 100 & + 0.73 & + 0.73 & & & @xmath75 18 & 17 & 12 & 16 & 12 & 10 & 0.18 & 0.18 & & & & & & & & & & & & & & & & & & & & disk-1 & 44 & 0.66 & + 4 & 34 & 18 & 45 & 43 & 25 & + 1.24 & + 1.08 & & & @xmath75 8 & 8 & 6 & 8 & 8 & 6 & 0.34 & 0.34 & & & & & & & & & & disk-2 & 38 & 0.58 & + 8 & 43 & 19 & 51 & 47 & 25 & + 1.15 & + 1.01 & & & @xmath75 9 & 10 & 6 & 9 & 9 & 6 & 0.35 & 0.35 & & & & & & & & & & disk-3 & 51 & 0.76 & + 6 & 45 & 16 & 52 & 48 & 29 & + 0.94 & + 0.79 & & & @xmath75 8 & 9 & 6 & 8 & 8 & 5 & 0.30 & 0.30 & & & & & & & & & & all stars & 213 & 1.40 & + 7 & 169 & 14 & 147 & 115 & 85 & + 0.73 & + 0.73 & & & @xmath75 10 & 10 & 6 & 10 & 7 & 6 & 0.11 & 0.11 | we present new statistical parallax solutions for the absolute magnitude and kinematics of rr lyrae stars .
we have combined new proper motions from the lick northern proper motion program with new radial velocity and abundance measures to produce a data set that is 50% larger , and of higher quality , than the data sets employed by previous analyses . based on an _ a priori _ kinematic study , we separated the stars into halo and thick disk sub - populations .
we performed statistical parallax solutions on these sub - samples , and found @xmath0 at @xmath1[fe / h]@xmath2 = 1.61 for the halo ( 162 stars ) , and @xmath3 at @xmath1[fe / h]@xmath2 = 0.76 for the thick disk ( 51 stars ) .
the solutions yielded a solar motion @xmath4 km s@xmath5 and velocity ellipsoid @xmath6 km s@xmath5 for the halo .
the values were @xmath7 km s@xmath5 and @xmath8 km s@xmath5 for the thick disk .
both are in good agreement with estimates of the halo and thick disk kinematics derived from both rr lyrae stars and other stellar tracers .
monte carlo simulations indicated that the solutions are accurate , and that the errors may be smaller than the estimates above .
the simulations revealed a small bias in the disk solutions , and appropriate corrections were derived .
the large uncertainty in the disk @xmath9 prevents ascertaining the slope of the @xmath9[fe / h ] relation . using a zero point defined by our halo solution and adopting a slope of 0.15 mag dex@xmath5
, we find that ( 1 ) the distance to the galactic center is @xmath10 kpc ; ( 2 ) the mean age of the 17 oldest galactic globular clusters is @xmath11 gyr ; and ( 3 ) the distance modulus of the lmc is @xmath12 mag .
estimates of @xmath13 which are based on an lmc distance modulus of 18.50 ( e.g. , cepheid studies ) increase by 10% if they are recalibrated to match our lmc distance modulus . |
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quantum entanglement @xcite describes a scenario where the quantum states of two objects separated in space are strongly correlated . these correlations can be exploited in emerging technologies such as quantum computing , should one be able to spatially separate the entangled objects without destroying the correlations . in a broader context , quantum entanglement could prove to be of practical importance in the fields of spintronics @xcite and information cryptography @xcite . it also holds a considerable interest from a purely fundamental physics point of view , prompting some of the more philosophically inclined discussions related to quantum theory and causality . superconductors have been proposed as natural sources for entangled electrons @xcite , as cooper pairs consist of two electrons that are both spin and momentum - entangled . the cooper pair can be spatially deformed by means of the crossed andreev reflection ( car ) process in superconducting heterostructures . in this scenario , an electron and hole excitation are two separate metallic leads are coupled by means of andreev scattering processes at two spatially distinct interfaces . unfortunately , the signatures of car are often completely masked by a competing process known as elastic co - tunneling ( ct ) which occur in the same type of heterostructures . in fact , the conductances stemming from ct and car may cancel each other completely @xcite , thus necessitating the usage of noise - measurements to find fingerprints of the car process in such superconducting heterostructures . recently , graphene @xcite has been studied as a possible arena for car - processes . in ref . @xcite , it was shown how a three - terminal graphene sheet containing @xmath0-doped , @xmath1-doped , and superconducting regions could be constructed to produce perfect car for one particular resonant bias voltage . also , the signatures of the car process in the noise - correlations of a similar device were studied in ref . however , the role played by the spin degree of freedom in graphene devices probing non - local transport has not been addressed so far . this is a crucial point since it might be possible to manipulate the spin - properties of the system to interact with the spin - singlet symmetry of the cooper pair in a fashion favoring car . in this paper , we show that precisely such an opportunity exists it is possible to obtain a spin - switch effect between virtually perfect car and perfect ct in a superconducting graphene spin valve . in contrast to ref . @xcite , this effect is seen for all bias voltages in the low - energy regime rather than just at one particular applied voltage difference . the key observation is that the possibility of tuning the local fermi - level to values equivalent to a weak , magnetic exchange splitting in graphene renders both the usual andreev reflection process and ct impossible . in contrast , this opportunity does not exist in conventional conductors where the fermi energy is large and of order @xmath2(ev ) . we show that graphene spin valves provide a possibility for a unique combination of non - local andreev reflection and spin - dependent klein tunneling @xcite . our model is shown in fig . [ fig : model ] , where ferromagnetism and superconductivity are assumed to be induced by means of the proximity effect @xcite to leads with the desired properties . a similar setup was considered in ref . @xcite , where the magnetoresistance of the system was studied . we organize this work as follows . in sec . [ sec : theory ] , we establish the theoretical framework which will be used to obtain the results . in sec . [ sec : results ] , we present our main findings for the non - local conductance in the graphene superconducting spin - valve with a belonging discussion of them . finally , we summarize in sec . [ sec : summary ] . we consider a ballistic , two - dimensional graphene structure as shown in fig . [ fig : model ] . in the left ferromagnetic region @xmath3 , the exchange field is @xmath4 , while it is @xmath5 in the right ferromagnetic region @xmath6 . in the superconducting region @xmath7 , the order parameter is taken to be constant with a real gauge @xmath8 . to proceed analytically , we make the usual approximation of a step - function behavior at the interfaces for all energy scales , i.e. the chemical potentials @xmath9 , the exchange field @xmath10 , and superconducting gap @xmath11 . this assumption is expected to be good when there is a substantial fermi - vector mismatch between the f and s regions , as in the present case . to make contact with the experimentally relevant situation , we assume a heavily doped s region satisfying @xmath12 . we use the dirac - bogoliubov de gennes equations first employed in ref . @xcite . for quasiparticles with spin @xmath13 , one obtains in an f@xmath14s graphene junction : @xcite @xmath15 where @xmath16 \hat{1}\end{aligned}\ ] ] and @xmath17 denotes a @xmath18 matrix . here , we have made use of the valley degeneracy and @xmath19 is the momentum vector in the graphene plane while @xmath20 is the vector of pauli matrices in the pseudospin space representing the two a , b sublattices of graphene hexagonal structure . the superconducting order parameter @xmath21 couples electron- and hole - excitations in the two valleys ( @xmath22 ) located at the two inequivalent corners of the hexagonal brillouin zone . the @xmath23 spinor describes the electron - like part of the total wavefunction @xmath24 and in this case reads @xmath25 while @xmath26 . here , @xmath27 denotes the transpose while @xmath28 is the time - reversal operator . from eq . ( [ eq : bdg ] ) , one may now construct the quasiparticle wavefunctions that participate in the scattering processes @xcite . we consider positive excitation energies @xmath29 with incoming electrons of @xmath0-type , i.e. from the conduction band @xmath30 ( we set @xmath31 from now on ) . the incoming electron from the left ferromagnet may either be reflected normally or andreev - reflection ( ar ) . in the latter process , it tunnels into the superconductor with another electron situated at @xmath32 , leaving behind a hole excitation with energy @xmath33 . the scattering coefficients for these two processes are @xmath34 and @xmath35 , respectively , and the total wavefunction may thus be written as : @xmath36 where we have defined the wavevectors @xmath37 we have omitted a common factor @xmath38 for all wavefunctions . similarly , assuming that the charge carriers in the right ferromagnetic region are also of the @xmath0-type , we obtain : @xmath39 it should be noted that the ar hole is generated in the conduction band if @xmath40 ( retro - ar ) , whereas it is generated in the valence band otherwise ( specular - ar ) . the @xmath41 sign above refers to parallell / antiparallell ( p / ap ) magnetization configuration . we assume that the superconducting region is heavily doped , @xmath42 , which causes the propagating quasiparticles to travel along the @xmath43-axis since the scattering angle in the superconductor satisfies @xmath44 . we obtain the following wavefunction ( @xmath45 ) : @xmath46 where @xmath47 while @xmath48 for subgap energies @xmath49 and @xmath50 for supergap energies @xmath51 . it is important to consider carefully the scattering angles in the problem . since we assume translational invariance in the @xmath52-direction , the @xmath52-component of the momentum is conserved . this gives us @xmath53 it is clear that the angle of transmission for the electrons in the right ferromagnet is equal to the angle of incidence when the magnetizations are p , i.e. @xmath54 . also , one infers that there exists a critical angle above which the scattered waves become evanescent , i.e. decaying exponentially . this may be seen by observing that the scattering angles exceed @xmath55 ( thus becoming imaginary ) above a certain angle of incidence @xmath56 . for instance , the ar wave in the left ferromagnetic region becomes evanescent for angles of incidence @xmath57 , where the critical angle @xmath58 is obtained by setting @xmath59 in the equation @xmath60 expressing conservation of momentum perpendicular to the interface . one finds that : @xmath61|.\end{aligned}\ ] ] thus , ar waves in the regime @xmath62 do not contribute to any transport of charge . a similar argument can be made for the transmitted electron wave - function in the right ferromagnetic region , corresponding to the ct process , where the critical angle for this process becomes @xmath63|.\end{aligned}\ ] ] in the p configuration , the ct process thus always contributes to the transport of charge . finally , the contribution to transport of charge from car comes from the hole - wave function in the right ferromagnetic region , which becomes evanescent for angles of incidence above the critical angle @xmath64|.\end{aligned}\ ] ] in the p configuration , this criteria is the same as the vanishing of local ar expressed by eq . ( [ eq : criticalar ] ) . intuitively , one might expect that the most interesting phenomena occur when the exchange field @xmath10 is comparable in magnitude to the chemical potential @xmath65 . if @xmath66 , the effect of the exchange field should be minor and the ar is never specular . in contrast , the situation becomes quite fascinating when we consider the case @xmath67 under the assumption of a doped situation @xmath68 . first of all , the incoming quasiparticles from the left ferromagnetic region are completely dominated by the majority spin carriers @xmath69 , since the density of states ( dos ) for @xmath70 electrons vanishes at the fermi level . since @xmath67 , the ar process is suppressed for all incoming waves as @xmath71 . we now show how the fate of the cross - conductance in the right ferromagnetic region depends crucially on whether the magnetization configuration is p or ap . in the p configuration , we see that @xmath72 , which means that the transport is purely governed by the ct process . in the ap configuration , we see that @xmath73 , which means that the transport is mediated purely by the car process . this suggests a remarkable spin - switch effect by reversing the direction of the field in the right ferromagnet , one obtains an abrupt change from pure ct to pure car processes mediating the transport of charge . in each case , there is no local ar in the left ferromagnetic region . in the standard metallic case , the distinct signatures for the ct and car contributions are masked by each other , and it becomes necessary to resort to noise - measurements in order to say something about the contribution from each process . in the present scenario , we have showed how it is possible to separate the two contributions directly by a simple spin - switch effect which is commonly employed in experimental work on f@xmath14s heterostructures . ( color online ) plot of the conductance for ct processes @xmath74 versus bias voltage in the upper panel and versus length of the s region in the lower panel . here , we consider the p alignment and @xmath67 such that @xmath75 . , width=302 ] let us now evaluate the conductance in the p and ap configuration quantitatively by using @xmath76 where we have introduced @xmath77 as the spin-@xmath13 normal - state conductance that takes into account the valley degeneracy , in addition to @xmath78 the density of states is determined by @xmath79 where @xmath80 is the width of the junction . the expression for @xmath81 is obtained by replacing @xmath82 with @xmath83 in eq . ( [ eq : conductance ] ) . since we here consider the case @xmath67 and @xmath84 , the formulas for the @xmath85 and @xmath81 may be simplified since @xmath86 . also , since the dos vanishes for minority spins for the injected electrons , only @xmath69 contributes for incoming electrons . the crucial point here is that in the p alignment , @xmath87 and @xmath88 such that @xmath89 and @xmath90 such that @xmath91 and @xmath92 . assuming a value of @xmath93 mev for the proximity - induced gap , this corresponds to an exchange splitting of @xmath94 mev in the f regions and a doping level @xmath95 mev in the s region , which should be experimentally feasible @xcite and well within the range of the validity for the linear dispersion relation in graphene . in fig . [ fig : p ] , we plot the cross - conductance @xmath96 in the p alignment both as a function of bias voltage and width of the s region . the same thing is done for @xmath97 in the ap alignment in fig . [ fig : ap ] . in both cases , the magnitude of the conductance varies strongly when considering different widths @xmath98 due to the fast oscillations which pertain to the formation of resonant transmission levels inside the superconductor . also , it is seen that while the ct process is favored for short junctions @xmath99 , the car process is suppressed in this regime in favor of normal reflection . upon increasing the junction width , the ct conductance drops while the car conductance peaks at widths @xmath100 . the remarkable aspect is that it is possible to switch between these two scenarios of exclusive ct and exclusive car simply by reversing the direction of magnetization in one of the ferromagnetic layers . ( color online ) plot of the conductance for car processes @xmath101 versus bias voltage in the upper panel and versus length of the s region in the lower panel . here , we consider the ap alignment and @xmath67 such that @xmath102 . , width=302 ] in order to obtain analytical results , we have assumed that the coulomb interaction and charge inhomogeneities may be neglected . it would be challenging to obtain a truly homogeneous chemical potential in a graphene sheet , and electron - hole puddles appear to be an intrinsic feature of graphene sheets @xcite . moreover , it has been speculated that such charge inhomogeneities may play an important role with regard to limiting the transport characteristics of graphene @xcite near the dirac points . however , for our purposes this is actually beneficial it is precisely the suppression of charge and spin transport at fermi level for the andreev reflection and co - tunneling process which renders possible the spin - switch effect . therefore , we do not expect that the inclusion of charge inhomogeneities should alter our results qualitatively . finally , we note that since the spin of the charge - carriers in each of the non - superconducting graphene sheets are practically speaking fixed due to the vanishing dos for minority spins , the spin - switch effect for car and ec predicted in this paper can not be directly related to entanglement . nevertheless , it constitutes a clear non - local signal for quantum transport which can be probed experimentally , and should be helpful in identifying clear - signatures of the mesoscopic car phenomenon . to summarize , we have considered non - local quantum transport in a graphene superconducting spin - valve . we have shown how one may create a spin - switch effect between perfect elastic co - tunneling and perfect crossed andreev - reflection for all applied bias voltages by reversing the magnetization direction in one of the ferromagnetic layers . the basic mechanism behind this effect is that the local fermi - level in graphene may be tuned so that the fermi surface for minority spins reduces to a single point in the presence of a weak , magnetic exchange splitting . this is very distinct from the equivalent spin valve structures in conventional metallic systems , where noise - measurements are required to clearly distinguish between these processes . h. b. heersche , p. jarillo - herrero , j. b. oostinga , l. m. k. vandersypen , and a. f. morpurgo , nature ( london ) * 446 * , 56 ( 2006 ) ; a. shailos , w. nativel , a. kasumov , c. collet , m. ferrier , s. gueron , r. deblock , and h. bouchiat , europhys . lett . * 79 * , 57008 ( 2007 ) . | we consider the non - local quantum transport properties of a graphene superconducting spin - valve .
it is shown that one may create a spin - switch effect between perfect elastic co - tunneling ( ct ) and perfect crossed andreev - reflection ( car ) for all bias voltages in the low - energy regime by reversing the magnetization direction in one of the ferromagnetic layers .
this opportunity arises due the possibility of tuning the local fermi - level in graphene to values equivalent to a weak , magnetic exchange splitting , thus reducing the fermi surface for minority spins to a single point and rendering graphene to be half - metallic .
such an effect is not attainable in a conventional metallic spin - valve setup , where the contributions from ct and car tend to cancel each other and noise - measurements are necessary to distinguish these processes . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
supersymmetry is expected to be one of the key ingredients to describe physics beyond the standard model ( sm ) . while tree - level supersymmetry breaking within the sm sector leads to light sfermions , the breaking sector is separated from the sm one and is mediated by some effective operators or quantum effects . among various mechanisms for realizing this scenario , the gauge mediation , relevant to this paper , is one of the most promising candidates with a strong prophetic power ( for a review , see @xcite ) . it is known that pseudomoduli directions are present in the supersymmetry - breaking vacuum of oraifeartaigh - like models with the canonical khler potential @xcite . an important implication of this result is that , if such models are used as the hidden sector of gauge mediation , gaugino masses are generally suppressed or the vacuum is unstable somewhere along the pseudomoduli @xcite . there have been various ways in the literature to avoid such a phenomenologically unfavorable situation , such as including nonminimal terms in the potential @xcite , quantum effects from specific scalar and/or vector multiplets @xcite , or accepting metastable vacua @xcite . another way , as we discussed before , is to introduce gauge multiplets and take non - negligible d term into account for making the vacuum stable without pseudomoduli . in our previous paper , we classified supersymmetry - breaking models with nonvanishing f and d terms @xcite . first , the models are divided into two categories based on whether the f - term potential has a supersymmetric minimum ( at finite field configuration ) . we then add the d term by gauging flavor symmetry and analyze the vacuum of the full scalar potential in each category . for models that do not satisfy the f - flatness conditions , we found that the full potential generally shows runaway behavior . on the other hand , when the f - flatness conditions are satisfied , supersymmetry can be broken without pseudomoduli only in the presence of the fayet - iliopoulos ( fi ) term . by using the latter class of models , we constructed a model of gauge mediation where gaugino masses are generated at the one - loop order . in this paper , we discuss another possibility for the classification : the f - term potential is minimized at some infinite field configuration , i.e. , it shows a runaway behavior . it is found that the runaway direction of the f - term potential can be uplifted by the d term , and a supersymmetry - breaking vacuum emerges at finite field configuration . there are several reasons to explore this class of models in detail . first of all , contrary to our previous result , there is no need to add the fi term for supersymmetry breaking . secondly , the vacuum automatically suppresses pseudomoduli directions associated with the f - term potential , since it has a runaway behavior and is stabilized by the d - term potential . we propose a minimal model with such properties and couple it with an appropriate messenger sector . in this model including the messenger sector , r symmetry is spontaneously broken at the tree level even though the model contains chiral superfields with u(1)@xmath0 charge 0 or 2 only . we notice that r symmetry breaking does not occur for oraifeartaigh - like models with such a u(1)@xmath0 charge assignment @xcite . this class of supersymmetry breaking can therefore provide a realistic model for gauge mediation , where leading - order gaugino masses are obtained at the stable vacuum . the outline of this paper is as follows . in sec . [ sec : review ] , we briefly review our classification of f- and d - term supersymmetry breaking . in sec . [ sec : runaway ] , we discuss the case in which the f - term potential shows runaway behaviors . after some general arguments , a minimal model is presented to realize the vacuum property listed above . further , appropriate messenger sectors are discussed and shown to be viable for generating gaugino masses . section [ sec : conclusion ] is devoted to summarizing our results and discussions on future directions . in the appendix , we show the potential analysis of the model given in sec . [ sec : runaway ] . we first review our previous result of the classification of supersymmetry breaking with both f and ( abelian ) d terms @xcite . throughout this paper , we assume the khler potential is canonical . the superpotential @xmath1 has a polynomial form of chiral superfields @xmath2 with u(1 ) charges @xmath3 ( the latin indices label their species ) . the scalar potential @xmath4 is then given by @xmath5 where @xmath6 and @xmath7 are the contributions from f and d terms : @xmath8 here @xmath9 is the u(1 ) gauge coupling constant and @xmath10 is the coefficient of the possible fayet - iliopoulos term @xcite . in the following , we abbreviate field derivatives of the superpotential as @xmath11 ( @xmath12 ) . we first divide models into two categories . the criterion for the classification is whether the f - flatness condition , @xmath13 , is satisfied or not at its minimum defined by @xmath14 . if the f - flatness condition is ( not ) satisfied , a model is called in the second ( first ) class . we then add the u(1 ) d term and analyze the tree - level behavior of the full scalar potential . for the first class of models , the scalar potential shows supersymmetric runaway behaviors when the d - term contribution is included . for the second class of models , on the other hand , supersymmetry can be broken without any pseudomoduli . that is , however , realized only in the presence of the fi term . ( see table [ table : class ] for the classification . ) .classification of supersymmetry breaking with f and d terms . [ cols="<,^,^,^",options="header " , ] + the superpotential is @xmath15 this form can be the most generic , renormalizable one if @xmath16 and @xmath17 have the charges @xmath18 and @xmath19 under an additional @xmath20 symmetry ( @xmath21 ) . we here comment on the role of each term in the superpotential : the first and second terms make the origin unstable along the meson @xmath22 . the third term lifts up the @xmath23 direction , and naively supersymmetry is broken with the f - term potential coming from the first three terms . however , the potential minimum is found to run away to infinity of the moduli space along the @xmath24 direction ( with a finite value of @xmath22 ) . it is further noticed that anomaly cancellation requires the existence of a negatively charged field ( @xmath17 ) . without the last fourth term , the value of @xmath17 is free and the direction @xmath25 becomes d flat , along which the potential minimum goes to infinity and supersymmetry is recovered . in this way , the superpotential is regarded as minimal one for the present purpose . the scalar potential @xmath26 is explicitly given by @xmath27 & & v_d \,=\ , \frac{g^2}{2}d^2 \,=\ , \frac{g^2}{2}\big(|x_+|^2+|\varphi_+|^2-|x_-|^2-|\varphi_-|^2\big)^2 \ , . \label{dterm_pot}\end{aligned}\ ] ] the f - term potential @xmath6 has the u(1 ) runaway directions discussed in the previous section . we find along the following direction @xmath28 and all the f terms vanish except for the positively charged field @xmath29 , @xmath30 the runaway direction is parametrized by @xmath31 . as @xmath32 , the positively charged field @xmath24 goes to infinity and the f - term potential ( @xmath29 ) approaches to zero . furthermore , no u(1)@xmath0 runaway is expected since all the scalar fields in the present model have r charges 0 or 2 @xcite . it is easily seen that the f - term runaway direction is stabilized by the d - term contribution because , along this direction , the d term increases as @xmath33 ( see also fig . [ fig : uplift ] . ) we again note that even when the runaway is lifted by the d term , it does not necessarily mean that the vacuum of the scalar potential is in the direction . ( dashed line ) and @xmath4 ( solid line ) along the runaway direction . the runaway direction of @xmath6 is uplifted by the d - term contribution @xmath7.,width=245 ] we then analyze the scalar potential in detail to confirm that a stable supersymmetry - breaking vacuum is obtained in some parameter region . the parameters appearing in the superpotential are assumed to be real and positive without loss of generality . the vacuum is identified by solving the stationary conditions for the full scalar potential @xmath4 . a trivial configuration satisfying the stationary conditions is the origin at which all the scalar fields vanish . this point however is unstable and we do not consider it in the following . by some calculation ( the detail is summarized in the appendix ) , we can show that the vacuum satisfies @xmath34 where several f terms vanish ; @xmath35 then the stationary conditions for @xmath36 , @xmath37 , and @xmath16 automatically hold , and the scalar potential simplifies to @xmath38 with @xmath39 . with this reduced potential , the stationary condition for @xmath17 reads @xmath40 indicating @xmath41 or @xmath42 . we discuss these two cases separately below , solving the remaining stationary conditions for @xmath43 . : in this case , for a large mass ( @xmath44 ) or a small one ( @xmath45 ) , we can approximately write down the analytic solution to the stationary conditions for @xmath43 . for the large mass regime , the solution is given by @xmath46 where the f and d components other than become @xmath47 at the leading order . therefore , the scalar potential is dominated by @xmath48 , i.e. , @xmath49 . on the other hand , for the small mass regime , we have @xmath50 where the f and d components are @xmath51 and the scalar potential is found to be dominated by @xmath29 , i.e. , @xmath52 . notice that , in both regimes of @xmath53 , the r symmetry is unbroken as @xmath54 . the stability of these vacua is read off from the eigenvalues of the squared mass matrix for the scalar fields . we find that all eigenvalues except for @xmath17 are positive semidefinite . along the @xmath17 direction , the eigenvalue is given by @xmath55 this eigenvalue is positive when @xmath56 for the large mass regime and @xmath57 for the small mass regime . therefore the supersymmetry - breaking vacuum realized is stable for these parameters . otherwise , @xmath58 is a saddle point and the true vacuum is given by the following second case : : in this case , the stationary conditions are complicated and we only present a typical numerical solution . for example , when the model parameters are set as @xmath59 , which do not satisfy the above condition for the stability of @xmath58 , the vacuum is located at @xmath60 where nonvanishing f and d components are @xmath61 all the f and d terms become comparable to each other and contribute to the scalar potential @xmath62 . the stability of this vacuum is confirmed numerically . we have also checked that , for a wider parameter region , the scalar potential has a similar behavior . in fig . [ fig : vac - height ] , we show the normalized potential @xmath63 characterizing supersymmetry breaking at the vacuum as a function of @xmath64 for the large mass regime ( @xmath65 . without the @xmath64 term , supersymmetry recovers at infinity of the moduli space , as we mentioned before . ( for the @xmath66 case ) . the other parameters are fixed to @xmath67 . in the limit @xmath68 , supersymmetry is recovered , but some expectation values run away to infinity.,width=245 ] as @xmath64 becomes larger , @xmath17 is stabilized for @xmath56 and the vacuum is shifted up to , where @xmath69 . we then discuss the gauge - mediation scenario by employing the above model as a supersymmetry breaking sector . appropriate messenger fields and their superpotential are identified for generating one - loop - order gaugino masses for the two cases , @xmath58 and @xmath66 , separately . : as we have shown , the f and d terms have different behaviors depending of whether the mass parameter @xmath53 is large or small . for the large @xmath53 case , , the supersymmetry - breaking scale is governed by @xmath48 from which sfermions are expected to receive soft masses . for generating a similar size of gaugino masses , a simple way is to introduce the messenger fields @xmath70 and @xmath71 which are vectorlike under the sm gauge symmetry and have the superpotential , @xmath72 for @xmath73 , the standard one - loop diagram of messenger fields generates a gaugino mass @xmath74 ; @xmath75 where @xmath76 is the sm gauge coupling , and @xmath77 is the dynkin index of the sm gauge symmetry for @xmath70 and @xmath71 . the gaugino mass is given at the vacuum discussed in the previous section . it is , however , noticed that the r symmetry is softly broken by the parameter @xmath78 that makes @xmath79 a local minimum . another way of obtaining gaugino masses would be to consider the direct gauge mediation , i.e. , to generalize u(1 ) to non - abelian symmetry containing the sm one . by adding a small supersymmetric mass for @xmath43 , they behave as messengers and would induce the sm gaugino masses without introducing additional multiplets . for the small mass case , , the charged f term @xmath29 dominates the supersymmetry - breaking scale . to split messenger masses with @xmath29 , we must introduce two pairs of vectorlike messengers with u(1 ) charges @xmath80 and @xmath81 and couple them with @xmath36 in the superpotential . for further details of the messenger sector , the stability of the vacuum , and gaugino mass generation , see ref . @xcite . : in the vacuum , the u(1)@xmath0 symmetry given in table [ table : assign ] is broken by the f term of @xmath16 which is neutral under both u(1 ) and u(1)@xmath0 . we consider the following messenger superpotential @xmath82 where @xmath70 and @xmath71 are vectorlike multiplets , charged under the sm gauge symmetry . it is noticed that the @xmath20 symmetry is softly broken in this messenger sector . note that the superpotential ( [ example_superpotential ] ) has an anomalous u(1) symmetry , under which @xmath16 and @xmath17 are charged , but it is explicitly broken in ( [ messenger ] ) . the r symmetry is spontaneously broken in the full potential with both ( [ example_superpotential ] ) and ( [ messenger ] ) , if the vacuum in the previous section is stable . the vacuum stability is ensured by taking the parameter @xmath78 sufficiently large . the gaugino mass is evaluated for @xmath83 as @xmath84 which comes from a one - loop diagram , where @xmath70 , @xmath71 circulate in the loop . from these observations , we conclude that this class of supersymmetry - breaking models is useful to build a simple , realistic hidden sector of gauge mediation . we have studied supersymmetry - breaking models with both f and d terms being nonvanishing . in particular , we have focused on the case that the f - term potential shows runaway behaviors that originate from symmetries of theory considered . the runaway directions are uplifted by the d term and a supersymmetry - breaking vacuum is realized at finite field configuration . an interesting property of this approach is that ( phenomenologically disfavored ) pseudomoduli are absent in the vacuum since they are related to the minimization of the f - term potential . moreover , there is no need to add the fi term for supersymmetry breaking . along this line , a minimal renormalizable model has been presented where supersymmetry is broken . for an application to gauge mediation , we have introduced appropriate messenger sectors and confirmed that r symmetry is spontaneously broken , and gaugino masses in the visible sector are generated at the comparable order of sfermion masses . this class of models may open up a new way to build realistic models of gauge mediation , circumventing the lemma proved by komargodski and shih @xcite . as remarked , the d - term lifted runaway might be destabilized along other orthogonal directions such as the u(1)@xmath0 runaway . we do not have any criteria to ensure that the d - term uplifting of runaway directions can lead to the stable and global minimum of the scalar potential . it would be interesting if one could carry out a general argument on this issue . we would like to thank l.b . anderson for useful advice on the numerical analysis . thanks the yitp workshop on string theory and field theory and the 2012 simons workshop on mathematics and physics , where this work was partly done , for hospitality . is in part supported by the jsps postdoctoral fellowship for research abroad and is grateful to the center for the fundamental laws of nature at harvard university for support . k.y . is supported in part by the grant - in - aid for scientific research no . 23740187 and also in part by keio gijuku academic development funds . this work is supported by the next generation of physics , spun from universality and emergence , " the gcoe program from the ministry of education , culture , sports , science and technology of japan . in this appendix , we give some details of the potential analysis for the model given in sec . [ subsec : example ] . in particular , we explain the derivation of the vacuum expectation values . the field derivatives of the superpotential are given by @xmath85 w_{\varphi_+ } = \lambda x_0\varphi_- + \lambda'x_-\varphi_0 \ , , \qquad w_{\varphi_- } = \lambda x_0\varphi_+ + mx_+ \ , , \qquad w_{\varphi_0 } = \lambda'x_-\varphi_+ \,.\end{gathered}\ ] ] the scalar potential @xmath4 is the sum of the contributions from f and d terms , @xmath6 and @xmath7 , and these explicit forms are written down in and . the stationary conditions for @xmath4 are then given by @xmath86 first , by using and , we express @xmath37 and @xmath16 in terms of the other fields : @xmath87 \varphi_0 & \,=\ ; \frac{m}{\lambda'}\ , \frac{x_+x_-^*\varphi_-}{\varphi_+\big(|x_-|^2 + |\varphi_+|^2+|\varphi_-|^2\big ) } \ , . \label{ph02}\end{aligned}\ ] ] if @xmath88 , we find from and , @xmath89 this implies that @xmath90 is negative at the stationary point ( if @xmath88 ) . on the other hand , from and , we have @xmath91 while the equation @xmath92 gives @xmath93 & & \qquad -\varphi_-\lambda x_0 ( \lambda x_0\varphi_- + \lambda'x_-\varphi_0)^ * + \lambda x_0(\lambda x_0\varphi_+ + mx_+)^ * \,=\ , 0 \ , . \label{ph+ph-}\end{aligned}\ ] ] by using the relations , @xmath94 following from and , we find another expression for @xmath90 from and : @xmath95 \ , . \label{dpos}\ ] ] that implies @xmath90 is positive at the stationary point . for both conditions and to be true , @xmath96 is the only solution , but it can not be satisfied because the origin of meson direction @xmath22 is destabilized by the f - term potential . therefore @xmath88 , which is the assumption for , should not be realized . in the end , from and , we find the following vacuum expectation values : @xmath97 l. g. aldrovandi and d. marques , jhep * 0805 * ( 2008 ) 022 [ arxiv:0803.4163 [ hep - th ] ] ; y. nakai and y. ookouchi , jhep * 1101 * ( 2011 ) 093 [ arxiv:1010.5540 [ hep - th ] ] ; t. s. ray , phys . rev . d * 85 * ( 2012 ) 035003 [ arxiv:1111.4266 [ hep - ph ] ] . m. dine and j. mason , phys . d * 77 * ( 2008 ) 016005 [ hep - ph/0611312 ] ; k. intriligator and m. sudano , jhep * 1006 * ( 2010 ) 047 [ arxiv:1001.5443 [ hep - ph ] ] ; e. dudas , s. lavignac and j. parmentier , phys . b * 698 * ( 2011 ) 162 [ arxiv:1011.4001 [ hep - th ] ] . r. kitano , h. ooguri and y. ookouchi , phys . d * 75 * ( 2007 ) 045022 [ arxiv : hep - ph/0612139 ] ; b. k. zur , l. mazzucato and y. oz , jhep * 0810 * ( 2008 ) 099 [ arxiv:0807.4543 [ hep - ph ] ] ; a. giveon , a. katz and z. komargodski , jhep * 0907 * ( 2009 ) 099 [ arxiv:0905.3387 [ hep - th ] ] ; s. a. abel , j. jaeckel and v. v. khoze , phys . b * 682 * ( 2010 ) 441 [ arxiv:0907.0658 [ hep - ph ] ] ; m. bertolini , l. di pietro and f. porri , jhep * 1201 * , 158 ( 2012 ) [ arxiv:1111.2307 [ hep - th ] ] . t. azeyanagi , t. kobayashi , a. ogasahara and k. yoshioka , jhep * 1109 * ( 2011 ) 112 [ arxiv:1106.2956 [ hep - ph ] ] . | we study the d - term effect on runaway directions of the f - term scalar potential .
a minimal renormalizable model is presented where supersymmetry is broken without any pseudomoduli .
the model is applied to the hidden sector of gauge mediation for spontaneously breaking r symmetry and generating nonvanishing gaugino masses at the one - loop order . |
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the spectra of quasars show a @xmath0forest `` of absorption lines blueward of the ly-@xmath1 emission line ( lynds 1971 , sargent et al . 1980 , weymann , carswell , & smith 1981 ) . observational and theoretical work in recent years has shown that most of this absorption can be attributed to neutral hydrogen in galaxies and large - scale structure along the line of sight ( cen et al . 1994 , lanzetta et al . 1995,1996 , zhang et al . 1995 , hernquist et al . 1996 , miralda - escud et al . 1996 , bi & davidsen 1997 , chen et al . 1998 , theuns et al . 1998 , ortiz - gil et al . 1999 , impey , petry , & flint 1999 , dav et al . 1999 , bryan et al . 1999 ) . in aggregate , qso spectra show an increasing line density with increasing redshift such that @xmath19 ( sargent et al . 1980 , weymann , carswell , & smith 1981 , young et al . 1982 , murdoch et al . 1986 , lu , wolfe , & turnshek , 1991 , bechtold 1994 , kim et al . but the line density within an individual quasar spectrum decreases with proximity to the ly-@xmath1 emission line ( weymann , carswell , & smith 1981 , murdoch et al . this is generally thought to be due to enhanced ionization of neutral hydrogen in the vicinity of the quasar due to ionizing photons from the quasar itself . this @xmath0proximity effect '' can be used to measure the mean intensity of the uv background , denoted @xmath3 ( carswell et al . 1987 , bajtlik , duncan , & ostriker 1988 , hereafter bdo ) . @xmath3 has been measured at @xmath20 by a variety of authors ( bdo , lu , wolfe , & turnshek 1991 , giallongo et al . 1993,1996 , bechtold 1994 , williger et al . 1994 , cristiani et al . 1995 , fernndez - soto et al . 1995 , lu et al . 1996 , savaglio et al . 1997 , cooke et al . 1997 , scott et al . the results , summarized in paper ii of this series ( scott et al . 2000b ) , are in general agreement with the predictions of models of the uv background which integrate the contribution from known population of quasars and include reprocessing effects in an inhomogeneous intergalactic medium ( haardt & madau 1996 , hereafter hm96 , fardal et al . 1998 ) . in paper ii , the mean intensity of the ionizing background was found to be @xmath21 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath22 . the decline of the quasar space density from @xmath17 to the present is expected to drive a corresponding decline in the intensity of the uv background . kulkarni & fall ( 1993 , hereafter kf93 ) measured @xmath23 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath24 from a subset of the now complete hst quasar absorption line key project sample presented by bahcall et al . ( 1993 ) . much of this previous work has relied upon the technique for measuring @xmath3 outlined by bdo . this technique requires the entire sample of absorption lines to be binned according to the ratio of the quasar flux at the physical position of the absorber to the background flux . this is done for several initial guesses of the background intensity ; and the value that gives the lowest @xmath25 between the binned data and the ionization model is chosen as the best fit @xmath3 . however , this is not the optimal technique to use at low redshift where absorption line densities are low . kf93 developed a maximum likelihood technique to address this issue and used it in their measurement of @xmath3 at @xmath24 . however , their measurement was based upon a sample of only 13 qsos and less than 100 lines , and has correspondingly large error bars . in addition , the value these authors find is lower than the predictions of the models of haardt & madau ( 1996 ) , though consistent within the uncertainties , as shown in figure 13 of paper ii and in figure [ fig : lowzcomp ] of this paper . given the importance of the value of the hi ionization rate to the hydrodynamical evolution of the low redshift universe , performing this measurement with a much larger line sample is worthwhile . the low redshift hydrodynamic simulations of theuns et al . ( 1998 ) and dav et al . ( 1999 ) indicate that the evolution of the ionizing background is the primary driver behind the change of character of the ly-@xmath1 forest from high redshift to low redshift , specifically , the break in the number distribution of ly-@xmath1 lines at @xmath26 ( morris et al . 1991 , bahcall et al . 1991 , weymann et al . the growth of structure pulling gas from low density regions into high density regions also contributes to this and other attributes of the evolution of the ly-@xmath1 forest . shull et al . ( 1999 ) estimate the local ionizing background including contributions to the background from starburst galaxies as well as seyferts and qsos . their models include a treatment of the opacity of the low redshift ly-@xmath1 forest using information drawn from recent observational work ( weymann et al . 1998 , penton et al . they find that starbursts and agn could contribute approximately equally to the ionizing background at low redshift , each @xmath27 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 . the full hst / fos archival data set is presented in paper iii and can also be found at or , so we describe the data used in this paper only briefly in [ sec - data ] . in [ sec - zsys ] and [ sec - flux ] , we discuss our treatment of two parameters of each sample object which are integral to the proximity effect analysis , systemic redshifts and lyman limit fluxes . we outline the proximity effect analysis in [ sec - analysis ] and we present our results in [ sec - disc ] . we discuss the recovery of @xmath3 from simulated ly-@xmath1 forest spectra in [ sec - sims ] . a comparison of the results from radio loud and radio quiet qsos is given in [ sec - rl ] . the maximum likelihood solutions for @xmath3 found when allowing for an equivalent width threshold that varies across each sample spectrum are discussed in [ sec - varthr ] . solutions for the hi ionization rate are given in [ sec - gam ] . the effect of a non - zero @xmath28 on our calculations is discussed in [ sec - omegal ] , and the effect of the ionizing background on the ly-@xmath1 forest line density is discussed in [ sec - dndz ] . we provide comparisons with previous observational work on the low redshift uv background in [ sec - prevres ] and with models of this background in [ sec - models ] . a discussion of possible systematic effects on this analysis is given in [ sec - systematics ] . we conclude with a summary of the results in [ sec - summary ] . the reduction of the fos data is described in paper iii . table [ tab - z ] lists the objects used in the proximity effect analysis along with the object s redshift and classification in the nasa extragalactic database . for the reasons outlined in scott et al . ( 2000b ) we have removed from the full fos sample of paper iii the spectra of quasars known to be lensed , as well as those that show damped ly-@xmath1 absorption , associated absorption , or broad intrinsic absorption . for our primary proximity effect sample , we also remove objects classified as blazars ( bl lacs and optically violent variables ) on the grounds that their continua are highly variable . however , we also perform the proximity effect analysis with associated absorbers , damped ly-@xmath1 absorbers , and blazars included in order to determine if they affect the results obtained . as discussed in paper iii , objects observed only in the period before the costar upgrade to the hst optics and with the a-1 fos aperture are particularly subject to irregular line spread functions . we have omitted those data from this analysis as well . the distributions in redshift of the qsos and absorption lines used in this paper are shown in figure [ fig - samphist ] . qso redshifts based on the ly-@xmath1 emission line have been shown to be blueshifted from the systemic redshift based on narrow emission lines by up to @xmath29 km s@xmath5 . generally , the forbidden oiii doublet at 4959 , 5007 is taken to be the most reliable indicator of the qso systemic redshift ; though other lines such as mg ii @xmath302796,2803 and balmer lines have been shown to trace the systemic redshift as well , with some spread . ( zheng & sulentic 1990 , tytler & fan 1992 , laor et al . 1994,1995 , corbin & boroson 1996 ) however , the results of nishihara et al . ( 1997 ) , m@xmath31intosh et al . ( 1999 ) , and scott et al . ( 2000b ) indicate that in fact h@xmath32 may not reflect the systemic redshift of high redshift qsos . spectra of the emission lines h@xmath32 , [ oiii]@xmath335007 , or mg ii were obtained for several objects in our total proximity effect sample . the observations were carried out on the nights of 19 december 1995 , 14 january 1996 , 20 and 21 april 1996 , 12 and 13 december 1996 , and 2 february 1997 . these observations are summarized in table [ tab - zhst ] . the 19 december 1995 and 13 december 1996 observations were made using the 1.5 meter tillinghast telescope at the fred lawrence whipple observatory using the fast spectrograph ( fabricant et al . 1998 ) and a thinned loral 512x2688 ccd chip ( gain = 1.06 , read noise = 7.9 e@xmath34 ) binned by a factor of 4 in the cross - dispersion direction . observations were made using a 300 lines mm@xmath5 grating blazed at 4750 and a 3@xmath35 slit . these spectra cover a wavelength range of 36607540 . this is listed as set - up ( 1 ) in table [ tab - zhst ] . the january , april , and december 10 and 12 , 1996 observations were made using the steward observatory bok 90 inch telescope using the boller and chivens spectrograph with a 600 l mm@xmath5 grating blazed at 6681 in the first order , a 1.5@xmath35 slit , and a 1200 x 800 ccd array with a gain of 2.2 e@xmath34 adu@xmath5 and a read noise of 7.7 e@xmath34 , binned 1x1 . for the january 1996 observations , the data were obtained with one of two grating tilts , one resulting in wavelength coverages of 36005825 and 68709140 . for the april 1996 data , the wavelength ranges were 41406370 and 52807550 . two grating tilts were also used for the december 1996 data , giving wavelength coverages of 45006700 and 56107860 . the spectrum of one object , 0827 + 2421 , was obtained on 15 february 1997 at the multiple mirror telescope with the blue channel spectrograph , a 2 @xmath35 slit , the 3k x 1k ccd array , and the 800 l mm@xmath5 grating blazed at 4050 with spectral coverage of 43656665 . the spectra are shown in figure [ fig - zspec ] and the lines used for redshift measurements are labeled . taking a simple cursor measurement of each line centroid , we find a mean [ oiii]-balmer line @xmath36v of -30 @xmath37 1010 km s@xmath5 for 31 objects and a mean [ oiii]-mg ii @xmath36v of 58 @xmath37 576 km s@xmath5 for 31 objects . the mean blueshift of the ly-@xmath1 emission line with respect to [ oiii ] is 289 @xmath37 727 km s@xmath5 based on 51 measurements . the redshifts measured for each object in our sample are shown in table [ tab - z ] ; and the results are shown in figure [ fig - zhist ] . gaussian fits to the lines give similar results . we therefore treat both balmer lines and mg ii in addition to [ oiii ] as good systemic redshift indicators for these low redshift objects . in the case of a qso for which we have only a ly-@xmath1 emission line measurement of the redshift , we add 300 km s@xmath5 to this value to estimate its systemic redshift . our method for estimating lyman limit fluxes for each qso is the same as that described in paper ii . for objects with spectral coverage between the ly-@xmath1 and civ emission lines , we extrapolate the flux from 1450 in the quasar s rest frame to 912 using @xmath38 and a spectral index @xmath1 measured primarily from the spectral region between the ly-@xmath1 and c iv emission lines . figure [ fig - fnus ] shows the fos spectra for which these fits were made along with the power law fits themselves . in some cases , @xmath1 is poorly constrained from these fits , especially if there was little spectral coverage redward of ly-@xmath1 emission in the data . if another measurement of the spectral index was available in the literature for these objects , we used it ; otherwise , we used our measurement . table [ tab - flux ] lists the lyman limit flux for each object in this proximity effect sample and either a ) the flux at 1450 , or some other appropriate wavelength free of emission features , measured from the fos data , or b ) a directly measured lyman limit flux and the reference . if available from the extracted archive data , red spectra and the fits to them are presented for objects which were observed only with pre - costar fos and a-1 aperture , though these data were not subsequently used for any ly-@xmath1 forest studies . see table 4 of paper iii . in figure [ fig - zl ] , we show qso lyman limit luminosities versus emission redshift for this hst / fos sample combined with the high redshift objects presented in papers i and ii . only at the lowest redshifts is there any trend of luminosity with redshift . the distribution of ly-@xmath1 lines in redshift and equivalent width is given by : @xmath39 the distribution in redshift and hi column density , n , is : @xmath40 the parameter @xmath41 is the redshift distribution parameter . the quantities @xmath42 in equ . [ eq : dndzdw ] and @xmath32 in equ . [ eq : dndzdnh1 ] are the line rest equivalenth width and column density distribution parameters , respectively . the quantities @xmath43 and @xmath44 are normalizations . the bdo method for measuring @xmath3 consists of binning all lines in the sample in the parameter @xmath45 , the ratio of qso to background lyman limit flux density at the physical location of the absorber : @xmath46 for various values of @xmath3 . the value of @xmath3 that results in the lowest @xmath25 between the binned data and the ionization model , @xmath47^{-(\beta-1 ) } , \label{eq : dndx}\ ] ] is considered to be the optimal value . this ionization model follows from the assumption that the column densities of lines are modified by the presence of the qso according to @xmath48 where @xmath49 is the column density a given line would have in the absence of the qso . the 1@xmath50 errors are found from @xmath51 for 7 degrees of freedom ( press et al . 1992 ) . the value of @xmath45 for each line in a given sample depends not only upon the value of @xmath3 assumed , but also on the cosmological model , as @xmath52 and @xmath53 where @xmath54 is the luminosity distance of an individual absorber from the qso and @xmath55 luminosity distance to the qso from the observer . the luminosity distance between two objects at different redshifts can be calculated analytically for cosmological models in which @xmath16 . we return to this point in section [ sec - omegal ] below . if the proximity effect is indeed caused by enhanced ionization of the igm in the vicinity of qsos , one may expect to observe a larger deficit of lines relative to the ly-@xmath1 forest near high luminosity qsos than near low luminosity qsos . in figure [ fig - npred](a ) , we plot the fractional deficit of lines with respect to the number predicted by equ . [ eq : dndzdw ] versus distance from the qso for this hst / fos sample combined with the high redshift objects observed with the multiple mirror telescope ( mmt ) presented in papers i and ii . we divide our qso sample into high and low luminosity objects at the median lyman limit luminosity of the combined mmt and hst / fos sample , log(l@xmath56 ) @xmath57 31 . high luminosity objects show a marginally more pronounced proximity effect than low luminosity objects : 4.9@xmath50 for qsos with log(l@xmath56 ) @xmath58 31 versus 3.2@xmath50 for qsos with log(l@xmath56 ) @xmath59 31 . in panel ( b ) , we plot the line deficit within 2 h@xmath60 mpc as a function of log(l@xmath56 ) . the lack of a significant difference in the line deficit between high and low luminosity qsos may indicate the presence of clustering , if absorption features cluster more strongly around more luminous qsos with deeper potential wells . we will address the issue of clustering further below . the bdo method of measuring the background can result in poor statistics at low redshift due to the low line density in the low redshift ly-@xmath1 forest . we will quote results from this method , but we will generally the maximum likelihood method for measuring @xmath3 as presented by kf93 , which consists of constructing a likelihood function of the form @xmath61 \label{eq : maxlike},\ ] ] where @xmath62^{-(\beta-1 ) } , \label{eq : fnz}\ ] ] and the indicies @xmath63 and @xmath64 denote sample absorption lines and quasars , respectively . using the values of @xmath41 and @xmath43 from a separate maximum likelihood analysis on the ly-@xmath1 forest excluding regions of the spectra affected by the proximity effect ( dobrzycki et al . 2001 , paper iv ) , and a value of @xmath32 from studies with high resolution data , eg . @xmath65 from hu et al . ( 1995 ) , the search for the best - fit value of @xmath3 consists of finding the value that maximizes this function , fixing the other parameters . if the line density is low throughout a single ly-@xmath1 forest spectrum , it becomes difficult to distinguish any proximity effect , even in a large sample of spectra . the absence of a proximity effect in this model formally translates into the limit @xmath66 because in this scenario , the qso has no additional effect on its surroundings and therefore generates no relative line underdensity . the errors quoted in the values of log[@xmath3 ] are found from the fact that in solving for log[@xmath3 ] alone , the logarithm of the likelihood function , @xmath68 , is distributed as @xmath25 with one degree of freedom . in the case of an ill - defined solution , the likelihood function is very broad and the formal error approaches infinity . if a proximity effect is weak but not absent in the data , a maximum likelihood solution is sometimes possible , but with no well - defined 1@xmath50 upper limit on log[@xmath3 ] . in other words , if an upper limit of infinity is quoted , the data can not rule out the nonexistence of a proximity effect to within 1@xmath50 confidence . using a constant equivalent width threshold results in the loss of a large amount of spectral information . in the case of a large equivalent width threshold , of course , many weak lines are discarded ; and in the case of a small threshold , regions of spectra where the signal - to - noise ratio ( s / n ) does not permit the detection of lines all the way down to the specified threshold are lost and only the highest s / n spectral regions are used . the technique of measuring the statistics @xmath41 and @xmath42 has been expanded to allow for a threshold that varies with s / n across each qso spectrum ( bahcall et al . 1993,1996 , weymann et al . 1998 , scott et al . we will use this variable threshold information to measure @xmath3 as well . the results of this analysis are given in table [ table - jnu ] . before we begin the discussion of the results , some words about the normalization values listed in table [ table - jnu ] are in order . in the bdo method for measuring @xmath3 , lines are binned in @xmath45 and compared to the ionization model given by equ . [ eq : dndx ] , for an assumed value of @xmath32 . in this case , the normalization listed in table [ table - jnu ] is the parameter in equ . [ eq : dndzdw ] , found from the number of lines in the sample and the maximum likelihood value of @xmath41 : @xmath69 where @xmath70 is the total number of lines observed with rest equivalent width greater than @xmath71 , the limiting equivalent width of the line sample . for the maximum likelihood solutions for @xmath3 , we convert line equivalent widths to column densities using the ly-@xmath1 curve of growth and an assumed value of @xmath72 , the characteristic doppler parameter of the lines . as we will demonstrate , different values of @xmath32 and @xmath72 have only a small effect on the value of @xmath3 found . the normalization is given by @xmath73 where @xmath74 is the limiting column density across each qso spectrum corresponding to a limiting equivalent width . this quantity can be held constant , as in the bdo method , or it can be allowed to vary across each qso spectrum . in both of these formulations for the normalization , a proximity region around the qso is neglected and that proximity region is either defined by a velocity cut , eg . @xmath75 - 3000 km s@xmath5 , or by a cut in @xmath45 , eg . @xmath76 . we also use the standard bdo method to find @xmath77= -22.04^{+0.43}_{-1.11}$ ] and @xmath78 for equivalent width thresholds of 0.32 and 0.24 respectively . figures [ fig : chi2](a ) and ( d ) illustrate the @xmath25 of the binned data compared to the bdo ionization model as a function of assumed @xmath79 ) for these two thresholds . the bdo ionization model is expressed in terms of the number of lines per coevolving coordinate : @xmath80 where @xmath81 . this @xmath25 curve is very broad , which is reflected in the large error bars and indicates the difficulty in isolating the optimal mean intensity of a weak background using this technique . figures [ fig : dndx](a ) and ( d ) show the binned data and the ionization model for the values of @xmath79 ) listed above , those that give the lowest @xmath25 between the binned data and the model , ie . the minima of the curves in figures [ fig : chi2](a ) and ( d ) . we executed the maximum likelihood search for @xmath3 , using two different fixed equivalent width thresholds , 0.24 and 0.32 as well as for the case of a variable threshold across all the spectra . the uncertainty in @xmath41 does not translate directly into a large uncertainty in @xmath3 . changing the value of @xmath41 alters the maximum likelihood normalization , @xmath44 , according to equ . [ eq : mlnorm ] . from the sample of lines with rest equivalent widths greater than 0.32 we find @xmath77= -22.11^{+0.51}_{-0.40}$ ] for @xmath82 . varying @xmath41 by @xmath83 gives @xmath77= -22.21 $ ] and @xmath84 with similar uncertainties . the data used here are not of sufficient resolution to fit voigt profiles to the absorption features and derive hi column densities and dopper parameters . we therefore choose to fix the values of @xmath32 and @xmath72 to those found from work on high resolution data , rather than allow them to freely vary in our analysis . for the 0.32 fixed equivalent width threshold , we tested several pairs of values of @xmath85 where @xmath72 is in km s@xmath5 : ( 1.46,35 ) and ( 1.46,25 ) where the value of @xmath32 is taken from hu et al . ( 1995 ) ; as well as ( 1.45,25 ) and ( 1.70,30 ) found from low redshift ly-@xmath1 forest spectra taken with the goddard high resolution spectrograph ( ghrs ) on hst by penton et al . ( 2000a , b ) . in addition , dav & tripp ( 2001 ) have found some evidence for @xmath32 increasing to 2.04 at @xmath86 from high resolution echelle data from the space telescope imaging spectrograph aboard the hst . we test this value as well . the likelihood functions for the maximum likelihood solutions listed in rows 2 - 6 , 8 - 12 , 14 , and 18 of table [ table - jnu ] are shown in figure [ fig : like1 ] . the binned data and ionization models are plotted in figure [ fig : dndxl1 ] . the values of @xmath3 derived for these various pairs of values of @xmath32 and @xmath72 are not significantly different from one another , though the results in table [ table - jnu ] indicate that varying @xmath32 has a larger impact on the inferred @xmath3 than does varying @xmath72 . the solution for @xmath87 differs from the @xmath65 solution by @xmath88 . in the analysis that follows , we adopt the values 1.46 and 35 km s@xmath5 . the models of haardt & madau ( 1996 ) predict that the uv background arising from qsos drops by over an order of magnitude from @xmath89 to @xmath90 . we therefore divide the sample into low and high redshift subsamples at @xmath91 and use both the bdo method and the maximum likelihood method for finding @xmath3 . these results , also listed in table [ table - jnu ] , confirm some evolution in @xmath3 , though not at a high level of significance . for the bdo solutions , we find log[@xmath3 ] at @xmath8 is equal to @xmath92 and log[@xmath3 ] at @xmath9 is equal to @xmath93 . the restrictive 1@xmath50 upper limit for log[@xmath3 ] at @xmath9 arises from the steeply rising @xmath25 as a function of log[@xmath3 ] shown in figure [ fig : chi2 ] . this , in turn arises from the single line in the highest @xmath94 bin moving to the next bin for larger values of @xmath3 , resulting in a drastic change in the @xmath25 with respect to the photoionization model . we do not consider this to be a reliable indicator of the uncertainty in @xmath3 at @xmath9 . the maximum likelihood technique gives more robust estimates of the uncertainties . from this analysis , we find log[@xmath3 ] at @xmath8 is found to be @xmath95 , while at @xmath9 it is -21.98@xmath96 . these results are shown in figures [ fig : lowzcomp](a ) and [ fig : allzcomp ] . including associated absorbers , damped ly-@xmath1 absorbers , or blazars in the proximity effect analysis appears to have little effect on the results . one might expect associated absorbers to reduce the magnitude of the observed proximity effect and hence cause @xmath3 to be overestimated . the value found including the 45 associated absorbers in our sample is indeed larger , log[@xmath3]@xmath97 , versus log[@xmath3]@xmath98 , but not significantly so . likewise , if the intervening dust extinction in damped ly-@xmath1 absorbers is significant , including these objects in our analysis could cause us to overestimate the magnitude of the proximity effect and hence underestimate @xmath3 . however , the inclusion of these 7 objects only negligibly reduces the value of @xmath3 derived . qso variability on timescales less than @xmath99 years would be expected to smooth out the proximity effect distribution ( bdo ) . however , the inclusion of 6 blazars in the sample , all at @xmath8 , resulted in no discernible change in @xmath3 . the sample used in the analysis of hi ionization rates discussed below includes all of these objects . for each solution , we calculate the @xmath25 with respect to the ionization model expressed by equ . [ eq : dndx ] , and the probability that the observed @xmath25 will exceed the value listed by chance for a correct model , q@xmath100 ( press et al . we also execute a kolmogorov - smirnov ( ks ) test for each solution . the ks test provides a measure of how well the assumed parent distribution of lines with respect to redshift , given by equ . [ eq : fnz ] , reflects the true redshift distribution of lines ( cf . murdoch et al . 1986 , press et al . 1992 ) . the ks probability , q@xmath101 , indicates the probability that a value of the ks statistic larger than the one calculated could have occurred by chance if the assumed parent is correct . the ks probability associated with each solution for @xmath3 is listed in column 10 of table [ table - jnu ] . we tested our maximum likelihood methods , including our treatment of the variable equivalent width thresholds by running our analysis on a simulated data set . each of the 151 spectra in this simulated data set had a redshift equal to that of an object in our data set . all objects including those showing associated absorption , damped ly-@xmath1 absorption , or blazar activity are included in this simulated set . each spectrum is created using a monte carlo technique by which lines are placed in redshift and column density space according to equ . [ eq : dndzdnh1 ] . a background of known mean intensity modifies the column densities of the lines according to the bdo formulation given by equ . [ eq : column ] . the same analysis done on the data , consisting of the line - finding algorithm and the maximum likelihood searches for @xmath41 and @xmath3 , is then used on the simulated spectra in order to recover the input @xmath79 ) . three different values of log[@xmath3 ] are input , -21 , -22 , and -23 , and the results are listed in table [ table - sim ] . in order to understand the possible range of recovered log[@xmath3 ] , we repeated the input log[@xmath3]@xmath102 simulation in the constant threshold case nine additional times , resulting in @xmath103 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 . in addition , since we observe the background to evolve with redshift from @xmath91 to @xmath90 , we implement a model in which @xmath3 varies as a power law in @xmath104 over the redshift range of the data . this relationship is defined by the best fit to a power law variation of @xmath3 with redshift : @xmath77= 0.017\log(1+z)-21.87 $ ] . we recover this using both the constant threshold and the variable threshold analyses , at all redshifts and at @xmath8 and @xmath9 separately . the results of this exercise are shown in table [ table - sim ] and in figure [ fig : sims ] . these simulation results indicate that both the constant and variable threshold analyses can overestimate the background by up to a factor of 3 - 5 , though the uncertainties for the variable threshold solutions are consistently lower , as a factor of @xmath105 more lines are used in these solutions . we separated the first of the input log[@xmath3]@xmath102 simulated data samples into high and low redshift subsamples at @xmath91 , in order to determine if the change in @xmath3 as a function of redshift could be falsely introduced in a case there the input background is constant with redshift . for both the constant and variable threshold treatments , this is not the case . the value found for the low redshift subsample is actually larger than the value found for the high redshift subsample in both treatments . in the case of the varying input @xmath77 $ ] , the values recovered for the high redshift subsample and for the entire redshift range of the data are overestimates . the slope of the linear relationship between @xmath77 $ ] and @xmath106 is quite small , 0.017 , resulting in a variable input @xmath77 $ ] that is actually nearly constant with redshift . the solution for @xmath8 matches the input well for both the constant and variable threshold cases . at @xmath9 , the variable threshold solution overestimates the input by a larger factor , @xmath573 , or 1.6@xmath50 , than does the constant threshold solution , @xmath572 , or less than 1@xmath50 . in paper ii , we argued that curve - of - growth effects are likely to come into play in the proximity effect analysis and to play a larger role for cases in which @xmath3 is large and the proximity effect signature is small . here we find that the input @xmath3 is recovered most effectively by the constant and variable threshold cases for the largest input value of @xmath77 $ ] , @xmath107 . however , nearly every case tested with these simulations results in a value of @xmath3 larger than the input value , especially when a variable equivalent width threshold is used . the only case where the difference is significant is the input @xmath77=-23 $ ] , variable threshold case . the recovered @xmath77 $ ] , -22.47 , is 4@xmath50 larger than the input . we will return to the discussion of the variable threshold in section [ sec - varthr ] below . as described in paper ii , solving for the hi ionization rate , @xmath108 instead of @xmath3 avoids the assumption that the spectral indicies of the qsos and the background are identical . we modified our maximum likelihood code to use the values of @xmath1 for each qso listed in table [ tab - flux ] to measure this quantity and the results are listed in table [ table - gamma ] . for objects with no available measured value of @xmath1 , we use @xmath109 , the extreme ultraviolet spectral index measured from a composite spectrum of 101 hst / fos qso spectra by zheng et al . ( the result for lines above a constant 0.32 rest equivalent width threshold is @xmath110 . this result is not substantially changed if we instead use @xmath111 , the value found from a composite of 184 qso spectra from hst / fos , ghrs , and stis by telfer et al . ( 2001 ) , giving @xmath112 . we also find little change in the result if we assume @xmath109 or @xmath111 for all qsos . the variable threshold data result in a high hi ionization rate , and this is discussed further in the following section . the constant threshold result is plotted in figure [ fig : gam ] . evolution in the uv background is more apparent in the hi ionization rate than in the solutions for @xmath3 . the result at @xmath9 is 6.5 times larger than that at @xmath8 . the values of @xmath3 implied by these solutions for @xmath113 and a global source spectral index @xmath114 are also listed in table [ table - gamma ] . we also parametrize the evolution of the hi ionization rate as a power law : @xmath115 and solve for the parameters @xmath116 and @xmath117 in both the constant and variable threshold cases . the values we find are shown as the dashed line in figure [ fig : gam ] also listed in table [ table - gamma ] . hm96 parametrize their models of the hi ionization rate with the function : @xmath118 we combine our data set with that of scott et al . 2000b to solve for the parameters @xmath119 , @xmath120 , @xmath121 , and @xmath122 . we find @xmath123= ( @xmath124 , 0.35 , 2.07 , 1.77 ) for @xmath32=1.46 and @xmath123 = ( @xmath125 , 1.45 , 2.13 , 1.42 ) for @xmath32=1.7 , while the parameters found by hm96 for @xmath126 are ( @xmath12 , 0.43 , 2.30 , 1.95 ) . these results are also represented by the solid curves in figure [ fig : gam ] , while the hm96 parametrization is shown by the dotted line for comparison . the variable threshold analysis yielded some unexpected results . as seen in the majority of the simulations , the values of @xmath79 ) found were consistently larger than the values found using a constant equivalent width threshold , indicating that the inclusion of weaker lines suppresses the proximity effect . this is to be expected if clustering is occurring ( loeb & eisenstein 1995 ) , which in itself is to be expected to be more prominent at low redshift than at high redshift . however , the suppression of the proximity effect by the inclusion of weak lines is somewhat counterintuitive from the perspective of the curve of growth . most of the lines included in a constant threshold solution are on the flat part of the curve of growth . therefore , though the ionizing influence of the quasar may be translated directly into a change in the hi column density , as predicted by the bdo photoionization model , this will not necessarily result in a corresponding change in the line equivalent width . the solution for @xmath8 is nearly a factor of 3 larger than the the solution found in the case of a constant , 0.32 equivalent width threshold . the solution for @xmath9 is a factor of @xmath127 larger than the constant threshold solution , with no well - defined 1@xmath50 upper limit due to the flattening of the likelihood function towards high @xmath79 ) this likelihood function for the total sample shows two peaks , the most prominent at log[@xmath79)]@xmath128 , the solution listed in table [ table - jnu ] , and a secondary peak at log[@xmath79 ) ] @xmath129 . this behavior is also exhibited , even more dramatically , in the solutions for the hi ionization rate , as discussed above . we conducted a jackknife resampling experiment ( babu & feigelson 1996 , efron 1982 ) to determine the source of these likelihood function peaks at large log(@xmath113 ) , or log[@xmath79 ) ] . two objects , 0743 - 6719 ( @xmath130 ) and 0302 - 2223 ( @xmath131 ) , are found from jackknife experiments to produce all of this effect . in the jackknife experiment , we perform the maximum likelihood calculation of @xmath79 ) n times , where n is the number of objects in the high redshift subsample . in each calculation , one object from the total sample is removed . the results of this experiment are shown in the histogram in figure [ fig : jack ] . the removal of 0743 - 6719 or 0302 - 2223 results in the two values of @xmath113 that are well - defined and that are in reasonable agreement with the value calculated at high redshift in the constant threshold case . removing only the one line from 0743 - 6719 nearest the ly-@xmath1 emission line with @xmath132 and observed equivalent width equal to 0.23 results in @xmath133 s@xmath5 . this object was part of the hst key project sample ( jannuzi et al . 1998 ) and they cite no evidence of associated aborption in its spectrum . removing only the one line from 0302 - 2223 nearest the ly-@xmath1 emission line with @xmath134 and observed equivalent width equal to 0.27 results in @xmath135 s@xmath5 . this object shows an absorption system at @xmath136 and is classified as an associated absorber . no metal absorption is seen at @xmath134 , though this absorber is within 5000 km s@xmath5 of the qso , the canonical associated absorber region . removing both of these lines gives @xmath137 s@xmath5 . due to the small equivalent widths of both of these lines they are not included in the constant threshold analysis , and the solutions for @xmath3 and @xmath113 for @xmath9 are well - defined . it appears that this method has some trouble reliably recovering the background from a sample of absorption lines above an equivalent width threshold allowed to vary with s / n . as the method works well for the constant threshold case , we contend that the photoionization model , expressed in equ . [ eq : dndx ] , used to create the likelihood function must not be an adequate model for the proximity effect when weak lines are included in the analysis . liske & williger ( 2001 ) introduce a method for extracting @xmath3 from qso spectra based on flux statistics . we shall return to this topic in future work . as the results listed in table [ table - jnu ] indicate , the inclusion of the four blazars and one bl lac object , all at @xmath8 , in our sample does not change the result significantly . however , there is much observational evidence that radio loud and radio quiet quasars inhabit different environments , namely that radio loud quasars reside in rich clusters while radio quiet quasars exist in galaxy environments consistent with the field ( stockton 1982 , yee & green 1984 , 1987 , yee 1987 , yates , miller , & peacock 1989 , ellingson , yee , & green 1991 , yee & ellingson 1993 , wold et al . 2000 , smith , boyle , & maddox 2000 ) . if there is a corresponding increase in the number of ly-@xmath1 absorption lines in the spectra of radio loud objects , this could cause the proximity effect to be suppressed , and the measured log[@xmath3 ] to be artificially large . we have therefore divided our sample into radio loud and radio quiet subsamples using the ratio of radio to uv flux to characterize the radio loudness , @xmath138/log[s(1450 } \ ; \mbox{\aa})].\ ] ] the value of rl for each object in our sample is listed in table [ tab - flux ] . a histogram of these values and the distribution of rl with @xmath139 for the sample objects are shown in figure [ fig : rl ] . the division between radio loud and radio quiet was chosen to be rl=1.0 . the resulting values of log[@xmath3 ] for these subsamples are listed in table [ table - jnu ] . there is no significant trend for log[@xmath3 ] to appear larger for radio loud objects than for radio quiet objects . we performed the maximum likelihood calculation for the case of a non - zero cosmological constant . this means that the observer - qso and absorber - qso luminosity distances that appear in the relationship between @xmath140 and @xmath139 ( bdo ) must be calculated numerically from the expression : @xmath141 where @xmath142 ( peebles , 1993 ) as this integral can not be reduced to an analytical form for @xmath143 . the calculations in the sections above assume ( @xmath144,@xmath28 ) = ( 1.0,0.0 ) . here , we perform the maximum likelihood search for @xmath79 ) using ( @xmath144,@xmath28 ) = ( 0.3,0.7 ) . for a qso at @xmath145 with a lyman limit flux density of 0.1 @xmath146jy , an absorber at @xmath147 , and an assumed background of log[@xmath3]@xmath102 . , this ( @xmath144,@xmath28 ) results in a value of @xmath140 that is @xmath148% smaller than that inferred in the @xmath16 case . unlike all the other solutions performed , we ignore redshift path associated with metal lines and use all redshifts between @xmath149 and @xmath150 . this does not change the results significantly , but cuts down the computation time substantially . the results are listed in table [ table - jnu ] and are plotted in figure [ fig : lowzcomp ] . for comparison , we also give the solutions for @xmath3 found using the standard parameters , ( @xmath144,@xmath28 ) = ( 1.0,0.0 ) , with this redshift path neglected . we find that ( @xmath144,@xmath28 ) = ( 0.3,0.7 ) , does not change the value of @xmath3 derived significantly from the value found using ( @xmath144,@xmath28 ) = ( 1.0,0.0 ) . we performed a slightly modified re - analysis of the scott et al . ( 2000b ) sample of objects at @xmath17 and found little effect at high redshift as well . the solution found for ( @xmath144,@xmath28 ) = ( 1.0,0.0 ) was log[@xmath3]@xmath151 , while for ( @xmath144,@xmath28 ) = ( 0.3,0.7 ) , we find log[@xmath3]@xmath152 for these data . in the case of a size distribution of ly-@xmath1 absorbers that is constant in redshift , the evolution of the number of ly-@xmath1 absorption lines per unit redshift is given by : @xmath154^{-0.5 } , \label{equ : noevol}\ ] ] ( sargent et al . 1980 ) where @xmath155 equals the absorber cross section times the absorber comoving number density times the hubble distance , @xmath156 . a plot of @xmath153 versus @xmath139 for non - evolving ly-@xmath1 absorbers in ( @xmath157,@xmath28 ) = ( 1.0,0.0 ) and ( 0.3,0.7 ) cosmologies is shown in figure [ fig : noevol ] . it is clear that non - evolving models are too shallow to fit points at @xmath20 , so the normalization is found from a fit to the fos data . the fos data at @xmath158 are consistent with a non - evolving population for @xmath159 . the data are less consistent with a non - evolving concordance model in which @xmath160 , though not significantly so . the number density evolution of ly-@xmath1 absorbers over the redshift range @xmath161 can not be approximated with a single power law . there is a significant break in the slope of the line number density with respect to redshift , near @xmath26 ( weymann et al . 1998 , paper iv ) though kim , cristiani , & dodorico ( 2001 ) argue that the break occurs at @xmath162 . dav et al . ( 1999 ) show from hydrodynamical simulations of the low redshift ly-@xmath1 forest , that the evolution of the line density is sensitive mainly to the hi photoionization rate , but also to the evolution of structure ( cf . their figure 7 ) . the flattening of @xmath153 observed by weymann et al . ( 1998 ) is mostly attributed to a dramatic decline in @xmath163 with decreasing @xmath139 . dav et al . ( 1999 ) derive an expression for the density of ly-@xmath1 forest lines per unit redshift as a function of the hi photoionization rate : @xmath164^{\beta-1 } h^{-1}(z ) , \label{equ : dave}\ ] ] where @xmath165 is the normalization at some fiducial redshift which we choose to be @xmath90 and @xmath166 can be expressed by equ . [ equ : hmgam ] . we fit the fos and mmt absorption line data , binned in @xmath153 as presented in paper iv and scott et al . ( 2000a , paper i ) , to this function in order to derive the parameters describing @xmath166 implied by the evolution in ly-@xmath1 forest line density . we observe flattening of @xmath153 at @xmath158 , but not to the degree seen by weymann et al . ( 1998 ) in the key project data . as described in paper iv , we find @xmath167 , for lines above a 0.24 threshold , while weymann et al . ( 1998 ) measure @xmath168 . see paper iv for more discussion of the significance and underlying causes of this difference . we find @xmath169 and @xmath170 for @xmath171 and lines with rest equivalent widths above 0.24 and 0.32 respectively . these fits to equ . [ equ : dave ] are shown in figure [ fig : dndz](a ) . in panel ( b ) , we plot @xmath166 , as expressed in equ . [ equ : hmgam ] , evaluated using the parameters found from the fit to equ . [ equ : dave ] above . the hm96 solution and the solution derived from the full fos and mmt data sets are represented by the thick and thin solid lines respectively . the small values of @xmath121 derived from @xmath153 above translate into ionization rates that do not decrease dramatically with decreasing redshift and result from the less pronounced flattening of @xmath153 relative to the key project . these fits are particularly insensitive to the normalization , @xmath119 , so the errors on this parameter are large . these fits should therefore not be interpreted as measurements of @xmath166 as reliable as those found directly from the absorption line data . but we find them instructive nonetheless . the observed @xmath166 falls short of the ionization rate needed to fully account for the change in the ly-@xmath1 line density with redshift , indicating that if the value of @xmath41 at low redshift is indeed slightly larger than that found by the key project , @xmath153 may still be consistent with a non - evolving population of ly-@xmath1 absorbers in the sense noted above , but the formation of structure in the low redshift universe must play a significant role in determining the character of the ly-@xmath1 forest line density . kf93 performed a similar measurement with a small subsample of this total sample- the hst quasar absorption line key project data of bahcall et al . we compare our result to that from sample 2 of kf93 , which was constructed from the bahcall et al . ( 1993 ) data excluding one bal quasar and all heavy element absorption systems . the key project sample has since been supplemented ( bahcall et al . 1996 , jannuzi et al . 1998 ) and those data have been included when appropriate in the complete archival sample of fos spectra presented in paper iii . the mean intensity kf93 derive from their sample 2 ( @xmath172 km s@xmath5 , @xmath32=1.48 , @xmath41=0.21 ) is @xmath173 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 . this result is lower than ours for @xmath8 by a factor of @xmath174 , though the errors are large on both results are large enough that they are consistent . we use 162 lines in our low redshift solution for @xmath3 , 65 more than kf93 . several authors have examined the sharp cutoffs observed in the hi disks of galaxies in the context of using these signatures to infer the local ionizing background ( maloney 1993 , corbelli & salpeter 1993 , dove & shull 1994 ) . the truncations are modeled as arising primarily from photoionization of the disk gas by the local extragalactic background radiation field . using 21 cm observations ( corbelli , scheider , & salpeter 1989 , van gorkom 1993 ) to constrain these models , limits on the local ionizing background are placed at @xmath175 @xmath6 s@xmath5 , where @xmath176 and where @xmath177 for an isotropic radiation field . additionally , narrow - band and fabry - perot observations of h@xmath1 emission from intergalactic clouds ( stocke et al . 1991 , bland - hawthorn et al . 1994 , vogel et al . 1995 , donahue , aldering , & stocke 1995 ) place limits of @xmath178 @xmath6 s@xmath5 , or @xmath179 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 for @xmath180 , while results from measurements of galactic high velocity clouds ( kutyrev & reynolds 1989 , songaila , bryant , & cowie 1989 , tufte , reynolds , & haffner 1998 ) imply @xmath181 @xmath6 s@xmath5 , though the ionization of high velocity clouds may be contaminated by a galactic stellar contribution . tumlinson et al . ( 1999 ) have reanalyzed the 3c273/ngc3067 field using the h@xmath1 imaging data from stocke et al . ( 1991 ) as well as new ghrs spectra of 3c273 , in order to model the ionization balance in the absorbing gas in the halo of ngc3067 . from this analysis , they derive the limits , @xmath182 @xmath6 s@xmath5 , or @xmath183 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath184 . weymann et al . ( 2001 ) have recently reported an upper limit of @xmath185 @xmath6 s@xmath5 , or @xmath186 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 from fabry - perot observations of the intergalactic hi cloud , 1225 + 01 , for a face - on disk geometry . if an inclined disk geometry is assumed , this lower limit becomes @xmath187 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 . these results are summarized in figure [ fig : allzcomp ] . it is encouraging that the proximity effect value is consistent with the limits on the background set by these more direct estimates which are possible locally . haardt & madau ( 1996 ) calculated the spectrum of the uv background as a function of frequency and redshift using a model based on the integrated emission from qsos alone . the qso luminosity function is drawn from pei ( 1995 ) . the opacity of the intergalactic medium is computed from the observed redshift and column density distributions of ly-@xmath1 absorbers given by equ . [ eq : dndzdnh1 ] . the effects of attenuation and reemission of radiation by hydrogen and helium in ly-@xmath1 absorbers are included in these models . their result for @xmath126 and @xmath180 at @xmath90 is @xmath188 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 . fardal et al . ( 1998 ) compute opacity models for the intergalactic medium ( igm ) based on high resolution observations of the high redshift ly-@xmath1 forest from several authors . shull et al . ( 1999 ) extend the models of fardal et al . ( 1998 ) to @xmath90 , treating opacity of low redshift ly-@xmath1 forest from observations made with hst / ghrs ( penton et al . 2000a , b ) and with hst / fos ( weymann et al . 1998 ) . like haardt & madau ( 1996 ) , they also incorporate the observed redshift distribution of lyman limit systems with log(n@xmath189 ) @xmath190 ( stengler - larrea et al . 1995 , storrie - lombardi et al . their models also allow for a contribution from star formation in galaxies in addition to agn . the qso luminosity function again is taken to follow the form given by pei ( 1995 ) with upper / lower cutoffs at 0.01/10 l@xmath191 . qso uv spectral indicies are assumed to equal 0.86 , while the ionizing spectrum at @xmath192 has @xmath180 . the contribution to the background from stars was normalized to the h@xmath1 luminosity function observed by gallego et al . ( 1995 ) and the escape fraction of photons of all energies from galaxies was taken to be @xmath193 . the full radiative transfer model described in fardal et al . ( 1998 ) was used to calculate the contribution to the mean intensity by agn , but not the contribution from stars , as they were assumed to contribute no flux above 4 ryd , the energies at which the effects of igm reprocessing become important . these authors find @xmath194 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath195 , with approximately equal contributions from agn and stars , a value somewhat lower than our result for @xmath8 , but which is allowed within the errors . we estimate the contribution to the uv background from star - forming galaxies using the galaxy luminosity function of the canada - france redshift survey ( lilly et al . 1995 ) . at @xmath24 , we derive @xmath196 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 , assuming @xmath197 . the hm96 models for the qso contribution give @xmath198 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath24 . these estimates , and the range of measured @xmath3 in this paper , @xmath199 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 imply an escape fraction of uv photons from galaxies between 4% and 70% . the @xmath3 inferred from @xmath153 in section [ sec - dndz ] implies escape fractions well over 100% . bianchi et al . ( 2001 ) make updated estimates of the mean intensity of the background with contributions from both qsos and star - forming galaxies . their models incorporate various values of the escape fraction of lyman continuum photons from galaxies which are constant with redshift and wavelength . our new results at @xmath158 are most consistent with their models of the qso contribution alone , though some contribution from galaxies , ie . a small f@xmath200 , is allowed within the uncertainties . at @xmath201 , recent results from steidel , pettini , & adelberger ( 2001 ) on the lyman - continuum radiation from high redshift galaxies suggest that these sources become a more important component of the uv background at high redshift . drawing on lessons learned from our work on high redshift objects in paper ii , we have made corrections for quasar systemic redshifts before performing the proximity effect analysis , as discussed in [ sec - zsys ] . this correction , @xmath202 km s@xmath5 , was made to qso redshifts measured from ly-@xmath1 emission for objects for which no systemic redshift measurement was available . for the low redshifts considered in this paper , redshifts measured from [ oiii ] , mgii , or balmer emission lines were deemed suitable as qso systemic redshift measurements . we have removed known gravitational lenses from the sample . as discussed above , we perform the proximity effect analysis omitting and including spectra that show associated absorption and damped ly-@xmath1 absorption and determined that neither of these populations significantly biases our results . because we are working with low redshift data where line densities are low , we expect that blending has not contributed as strong a systematic effect as in the high redshift sample of paper ii . the curve - of - growth effects discussed in paper ii may still be present , since many lines in the sample have equivalent widths which place them on the flat part of the curve of growth . however , the effects of clustering may be even more important at low redshift than at high redshift . loeb & eisenstein ( 1995 ) showed how the fact that quasars reside in the dark matter potentials of galaxies and small groups of galaxies can influence the proximity effect signature . the peculiar velocities of matter clustered in these potentials can result in ly-@xmath1 absorption at redshifts greater than the quasar emission redshift . we found that including associated absorbers in our sample did not significantly change our results . recently , pascarelle et al . ( 2001 ) report evidence for a lower incidence of ly-@xmath1 absorption lines arising in the gaseous halos of galaxies in the vicinities of qsos than in regions far from qsos . they argue that galaxy - qso clustering may lead proximity effect measurements to overestimate @xmath3 at @xmath8 by a up to a factor of 20 . while we agree that most systematic effects in this type of analysis , including clustering , will lead to overestimates of @xmath3 , the agreement between our results and the direct measurements discussed in section [ sec - direct ] give us confidence that our results are not biased by this large a factor . the hydrodynamic simulations of the low redshift ly-@xmath1 forest of dav et al . ( 1999 ) indicate that , at low redshift , structures of the same column density correspond to larger overdensities and more advanced dynamical states than at high redshift . for a @xmath203 cosmology , an equivalent width limit of 0.32 corresponds to an overdensity of @xmath204 at @xmath22 , while at @xmath205 , this limit corresponds to @xmath206 . this may have implications on the clustering of ly-@xmath1 absorption lines around qsos and hence on the values of @xmath3 derived from the proximity effect . it is possible that we are seeing this clustering effect in the variable threshold solution at @xmath9 , in which the two highest @xmath45 lines in the sample are responsible for the inability to isolate a reasonable maximum likelihood @xmath3 . we have analyzed a set of 151 qsos and 906 ly-@xmath1 absorption lines , the subset of the total data set presented in paper iii that is appropriate for the proximity effect . the primary results of this paper are as follows : \(2 ) the value of @xmath3 is observed to increase with redshift over the redshift range of the sample data , @xmath7 . dividing the sample at @xmath207 , we find @xmath208 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 , at low redshift and @xmath209 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at high redshift . \(3 ) the inclusion of blazars at @xmath8 has no significant effect on the result . there is no significant difference between the values of @xmath3 derived from radio loud ( rl @xmath58 1.0 ) and radio quiet ( rl @xmath59 1.0 ) objects , indicating that the observed richness of quasar environments does not distinctly bias the proximity effect analysis . \(4 ) using information measured and gathered from the literature on each qso s uv spectral index and solving for the hi ionization rate , yields @xmath13 s@xmath5 for @xmath8 and @xmath14 s@xmath5 for and @xmath9 . solving directly for the parameters @xmath123 in the hm96 parametrization of @xmath163 using the hst / fos data presented by bechtold et al . ( 2001 ) combined with the high redshift , ground - based data presented by scott et al . ( 2000a , b ) results in @xmath123 = ( @xmath210 , 0.35 , 2.07 , 1.77 ) for @xmath65 and @xmath211 , 1.45 , 2.13 , 1.42 ) for @xmath212 for @xmath213 . \(5 ) allowing for a varying equivalent width threshold across each qso spectrum results in consistently higher values of @xmath3 than are found from the constant threshold treatments . at @xmath9 , the variable threshold solution is not well - constrained . jackknife experiments indicate that this is due the objects 0743 - 6719 and 0302 - 2223 , namely the highest @xmath45 absorption lines in each of their spectra . \(7 ) the @xmath8 result is in agreement with the range of values of the mean intensity of the hydrogen - ionizing background allowed by a variety of local estimates , including h@xmath1 imaging and modeling of galaxy hi disk truncations . to within the uncertainty in the measurement , this result agrees with the one previous proximity effect measurement of the low redshift uv background ( kf93 ) . these results are consistent with calculated models based upon the integrated emission from qsos alone ( hm96 ) and with models which include both qsos and starburst galaxies ( shull et al . the uncertainties do not make a distinction between these two models possible . \(8 ) the results presented here tentatively confirm the igm evolution scenario provided by large scale hydrodynamic simulations ( dav et al . this scenario , which is successful in describing many observed properties of the low redshift igm , is dependent upon an evolving @xmath3 which decreases from @xmath215 to @xmath216 . however , the low redshift uv background required to match the observations of the evolution of the ly-@xmath1 forest line density is larger than found from the data , indicating that structure formation is playing a role in this evolution as well . our results and the work of others are summarized in figure [ fig : allzcomp ] . we find some evidence of evolution in @xmath3 , though it appears that even larger data sets , especially at @xmath8 and/or improved proximity effect ionization models will be required to improve the significance . the authors thank the anonymous referee for a careful review of the paper and for helpful suggestions . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this project was supported by stsci grant no.ar-05785.02-94a and stsci grant no . go 066060195a . js acknowledges support of the national science foundation graduate research fellowship and the zonta foundation amelia earhart fellowship . js , jb , and mm received financial support from nsf grant ast-9617060b . ad acknowledges support from nasa contract no.nas8-39073 ( cxc ) . vpk acknowledges partial support from an award from the william f. lucas foundation and the san diego astronomers association . aldcroft , t. l. , bechtold , j. , & elvis , m. 1994 , , 93 , 1 appenzeller , i. , krautter , j. , mandel , h. , bowyer , s. , dixon , w. v. , hurwitz , m. , barnstedt , j. , grewing , m. , kappelmann , n. , & krmer , g. 1998 , , 500 , l9 babu , g. j. & feigelson , e. d. 1996 , astrostatistics , ( london : chapman & hall ) bahcall , j. n. , jannuzi , b. t. , schneider , d. p. , hartig , g. f. , bohlin , r. , junkkarinen , v. 1991 , , 377 , l5 bahcall , j. n. , bergeron , j. , boksenberg , a. , hartig , g. f. , jannuzi , b. t. , kirhakos , s. , sargent , w. l. w. , savage , b. d. , schneider , d. p. , turnshek , d. a. , weymann , r. j. , & wolfe , a. m. 1993 , , 87 , 1 bahcall , j. n. , bergeron , j. , boksenberg , a. , hartig , g. f. , jannuzi , b. t. , kirhakos , s. , sargent , w. l. w. , savage , b. d. , schneider , d. p. , turnshek , d. a. , weymann , r. j. , & wolfe , a. m. 1996 , , 457 , 19 bajtlik , s. , duncan , r. c. , & ostriker , j. p. 1988 , , 327 , 570 ( bdo ) barthel , p. d. , tytler , d. , & thompson , b. 1990 , , 82 , 339 basu , d. 1994 , ap&ss , 222 , 91 bechtold , j. 1994 , , 91 , 1 ( b94 ) bechtold , j. , dobrzycki , a. , wilden , b. , morita , m. , scott , j. , dobrzycka , d. , & tran , k.- v. , aldcroft , t. l. 2001 , , in press ( paper iii , astro - 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lombardi , l. j. 1994 , , 428 , 574 wills , d. & wills , b. j. 1976 , , 31 , 143 wold , m. , lacy , m. , lilje , p. b. , & serjeant , s. 2000 , , 316 , 267 yates , m. g. , miller , l. , & peacock , j. a. 1989 , , 240 , 129 yee , h. k. c. & ellingson , e. 1993 , , 411 , 43 yee , h. k. c. 1987 , , 94 , 1461 yee , h. k. c. & green r. f. 1987 , , 94 , 618 yee , h. k. c. & green r. f. 1984 , , 280 , 79 young , p. , sargent , w. l. w. , & boksenberg , a. 1982 , , 252 , 10 zhang , y. , anninos , p. , & norman , m. l. 1995 , , 453 , l57 zheng , w. & sulentic , j. w. 1990 , , 350 , 512 zheng , w. & malkan , m. a. 1993 , , 415 , 517 zheng , w. , kriss , g. a. , davidsen , a. f. , lee , g. , code , a. d. , bjorkman , k. s. , smith , p. s. , weistrop , d. , malkan , m. a. , baganoff , f. k. , & peterson , b. m. 1995 , , 444 , 632 zheng , w. , kriss , g. a. , telfer , r. c. , grimes , j. p. , & davidsen , a. f. 1997 , , 475 , 469 zotov , n. 1985 , , 295 , 94 llcccccccc 0003 + 1553 & opt.var . & 0.4497 & 0.4502 & 0.4503 & & ( 1 ) & ( 2 ) & ( 3 ) & + 0003 + 1955 & opt.var . & 0.0264 & 0.0264 & 0.0261 & & ( 1 ) & ( 1 ) & ( 4 ) & + 0007 + 1041 & opt.var . & 0.0902 & 0.0890 & 0.089 & 0.0895 & ( 1 ) & ( 1 ) & ( 5 ) & ( 6 ) + 0015 + 1612 & rqq & 0.5492 & & & & ( 1 ) & & & + 0017 + 0209 & liner & 0.3994 & & & & ( 1 ) & & & + 0024 + 2225 & & 1.1081 & 1.1096 & & & ( 1 ) & ( 7 ) & & + 0026 + 1259 & sy1 & 0.1453 & 0.1463 & 0.1452 & 0.1458 & ( 1 ) & ( 1 ) & ( 5 ) & ( 6 ) + 0042 + 1010 & & 0.5854 & 0.583 & 0.586 & 0.584 & ( 1 ) & ( 8 ) & ( 8 ) & ( 8 ) + 0043 + 0354 & bal ? & 0.3803 & & & & ( 1 ) & & & + 0044 + 0303 & sy1?&0.6219 & 0.6222 & & & ( 1 ) & ( 2 ) & & + 0050 + 1225 & compact , sy1 & 0.0594 & & & & ( 1 ) & & & + 0100 + 0205 & opt.var . & 0.3937 & & 0.3936 & & ( 1 ) & & ( 3 ) & + 0102 - 2713 & & 0.7763 & & & & ( 1 ) & & & + 0107 - 1537 & & 0.8574 & & & & ( 1 ) & & & + 0112 - 0142 & & 1.3739 & 1.3727 & & & ( 1 ) & ( 1 ) & & + 0115 + 0242 & opt.var . & 0.6652 & 0.6700 & & & ( 1 ) & ( 9 ) & & + 0117 + 2118 & & 1.4925 & 1.499 & 1.504 & 1.499 & ( 1 ) & ( 10)&(11)&(11 ) + 0121 - 5903 & sy1 & 0.0461 & 0.0462 & 0.044 & & ( 1 ) & ( 1 ) & ( 5 ) & + 0122 - 0021 & opt.var.,lpq & 1.0710 & 1.0895 & & & ( 1 ) & ( 12 ) & & + 0137 + 0116 & opt.var . & 0.2622 & & 0.2631 & 0.2644 & ( 1 ) & & ( 1 ) & ( 1 ) + 0159 - 1147 & opt.var.,sy1 & 0.6683 & 0.6696 & & & ( 1 ) & ( 13 ) & & + 0214 + 1050 & opt.var . & 0.4068 & & 0.407 & & ( 1 ) & & ( 14 ) & + 0232 - 0415 & opt.var . & 1.4391 & 1.4434 & & & ( 1 ) & ( 1 ) & & + 0253 - 0138 & & 0.8756 & & & & ( 1 ) & & & + 0254 - 3327b & opt.var . & 1.916 & & & & ( 15 ) & & & + 0302 - 2223 & dlas & 1.4021 & & & & ( 1 ) & & & + 0333 + 3208 & opt.var.,lpq & 1.2642 & 1.264 & & & ( 1 ) & ( 7 ) & & + 0334 - 3617 & & 1.1085 & & & & ( 1 ) & & & + 0349 - 1438 & & 0.6155 & 0.615 & & 0.6206 & ( 1 ) & ( 16 ) & & ( 1 ) + 0355 - 4820 & & 1.0058 & 1.005 & & & ( 1 ) & ( 2 ) & & + 0403 - 1316 & opt.var.,hpq & 0.5705 & & 0.571&&(1 ) & & ( 14 ) & + 0405 - 1219 & opt.var.,hpq&0.5717 & 0.5730 & 0.573 & 0.5731 & ( 1 ) & ( 16)&(14)&(16 ) + 0414 - 0601 & opt.var . & 0.7739 & 0.773 & 0.774 & & ( 1 ) & ( 2 ) & ( 5 ) & + 0420 - 0127 & blazar , hpq & 0.9122 & 0.9162 & & & ( 1 ) & ( 13 ) & & + 0439 - 4319 & & 0.5932 & & & & ( 1 ) & & & + 0454 - 2203 & dlas , lpq & 0.5327 & 0.5350 & 0.534 & & ( 1 ) & ( 2 ) & ( 14 ) & + 0454 + 0356 & dlas & 1.3413 & 1.3490 & & & ( 1 ) & ( 10 ) & & + 0518 - 4549 & sy1 & 0.0355 & 0.0341 & & 0.0339 & ( 1 ) & ( 1 ) & & ( 17 ) + 0537 - 4406 & bl lac , hpq & 0.8976 & 0.8926 & & & ( 1)&(18 ) & & + 0624 + 6907 & & 0.3663 & 0.3687 & 0.3710 & 0.3698 & ( 1 ) & ( 1 ) & ( 1 ) & ( 1 ) + 0637 - 7513 & sy1 & 0.6522 & 0.6565 & & 0.6570 & ( 1 ) & ( 18 ) & & ( 18 ) + 0710 + 1151 & opt.var . & 0.7712 & & & & ( 1 ) & & & + 0742 + 3150 & sy1 & 0.4589 & 0.462 & 0.461 & 0.4620 & ( 1 ) & ( 19)&(14)&(10 ) + 0743 - 6719 & opt.var . & 1.5109 & 1.5089 & & 1.511 & ( 1 ) & ( 20 ) & & ( 21 ) + 0827 + 2421 & blazar , hpq & 0.9363 & 0.94 & & 0.942 & ( 1 ) & ( 7 ) & & ( 7 ) + 0844 + 3456 & sy1 & 0.0637 & 0.0646 & 0.064 & & ( 1 ) & ( 1 ) & ( 5 ) & + 0848 + 1623 & opt.var . & & 1.9220 & & & & ( 7 ) & & + 0850 + 4400 & & 0.5132 & 0.5142 & & 0.5150 & & ( 1 ) & & ( 1 ) + 0859 - 1403 & blazar & 1.3338 & 1.3381 & & 1.341&(1 ) & ( 13 ) & & ( 21 ) + 0903 + 1658 & opt.var . & 0.4108 & 0.4106 & 0.4114 & & ( 1 ) & ( 22)&(22 ) & + 0907 - 0920 & & 0.630 & & & & & & & + 0916 + 5118 & & 0.5520 & 0.5525 & & 0.5536 & & ( 1 ) & & ( 1 ) + 0923 + 3915 & opt.var.,sy1,lpq & 0.6986 & 0.6990 & & & ( 1 ) & ( 24 ) & & + 0935 + 4141 & & 1.937 & & & & & & & + 0945 + 4053 & lpq & 1.2479 & 1.2506 & & & ( 1 ) & ( 19 ) & & + 0947 + 3940 & sy1 & 0.2057 & & 0.2059 & & ( 1 ) & & ( 25 ) & + 0953 + 4129 & sy1 ? & 0.2331 & & 0.247 & 0.2326 & ( 1 ) & & ( 25)&(25 ) + 0954 + 5537 & blazar , hpq & 0.9005 & 0.9025 & & & ( 1 ) & ( 1 ) & & + 0955 + 3238 & opt.var.,sy1.8 & 0.5281 & & 0.531 & 0.5309 & & & ( 14)&(10 ) + 0958 + 5509 & & 1.7569 & 1.7582 & & & ( 10)&(7 ) & & + 0959 + 6827 & & 0.7663 & 0.7724 & & & ( 1 ) & ( 1 ) & & + 1001 + 0527 & & 0.1589 & 0.1605 & & 0.160 & ( 1 ) & ( 1 ) & & ( 25 ) + 1001 + 2239 & & 0.9766 & & & & ( 1 ) & & & + 1001 + 2910 & agn & 0.3285 & & & 0.3293 & ( 1 ) & & & ( 1 ) + 1007 + 4147 & & 0.6110 & 0.6125 & & & ( 1 ) & ( 13 ) & & + 1008 + 1319 & & 1.3012 & 1.2968 & & & ( 1 ) & ( 1 ) & & + 1010 + 3606 & sy1 & 0.0785 & & 0.079 & & ( 1 ) & & ( 5 ) & + 1026 - 004a & & 1.4349 & & & & ( 1 ) & & & + 1026 - 004b & & 1.5253 & & & & ( 1 ) & & & + 1038 + 0625 & opt.var.,lpq & 1.2667 & 1.272 & & & ( 1 ) & ( 7 ) & & + 1049 - 0035 & sy1 & 0.3580 & 0.360 & & 0.3605 & ( 1 ) & ( 5 ) & & ( 10 ) + 1055 + 2007 & opt.var . & 1.1136 & 1.1165 & & & ( 1 ) & ( 13 ) & & + 1100 + 7715 & opt.var.,agn & 0.3120 & & 0.324 & 0.339 & ( 1 ) & & ( 25)&(25 ) + 1104 + 1644 & opt.var.,sy1 & 0.6294 & & 0.630 & 0.6307 & ( 1 ) & & ( 5 ) & ( 6 ) + 1114 + 4429 & sy1 & 0.1448 & 0.1442 & 0.143 & & ( 1 ) & ( 1 ) & ( 25 ) & + 1115 + 4042 & sy1 & 0.1545 & 0.1552 & & 0.156 & ( 1 ) & ( 1 ) & & ( 25 ) + 1116 + 2135 & e2,sy1 ? & & & 0.1768 & 0.1756 & & & ( 25)&(25 ) + 1118 + 1252 & opt.var . & 0.6823 & & & & ( 1 ) & & & + 1127 - 1432 & blazar , lpq & 1.1824 & 1.2121 & & & ( 1 ) & ( 18 ) & & + 1130 + 1108 & & 0.5065 & & 0.5110 & 0.5104 & ( 1 ) & & ( 1 ) & ( 1 ) + 1136 - 1334 & sy1 & 0.5551 & 0.5571 & & 0.5604 & ( 1 ) & ( 18 ) & & ( 18 ) + 1137 + 6604 & opt.var.,lpq & 0.6449 & 0.6448 & 0.646 & & ( 1 ) & ( 13)&(5 ) & + 1138 + 0204 & & 0.3789 & & 0.3820 & 0.3831 & ( 1 ) & & ( 1 ) & ( 1 ) + 1148 + 5454 & opt.var . & 0.9688 & 0.9777 & & & ( 1 ) & ( 10 ) & & + 1150 + 4947 & opt.var . & 0.3334 & 0.333 & 0.333 & 0.333 & ( 1 ) & ( 26)&(26)&(26 ) + 1156 + 2123 & & 0.3464 & & 0.3475 & 0.3459 & ( 1 ) & & ( 1 ) & ( 1 ) + 1156 + 2931 & blazar , hpq & 0.7225 & 0.7281 & & & ( 1 ) & ( 1 ) & & + 1206 + 4557 & & 1.1596 & 1.164 & & & ( 1 ) & ( 7 ) & & + 1211 + 1419 & rqq , sy1 & 0.0802 & 0.0805 & 0.0807 & 0.0810 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1214 + 1804 & & 0.3719 & & & 0.3726 & ( 1 ) & & & ( 1 ) + 1215 + 6423 & & 1.2981 & & & & ( 1 ) & & & + 1216 + 0655 & opt.var . & 0.3312 & 0.3302 & 0.334 & 0.3374 & ( 1 ) & ( 25)&(5 ) & ( 25 ) + 1219 + 0447 & agn & 0.0953 & 0.0931 & & & ( 1 ) & ( 1 ) & & + 1219 + 7535 & sb(r)ab pec , sy1 & 0.0701 & 0.0713 & 0.071&&(1 ) & ( 1 ) & ( 5 ) & + 1226 + 0219 & blazar , sy1,lpq & 0.156 & & 0.157&0.158 & ( 1 ) & & ( 27)&(27 ) + 1229 - 0207 & dlas , blazar , lpq & 1.0406 & 1.0439 & & & ( 1 ) & ( 13 ) & & + 1230 + 0947 & & 0.4176 & & 0.4162 & 0.4153 & ( 1 ) & & ( 1 ) & ( 1 ) + 1241 + 1737 & & 1.2807 & 1.282 & & & ( 1 ) & ( 7 ) & & + 1247 + 2647 & agn & 2.0394 & & & & ( 10 ) & & & + 1248 + 3032 & & 1.0607 & & & & ( 1 ) & & & + 1248 + 3142 & & & 1.029 & & & & ( 28 ) & & + 1248 + 4007 & & 1.0256 & 1.033 & & & ( 1 ) & ( 7 ) & & + 1249 + 2929 & & 0.8205 & & & & ( 1 ) & & & + 1250 + 3122 & & 0.7779 & & & & ( 1 ) & & & + 1252 + 1157 & opt.var . & 0.8701 & & & & ( 1 ) & & & + 1253 - 0531 & bl lac , hpq & 0.5367 & 0.5366 & 0.5356 & 0.536&(1 ) & ( 29)&(29)&(29 ) + 1257 + 3439 & opt.var . & 1.3760 & 1.376 & & & ( 1 ) & ( 7 ) & & + 1258 + 2835 & & 1.3611 & & & & ( 1 ) & & & + 1259 + 5918 & & 0.4679 & 0.4717 & & 0.4853 & ( 1 ) & ( 25 ) & & ( 25 ) + 1302 - 1017 & e4?,opt.var . & 0.2770 & 0.2867 & 0.278 & 0.2868 & ( 1 ) & ( 12)&(5 ) & ( 6 ) + 1305 + 0658 & & 0.6009 & 0.5999 & & & ( 1 ) & ( 1 ) & & + 1309 + 3531 & sab , sy1 & 0.1841 & & 0.184 & 0.183 & ( 1 ) & & ( 25)&(25 ) + 1317 + 2743 & & 1.0082 & 1.016 & & & ( 1 ) & ( 7 ) & & + 1317 + 5203 & blazar & 1.0550 & 1.0555 & & & ( 1)&(7 ) & & + 1318 + 2903 & opt.var . & 0.5469 & & & & ( 1 ) & & & + 1320 + 2925 & & 0.9601 & 0.972 & & & ( 1 ) & ( 7 ) & & + 1322 + 6557 & sy1 & 0.1676 & & & 0.1684 & ( 1 ) & & & ( 25 ) + 1323 + 6530 & & 1.6227 & 1.6233 & & & ( 1 ) & ( 30 ) & & + 1327 - 2040 & & 1.1682 & 1.170 & & & ( 1 ) & ( 18 ) & & + 1328 + 3045 & dlas & 0.8466 & 0.8508 & & & ( 1 ) & ( 13 ) & & + 1329 + 4117 & & 1.9351 & & & & ( 10 ) & & & + 1333 + 1740 & & 0.5464 & 0.5546 & & & ( 1 ) & ( 25 ) & & + 1351 + 3153 & & 1.3170 & 1.3382 & & & ( 1 ) & ( 31 ) & & + 1351 + 6400 & sy1 & 0.0886 & 0.0884 & 0.087 & 0.089 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1352 + 0106 & & 1.1200 & & & & ( 1 ) & & & + 1352 + 1819 & sy1 & 0.1508 & 0.1514 & 0.1572 & 0.1538 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1354 + 1933 & opt.var . & 0.7190 & 0.718 & 0.719 & & ( 1 ) & ( 7 ) & ( 5 ) & + 1356 + 5806 & & 1.3741 & 1.370&&&(1 ) & ( 7 ) & & + 1401 + 0952 & & 0.4363 & & & & ( 1 ) & & & + 1404 + 2238 & sy & 0.0966 & 0.0978 & & 0.098 & ( 1 ) & ( 1 ) & & ( 25 ) + 1407 + 2632 & & 0.95 & 0.946 & & 0.958 & ( 1 ) & ( 32 ) & & ( 32 ) + 1415 + 4509 & & 0.1145 & 0.1142 & 0.1143 & 0.1139 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1416 + 0642 & & 1.4339 & & & 1.442 & ( 1 ) & & & ( 21 ) + 1424 - 1150 & & 0.8033 & 0.8037 & & & ( 1 ) & ( 18 ) & & + 1425 + 2645 & opt.var . & 0.3634 & & & 0.3644 & ( 1 ) & & & ( 10 ) + 1427 + 4800 & sy1 & 0.2215 & & 0.2203 & 0.2246 & ( 1 ) & & ( 25)&(25 ) + 1435 - 0134 & & 1.3099 & & & & ( 1 ) & & & + 1440 + 3539 & compact & 0.0764 & 0.0772 & 0.0777 & 0.0772 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1444 + 4047 & e1 ? & 0.2659 & & 0.2672 & 0.267 & ( 1 ) & & ( 3 ) & ( 5 ) + 1512 + 3701 & sy1 ? & 0.3704 & 0.3734 & 0.371 & 0.3715 & ( 1 ) & ( 2 ) & ( 5 ) & ( 6 ) + 1517 + 2356 & & 1.9037 & & & & ( 10 ) & & & + 1517 + 2357 & & 1.834 & & & & & & & + 1521 + 1009 & & 1.3210 & 1.332 & & & ( 1 ) & ( 7 ) & & + 1538 + 4745 & & 0.7704 & 0.7711 & & & ( 1 ) & ( 7 ) & & + 1544 + 4855 & & 0.3985 & & & 0.4010 & ( 1 ) & & & ( 2 ) + 1555 + 3313 & & 0.9402 & 0.9427 & & & ( 1 ) & ( 31 ) & & + 1611 + 3420 & blazar , lpq & 1.3968 & 1.3997 & & & ( 1 ) & ( 33 ) & & + 1618 + 1743 & opt.var . & 0.5549 & 0.5560 & 0.555 & & ( 1 ) & ( 14)&(13 ) & + 1622 + 2352 & opt.var . & 0.9258 & 0.925 & & & ( 1 ) & ( 7 ) & & + 1626 + 5529 & sy1 & 0.1315 & 0.1325 & 0.132 & 0.133 & ( 1 ) & ( 1 ) & ( 25)&(25 ) + 1630 + 3744 & & 1.4712 & 1.478 & 1.474 & 1.478 & ( 1 ) & ( 10)&(11)&(27 ) + 1634 + 7037 & & 1.3338 & 1.338 & 1.336 & 1.342 & ( 1 ) & ( 10)&(11)&(27 ) + 1637 + 5726 & lpq & 0.7499 & 0.750&&0.751 & ( 1 ) & ( 7 ) & & ( 5 ) + 1641 + 3954 & opt.var.,hpq & 0.5946 & 0.5954 & 0.593&&(1 ) & ( 14)&(2 ) & + 1704 + 6048 & opt.var . & 0.3694 & 0.3704 & 0.372 & & ( 1 ) & ( 2 ) & ( 5 ) & + 1715 + 5331 & & 1.9371 & 1.932 & & & ( 10)&(7 ) & & + 1718 + 4807 & & 1.0809 & 1.0828 & & & ( 1 ) & ( 7 ) & & + 1803 + 7827 & bl lac & 0.6840 & & 0.6797 & & ( 1 ) & & & ( 23 ) + 1821 + 6419 & sy1 & 0.2957 & & 0.297 & & ( 1 ) & & ( 5 ) & + 1845 + 7943 & opt.var.,blrg,sy1 & 0.0567 & 0.0548 & & & ( 1 ) & ( 1 ) & & + 2112 + 0556 & & 0.4585 & & & 0.460 & ( 1 ) & & & ( 5 ) + 2128 - 1220 & opt.var.,lpq,sy1 & 0.4988 & 0.5000 & 0.499 & 0.5028 & ( 1 ) & ( 2 ) & ( 14)&(6 ) + 2135 - 1446 & e1,opt.var.,sy1 & 0.2016 & & 0.200 & 0.199 & ( 1 ) & & ( 14)&(34 ) + 2141 + 1730 & opt.var.,lpq,sy1 & 0.2124 & & 0.211 & & ( 1 ) & & ( 14 ) & + 2145 + 0643 & opt.var.,lpq & 0.9997 & 1.000 & & & ( 1 ) & ( 7 ) & & + 2155 - 3027 & opt.var.,bl lac & 0.116 & & & & & & & + 2201 + 3131 & lpq & 0.2953 & 0.2981 & 0.295 & 0.2979 & ( 1 ) & ( 16)&(5 ) & ( 16 ) + 2216 - 0350 & opt.var.,lpq & 0.8997 & 0.900 & & & ( 1)&(7 ) & & + 2223 - 0512 & opt.var.,hpq,bl lac & 1.4037 & & & & ( 1 ) & & & + 2230 + 1128 & blazar , hpq & 1.0367 & 1.0379 & & & ( 1 ) & ( 13 ) & & + 2243 - 1222 & opt.var.,hpq & 0.6257 & 0.6297 & & & ( 1 ) & ( 17 ) & & + 2251 + 1120 & opt.var . & & 0.322 & 0.326 & 0.3255 & & ( 34)&(5 ) & ( 10 ) + 2251 + 1552 & blazar , hpq & 0.8557 & & & & ( 1 ) & & & + 2251 - 1750 & opt.var.,sy1 & 0.0651 & 0.0637 & 0.064 & & ( 1 ) & ( 1 ) & ( 5 ) & + 2300 - 6823 & & 0.5149 & 0.511 & 0.516 & 0.512 & ( 1 ) & ( 35)&(35)&(35 ) + 2340 - 0339 & & 0.8948 & 0.893 & & & ( 1 ) & ( 7 ) & & + 2344 + 0914 & opt.var.,sy1 & 0.6710 & 0.6722 & 0.673 & 0.6731 & ( 1 ) & ( 16 ) & ( 5 ) & ( 16 ) + 2352 - 3414 & opt.var . & 0.7060 & 0.7063 & & & ( 1 ) & ( 2 ) & & + lcccc 0112 - 0142 & 18.0 & 1 & 13dec1996 & 1200 + 0137 + 0116 & 17.1 & 1 & 13dec1996 & 1200 + 0232 - 0415 & 16.4 & 1 & 13dec1996 & 1200 + 0349 - 1438 & 16.2 & 1 & 12dec1996 & 900 + 0414 - 0601 & 15.9 & 1 & 19dec1995 & 400 + 0454 - 2203 & 16.1 & 1 & 19dec1995 & 400 + 0624 + 6907 & 14.2 & 1 & 19dec1995 & 465 + 0827 + 2421 & 17.2 & 3 & 15feb1997 & 1200 + 0850 + 4400 & 16.4 & 1 & 19dec1995 & 300 + 0859 - 1403 & 16.6 & 2a & 12dec1996 & 3600 + 0916 + 5118 & 16.5 & 1 & 19dec1995 & 350 + 0923 + 3915 & 17.9 & 2b & 14jan1996 & 1800 + 0954 + 5537 & 17.7 & 2c & 20apr1996 & 3600 + 0959 + 6827 & 16.4 & 2b & 14jan1996 & 1800 + 1001 + 2910 & 15.5 & 2a & 12dec1996 & 3600 + 1008 + 1319 & 16.2 & 2a & 10dec1996 & 1800 + 1130 + 1108 & 16.9 & 2d & 14jan1996 & 3600 + 1138 + 0204 & 17.6 & 2e & 12dec1996 & 2400 + 1156 + 2123 & 17.5 & 2e & 12dec1996 & 1800 + 1156 + 2931 & 17.0 & 2a & 10dec1996 & 1800 + 1214 + 1804 & 17.5 & 2f & 21apr1996 & 1800 + 1230 + 0947 & 16.1 & 2f & 21apr1996 & 3600 + 1305 + 0658 & 17.0 & 2c & 20apr1996 & 3600 + lcccccrcccc 0003 + 1553 & 3.88 & 0.46 & [email protected] & [email protected] & 1.94 ( 1450 ) & 2.24 & ( 2)&(1b)&(1b)&(1b ) + 0003 + 1955 & 3.99 & 2.04 & [email protected] & [email protected] & 8.43 ( 1450)&-0.44 & ( 3)&(1a)&(1a)&(1a ) + 0007 + 1041 & 5.62 & & [email protected] & [email protected] & 1.47 ( 1450 ) & 0.00 & & ( 1a)&(1a)&(1a ) + 0015 + 1612 & 4.07 & & [email protected] & [email protected] & 0.11 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0017 + 0209 & 3.05 & & [email protected] & [email protected] & 0.31 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0024 + 2225&3.60 & & [email protected] & [email protected] & 0.79 ( 1450 ) & 2.40 & & ( 1c)&(1c)&(1c ) + 0026 + 1259 & 4.56 & & [email protected] & [email protected] & 2.22 ( 1450)&-0.04 & & ( 1b)&(1b)&(1b ) + 0042 + 1010 & 5.52 & & [email protected] & [email protected] & 0.09 ( 1450 ) & 2.99 & & ( 1c)&(1c)&(1c ) + 0043 + 0354&3.18 & & [email protected] & [email protected] & 0.97 ( 2093 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0044 + 0303 & 2.88 & 1.16 & [email protected] & [email protected] & 0.79 ( 1450 ) & 1.94 & ( 2)&(1c)&(1c)&(1c ) + 0050 + 1225&1.46 & & [email protected] & [email protected] & 2.56 ( 1450 ) & 0.06 & & ( 1a)&(1a)&(1a ) + 0100 + 0205 & 2.92 & & [email protected] & [email protected] & 0.45 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0102 - 2713 & 1.93 & & & 0.18 & 0.29 ( 1285 ) & 0.00 & & ( 4 ) & & ( 1b ) + 0107 - 1537 & 1.73 & & [email protected] & [email protected] & 0.16 ( 1450 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0112 - 0142&4.32 & & & 0.17 & 0.29 ( 1326 ) & 3.83 & & ( 4 ) & & ( 1c ) + 0115 + 0242&3.32 & & [email protected] & [email protected] & 0.08 ( 1450 ) & 4.08 & & ( 1c)&(1c)&(1c ) + 0117 + 2118&4.75 & 0.39 & [email protected] & [email protected] & 1.88 ( 1307 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 0121 - 5903 & 3.05 & & [email protected] & [email protected] & 2.91 ( 1450 ) & 0.00 & & ( 1a)&(1a)&(1a ) + 0122 - 0021 & 3.57 & & [email protected] & [email protected] & 0.86 ( 1450 ) & 3.13 & & ( 1c)&(1c)&(1c ) + 0137 + 0116 & 3.00 & & [email protected] & [email protected] & 0.07 ( 1450 ) & 3.97 & & ( 1b)&(1b)&(1b ) + 0159 - 1147 & 1.77 & & [email protected] & [email protected] & 1.33 ( 1450 ) & 3.01 & & ( 1c)&(1c)&(1c ) + 0214 + 1050 & 6.96 & & [email protected] & [email protected] & 1.22 ( 1450 ) & 2.57 & & ( 1b)&(1b)&(1b ) + 0232 - 0415 & 2.42 & 0.59 & & & & 2.73 & ( 2 ) & & & + 0253 - 0138&5.61 & & [email protected] & [email protected] & 0.78 ( 1450 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0254 - 3327b&2.32 & & & 0.28 & & 3.08 & & ( 4 ) & & + 0302 - 2223&1.87&0.31&[email protected] & [email protected] & 0.88 ( 1318 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 0333 + 3208 & 13.5 & & [email protected] & [email protected] & 0.81 ( 1450 ) & 3.38 & & ( 1c)&(1c)&(1c ) + 0334 - 3617&1.40 & & [email protected] & [email protected] & 0.14 ( 1450 ) & & & ( 1c)&(1c)&(1c ) + 0349 - 1438 & 3.83 & & [email protected] & [email protected] & 2.11 ( 1450 ) & 2.53 & & ( 1c)&(1c)&(1c ) + 0355 - 4820&1.16 & 0.39 & [email protected] & [email protected] & 0.70 ( 1450 ) & 2.91 & ( 5)&(1c)&(1c)&(1c ) + 0403 - 1316&3.65 & & [email protected] & [email protected] & 0.39 ( 1450 ) & 4.35 & & ( 1c)&(1c)&(1c ) + 0405 - 1219 & 3.74 & 2.05 & [email protected] & [email protected] & 4.14 ( 1450 ) & 2.68 & ( 2)&(1c)&(1c)&(1c ) + 0414 - 0601 & 5.14 & 0.34 & [email protected] & [email protected] & 0.70 ( 1450 ) & 2.66 & ( 2)&(1c)&(1c)&(1c ) + 0420 - 0127&7.10 & & [email protected] & [email protected] & 0.20 ( 1450 ) & 3.89 & & ( 1c)&(1c)&(1c ) + 0439 - 4319 & 2.30 & & [email protected] & [email protected] & 0.33 ( 1450 ) & 2.95 & & ( 1c)&(1c)&(1c ) + 0454 + 0356&7.39 & 0.38 & [email protected] & [email protected] & 1.26 ( 1336 ) & 2.50 & ( 2)&(1c)&(1c)&(1c ) + 0454 - 2203 & 2.99 & 0.38 & [email protected] & [email protected] & 1.28 ( 1450 ) & 2.77 & & ( 1b)&(1b)&(1b ) + 0518 - 4549 & 4.12 & & [email protected] & [email protected] & 0.13 ( 1450 ) & 5.06 & & ( 1a)&(1a)&(1a ) + 0537 - 4406&4.02 & 0.05 & [email protected] & [email protected] & 0.36 ( 1450 ) & 4.05 & ( 2)&(1c)&(1c)&(1c ) + 0624 + 6907 & 7.01 & & [email protected] & [email protected] & 5.26 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0637 - 7513 & 9.22 & 0.53 & [email protected] & [email protected] & 0.49 ( 1450 ) & 4.10 & ( 2)&(1c)&(1c)&(1c ) + 0710 + 1151&11.0 & & [email protected] & [email protected] & 1.22 ( 1450 ) & 4.12 & & ( 1c)&(1c)&(1c ) + 0742 + 3150&4.89 & 0.35 & [email protected] & [email protected] & 1.03 ( 1450 ) & 2.96 & ( 2)&(1b)&(1b)&(1b ) + 0743 - 6719 & 11.9 & 0.24 & & & & 3.46 & ( 2 ) & & & + 0827 + 2421&3.51 & & [email protected] & [email protected] & 0.59 ( 1450 ) & 3.17 & & ( 1c)&(1c)&(1c ) + 0844 + 3456&3.31 & & [email protected] & [email protected] & 4.94 ( 2495 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0848 + 1623&29.7 & & 0.46 & 0.15 & 0.19 ( 1450 ) & 0.00 & & ( 6 ) & & ( 11 ) + 0850 + 4400 & 2.53 & & [email protected] & [email protected] & 0.56 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0859 - 1403&5.71 & 0.60 & & & & 3.29 & ( 2 ) & & & + 0903 + 1658&3.61 & & [email protected] & [email protected] & 0.17 ( 1450 ) & 2.79 & & ( 1b)&(1b)&(1b ) + 0907 - 0920&4.57 & & [email protected] & [email protected]&0.11 ( 1822 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0916 + 5118 & 1.40 & & [email protected] & [email protected] & 0.82 ( 1450 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 0923 + 3915&1.53 & & [email protected] & [email protected] & 0.77 ( 1450 ) & 4.83 & & ( 1c)&(1c)&(1c ) + 0935 + 4141&1.32 & & & 0.55 & & 0.00 & & ( 4 ) & & + 0945 + 4053 & 1.44 & & [email protected] & [email protected] & 0.15 ( 1450 ) & 4.07 & & ( 1c)&(1c)&(1c ) + 0947 + 3940 & 1.61 & & [email protected] & [email protected] & 1.25 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 0953 + 4129 & 1.28 & & [email protected] & [email protected] & 1.58 ( 1450 ) & 0.10 & & ( 1b)&(1b)&(1b ) + 0954 + 5537&0.94 & & [email protected] & [email protected] & 0.18 ( 1450 ) & 3.51 & & ( 1c)&(1c)&(1c ) + 0955 + 3238&1.62 & 0.38 & [email protected] & [email protected] & 0.87 ( 1774 ) & 2.99 & ( 2)&(1c)&(1c)&(1c ) + 0958 + 5509&0.84 & 0.31 & & & & 0.00 & ( 2 ) & & & + 0959 + 6827 & 3.93 & & [email protected] & [email protected] & 1.10 ( 1720 ) & 1.99 & & ( 1c)&(1c)&(1c ) + 1001 + 0527&2.41 & & [email protected] & [email protected] & 0.55 ( 1450 ) & 0.26 & & ( 1b)&(1b)&(1b ) + 1001 + 2239 & 2.82 & & [email protected] & [email protected] & 0.12 ( 1450 ) & 3.17 & & ( 1c)&(1c)&(1c ) + 1001 + 2910 & 1.93 & & [email protected] & [email protected] & 1.88 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1007 + 4147 & 1.23 & 0.72 & [email protected] & [email protected] & 1.02 ( 1450 ) & 2.92 & ( 2)&(1c)&(1c)&(1c ) + 1008 + 1319&3.79 & 0.58 & & & & 0.00 & ( 2 ) & & & + 1010 + 3606 & 1.24 & & [email protected] & [email protected] & 1.00 ( 1450 ) & 0.00 & & ( 1a)&(1a)&(1a ) + 1026 - 004a & 4.85 & & & 0.11 & 0.19 ( 1328 ) & 0.00 & & ( 4 ) & & ( 1c ) + 1026 - 004b & 4.85 & & & 0.15 & 0.24 ( 1285 ) & 0.00 & & ( 4 ) & & ( 1c ) + 1038 + 0625&2.81 & & [email protected] & [email protected] & 1.00 ( 1361 ) & 3.09 & & ( 1c)&(1c)&(1c ) + 1049 - 0035&3.87 & 0.35 & [email protected] & [email protected] & 1.07 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 1055 + 2007 & 1.94 & & [email protected] & [email protected] & 0.34 ( 1450 ) & 3.64 & & ( 1c)&(1c)&(1c ) + 1100 + 7715&3.04 & & [email protected] & [email protected] & 1.33 ( 1450 ) & 2.76 & & ( 1b)&(1b)&(1b ) + 1104 + 1644 & 1.55 & & [email protected] & [email protected] & 1.22 ( 1450 ) & 2.66 & & ( 1c)&(1c)&(1c ) + 1114 + 4429&1.80 & & [email protected] & [email protected] & 0.35 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1115 + 4042&1.86 & & [email protected] & [email protected] & 1.35 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1116 + 2135 & 1.27 & & [email protected] & [email protected] & 2.87 ( 1450 ) & 0.01 & & ( 1b)&(1b)&(1b ) + 1118 + 1252&2.28 & & [email protected] & [email protected] & 0.14 ( 1450 ) & 2.75 & & ( 1c)&(1c)&(1c ) + 1127 - 1432&4.07 & & [email protected] & [email protected] & 0.49 ( 1450 ) & 4.78 & & ( 1c)&(1c)&(1c ) + 1130 + 1108&3.47 & & [email protected] & [email protected] & 0.62 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1136 - 1334 & 3.51 & 0.60 & [email protected] & [email protected] & 0.83 ( 1450 ) & 3.36 & ( 2)&(1b)&(1b)&(1b ) + 1137 + 6604&1.00 & 1.05 & [email protected] & [email protected] & 1.17 ( 1450 ) & 2.98 & ( 2)&(1c)&(1c)&(1c ) + 1138 + 0204&2.37 & & [email protected] & [email protected] & 0.35 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1148 + 5454 & 1.19 & 0.97 & [email protected] & [email protected] & 1.35 ( 1450)&-0.13 & ( 2)&(1c)&(1c)&(1c ) + 1150 + 4947 & 2.01 & & [email protected] & [email protected] & 0.26 ( 1450 ) & 3.44 & & ( 1b)&(1b)&(1b ) + 1156 + 2123 & 2.56 & & [email protected] & [email protected] & 0.49 ( 1450 ) & 2.23 & & ( 1b)&(1b)&(1b ) + 1156 + 2931 & 1.58 & 0.57 & [email protected] & [email protected] & 1.33 ( 1450 ) & 3.04 & ( 2)&(1c)&(1c)&(1c ) + 1206 + 4557 & 1.27 & 0.45 & [email protected] & [email protected] & 1.69 ( 1450 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 1211 + 1419 & 2.70 & & [email protected] & [email protected] & 2.37 ( 1450)&-0.37 & & ( 1a)&(1a)&(1a ) + 1214 + 1804&2.74 & & [email protected] & [email protected] & 0.52 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1215 + 6423&2.10 & & [email protected] & [email protected] & 0.18 ( 1340 ) & 3.18 & & ( 1c)&(1c)&(1c ) + 1216 + 0655 & 1.57 & & [email protected] & [email protected] & 1.44 ( 1450 ) & 0.44 & & ( 1b)&(1b)&(1b ) + 1216 + 503a&1.87 & & & 0.35 & 0.58 ( 1326 ) & 0.00 & & ( 4 ) & ( 1c)&(1c ) + 1219 + 0447&1.68 & & [email protected] & [email protected]&0.15 ( 2457 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 1219 + 7535&3.13 & & [email protected] & [email protected] & 2.21 ( 1450 ) & 0.45 & & ( 1a)&(1a)&(1a ) + 1226 + 0219&1.81 & 7.40 & [email protected] & [email protected] & 26.9 ( 1330 ) & 4.26 & ( 7)&(1a)&(1a)&(1a ) + 1229 - 0207&2.34 & 0.23 & [email protected] & [email protected] & 0.57 ( 1450 ) & 3.25 & & ( 1c)&(1c)&(1c ) + 1230 + 0947&1.81 & & [email protected] & [email protected] & 0.96 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1241 + 1737 & 1.81 & 0.25 & & & & 2.16 & ( 2 ) & & & + 1247 + 2647&1.03 & 0.76 & & & & -0.07 & ( 2 ) & & & + 1248 + 3032 & 1.23 & & [email protected] & [email protected] & 0.09 ( 1450 ) & 3.19 & & ( 1c)&(1c)&(1c ) + 1248 + 3142&1.27 & & & 0.26 & & 0.00 & & ( 4 ) & & ( 8 ) + 1248 + 4007 & 1.44 & 0.57 & [email protected] & [email protected] & 0.65 ( 1450 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 1249 + 2929&1.14 & & & 0.22 & & 0.00 & & ( 4 ) & & ( 8 ) + 1250 + 3122 & 1.24 & & & 0.33 & 0.54 ( 1279 ) & 0.00 & & ( 4 ) & ( 1b)&(1b ) + 1252 + 1157 & 2.34 & & [email protected] & [email protected] & 0.54 ( 1450 ) & 3.12 & & ( 1c)&(1c)&(1c ) + 1253 - 0531&2.12 & 1.43 & [email protected] & [email protected] & 0.30 ( 1450 ) & 4.47 & ( 2)&(1c)&(1c)&(1c ) + 1257 + 3439&1.13 & & & 0.51 & 0.94 ( 1450 ) & 1.14 & & ( 4 ) & & ( 9 ) + 1258 + 2835&0.93 & & [email protected] & [email protected] & 0.34 ( 1331 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 1259 + 5918 & 1.37 & 1.02 & [email protected] & [email protected] & 1.63 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 1302 - 1017 & 3.37 & 0.99 & [email protected] & [email protected] & 3.47 ( 1450 ) & 2.34 & ( 2)&(1b)&(1b)&(1b ) + 1305 + 0658 & 2.16 & & [email protected] & [email protected] & 0.23 ( 1450 ) & 3.13 & & ( 1c)&(1c)&(1c ) + 1309 + 3531&2.55 & & [email protected] & [email protected] & 1.12 ( 1450 ) & 1.58 & & ( 1b)&(1b)&(1b ) + 1317 + 2743 & 1.18 & 0.73 & [email protected] & [email protected] & 1.40 ( 1450 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 1317 + 5203&1.90 & & [email protected] & [email protected] & 0.66 ( 1450 ) & 2.70 & & ( 1c)&(1c)&(1c ) + 1318 + 2903 & 1.14 & 0.26 & [email protected] & [email protected] & 0.56 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 1320 + 2925 & 1.17 & & [email protected] & [email protected] & 0.36 ( 1450 ) & 0.00 & & ( 1c)&(1c)&(1c ) + 1322 + 6557 & 1.92 & & [email protected] & [email protected] & 1.01 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1323 + 6530&1.99 & & & 0.11 & & 3.02 & & ( 4 ) & & + 1327 - 2040&7.53 & 0.19 & [email protected] & [email protected] & 0.82 ( 1450 ) & 2.62 & ( 2)&(1c)&(1c)&(1c ) + 1328 + 3045&1.16 & & [email protected] & [email protected] & 0.24 ( 1450 ) & 4.49 & & ( 1c)&(1c)&(1c ) + 1329 + 4117&0.97 & 0.95 & & & & 0.00 & ( 2 ) & & & + 1333 + 1740 & 1.75 & 0.51 & [email protected] & [email protected] & 1.01 ( 1450 ) & 1.39 & ( 2)&(1b)&(1b)&(1b ) + 1351 + 3153&1.29 & & [email protected] & [email protected] & 0.11 ( 1319 ) & 2.88 & & ( 1c)&(1c)&(1c ) + 1351 + 6400&2.10 & & [email protected] & [email protected] & 4.36 ( 2531 ) & 1.10 & & ( 1c)&(1c)&(1c ) + 1352 + 0106 & 2.25 & 0.07 & [email protected] & [email protected] & 1.05 ( 1450 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 1352 + 1819 & 2.03 & & [email protected] & [email protected] & 0.71 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1354 + 1933 & 2.21 & 0.40 & [email protected] & [email protected] & 0.77 ( 1450 ) & 3.53 & ( 2)&(1c)&(1c)&(1c ) + 1356 + 5806&1.40 & & [email protected] & [email protected] & 0.59 ( 1344 ) & 2.34 & & ( 1c)&(1c)&(1c ) + 1401 + 0952&1.96 & & [email protected] & [email protected] & 0.31 ( 1450 ) & 0.72 & & ( 1b)&(1b)&(1b ) + 1404 + 2238&1.99 & & [email protected] & [email protected] & 0.86 ( 2413 ) & 0.29 & & ( 1c)&(1c)&(1c ) + 1407 + 2632 & 1.47 & 0.83 & [email protected] & [email protected] & 1.38 ( 1450 ) & 0.00 & ( 2)&(1c)&(1c)&(1c ) + 1415 + 4509 & 1.13 & & [email protected] & [email protected] & 1.32 ( 1790 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1416 + 0642&6.24 & & [email protected] & [email protected] & 0.40 ( 1308 ) & 3.67 & & ( 1c)&(1c)&(1c ) + 1424 - 1150 & 7.54 & & [email protected] & [email protected] & 0.83 ( 1450 ) & 2.59 & & ( 1c)&(1c)&(1c ) + 1425 + 2645&2.55 & 0.15 & [email protected] & [email protected] & 0.48 ( 1450 ) & 2.43 & ( 2)&(1b)&(1b)&(1b ) + 1427 + 4800 & 1.88 & & [email protected] & [email protected] & 0.86 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1435 - 0134 & 3.66 & 0.82 & & & & 0.00 & ( 5 ) & & & + 1440 + 3539&1.00 & & [email protected] & [email protected] & 4.96 ( 1857)&-0.58 & & ( 1b)&(1b)&(1b ) + 1444 + 4047 & 1.27 & 0.89 & [email protected] & [email protected] & 1.59 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 1512 + 3701 & 1.39 & 0.57 & [email protected] & [email protected] & 0.95 ( 1450 ) & 2.75 & ( 2)&(1b)&(1b)&(1b ) + 1517 + 2356&3.91 & & & 0.51 & & 0.00 & & ( 4 ) & & + 1517 + 2357&3.91 & & & 0.08 & & 0.00 & & ( 4 ) & & + 1521 + 1009 & 2.88 & 1.65 & & & & 0.00 & ( 2 ) & & & + 1538 + 4745&1.64 & 0.34 & [email protected] & [email protected] & 1.34 ( 1450 ) & 1.28 & ( 2)&(1c)&(1c)&(1c ) + 1544 + 4855 & 1.60 & 0.10 & [email protected] & [email protected] & 0.95 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 1555 + 3313&2.35 & & [email protected] & [email protected]&0.07 ( 1450 ) & 3.03 & & ( 1c)&(1c)&(1c ) + 1611 + 3420&1.65 & & & 0.18 & 0.30 ( 1322 ) & 4.88 & & ( 4 ) & ( 1c)&(1c ) + 1618 + 1743 & 4.14 & & [email protected] & [email protected] & 1.13 ( 1450 ) & 2.70 & & ( 1b)&(1b)&(1b ) + 1622 + 2352 & 4.46 & & [email protected] & [email protected] & 0.21 ( 1450 ) & 3.54 & & ( 1c)&(1c)&(1c ) + 1626 + 5529&1.83 & & [email protected] & [email protected] & 1.30 ( 1450 ) & 0.00 & & ( 1b)&(1b)&(1b ) + 1630 + 3744 & 1.07 & 0.84 & & & & 0.00 & ( 2 ) & & & + 1634 + 7037 & 4.55 & 1.96 & & & & 0.00 & ( 2 ) & & & + 1637 + 5726&1.90 & & [email protected] & [email protected] & 0.70 ( 1450 ) & 3.98 & & ( 1c)&(1c)&(1c ) + 1641 + 3954&1.02 & 0.61 & [email protected] & [email protected] & 0.67 ( 1450 ) & 3.92 & ( 2)&(1c)&(1c)&(1c ) + 1704 + 6048&2.32 & 0.90 & [email protected] & [email protected] & 1.68 ( 1450 ) & 2.86 & ( 2)&(1b)&(1b)&(1b ) + 1715 + 5331 & 2.71 & & 0.43 & 0.58 & 0.29 ( 1450 ) & 0.53 & & ( 10 ) & & ( 2 ) + 1718 + 4807&2.27 & & [email protected] & [email protected] & 4.09 ( 1450 ) & 1.52 & & ( 1c)&(1c)&(1c ) + 1803 + 7827&3.92 & & [email protected] & [email protected] & 1.16 ( 1450 ) & 3.35 & & ( 1c)&(1c)&(1c ) + 1821 + 6419&3.98 & 1.86 & [email protected] & [email protected] & 8.37 ( 2204 ) & 1.10 & ( 2)&(1c)&(1c)&(1c ) + 1845 + 7943&4.17 & & [email protected] & [email protected] & 0.58 ( 1450 ) & 3.88 & & ( 1a)&(1a)&(1a ) + 2112 + 0556 & 6.48 & 0.29 & [email protected] & [email protected] & 0.67 ( 1450 ) & 0.00 & ( 2)&(1b)&(1b)&(1b ) + 2128 - 1220 & 4.75 & 0.35 & [email protected] & [email protected] & 2.02 ( 1450 ) & 2.99 & ( 11)&(1b)&(1b)&(1b ) + 2135 - 1446&4.71 & & [email protected] & [email protected] & 0.88 ( 1450 ) & 3.17 & & ( 1b)&(1b)&(1b ) + 2141 + 1730&8.20 & & [email protected] & [email protected] & 1.43 ( 1450 ) & 2.84 & & ( 1b)&(1b)&(1b ) + 2145 + 0643 & 4.90 & & [email protected] & [email protected] & 1.14 ( 1450 ) & 3.58 & & ( 1c)&(1c)&(1c ) + 2201 + 3131&9.02 & 0.60 & [email protected] & [email protected] & 4.93 ( 1450 ) & 3.64 & ( 2)&(1b)&(1b)&(1b ) + 2216 - 0350&5.66 & 0.18 & [email protected] & [email protected] & 0.71 ( 1450 ) & 3.43 & ( 2)&(1c)&(1c)&(1c ) + 2223 - 0512&5.47 & 0.16 & & & & 4.35 & ( 2 ) & & & + 2230 + 1128&5.42 & & [email protected] & [email protected] & 0.64 ( 1450 ) & 4.39 & & ( 1c)&(1c)&(1c ) + 2243 - 1222 & 4.94 & & [email protected] & [email protected] & 1.25 ( 1450 ) & 3.32 & & ( 1c)&(1c)&(1c ) + 2251 + 1120&5.08 & & 1.2 & 1.46 & 0.49 ( 1450 ) & 3.06 & & ( 12 ) & & ( 2 ) + 2251 + 1552 & 6.38 & 0.09 & [email protected] & [email protected] & 1.15 ( 1450 ) & 3.94 & ( 2)&(1c)&(1c)&(1c ) + 2251 - 1750&2.77 & & [email protected] & [email protected] & 4.32 ( 2507 ) & 0.07 & & ( 1c)&(1c)&(1c ) + 2300 - 6823 & 3.69 & & [email protected] & [email protected] & 0.22 ( 1450 ) & 3.18 & & ( 1b)&(1b)&(1b ) + 2340 - 0339 & 3.61 & & [email protected] & [email protected] & 1.36 ( 1450 ) & 2.24 & & ( 1c)&(1c)&(1c ) + 2344 + 0914 & 5.76 & 0.34 & & 0.22 & 0.41 ( 1450 ) & 3.52 & ( 2)&(4 ) & ( 9 ) & ( 9 ) + 2352 - 3414 & 1.08 & & [email protected] & [email protected] & 0.77 ( 1450 ) & 2.70 & & ( 1c)&(1c)&(1c ) + llclcccccc 1 & 259 & 0.82 , 13.6 & 1.46 & & bdo & -22.04@xmath217 & 2.13 & 0.95 & 0.80 + 1 & 259 & 0.82 , 6.73 & 1.46 & 35 & ml & -22.11@xmath218 & 1.21 & 0.29 & 0.80 + 1 & 259 & 0.82 , 9.61 & 1.46 & 25 & ml & -22.12@xmath219 & 1.01 & 0.41 & 0.80 + 1 & 259 & 0.82 , 9.31 & 1.45 & 25 & ml & -22.13@xmath220 & 0.78 & 0.58 & 0.80 + 1 & 259 & 0.82 , 11.8 & 1.70 & 30 & ml & -21.74@xmath221 & 1.34 & 0.23 & 0.80 + 1 & 259 & 0.82 , 38.0 & 2.04 & 25 & ml & -21.47@xmath222 & 1.10 & 0.35 & 0.80 + 2 & 289 & 0.15 , 31.3 & 1.46 & & bdo & -22.06@xmath223 & 2.62 & 0.91 & 0.30 + 2 & 289 & 0.15 , 12.0 & 1.46 & 35 & ml & -22.03@xmath224 & 1.32 & 0.24 & 0.30 + 2 & 289 & 0.15 , 13.9 & 1.46 & 25 & ml & -22.04@xmath221 & 1.34 & 0.23 & 0.30 + 2 & 289 & 0.15 , 13.6 & 1.45 & 25 & ml & -22.06@xmath225 & 1.48 & 0.18 & 0.30 + 2 & 289 & 0.15 , 17.6 & 1.70 & 30 & ml & -21.69@xmath226 & 1.47 & 0.18 & 0.30 + 2 & 289 & 0.15 , 31.1 & 2.04 & 25 & ml & -21.42@xmath227 & 0.88 & 0.50 & 0.30 + 1a & 162 & 1.50 , 10.1 & 1.46 & & bdo & -22.87@xmath228 & 1.51 & 0.98 & 0.64 + 1a & 162 & 1.50 , 4.92 & 1.46 & 35 & ml & -22.18@xmath229 & 0.17 & 0.98 & 0.64 + 1a & 162 & 1.50 , 3.67 & 1.46 & 35 & ml & -21.72@xmath230 & 1.02 & 0.40 & 0.62 + 1a & 162 & 1.50 , 3.71 & 1.46 & 35 & ml & -21.88@xmath231 & 0.98 & 0.43 & 0.62 + 1b & 97 & -0.87 , 53.0 & 1.46 & & bdo & -22.02@xmath232 & 2.44 & 0.87 & 0.98 + 1b & 97 & -0.87 , 26.1 & 1.46 & 35 & ml & -21.98@xmath96 & 2.25 & 0.03 & 0.98 + 1b & 97 & -0.87 , 21.5 & 1.46 & 35 & ml & -21.76@xmath233 & 1.31 & 0.24 & 0.95 + 1b & 97 & -0.87 , 21.5 & 1.46 & 35 & ml & -21.95@xmath234 & 1.27 & 0.26 & 0.95 + 3 & 214 & 0.28 , 9.97 & 1.46 & 35 & ml & -21.57@xmath235 & 0.47 & 0.82 & 0.70 + 4 & 208 & 1.04 , 5.76 & 1.46 & 35 & ml & -22.15@xmath236 & 1.47 & 0.19 & 0.65 + 5 & 373 & 0.60 , 7.93 & 1.46 & 35 & ml & -21.74@xmath237 & 0.97 & 0.44 & 0.96 + 6 & 301 & 0.89 , 6.57 & 1.46 & 35 & ml & -22.17@xmath224 & 0.98 & 0.43 & 0.97 + 7 & 415 & 0.67 , 7.72 & 1.46 & 35 & ml & -21.82@xmath238 & 0.93 & 0.46 & 0.98 + 7a & 213 & 0.79 , 7.28 & 1.46 & 35 & ml & -22.22@xmath239 & 0.29 & 0.94 & 0.64 + 7b & 202 & 0.72 , 7.29 & 1.46 & 35 & ml & -21.60@xmath240 & 1.15 & 0.33 & 0.98 + 8 & 422 & 0.69 , 7.64 & 1.46 & 35 & ml & -21.85@xmath241 & 0.82 & 0.55 & 0.97 + 8a & 220 & 0.84 , 7.10 & 1.46 & 35 & ml & -22.23@xmath242 & 0.46 & 0.83 & 0.56 + 8b & 202 & 0.72 , 7.29 & 1.46 & 35 & ml & -21.60@xmath240 & 1.15 & 0.33 & 0.98 + 9 & 906 & 0.61 , 9.26 & 1.46 & 35 & ml & -21.21@xmath243 & 0.55 & 0.76 & 0.91 + 9a & 474 & 0.63 , 9.23 & 1.46 & 35 & ml & -21.79@xmath244 & 0.76 & 0.59 & 0.87 + 9b & 432 & 1.05 , 6.40 & 1.46 & 35 & ml & -20.82@xmath245 & 0.33 & 0.91 & 0.71 + cclccc -23.0 & all & 0.32 & 1.41,7.81 & -22.74@xmath246 & 5.13 + -23.0 & all & variable & 1.15,8.94 & -22.47@xmath247 & 5.73 + -22.0 & all & 0.32 & 1.17,8.27 & -21.32@xmath248 & 13.6 + -22.0 & @xmath249 & 0.32 & 0.95,8.74 & -20.81@xmath250 & 9.53 + -22.0 & @xmath251 & 0.32 & 1.79,5.21 & -21.64@xmath252 & 6.70 + -22.0 & all & variable & 1.48,6.71 & -21.63@xmath253 & 3.21 + -22.0 & @xmath249 & variable & 0.75,10.0 & -21.34@xmath254 & 11.3 + -22.0 & @xmath251 & variable & 1.52,6.09 & -21.63@xmath255 & 1.30 + -21.0 & all & 0.32 & 1.44,7.25 & -20.81@xmath256 & 1.56 + -21.0 & all & variable & 1.13,8.72 & -20.81@xmath257 & 0.73 + @xmath258 & all & 0.32 & 0.99,9.46 & -21.54@xmath259 & 3.35 + @xmath258 & @xmath249 & 0.32 & 0.51,11.1 & -21.80@xmath260 & 4.63 + @xmath258 & @xmath251 & 0.32 & 1.90,4.74 & -21.54@xmath261 & 1.55 + @xmath258 & all & variable & 1.38,7.32 & -21.56@xmath262 & 0.57 + @xmath258 & @xmath249 & variable & 0.84,10.1 & -21.83@xmath263 & 3.28 + @xmath258 & @xmath251 & variable & 2.48,2.67 & -21.37@xmath264 & 0.82 + llcccccc 1 & 0.69,7.65 & 1.46 & 35 & -12.17@xmath265 & 0.49 & 0.81 & -21.56 + 1a & 0.85,7.11 & 1.46 & 35 & -12.70@xmath239 & 0.38 & 0.88 & -22.09 + 1b & 0.72,7.29 & 1.46 & 35 & -11.88@xmath266 & 0.48 & 0.81 & -21.28 + 2 & 0.61,9.27 & 1.46 & 35 & -11.27@xmath267 & 0.78 & 0.58 & -20.67 + 2a & 0.63,9.24 & 1.46 & 35 & -12.23@xmath268 & 1.17 & 0.31 & -21.62 + 2b & 1.05,6.40 & 1.46 & 35 & -9.089@xmath269 & 1.17 & 0.31 & -18.48 + 1 & 0.69 , 7.21 & 1.46 & 35 & -12.67,1.73 & 1.01 & 0.40 & + 2 & 0.61 , 9.04 & 1.46 & 35 & -10.86,3.04 & 0.47 & 0.82 & + | in paper iii of our series @xmath0a uniform analysis of the ly-@xmath1 forest at @xmath2 " , we presented a set of 270 quasar spectra from the archives of the faint object spectrograph ( fos ) on the hubble space telescope ( hst ) .
a total of 151 of these spectra , yielding 906 lines , are suitable for using the proximity effect signature to measure @xmath3 , the mean intensity of the hydrogen - ionizing background radiation field , at low redshift . using a maximum likelihood technique and the best estimates possible for each qso s lyman limit flux and systemic redshift
, we find @xmath4 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 at @xmath7 .
this is in good agreement with the mean intensity expected from models of the background which incorporate only the known quasar population .
when the sample is divided into two subsamples , consisting of lines with @xmath8 and @xmath9 , the values of @xmath3 found are 6.5@xmath10 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 , and 1.0@xmath11 ergs s@xmath5 @xmath6 hz@xmath5 sr@xmath5 , respectively , indicating that the mean intensity of the background is evolving over the redshift range of this data set . relaxing the assumption that the spectral shapes of the sample spectra and the background are identical , the best fit hi photoionization rates are found to be @xmath12 s@xmath5 for all redshifts , and @xmath13 s@xmath5 and @xmath14 s@xmath5 for @xmath8 and @xmath9 , respectively . the inclusion of blazars , associated absorbers , or damped ly-@xmath1 absorbers , or the consideration of a @xmath15 cdm cosmology rather than one in which @xmath16 has no significant effect on the results .
the result obtained using radio loud objects is not significantly different from that found using radio quiet objects .
allowing for a variable equivalent width threshold gives a consistently larger value of @xmath3 than the constant threshold treatment , though this is found to be sensitive to the inclusion of a small number of weak lines near the qso emission redshifts .
this work confirms that the evolution of the number density of ly-@xmath1 lines is driven by a decrease in the ionizing background from @xmath17 to @xmath18 as well as by the formation of structure in the intergalactic medium . |
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the search for supersymmetry ( susy ) is one of the main goals at present and future colliders @xcite as susy is generally accepted as one of the most promising concepts for physics beyond the standard model ( sm ) @xcite . a special feature of susy theories is the existence of the neutralinos , the spin1/2 majorana superpartners of the neutral gauge bosons and higgs bosons . in the mssm , the neutralinos are expected to be among the light supersymmetric particles that can be produced copiously at future high energy colliders @xcite . once several neutralino candidates are observed at such high energy colliders , it will be crucial to establish the majorana nature and cp properties of the neutralinos . in this light , many extensive studies of the general characteristics of the neutralinos in their production and decays @xcite as well as in the selectron pair production @xcite at @xmath4 and/or @xmath5 linear colliders have been performed . + in the present work , we analyze two body tree level decays of neutralinos into a neutralino plus a @xmath0 boson or a lightest neutral higgs boson @xmath1 in order to probe the majorana nature of the neutralinos and cp violation in the neutralino system . a comprehensive analysis of the two body decays of neutralinos as well as charginos was given previously in ref . we however note that a rather light higgs boson mass was assumed and no @xmath0 boson polarization was considered in the previous work . one powerful diagnostic tool in the present analysis is @xmath0 polarization , which can be reconstructed with great precision through @xmath0boson leptonic decays , @xmath6 , in particular , with @xmath7 . + it is possible that due to the masses of the relevant particles , no two body tree level decays are allowed , in which case the dominant decays would consist of three body tree level @xcite or two body one loop decays @xcite . however , a sufficiently heavy neutralino can decay via tree level two body channels containing a @xmath0 or @xmath1 with its mass less than 135 gev in the context of the mssm @xcite . if some sfermions are sufficiently light , two body tree level decays of neutralinos into a fermion and a sfermion may be also be important . however , neutralinos heavier than the squarks will be extremely difficult to isolate at hadron colliders , because the squarks and gluinos are strongly produced and they decay subsequently into lighter neutralinos and charginos . on the other hand , at @xmath4 colliders , squarks and sleptons , if they are kinematically accessible , are fairly easy to produce and study directly . with these phenomenological aspects in mind , we assume in the present work that all the sfermions are heavier than ( at least ) the second lightest neutralino @xmath8 . then , we investigate the mssm parameter space for the two body tree level decays of the neutralino @xmath8 and show how the majorana nature and cp properties of the neutralinos can be probed through the two body decays @xmath9 , once such two body decays are kinematically allowed . + the paper is organized as follows . section [ sec : mixing ] is devoted to a brief description of the mixing for the neutral gauginos and higgsinos in cp noninvariant theories with non vanishing phases . in sec . [ sec : two - body ] , after explaining the reconstruction of @xmath0boson polarization through the @xmath0 decays into two lepton pairs , we present the formal description of the ( polarized ) decay widths of the two body neutralino decays into a lightest neutralino @xmath10 plus a @xmath0 boson or a lightest higgs boson @xmath1 with special emphasis on the polarization of the @xmath0 boson . in sec . [ sec : analysis ] , we first investigate the region of the mssm parameter space where the two body neutralino decays are allowed and discuss the dependence of the branching ratios and decay widths on the relevant susy parameters . then , we give a simple numerical demonstration of how the majorana nature and cp properties of the neutralinos can be probed through the two body decays @xmath11 . finally , we conclude in sec . [ sec : conclusion ] . in the mssm , the mass matrix of the spin-1/2 partners of the neutral gauge bosons , @xmath12 and @xmath13 , and of the neutral higgs bosons , @xmath14 and @xmath15 , takes the form @xmath16 0 & m_2 & m_z c_\beta c_w & -m_z s_\beta c_w\\[1 mm ] -m_z c_\beta s_w & m_z c_\beta c_w & 0 & -\mu \\[1 mm ] m_z s_\beta s_w & -m_z s_\beta c_w & -\mu & 0 \end{array}\right)\ , , \label{eq : massmatrix}\end{aligned}\ ] ] in the @xmath17 basis . here @xmath18 and @xmath19 are the fundamental susy breaking u(1 ) and su(2 ) gaugino mass parameters , and @xmath20 is the higgsino mass parameter . as a result of electroweak symmetry breaking by the vacuum expectation values of the two neutral higgs fields @xmath21 and @xmath22 ( @xmath23 , @xmath24 where @xmath25 ) , non diagonal terms proportional to the @xmath0boson mass @xmath26 appear and the gauginos and higgsinos mix to form the four neutralino mass eigenstates @xmath27 ( @xmath28@xmath29 ) , ordered according to increasing mass . in general the mass parameters @xmath18 , @xmath19 and @xmath20 in the neutralino mass matrix ( [ eq : massmatrix ] ) can be complex . parameterization of the fields , @xmath19 can be taken real and positive , while the u(1 ) mass parameter @xmath18 is assigned the phase @xmath30 and the higgsino mass parameter @xmath20 the phase @xmath31 . for the sake of our latter discussion , it is worthwhile to note that in the limit of large @xmath32 the gaugino higgsino mixing becomes almost independent of @xmath32 and the neutralino sector itself becomes independent of the phase @xmath31 in this limit . + the neutralino mass eigenvalues @xmath33 ( @xmath28-@xmath29 ) can be chosen positive by a suitable definition of the mixing matrix @xmath34 , rotating the gauge eigenstate basis @xmath17 to the mass eigenstate basis of the majorana fields : @xmath35 . in general the mixing matrix @xmath34 involves 6 non trivial angles and 9 non trivial phases , which can be classified into three majorana phases and six dirac phases @xcite . the neutralino sector is cp conserving if @xmath20 and @xmath18 are real , which is equivalent to vanishing dirac phases ( mod @xmath36 ) and majorana phases ( mod @xmath37 ) . majorana phases of @xmath38 do not signal cp violation but merely indicate different intrinsic cp parities of the neutralino states in cp invariant theories @xcite . before describing the two body decays @xmath39 in detail , we explain how to reconstruct the @xmath0 polarization through the lepton angular distributions of the @xmath0boson leptonic decays , @xmath40 , particularly with @xmath7 . in the rest frame of the decaying @xmath0 boson , which can be reconstructed with great precision by measuring the lepton momenta , the lepton angular distributions are given by @xmath41}\ , \frac{d\gamma[z(\pm)\rightarrow l^+l^-]}{d\cos\theta_l } = \frac{3}{8}\left[1+\cos^2\theta_l\pm 2\,\xi_l \cos\theta_l\right]\ , , \nonumber\\ & & \frac{1}{\gamma[z\rightarrow l^+l^-]}\ , \frac{d\gamma[z(0)\rightarrow l^+l^-]}{d\cos\theta_l } = \frac{3}{4}\,\sin^2\theta_l\ , , \label{eq : polar_angle}\end{aligned}\ ] ] for the @xmath0boson helicities , @xmath42 and @xmath43 , respectively , where @xmath44 with @xmath45 and @xmath46 , and @xmath47 is the polar angle of the @xmath48 momentum with respect to the @xmath0 boson polarization direction . here , the decay width @xmath49 $ ] is the average of three polarized decay widths , @xmath50 = \frac{1}{3 } \left\{\ , \gamma[\ , z(+)\rightarrow l^+l^-]+\gamma[\ , z(0)\rightarrow l^+l^- ] + \gamma[\ , z(-)\rightarrow l^+l^-]\,\right\}.\end{aligned}\ ] ] we emphasize that the three polar angle distributions ( [ eq : polar_angle ] ) can be determined without knowing the full kinematics of the decay @xmath51 . in contrast , the distributions involving the interference of the amplitudes with different @xmath0 helicities are always accompanied with azimuthal angle dependent terms . as the lightest neutralino @xmath10 assumed to be the lightest susy particle ( lsp ) always escapes detection , the kinematics of the two body decay @xmath51 can not be fully reconstructed so that the azimuthal angle dependent distributions are not fully available . + the decay width of the decay @xmath52 producing a @xmath0 boson with its helicity , @xmath42 or @xmath43 , reads @xmath53 = \frac{g^2_z\,\lambda^{1/2}_z}{16\pi m^3_i}\,(|v|^2+|a|^2 ) \left[\,m^2_i+m^2_j - m^2_z- 2m_i m_j\ , { \cal a}_n \pm \frac{\lambda^{1/2}_z}{2 } { \cal a}_t\right]\ , , \nonumber\\ & & \gamma\left[\tilde{\chi}^0_i\rightarrow \tilde{\chi}^0_j\ , z(0)\right ] = \frac{g^2_z\,\lambda^{1/2}_z}{16\pi m^3_i}\,(|v|^2+|a|^2 ) \left[\frac{\lambda_z}{m^2_z}+m^2_i+m^2_j - m^2_z - 2\,m_i m_j\,{\cal a}_n\,\right]\ , , \label{eq : polarized_decay_width}\end{aligned}\ ] ] respectively , where the asymmetries @xmath54 and @xmath55 are defined in terms of the vector and axial vector couplings @xmath56 and @xmath57 of the @xmath0 boson to the neutralino current as @xmath58 with @xmath59 and the kinematical factor @xmath60[(m_i - m_j)^2-m^2_z]$ ] . combining the leptonic @xmath0boson decay distributions ( [ eq : polar_angle ] ) with the polarized decay widths ( [ eq : polarized_decay_width ] ) , we obtain the correlated polar angle distribution : @xmath61\,\bigg\ { \left(\gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j\ , z(+ ) ] + \gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j\ , z(- ) ] \right)\ , ( 1+\cos^2\theta_l)\nonumber\\ & & { } \hskip 2.cm + 2\left(\gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j\ , z(+ ) ] -\gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j\ , z(- ) ] \right)\ , \xi_l\,\cos\theta_l \nonumber\\ & & { } \hskip 2.cm + 2\,\gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j\ , z(0 ) ] \,\sin^2\theta_l\bigg\}\ , . \label{eq : correlated_polar_angle}\end{aligned}\ ] ] consequently , each polarized decay width can be extracted from the correlated polar angle distribution by projecting out the distribution with a proper lepton polar angle distribution . + the explicit forms of the vector and axial vector couplings @xmath56 and @xmath57 in eq . ( [ eq : asymmetry ] ) are given in terms of the @xmath62 neutralino diagonalization matrix @xmath34 in the mssm by @xmath63 note that _ the vector coupling @xmath56 is pure imaginary and the axial vector coupling @xmath57 is pure real . _ this characteristic property of the @xmath0-@xmath64-@xmath65 coupling due to the majorana nature of neutralinos leads to one important relation between the polarized decay widths with the @xmath0boson helicities , @xmath42 : @xmath66 \ , = \ , \gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j z(-)]\ , , \label{eq : majorana}\end{aligned}\ ] ] which is valid even in the cp non invariant theory . this relation can be checked by measuring the forward angle asymmetry of the correlated polar angle distribution ( [ eq : correlated_polar_angle ] ) . however , because of the small analyzing power @xmath67 , it will be necessary to have sufficient large number of decay events to measure the asymmetry with good precision . in addition to the relation ( [ eq : majorana ] ) , the relative intrinsic cp parity of two neutralinos in the cp invariant theory can be determined by measuring the ratio of the longitudinal decay width to the transverse decay width , which satisfies @xmath68 } { \gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j z(+ ) ] + \gamma[\,\tilde{\chi}^0_i\rightarrow\tilde{\chi}^0_j z(- ) ] } = \frac{(m_i\mp m_j)^2}{m^2_z}\ , , \label{eq : ratio r}\end{aligned}\ ] ] for the even / odd relative intrinsic cp parity with @xmath69/@xmath70 , _ i.e. _ @xmath71 , respectively . in the cp non invariant theory , both the vector and axial couplings are in general non vanishing , leading to the value of the asymmetry @xmath54 different from @xmath42 . therefore , any precise measurements of the asymmetry @xmath54 will provide us with an important probe of cp violation in the neutralino system under the assumption that _ the neutralino masses are measured with good precision , independently of the decay modes . _ + next , we give the decay formulas into final states containing a lightest neutral higgs boson @xmath1 . the explicit form of the decay width of the decay @xmath72 is written as @xmath73 = \frac{g^2\,\lambda^{1/2}_h}{16\pi m^3_i } \left[|s|^2\,((m_i+m_j)^2-m^2_h ) + |p|^2\ , ( ( m_i - m_j)^2-m^2_h)\right]\,,\end{aligned}\ ] ] with the kinematical factor @xmath74[(m_i - m_j)^2-m^2_h]$ ] . the scalar and pseudoscalar couplings , @xmath75 and @xmath76 , of the higgs boson @xmath1 to the neutralino current are defined in terms of the mixing matrix @xmath34 as @xmath77\,,\nonumber\\ & & p = \frac{i}{2}{\im { \rm m}}\left[(n_{j2}-t_w n_{j1})(s_\alpha n_{i3}+c_\alpha n_{i4 } ) + ( i\leftrightarrow j)\right]\,,\end{aligned}\ ] ] where @xmath78 , @xmath79 and @xmath80 for the neutral higgs mixing angle @xmath81 . if the charged higgs boson mass in the mssm is very large , @xmath82 this decoupling approximation of the cosine and sine of the mixing angle @xmath81 is very good if the charged higgs mass is larger than twice the @xmath0 boson mass @xcite . for the sake of discussion , we take the decoupling limit in the present work . + in some susy scenarios , the lightest neutralino @xmath10 is the lsp and the second lightest neutralino @xmath8 among the other three neutralino states are expected to be lighter than sfermions and gluino @xcite . then , the two body decays @xmath83 or @xmath1 as well as the two body decays of the heavier neutralinos @xmath84 @xcite will constitute the major decay modes of the neutralinos , respectively , once the two body tree level decay modes are kinematically allowed . in the following numerical analysis we will ignore all other modes except for the two body tree level decays of the neutralinos . for the branching ratio calculations for the two body decays @xmath85 , we assume that all the susy parameters are real , @xmath18 is related to @xmath19 by the gaugino mass unification condition @xmath86 and the higgs boson mass @xmath87 is 115 gev . in addition , we assume that the mssm higgs system is in the decoupling regime so that the characteristics of the lightest higgs boson @xmath1 is similar to the sm higgs boson to a good approximation @xcite . + + -0.5 cm figure [ fig : bratio ] shows the regions of the decays of @xmath8 on the @xmath88 plane . in the region denoted by three body decays " , no two body modes are kinematically allowed . in the @xmath0 region " ( red colored ) , only the two body decay into a @xmath0 is allowed and in the @xmath89 region " , both the two body decays @xmath90 are allowed . we divide the @xmath89 region " into three parts , according to @xmath91\leq 10\%$ ] , @xmath92\leq 20\%$ ] and @xmath91\geq 20\%$ ] . ( here , @xmath91\equiv { \rm br}[\tilde{\chi}^0_2\rightarrow\tilde{\chi}^0_1\,z]$ ] . ) in addition , as a reference , the region excluded by the experimental bound @xcite on the lighter chargino mass @xmath93 gev is displayed by the blue hatched region . + we first note that , if @xmath94 $ \sim$}}~}2m_z$ ] , the mass difference @xmath95 is less than @xmath26 for all @xmath20 and the mass difference is very small for @xmath96 . so , as clearly shown in fig . [ fig : bratio ] the two body decay @xmath97 is allowed only when @xmath98 $ \sim$}}~}m_2 { \raisebox{-0.13cm}{~\shortstack{$<$\\[-0.07 cm ] $ \sim$}}~}2 |\mu|$ ] under the assumption of the gaugino mass unification condition . in addition , we find from the figure that for the two body decays the magnitude of @xmath20 is required to be larger than about 270 gev and that , once the two body higgs mode @xmath99 is open kinematically , this two body decay mode dominates in most of the @xmath89 region . the region where the decay @xmath100 is appreciable is not symmetric between positive and negative @xmath20 in the @xmath89 region . the branching ratio @xmath91 $ ] is significant only in a small area of the positive @xmath20 region , but in a large area of the negative @xmath20 region . + on the other hand , we find numerically that , for the heavier neutralinos @xmath84 , the @xmath101 region for the two body decays @xmath102 expands drastically . a large region with small @xmath103 but large @xmath19 as well as with small @xmath19 but large @xmath103 also allows for the two body decays of the heavier neutralinos , @xmath102 . only in the wedge shaped band region of the width of about 100 gev around the line satisfying the relation @xmath104 no two body decays for the heavier neutralino @xmath105 are allowed , while the heaviest neutralino can still decay into @xmath10 and @xmath89 in the ( almost ) entire parameter space , possibly except for the region excluded by the experimental lighter chargino mass bound . + consequently , for most of the parameter space of the mssm the decays of the two heavier neutralinos are dominated by two body tree level processes of which the final state consists of a @xmath0 boson or @xmath1 boson together with one of the lighter neutralinos , or a @xmath106 boson and one of the charginos . furthermore , the two body decays of the second lightest neutralino @xmath8 can be significant in a large region of the parameter space of the mssm . + -0.3 cm in addition to the branching ratios , it is also crucial to analyze the absolute size of the decay width @xmath107 $ ] . depending on the values of the relevant couplings , the two body decay widths could be smaller than the three body decay widths involving virtual sfermion exchanges , unless the sfermions are too heavy . we exhibit in fig . [ fig : width ] the dependence of the decay width @xmath108 $ ] on the higgsino mass parameter @xmath20 , assuming again that @xmath20 is real and taking @xmath109 , @xmath110 gev and @xmath111 . the decay width decreases rapidly with increasing @xmath103 . this is because the couplings of the @xmath0 boson to the neutralino current are governed by the higgsino components of the neutralinos ( see eq . ( [ eq : va ] ) ) so that the @xmath0-@xmath64-@xmath65 couplings are strongly suppressed for large @xmath103 . therefore , for large @xmath103 , some three body decays could be more dominant than the two body decays . + -0.3 cm in the previous subsection , we restrict ourselves to the cp invariant case with real parameters , as the qualitative results obtained from the cp even quantities are not expected to change so significantly even if the parameters are complex . but , the parameters @xmath18 and @xmath20 are in general complex so that it is important to check whether they indeed have complex phases or not . the existence of the complex phases in the neutralino system , which in general cause cp violation , can be established by the measurements of the ratio @xmath112 with @xmath113 defined in eq . ( [ eq : ratio r ] ) as well as the neutralino masses @xmath114 and @xmath115 . the ratio @xmath116 is @xmath117 ( @xmath118 ) in the cp invariant theory for the positive ( negative ) relative intrinsic cp parity of the neutralinos , @xmath64 and @xmath65 , taking part in the decay @xmath119 , respectively . + to explicitly show the dependence of the ratio @xmath120 on the cp phases @xmath30 and @xmath31 , we chose a specific set of real parameters @xmath121 as a simple numerical example with @xmath122 , while varying the phases @xmath30 and @xmath31 . numerically , we find that the ratio @xmath116 is insensitive to the phase @xmath31 . this is because the phase dependence is always accompanied with @xmath123 for @xmath111 , which is already small , and the higgsino components of the neutralinos are small for large @xmath103 . so , we show in fig . [ fig : etacp ] the dependence of the ratio @xmath124 ( red solid ) as well as the asymmetry @xmath125 ( blue dot dashed ) only on the phase @xmath30 for the real parameter set for one fixed value of @xmath126 . clearly , in the cp invariant case with @xmath127 or @xmath128 , the absolute magnitude of the ratio @xmath116 as well as the asymmetry @xmath125 is 1 , but it is different from 1 in the cp non invariant case . in the given real parameter set , we find that the ratio @xmath116 is quite sensitive to the phase @xmath30 near @xmath129 , while it is not so sensitive to the phase near @xmath130 and @xmath128 . + in the present work the analysis for probing the majorana nature and cp violation in the neutralino system has been carried out at the tree level . however , it will be important to include loop corrections to the two body tree level decays because , if they are small , the tree level cp violation effects might be diluted by loop induced cp violation effects originating from other sectors of the mssm . + for a large portion of the mssm parameter space , the decay of the second lightest neutralino @xmath8 as well as the heavier neutralinos @xmath84 could be dominated by two body processes in which the final state consists of a @xmath0 or a lightest higgs boson @xmath1 together with a lightest neutralino @xmath10 , assumed to be the lightest supersymmetric particle . the main conclusion of the present work is that , unless the two body decay @xmath119 is strongly suppressed , the @xmath0 polarization , which can be reconstructed through great precision via the leptonic @xmath0boson decays @xmath131 , provides us with a powerful probe of the majorana nature of the neutralinos and cp violation in the neutralino system . the work of syc was supported in part by the korea research foundation grant ( krf2002070c00022 ) and in part by kosef through chep at kyungpook national university and the work of ygk was supported by the korean federation of science and technology societies through the brain pool program . tesla technical design report , part : iii physics at an @xmath4 linear collider , _ eds . _ heuer , d. miller , f. richard and p. zerwas , desy 2001 - 011 [ hep - ph/0106315 ] ; 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j.f . gunion and h.e . haber , phys . d * 67 * ( 2003 ) 075019 [ hep ph/0207010 ] . | in the minimal supersymmetric standard model ( mssm ) , the neutralinos , the spin1/2 majorana superpartners of the neutral gauge and higgs bosons , are expected to be among the light supersymmetric particles that can be produced copiously at future high energy colliders .
we analyze two body neutralino decays into a neutralino plus a @xmath0 boson or a lightest neutral higgs boson @xmath1 , allowing the relevant parameters to have complex phases .
we show that the two body tree level decays of neutralinos are kinematically allowed in a large region of the mssm parameter space and they can provide us with a powerful probe of the majorana nature and cp properties of the neutralinos through the @xmath0boson polarization measured from @xmath0boson leptonic decays .
= 100000 kias
p03079 + kupt0304 + hep - ph/0311037 + * analysis of the neutralino system in two body decays of neutralinos * + s. y. choi@xmath2 and y. g. kim@xmath3 0.5 cm 1 .
@xmath2 _ department of physics , chonbuk national university , chonju 561 - 756 , korea _ + 2 .
@xmath3 _ department of physics , korea university , seoul 136701 , korea _ |
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the current astrophysical observations of the type ia supernovae ( snia ) @xcite , the cosmic microwave background ( cmb ) @xcite and the large scale structure ( lss ) @xcite have revealed that the expansion of our universe is accelerated @xcite . this indicates that there exists some unknown energy , called dark energy , to realize the accelerated expansion . the simplest interpretation of dark energy is the cosmological constant . however , this model requires an incredible fine - tuning , since the observed cosmological constant is extremely small compared to the fundamental planck scale @xmath4 . also , this model suffers from the cosmic coincidence problem : why the cosmological constant and matter have comparable energy density today even though their time evolution is so different . among various attempts to solve these problems , we focus on the holographic dark energy ( hde ) models @xcite motivated by the holographic principle of quantum gravity @xcite . requiring that the total vacuum energy of a system with size @xmath5 would not exceed the mass of the black hole of the same size , the dark energy density is proposed as @xmath6 where @xmath7 is a dimensionless parameter , @xmath8 is the reduced planck mass . as for the size @xmath5 , which is regarded as an ir cutoff , various possibilities are discussed in literatures , such as the hubble parameter @xmath9 @xcite , the future event horizon @xmath10 @xcite , the age of our universe @xmath11 @xcite , and the ricci scalar curvature @xmath12 @xcite . in our previous work @xcite , we studied the ricci dark energy ( rde ) model with @xmath12 by introducing an interaction between dark energy and matter . it was shown that a nonvanishing interaction rate @xmath13 is favored by the observations @xcite . in this paper , we extend our previous analysis , and consider the interacting rde ( irde ) model in non - flat universe . this paper is organized as follows . in sec . [ irde ] , we describe the generalized irde model in non - flat universe , and obtain analytic expressions for cosmic time evolution . in sec . [ obs ] , we discuss the observational constraints on this model . we summarize our results in sec . we study the interaction ricci dark energy ( irde ) model in non - flat universe . the friedmann - robertson - walker metric non - flat univrerse is given by @xmath14 where @xmath15 , 0 , @xmath16 for closed , flat , and open geometries . the friedmann equation in non - flat univrerse takes the form @xmath17 where @xmath18 and @xmath19 represent energy density of dark energy , matter , radiation and curvature , respectively , and @xmath20 is the hubble parameter . we generalize the energy density of the ricci dark energy as @xmath21 where @xmath22 , @xmath23 and @xmath24 are dimensionless parameters and @xmath25 . in the case of @xmath26 , this model is reduced to the ordinary rde model @xcite . moreover , we introduce a phenomenological interaction between dark energy and matter . the energy densities @xmath27 and @xmath28 obey the following equations @xcite @xmath29 we adopt the interaction rate given by @xmath30 where @xmath31 is a dimensionless parameter @xcite . to solve @xmath32 , combining with eqs . ( [ rde ] ) and ( [ clm ] ) , the friedmann equation ( [ friedmann ] ) is transformed as @xmath33 the solution to eq . ( [ eoh ] ) is given by @xmath34 where @xmath35 @xmath36 , @xmath37 and @xmath38 . when @xmath39 can be imaginary for sufficiently large @xmath22 , @xmath23 and @xmath31 , @xmath40 has oscillatory behavior @xcite . the constants @xmath41 , @xmath42 and @xmath43 are obtained as @xmath44 @xmath45 @xmath46 in the case of @xmath47 , @xmath48 and @xmath49 , eq . ( [ solution ] ) reduces to the result obtained in our previous work @xcite . substituting eq . ( [ solution ] ) to eq . ( [ rde ] ) , the ricci dark energy density is obtained as @xmath50 where @xmath51 , @xmath52 , likewise , the matter density is obtained as @xmath53 to derive the equation of state parameter @xmath54 of the ricci dark energy , substituting eq . ( [ rdes ] ) into the following expression : @xmath55 in this section , we discuss cosmological constraints on the irde model in the non - flat universe ( @xmath56 ) obtained from snia , cmb , bao and the hubble parameter observations . the luminosity distance in the non - flat universe can be written as @xmath57 the snia observations measure the distance modulus @xmath58 of a supernova and its redshift @xmath59 . the distance modulus is given by @xmath60 we use the union data set of 580 snia @xcite to obtain limits on the relevant parameters @xmath61 and @xmath62 by minimizing @xmath63 @xcite . the cmb shift parameter @xmath64 is given by @xmath65 where @xmath66 is the redshift at recombination , and @xmath67 is the matter fraction at present . we use the value @xmath68 obtained from the wmap9 data @xcite . the cmb constraints are obtained by minimizing @xmath69 @xcite . the shift parameter gives a complementary bound to the snia data ( @xmath70 ) , since this parameter involves the large redshift behavior ( @xmath71 ) . signatures of the baryon acoustic oscillation ( bao ) are provided by the observations of large - scale galaxy clustering . the bao parameter @xmath72 is defined by @xmath73^{2/3 } , \end{aligned}\ ] ] where @xmath74 . we use the measurement of the bao peak in the distribution of luminous red galaxies ( lrgs ) observed in sdss @xcite : @xmath75 the bao constraints are obtained by minimizing @xmath76 @xcite . the hubble parameter constraints are given by minimizing @xmath77 where @xmath78 is the @xmath79 uncertainty of the observational @xmath80 data @xcite . in fig.[fig : w_a_gamma ] , we plot the equation of state parameter @xmath54 for @xmath81 ( blue ) , 0 ( red ) and 0.1 ( green ) as a function of the scale factor @xmath82 . the dotted and solid lines are the results for the case without interaction ( @xmath31=0 ) and with interaction ( @xmath83 ) , respectively . the dark energy parameter is fixed as @xmath84 . though the value @xmath85 is too large ( see fig . [ fig : contours ] ) , we use these values to visualize the effect of the curvature in the figure . the condition for accelerated expansion at present @xmath86 @xmath87 @xmath88 is satisfied for both cases . in fig.[fig : omega_a_gamma ] , we plot the evolution of the energy density fractions @xmath89 for radiation ( green ) , matter ( red ) and dark energy ( blue ) for @xmath84 . the dotted and solid lines are the results for the case without interaction ( @xmath31=0 ) and with interaction ( @xmath83 ) , respectively . the panels ( a ) , ( b ) and ( c ) corresponds to @xmath49 , @xmath90 and 0.1 , respectively . one can see that the effect of the curvature can be important only for @xmath82 @xmath91 1 . the fractions @xmath92 and @xmath93 for @xmath81 in the panel ( b ) ( @xmath94 in the panel ( c ) ) are slightly increased ( decreased ) around 0.1 @xmath1 @xmath82 @xmath1 1 compared to the flat case ( a ) . around radiation - matter equality , the radiation component @xmath95 is increased by several percents compared to the corresponding result without interaction ( dotted line ) , while the @xmath93 is decreased due to the interaction . ) -plane in the case without interactions ( @xmath96 ) . the @xmath79 , @xmath97 and @xmath98 contours are drawn with solid , dashed and dotted lines , respectively . the joined constraints using @xmath99 are shown as shaded contours . [ fig : contours ] , title="fig:",scaledwidth=95.0% ] + ( a ) ) -plane in the case without interactions ( @xmath96 ) . the @xmath79 , @xmath97 and @xmath98 contours are drawn with solid , dashed and dotted lines , respectively . the joined constraints using @xmath99 are shown as shaded contours . [ fig : contours ] , title="fig:",scaledwidth=95.0% ] + ( b ) versus @xmath82 for rde(@xmath100 ) the lines for k=-0.1 ( blue ) , 0 ( red ) and 0.1 ( green ) in the case without interactions ( dotted line ) and with interaction ( solid line ) . , width=364 ] versus @xmath82 for rde(@xmath101 ) . the lines for radiation ( green ) , matter ( red ) , dark energy ( blue ) and curvature ( black ) in the case without interactions ( dotted line ) and with interaction ( solid line ) . these figures describe in the case of @xmath49 ( fig . a ) , -0.1 ( fig . b ) and 0.1 ( fig . [ fig : omega_a_gamma ] , width=264 ] ( a ) versus @xmath82 for rde(@xmath101 ) . the lines for radiation ( green ) , matter ( red ) , dark energy ( blue ) and curvature ( black ) in the case without interactions ( dotted line ) and with interaction ( solid line ) . these figures describe in the case of @xmath49 ( fig . a ) , -0.1 ( fig . b ) and 0.1 ( fig . [ fig : omega_a_gamma ] , width=268 ] ( b ) versus @xmath82 for rde(@xmath101 ) . the lines for radiation ( green ) , matter ( red ) , dark energy ( blue ) and curvature ( black ) in the case without interactions ( dotted line ) and with interaction ( solid line ) . these figures describe in the case of @xmath49 ( fig . a ) , -0.1 ( fig . b ) and 0.1 ( fig . c ) . [ fig : omega_a_gamma ] , width=264 ] ( c ) we have considered the irde model in the non - flat universe . we have derived the analytic solutions for the hubble parameter ( [ hubbleparameter ] ) , the dark energy density ( [ rdes ] ) and matter energy density ( [ rms ] ) . we have also studied astrophysical constraints on this model using the recent observations including snia , bao , cmb anisotropy , and the hubble parameter . we have shown that the allowed parameter range for the fractional energy density of the curvature is @xmath0 @xmath1 @xmath2 @xmath1 @xmath3 for @xmath31 @xmath102 0.15 . the best fit values with @xmath79 error are @xmath103 and @xmath104 with @xmath105 . we have shown that the irde model with a small curvature is allowed by observational constraints . without the interaction , the flat universe is observationally disfavored in this model . we would like to thank t. nihei for his valuable discussion , helpful advice and reading the manuscript . 99 a. g. riess et al . j. 116 ( 1998 ) 1009 ; s. perlmutter et al . , astrophys . j. 517 ( 1999 ) 565 . c. l. bennet et al . , astrophys . j. suppl . 148 ( 2003 ) 1 ; d. n. spergel et al . , astrophys . j. suppl . 148 ( 2003 ) 175 ; d. n. spergel et al . , astrophys . j. suppl . 170 ( 2007 ) 377 ; l. page et al . , astrophys . j. suppl . 170 ( 2007 ) 335 ; g. hinshaw et al . , astrophys . j. suppl . 170 ( 2007 ) 263 ; e. komatsu et al . , astrophys . j. suppl . 180 ( 2009 ) 330 [ arxiv:0803.0547 ] ; e. komatsu et al . 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we study astrophysical constraints on this model using the recent observations including the type ia supernovae ( snia ) , the baryon acoustic oscillation ( bao ) , the cosmic microwave background ( cmb ) anisotropy , and the hubble parameter .
it is shown that the allowed parameter range for the fractional energy density of the curvature is @xmath0 @xmath1 @xmath2 @xmath1 @xmath3 in the presence of the interactions between dark energy and matter . without the interaction , the flat universe is observationally disfavored in this model . |
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bose - einstein condensates ( becs ) located in different minima of an external potential created by magnetic or light forces and coupled by tunneling through a potential barrier have been the host to many exciting developments and discoveries in recent years @xcite . phenomena explored include analogs of the josephson effect in double or multiple quantum - well structures @xcite , gap solitons of repulsive becs @xcite , and quantum phase transitions @xcite . often these systems are modeled by considering just one mode per potential minimum and their linear coupling provided by tunnling through a separating barrier . these simplified models , which are usually labelled as two - mode or multiple - mode models , variants of the bose - hubbard model , or the discrete nonlinear schrdinger equation , are tailored to describe certain properties or aspects of the dynamics of the many - body system under investigation . although there is an abundance of literature on such models @xcite , there still appear to exist misconceptions about the nature of the effective model parameters , especially the sign of the tunnel coupling , as only few works attempt to calculate such parameters based on a more complete theoretical treatment @xcite . the sign of the tunnel coupling bears special significance in systems where the tunneling appears over an extended ( at least one - dimensional ) region of space . such systems have recently been analysed by bouchoule @xcite and kaurov and kuklov @xcite , who studied two parallel tunnel - coupled cigar - shaped becs . in another recent work , lesanovsky and von klitzing studied the stability of tunnel - coupled annular becs @xcite . the latter paper points to an interesting dynamical instability leading to the spontaneous formation of angular - momentum fluctuations . we will show further below that the sign on the tunnel coupling bears consequences on the nature and stability of the stationary states found in the mean - field treatment of tunnel - coupled becs . specifically , we find that the system studied by lesanovsky and von klitzing has a stable ground state for any value of the tunnel coupling and repulsive interactions . the instability of an _ excited _ state with @xmath0 phase difference between the condensates can be interpreted in terms of the familiar snake instability @xcite . the ground state of a rotating co - planar double - ring system is discussed in ref . @xcite . we examine the stationary states of double - ring becs in sec . [ sec : neg ] . a careful analysis of the sign of the tunnel coupling used in effective models for becs in double - well traps follows in sec . [ sec : sign ] . conclusions are presented in sec . [ sec : concl ] . the calculation performed in ref . @xcite starts from a number of generally reasonable assumptions . under the condition that radial excitations of the vertically stacked annular becs are suppressed by the trapping potentials and the only mechanism for coupling the two systems is via tunneling through a potential barrier , the gross - pitaevskii equation for the two - mode spinor wave function @xmath1 specialises to @xmath2 here @xmath3 is the condensate wave function for atoms in the upper ( lower ) ring . our equation ( [ chieqs ] ) agrees with eq . ( 2 ) of ref . @xcite , except that we explicitly indicate the negative sign of the tunnel coupling . we give detailed reasons for the relevance of the sign of the tunnel coupling below in sec . [ sec : sign ] , where we also show that the tunnel coupling is indeed negative . at this point , we only note that the tunnel coupling @xmath4 was assumed to be positive in ref . @xcite ( see their fig . 1 ) , in contradiction to our findings . the most general form of the polar - angle - dependent wave function can be written as a fourier series , @xmath5 . inserting this _ ansatz _ into eq . ( [ chieqs ] ) and equating coefficients of the orthogonal fourier components , we find @xmath6 this result differs from the corresponding eq . ( 3 ) in ref . @xcite in the tunnel coupling and the non - linear term . as a first approximation , it is reasonable to assume that only the @xmath7 mode is occupied in each of the two annuli . straightforward calculation yields the new ground and excited state of the coupled - annuli system , which are the symmetric and antisymmetric superpositions of single - well states having chemical potential @xmath8 , respectively . @xmath9 is defined in terms of the equal number of atoms @xmath10 in each well as in ref . in order to study the stability of these states , finite but small amplitudes in the @xmath11 modes are assumed : @xmath12 \quad .\ ] ] here the subscript @xmath13 distinguishes perturbations to the ground and excited states , respectively . inserting the perturbation ( [ peransatz ] ) into eq . ( [ alphaeqs ] ) and linearising in the small amplitudes @xmath14 yields @xmath15 the upper ( lower ) sign refers to the symmetric ground ( antisymmetric excited ) state . crucial differences between our eq . ( [ bdgeqsfin ] ) and eq . ( 5 ) in ref . @xcite result in markedly different excitation spectra . we find that both the symmetric ( ground ) state and antisymmetric ( excited ) state share one branch , @xmath16 whose frequency is independent of the tunnel coupling . this was also found in ref . in contrast to these authors , however , we find that the second branch differs for the two states : @xmath17 clearly , @xmath18 is always real for repulsive becs ( @xmath19 ) , implying stability of the symmetric ( ground ) state of the coupled annular condensates . in contrast , the antisymmetric ( excited ) state will become unstable for @xmath20 , signified by @xmath21 becoming imaginary in this range . our own numerical simulations of the time evolution of the antisymmetric state seeded with a small amount of noise show the development of angular - momentum josephson junctions similar to those shown in figs . 2 and 3 of ref . @xcite . in attractive condensates where @xmath22 , imaginary solutions of @xmath23 for @xmath24 indicate the well - known modulational instability towards the formation of localized peaks ( bright solitons ) in the individual rings . for the symmetric state , @xmath18 does not add new instabilities ( with imaginary solutions for @xmath25 ) . the antisymmetric state , however , is further destabilised by the tunnel coupling due to imaginary frequencies of @xmath21 at @xmath26 . in our analysis so far we have assumed that the sign of the coupling constant @xmath4 is negative . this lead to the symmetric state with @xmath27 and @xmath28 with @xmath29 being the ground state . let us now briefly consider the consequences of the ( hypothetical ) case of a positive coupling constant @xmath30 . the analysis of sec [ sec : neg ] can be carried out the same way as before , with the difference that @xmath31 should be replaced by @xmath32 in all formulae . it is easily seen that , in this case , the antisymmetric state with @xmath33 will be the ground state . since the sign change also affects eq . ( [ eqn : unstable ] ) , we find the antisymmetric state being stable ( for @xmath19 ) and the symmetric one becoming unstable . however , since the roles of these states have changed , we still find that the ground state is stable for repulsive becs . energy difference @xmath34 between the lowest antisymmetric and symmetric eigenstates of a quadratic - plus - quartic double - well potential , plotted as a function of the dimensionless effective interaction strength @xmath35 . the fact that @xmath36 indicates that the symmetric ( node - less ) state remains the ground state even in the limit where the atoms interact strongly . double - well parameters [ see eq . ( [ eq : dwell ] ) ] are @xmath37 and @xmath38 ( solid curve ) , 0.02 ( dashed curve ) , 0.05 ( dot - dashed curve).,width=288 ] in order to determine the correct sign and value of the coupling constant @xmath4 appearing in eq . ( [ chieqs ] ) , we briefly revisit the derivation of this model . for the purpose of finding @xmath4 , the azimuthal degree of freedom in the double - ring model of ref . @xcite is irrelevant and it suffices to consider the problem of a bec in a one - dimensional ( 1d ) double - well potential , as in refs . generalisation to multiple wells and different geometries ( coupled cigars or pancakes ) are straightforward . different derivations of effective two - mode models have been presented in the literature @xcite . the goal of a two - mode model is generally to correctly describe the ground and low - lying excited states of the system . the quantity that is obtainable from the 1d model and carries unambiguous information about the sign of the tunnel coupling is the energy difference @xmath39 between the antisymmetric state with one node and the node - less symmetric state . in the simplest case , the tunnel coupling @xmath4 is determined from the single - particle linear schrdinger equation . this approach is commonly used when deriving the fully quantum - mechanical bose - hubbard model @xcite and was the basis of ref . @xcite . in this case both the sign and the value of @xmath4 are completely independent of particle number or interaction strength . the node theorem of quantum mechanics @xcite guarantees that the node - less symmetric state in a one - dimensional double - well potential must be the ground state , thus @xmath36 and consequently , the correct sign of @xmath4 is negative . in a more general class of models based on mean - field theory , the parameters of the two - mode model are chosen in order to reproduce @xmath40 as found from a one - dimensional gp equation . the ordering of eigenvalues of the gp equation by the number of nodes in the wave function is now no longer guaranteed by the node theorem of linear quantum mechanics and we are not aware of a non - linear generalization of this theorem . however , we find by numerical calculation that the ordering is preserved under repulsive interactions . the main result of this section is the dependence of @xmath40 on the non - linear interaction strength @xmath35 , shown in fig . [ fig : sas ] . as can be seen from fig . [ fig : sas ] , the presence of a repulsive non - linear interaction does not change the sign of @xmath40 and therefore @xmath4 remains negative . we now present details of our calculation . starting from the three - dimensional gp equation for a bec in a double - well or double - ring trap and employing a separation ansatz , an effective 1d equation describing the dynamics in the direction perpendicular to the potential barrier can be derived : @xmath41 \phi(\xi ) \ , .\ ] ] here the energy scale @xmath42 and length scale @xmath43 defined by the trap are used as units for all energies and the spatial coordinate , respectively , and the condensate wave function @xmath44 is normalized to unity . we introduced the dimensionless interaction strength @xmath45 , where @xmath46 denotes the number of atoms in the trap and @xmath47 is the effective 1d interaction strength @xcite . to be specific , we use the double - well potential @xmath48 where @xmath49 parameterizes the barrier height between the two wells centered at @xmath50 . it is straightforward to solve eq . ( [ eq:1dgpe ] ) with the potential ( [ eq : dwell ] ) and find the lowest symmetric and antisymmetric eigenstates as well as their respective energies @xmath51 and @xmath52 . figure [ fig : wavefunct ] shows typical results obtained for low and high interactions strengths , respectively . as is apparent from the figure , the higher repulsive interaction strength is associated with more strongly delocalized double - well wave functions , indicating an effectively stronger tunnel coupling . this can be explained simply by noting that the nonlinear interaction energy for the two condensate fractions in each well shifts up their respective energies , thus effectively lowers the barrier and brings the condensates closer together . as a result , the effective tunnel coupling increases . most importantly , the energy difference between the lowest symmetric and antisymmetric eigenstates remains positive for any strength of repulsive interactions , which implies that the sign of the tunnel coupling @xmath4 entering eq . ( [ chieqs ] ) is negative . reference @xcite predicts a dynamical instability of a repulsively interacting bec in a double - ring trap against angular momentum fluctuations . we have carefully revisited the analysis of ref . @xcite and have recalculated the elementary excitation spectrum . this leads us to a different conclusion that makes physical sense . the ground state of a non - rotating condensate in the double - ring configuration is stable against spontaneous angular momentum oscillations . however , the antisymmetric state with its circular node between the two annular quantum wells can be viewed as the analog of a stationary 2d dark soliton , which is known to have a dynamical instability towards the formation of local vorticity ( " snake instability ) @xcite . | we revisit recent claims about the instability of non - rotating tunnel coupled annular bose - einstein condensates leading to the emergence of angular - momentum josephson oscillation [ phys .
rev .
lett . * 98 * , 050401 ( 2007 ) ] .
it was predicted that all stationary states with uniform density become unstable in certain parameter regimes . by careful analysis
, we arrive at a different conclusion .
we show that there is a stable non - rotating and uniform ground state for any value of the tunnel coupling and repulsive interactions .
the instability of an excited state with @xmath0 phase difference between the condensates can be interpreted in terms of the familiar snake instability .
we further discuss the sign of the tunnel coupling through a separating barrier , which carries significance for the nature of the stationary states .
it is found to always be negative for physical reasons . |
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dwarf novae a subclass of cataclysmic variable stars are quite well studied interacting binary systems composed of late - type red dwarf secondary and white dwarf primary stars ( warner 1995 , hellier 2001 ) . matter transferred from the red dwarf forms an accretion disc around the white dwarf . although in the last decade significant progress has been made in explaining the behaviour of dwarf novae light curves , some physical processes ongoing in these systems are still not fully understood ( see for example smak 2000 , schreiber and lasota 2007 ) . in particular , the thermal - tidal instability model of osaki ( 1996 , 2005 ) describing the phenomenon of superoutbursts and superhumps may be tested by examination of su uma - type dwarf novae light curves . additionally , objects near and inside the so called period gap are very important from an evolutionary point of view . those systems give us an unprecedented opportunity to study the evolution of dwarf novae . v419 lyr is a poorly studied cataclysmic variable discovered by kurochkin ( 1990 ) and originally classified as a z cam - type dwarf nova . later , nogami et al . ( 1998 ) caught this object in outburst and found superhumps in its light curve . detection of superhumps together with characteristic properties of the outburst allowed them to classify v419 lyr as a su uma - type dwarf nova , but short coverage of the eruption did not allow accurate determination of the superhump period . nevertheless , there was a strong suggestion that v419 lyr has one of the longest orbital periods known among su uma variables . this object has been monitored at various photometric bands by the variable star network ( vsnet ) ( see for example kato et al . the observations from that program enabled a tentative determination of the supercycle period to be about @xmath4 days ( katysheva and pavlenko 2003 ) . moreover , morales - rueda and marsh ( 2002 ) obtained a spectrum of v419 lyr during outburst showing a relatively broad absorption feature around 430 - 440 nm . in this work we present an analysis of photometric data collected during the 2006 july superoutburst of v419 lyr . the data are much richer than previous studies and provide us with an opportunity to determine parameters describing this system more precisely . the curious variable experiment ( curve ) team ( see for example olech et al . 2004 , 2006 ) , alerted by the vsnet mailing list , found v419 lyr in a very bright state on 2006 july 17/18 . subsequently the object was monitored on 13 consecutive nights ( with a gap on july 24/25 ) until its return to quiescence on august 2/3 . the observations were performed using a 0.6-m cassegrain telescope equipped with a tektronix tk512cb back - illuminated ccd camera . the image scale was @xmath5pixel providing a @xmath6 field of view ( udalski and pych , 1992 ) observations were made unfiltered for two reasons . first , due to lack of an autoguiding system , we wished to keep exposures short in order to minimize guiding errors . second , because our main goal was analysis of the temporal behaviour of the light curve , the use of filters might cause the object to be too faint to observe in quiescence . exposure times were 90 seconds during the bright state and 100 - 150 seconds at minimum light . all curve team data reductions were performed using a standard procedure based on iraf package and the profile photometry has been derived using the daophotii package ( stetson 1987 ) . during preliminary analysis of the data we found the aavso archive containing several ccd observations of the same superoutburst of v419 lyr made by observers from england ( d.b . ) and the united states ( r.k . ) . therefore we decided to combine our data to obtain better results . d.b . used a 0.35-m meade schmidt - cassegrain telescope with a starlight xpress sxv - h9 ccd camera . data were taken unfiltered . exposures were 40 , 50 or 60 sec depending on conditions . the aip4win package ( berry and burnell 2000 ) was used to dark - subtract and flat - field all images before measuring them using aperture photometry . the magnitude of v419 lyr was determined by differential photometry with respect to an ensemble of two nearby comparison stars . r.k . used 0.25-m meade lx-200 schmidt - cassegrain telescope equipped with an apogee ap-47 ccd camera and clear filter characterized by ir block at 700 nm . mpo canopus software ( warner 2007 ) was used for differential photometry of the variable with an average of four comparison stars . table 1 presents a journal of our ccd observations of v419 lyr . in total , we observed the star for more than 48 hours on 14 nights and collected 3179 exposures . figure 1 presents the photometric behaviour of v419 lyr in 2006 july and august . dots and open circles correspond to our ccd observations and visual estimates of aavso observers , respectively . the shape of the light curve corresponds to the standard picture of a superoutburst . first , the star rose rapidly from its quiescent level to a peak magnitude of around 14.5 mag . due to the lack of observations , the exact time of the rise is unknown . v419 lyr was seen by aavso observers in a bright state on july 16 . two nights later , during our first run , the star was still at the same brightness and showing fully developed superhumps with a peak - to - peak amplitude of 0.3 mag - a feature characteristic of the beginning of a superoutburst . thus we conclude that the july 2006 superoutburst started around july 15 . after reaching peak magnitude , v419 lyr entered the plateau stage lasting about 11 days with an average decline rate of about 0.1 mag d@xmath7 . the plateau stage ended rapidly on july 27 and during the next two days we observed the final decline stage with a change of brightness of about 1 mag d@xmath7 . on july 30 the star was again in quiescence showing a mean brightness of 18.1 mag . thus the entire superoutburst lasted about 15 days . the linear fit to the plateau stage in figure 1 is helpful in showing there was no rebrightening as has been observed in some su uma systems ( kato et al . figure 2 shows the light curves of v419 lyr during thirteen individual nights . filled circles correspond to the ostrowik observatory data , while open circles and squares represent d.b . and r.k . measurements respectively . the magnitudes have been transformed to a common @xmath8 system and detrended for purposes of fourier power spectrum analysis . the superhumps are clearly visible and have an initial amplitude of 0.3 mag . later the superhump amplitude gradually decreases and on august 2/3 is practically indistinguishable from noise . as we noted earlier , all light curves of v419 lyr in superoutburst were detrended by removing a fit based on a first or second order polynomial . then we analyzed them using anova statistics ( schwarzenberg - czerny 1996 ) . the resulting periodogram is shown in figure 3 . it shows a very clear and dominant peak at frequency @xmath9 c / d , which we interpret as due to superhumps and corresponds to the period @xmath10 days . the spectrum has almost no 1-day aliases due to good coverage and the use of data from three sites - two from europe and one from the united states . a small peak is also observed around 22 c / d , the first harmonic of the main frequency . we then prewhitened the light curve of v419 lyr during the superoutburst with the main period and its first harmonic . the power spectrum of the resulting light curve shows no other periodicities . to check the stability of the superhump period and to better determine its value , we constructed an @xmath11 diagram . because the maxima in our case were almost always clearly visible and easier to measure than minima , we decided to use the timings of the former . finally , we were able to determine 27 times of maxima which are listed in table 2 together with associated errors , cycle numbers @xmath12 , and @xmath11 values . .times of maxima observed in the light curve of v419 lyr during its 2006 superoutburst [ cols=">,^,^,>",options="header " , ] a least - squares linear fit to the data taken during the plateau phase of superoutburst gives the following ephemeris for the maxima : @xmath13 the above equation indicates that the mean superhump period was @xmath14 days , which agrees within errors with the determination based on anova statistics . combining these two measurements gives us our final estimate of the mean superhump period of v419 lyr during its 2006 july superoutburst which is @xmath15 days ( @xmath2 min ) . the @xmath11 values computed according to the ephemeris ( 1 ) are listed in table 2 and also shown in figure 4 . it is clear that v419 lyr , during its 2006 superoutburst , showed clear changes of superhump period . in the cycle range @xmath16 the period was quickly decreasing . a second - order polynomial fit to @xmath17 dependence in this range corresponds to the solid line in the bottom panel of figure 4 and is expressed by the following ephemeris : @xmath18 this equation indicates that the period derivative has the large value @xmath3 . this is the second largest negative value detected in su uma stars . figure 5 , taken from kato et al . ( 2003b ) and olech et al . ( 2003 ) , shows the position of v419 lyr on the @xmath19 diagram . only kk tel had a faster period decrease during its 2002 june superoutburst ( kato et al . it is interesting that both v419 lyr and kk tel are long superhump period dwarf novae with orbital periods very close or even within the period gap . it is worth commenting on the behaviour of the superhump period after cycle number @xmath20 . at that moment , the star was still in the plateau phase of the superoutburst , three days before entering the final decline , and superhumps were still clearly visible in the light curve . the @xmath11 values for cycle numbers from 70 to 111 can be roughly fitted with a straight line , which indicates that the period decrease had stopped and its value stabilized at @xmath21 days . the noisy maxima with cycle numbers above 120 are shifted by @xmath22 cycle with respect to the superhump maxima from earlier nights . this indicates that they may be connected with late superhumps or even with the orbital wave . it is interesting that our @xmath11 diagram could be also interpreted in diffrent way than `` common superhump followed by transition to late superhump '' scenario . fitting a 5th order polynomial to the moments of maxima appears to work with the data about as well as a quadratic followed by a linear trend as described previously . we are not arguing in favour of this interpretation , simply saying that there might be other interpretations of the data . the upper panel of figure 4 shows the evolution of the amplitude of superhumps through the entire superoutburst . in a typical su uma star , fully developed superhumps have an amplitude of about 0.3 mag and a characteristic tooth shape . interestingly , these properties seem to be completely independent of the inclination of the orbit of the binary . as the outburst progresses , the amplitude gradually decreases and the profile of the humps changes . a few days after maximum , so called secondary humps or interpulses become visible . in the beginning they are small but with time they may become as high as the main maxima - most probably evolving towards late superhumps . v419 lyr is an interesting case because it seems not to follow this scenario . on the nights of july 19/20 and 20/21 it clearly showed double maxima . however , these quickly disappeared . during subsequent nights there was hardly a trace of secondary humps . very weak humps could be seen only on july 23/24 and 25/26 and quite strong ones on july 27/28 . we were curious about the reason for such behaviour . the only property which strongly differs in v419 lyr from typical su uma stars is its long superhump / orbital period placing it within the period gap . we therefore reviewed the literature to investigate the occurrence of secondary humps among long period systems ( i.e. with superhump period @xmath23 days ) . the summary of our review is given in table 3 . l l c c c c c star & @xmath24 [ days ] & clear & poorly & invisible & ref + hs vir & 0.08077 & - & + & - & 1,2 + v359 cen & 0.08092 & - & + & - & 3,4 + v660 her & 0.081 & - & - & + & 5 + v503 cyg & 0.08101 & + & - & - & 6 + br lup & 0.08220 & + & - & - & 7,8 + v877 ara & 0.08411 & + & - & - & 9 + ab nor & 0.08438 & - & + & - & 10 + v369 peg & 0.08484 & - & - & + & 11 + hv aur & 0.08559 & + & - & - & 12 + ef peg & 0.08705 & - & + & - & 13,14 + ty psa & 0.08765 & - & + & - & 15,16 + bf ara & 0.08797 & - & - & + & 17 + kk tel & 0.08803 & - & - & + & 18 + dv uma & 0.08869 & - & + & + & 19,20 + v419 lyr & 0.0901 & - & + & - & this study,21 + uv gem & 0.0902 & - & + & - & 22 + v344 lyr & 0.09145 & - & - & + & 23 + yz cnc & 0.09204 & - & + & - & 24,25 + gx cas & 0.09297 & - & - & + & 26 + v725 aql & 0.09909 & - & + & - & 27 + mn dra & 0.1055 & - & - & + & 28,29 + + + + + + + + + among 21 reviewed stars only four show clear secondary humps . the rest of them show no interpulses at all or only a weak trace of them . the orbital period of v419 lyr is unknown . however it is possible to estimate its value using the relation in stolz and schoembs ( 1984 ) connecting the period excess @xmath25 defined as @xmath26 with the orbital period of the binary . this empirical relation is as follows : using the definition of @xmath25 and knowing @xmath24 for v419 lyr , we were able to estimate the orbital period as @xmath28 days . this is slightly longer than two hours which indicates that v419 lyr is a dwarf nova in the period gap . many characteristics of v419 lyr are typical of su uma stars . it goes into superoutburst every year or so , the eruption lasts about two weeks and has an amplitude of @xmath0 mag . superhumps appear shortly after the beginning of the superoutburst and have a maximum amplitude of 0.3 mag , which decreases to 0.1 mag at the end of the outburst . in addition to its long orbital period , v419 lyr is unusual in two other properties . its superhump period derivative has one of the largest negative values known and it shows only a weak trace of secondary humps in the final stages of the superoutburst . * acknowledgments . * we acknowledge generous allocation of the warsaw observatory 0.6-m telescope time . this work used the online service of the vsnet and aavso . we would like to thank prof . jzef smak for fruitful discussions . antipin , s. v. , & pavlenko , e. p. 2002 , _ astron . _ , 391 , 565 barwig , h. , kudritzki , r. p. , vogt , n. , & hunger , k. 1982 , _ astron . _ , 114 , l11 berry r. , burnell j. 2000 , the handbook of astronomical image processing , willmann - bell feline , w. j. , dhillon , v. s. , marsh , t. r. , & brinkworth , c. s. 2004 , _ mnras _ , 355 , 1 harvey , d. , skillman , d. r. , patterson , j. , & ringwald , f. a. 1995 , _ pasp _ , 107 , 551 hellier , c. 2001 , cataclysmic variable stars , springer howell , s. b. , schmidt , r. , deyoung , j. a. , fried , r. , schmeer , p. , & gritz , l. 1993 , _ pasp _ , 105 , 579 kato , t. 1993 , _ pasj _ , 45 , l67 kato , t. , nogami , d. , masuda , s. , & baba , h. 1998 , _ pasp _ , 110 , 1400 kato , t. , sekine , y. , & hirata , r. 2001 , _ pasj _ , 53 , 1191 kato , t. , & uemura , m. 2001 , informational bulletin on variable stars , 5158 , 1 kato , t. 2001 , informational bulletin on variable stars , 5104 , 1 kato , t. , et al . 2002 , _ astron . _ , 395 , 541 kato , t. 2002 , _ pasj _ , 54 , 87 kato , t. , nogami , d. , moilanen , m. , yamaoka , h. 2003a , _ pasj _ , 55 , 989 kato , t. , et al . 2003b , _ mnras _ , 339 , 861 kato , t. , bolt , g. , nelson , p. , monard , b. , stubbings , r. , pearce , a. , yamaoka , h. , richards , t. 2003c , _ mnras _ , 341 , 901 kato , t. , uemura , m. , ishioka , r. , nogami , d. , kunjaya , c , baba , h. , yamaoka , h. 2004a , _ pasj _ , 56 , 1 kato , t. , et al . 2004b , _ mnras _ , 347 , 861 katysheva , n.a . , pavlenko , e.p . 2003 , _ astophysics _ , 46 , 114 kuroshkin , n.e . 1990 , _ peremennye zvezdy _ , 17 , 186 mennickent , r. e. , & sterken , c. 1998 , _ pasp _ , 110 , 1032 nogami , d. , kato , t. , masuda , s. , & hirata , r. 1995 , informational bulletin on variable stars , 4163 , 1 nogami , d. , kato , t. , & masuda , s. 1998 , _ pasj _ , 50 , 411 nogami , d. , kato , t. , baba , h. , novk , r. , lockley , j. j. , somers , m. 2001 , _ mnras _ , 322 , 79 nogami , d. , et al . 2003 , _ astron . _ , 404 , 1067 odonoghue , d. 1987 , _ astrophys . space science _ , 136 , 247 olech , a. , schwarzenberg - czerny , a. , kedzierski , p. , zloczewski , k. , mularczyk , k. , wisniewski , m. 2003 , _ acta astron . _ , 53 , 175 olech , a. , zloczewski , k. , mularczyk , k. , kedzierski , p. , wisniewski , m. , stachowski , g. 2004 , _ acta asron . _ , 54 , 57 olech , a. , zloczewski , k. , cook , l. m. , mularczyk , k. , kedzierski , p. , wisniewski , m. 2005 , _ acta astron . _ , 55 , 237 olech , a. , mularczyk , k. , kedzierski , p. , zloczewski , k. , wisniewski , m. , szaruga , k. 2006 , _ astron . astrophys . _ , 452 , 933 osaki , y. 1996 , _ pasp _ , 108 , 39 osaki , y. 2005 , _ proc . of the japan academy _ , seires b , 81 , 291 patterson , j. 1979 , _ astron . j. _ , 84 , 804 schreiber , m.r , lasota , j .- 2007 , arxiv:0706.3888 schwarzenberg - czerny , a. 1996 , _ apj letters _ , 460 , l107 smak , j. 2000 , _ new astronomy _ , 44 , 171 stetson , p.b . 1987 , _ pasp _ , 99 , 191 stolz , b. , schoembs , r. 1984 , _ astron . _ , 132 , 187 udalski a. , pych w. 1992 , _ acta astron . _ , 42 , 285 uemura , m. , kato , t. , pavlenko , e. , baklanov , a. , & pietz , j. 2001 , _ pasj _ , 53 , 539 warner , b. , odonoghue , d. , & wargau , w. 1989 , _ mnras _ , 238 , 73 warner b. 1995 , cataclysmic variable stars , cambridge university press . warner b. 2007 , http://www.minorplanetobserver.com/mposoftware/mpocanopus.htm woudt , p. a. , & warner , b. 2001 , _ mnras _ , 328 , 159 | we report extensive photometry of the dwarf nova v419 lyr throughout its 2006 july superoutburst till quiescence . the superoutburst with amplitude of @xmath0 magnitude lasted at least 15 days and was characterized by the presence of clear superhumps with a mean period of @xmath1 days ( @xmath2 min ) . according to the stolz - schoembs relation
, this indicates that the orbital period of the binary should be around 0.086 days i.e. within the period gap . during the superoutburst the superhump period was decreasing with
the rate of @xmath3 , which is one of the highest values ever observed in su uma systems . at the end of the plateau phase ,
the superhump period stabilized at a value of 0.08983(8 ) days .
the superhump amplitude decreased from 0.3 mag at the beginning of the superoutburst to 0.1 mag at its end . in the case of v419 lyr
we have not observed clear secondary humps , which seems to be typical for long period systems .
* key words : * stars : individual : v419 lyr binaries : close novae , cataclysmic variables |
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heavy fermion ( hf ) compounds with elements from the lanthanide or actinide series share some rather general features in the fermi liquid phase , namely a strong enhancement of the effective carrier mass and a similarly enhanced pauli susceptibility with a wilson ratio typically of the order , but larger than one @xcite . in addition to this well - understood fermi liquid phase @xcite , hf systems also exhibit a variety of phase transitions , among them magnetic and superconducting phases . this aroused the strong interest of both experimentalists and theorists , as @xmath0-electrons conventionally tend to suppress superconductivity . the discovery of quantum critical phenomena @xcite eventually showed the intimate link between the latter two . in spite of the rather large collection of experimental results , an accepted theoretical description of superconductivity has not yet been established . moreover , the role of phonons on the low - energy properties of hf compounds and in particular their relevance for a microscopic theory of superconductivity in hf systems has not yet been addressed in detail @xcite . we present the first results of a study of the kondo lattice model with an effective attractive interaction among the conduction electrons . the latter can be thought to be obtained from an optical phonon mode treated in the antiadiabatic limit ( a more realistic description employing a true optical phonon in the calculation is the subject of ongoing investigations ) . our model is thus @xmath1 where @xmath2 annihilates ( creates ) a conduction electron at lattice site @xmath3 with spin @xmath4 , @xmath5 is the effective interaction between conduction electrons , and @xmath6 the kondo exchange . note that in our notation antiferromagnetic coupling means @xmath7 . finally , @xmath8 denotes the vector of pauli spin matrices . we solve the model with dynamical mean - field theory ( dmft ) @xcite and wilson s numerical renormalization group ( nrg ) @xcite . calculations were performed for a bethe lattice with infinite coordination number . in order to study superconductivity we allow for a corresponding symmetry broken phase @xcite . note that we can not include unconventional order parameters here , as dmft only allows for @xmath9-wave phases @xcite . ) at half filling with @xmath10 at @xmath11 . right : low - energy scale obtained from the width of the gap in the attractive case @xmath12 for @xmath10 . the inset shows the different gaps for small repulsive and attractive @xmath5 . , title="fig:",scaledwidth=65.0% ] ) at half filling with @xmath10 at @xmath11 . right : low - energy scale obtained from the width of the gap in the attractive case @xmath12 for @xmath10 . the inset shows the different gaps for small repulsive and attractive @xmath5 . , title="fig:",scaledwidth=33.0% ] it is necessary to stress that the model ( [ eq:1 ] ) does not show the usual symmetry @xmath13 under simultaneous exchange of spin and charge , i.e. its physics can not be inferred from the corresponding magnetic properties of the model with repulsive interaction . in fig . [ fig:1 ] we compare the two cases @xmath14 and @xmath12 . superficially , for weak interaction @xmath15 , both seem to be rather similar . however , as is evident from the inset to the right part and the behavior for larger interaction the insulator is much stronger for repulsive @xmath5 . it here originates from the formation of a local spin singlet rather than being of mott - hubbard type @xcite . for attractive @xmath12 , on the other hand , we find that kondo screening is strongly suppressed and the system eventually recovers mott - hubbard physics in the charge sector for large @xmath16 . this difference can be easily understood . for attractive @xmath5 the conduction system namely experiences strong correlations in the charge sector , while the kondo exchange tries to develop such feature in the spin sector . obviously , when @xmath17 , where @xmath18 is the kondo scale for @xmath19 , spin fluctuations will efficiently be suppressed , leading to the observed behaviour . the suppression of the kondo scale is actually stronger than exponential , as shown in the right part of fig . [ fig:1 ] . from these results we draw two conclusions : 1 ) phonons are clearly extremely important even to the paramagnetic phase of the kondo model and thus to properly account for the low - energy scale of hf systems ; 2 ) as the low - energy scale is efficiently reduced by an attractive interaction among the conduction electrons , we expect that @xmath9-wave superconductivity will actually prevail in a large part of the phase diagram , in particular for small kondo coupling @xmath6 . in order to allow the system to show superconductivity , we have to reformulate the dmft equations in nambu space and extend the nrg accordingly . the latter has been accomplished some time ago already ( for a review see and references therein ; the actual way to combine dmft and nrg has been extensively discussed by bauer @xcite ) . for ( a ) @xmath20 , ( b ) @xmath21 and ( c ) @xmath22 . , scaledwidth=65.0% ] for a very small kondo exchange interaction @xmath23 , the ground state of the model is dominated by superconductivity . this becomes apparent from fig . [ fig:3 ] , where the dos ( upper panels ) and the real part of the anomalous green s function ( lower panels ) is shown at half filling ( full curves ) as well as at finite filling @xmath24 ( dashed curves ) . as the low - energy scale of the model with @xmath19 is always largest at half filling @xcite , we can expect that this result remains stable for all fillings @xmath25 . note that in none of the cases one does observe a significant dependence of the gap on the fillings , i.e. local correlations due to kondo screening are frozen out here since @xmath26 . furthermore , @xmath27 and consequently one expects and observes a bcs like gap structure , only weakly smeared out by self - energy broadening . and ( a ) @xmath28 , ( b ) @xmath29 and ( c ) @xmath30.,scaledwidth=65.0% ] increasing @xmath31 has two effects . first , there appears a finite , critical @xmath32 below which no superconducting solution exists . this can be seen in fig . [ fig:4]a , where dos and real part of the anomalous green s function are shown for @xmath33 and a small @xmath28 . note that at half filling we find a kondo insulator , which has a gap in the dos . from that perspective the result is actually indistinguishable from the superconducting phase . the anomalous part , however , vanishes here , i.e. we have indeed a normal state and thus an insulator . for larger interactions , the superconducting phase reappears . compared to the case with small @xmath6 , we observe here visible reduction of the gap and also a broadening of the singularities at the gap edges . we attribute this behavior to the correlations induced by the kondo exchange . we have studied the kondo lattice model with an attractive interaction between the conduction electrons , which may arise in the presence of phonons , notably optical modes like breathing modes . we found a tremendous effect of an attractive interaction on the low - energy scale , drastically reducing it already for comparatively modest @xmath16 . this behavior can be understood in terms of a competition between a spin kondo effect and the charge fluctuations introduced by the attractive @xmath5 . as the latter also favour superconductivity we can expect , and indeed do find , that for experimentally relevant values of the kondo coupling and interaction parameters the model shows an @xmath9-wave type superconducting ground state . for real systems this underlines the importance of elastic degrees of freedom for a proper description of the physics of hf materials . | we study the kondo lattice model with additional attractive interaction between the conduction electrons within the dynamical mean - field theory using the numerical renormalization group to solve the effective quantum impurity problem .
in addition to normal - state and magnetic phases we also allow for the occurrence of a superconducting phase . in the normal phase we observe a very sensitive dependence of the low - energy scale on the conduction - electron interaction .
we discuss the dependence of the superconducting transition on the interplay between attractive interaction and kondo exchange . |
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let @xmath7 be the tropical semifield , where the tropical sum @xmath8 is taking the maximum @xmath9 , and the tropical product @xmath10 is taking the ordinary sum @xmath11 . let @xmath12 be the sub - semifield of @xmath13 . a tropical curve @xmath5 is a metric graph with possibly unbounded edges . equivalently , in a more formal form , a tropical curve is a compact topological space homeomorphic to a one - dimensional simplicial complex equipped with an integral affine structure over @xmath14 ( see @xcite ) . finite graphs are seen as a discrete version of tropical curves . in relation to the classical algebraic curves , tropical curves and finite graphs have been much studied recently . for example , the riemann - roch formula on finite graphs and tropical curves ( analogous to the classical riemann - roch formula on algebraic curves ) are established in @xcite . the clifford theorem is established in @xcite . in this article , we consider whether the analogy of the following classical theorem holds or not . let @xmath0 be a smooth complex projective curve of genus @xmath15 , and let @xmath1 be the canonical divisor on @xmath0 . let @xmath16 be the canonical ring . then : 1 . @xmath17 is finitely generated as a graded ring over @xmath18 . @xmath17 is generated in degree at most three . our first result is that for a finite graph @xmath3 , the analogous statement ( a ) holds , but that the degrees of generators can not be bounded by a universal constant . for a divisor @xmath19 on @xmath3 , let @xmath20 be the set of rational functions @xmath21 on @xmath3 such that @xmath22 is effective ( see @xcite for details ) . we also refer to [ fgdiv ] for terminology . we show that the direct sum @xmath23 has a graded semi - ring structure over @xmath24 for any finite graph @xmath3 and any divisor @xmath19 on @xmath3 ( lemma [ semiringforfg ] ) . then the following is the first result : [ thm : main:1 ] let @xmath3 be a finite graph and let @xmath25 be the canonical divisor on @xmath3 . we set @xmath26 . then : 1 . @xmath4 _ is _ finitely generated as a graded semi - ring over @xmath24 . 2 . for any integer @xmath27 , there exists a finite graph @xmath28 such that @xmath29 is _ not _ generated in degree at most @xmath30 . for ( a ) , we show that , in fact , the semi - ring @xmath23 is finitely generated as a graded semi - ring over @xmath24 for any divisor @xmath19 on @xmath3 . our next result is that for a tropical curve @xmath5 with integer edge - length , the analogous statement ( a ) does _ not _ hold in general ( hence neither ( b ) ) . we give a sufficient condition for non - finite generation of the canonical semi - ring of tropical curves . for a divisor @xmath19 on @xmath5 , let @xmath31 be the set of rational functions @xmath21 on @xmath5 such that @xmath22 is effective ( see @xcite for details ) . we also refer to [ tcdiv ] for terminology . we show that the direct sum @xmath32 has a graded semi - ring structure over @xmath13 for any tropical curve @xmath5 and any divisor @xmath19 on @xmath5 ( lemma [ semiring ] ) . then the following is the second result : [ thm : main:2 ] let @xmath5 be a @xmath33-tropical curve of genus @xmath15 , and let @xmath34 be the canonical divisor on @xmath5 . assume that there exist an edge @xmath35 of the canonical model of @xmath5 and a positive integer @xmath36 such that @xmath35 is not a bridge and @xmath37 is linearly equivalent to @xmath38 + n(g-1)[q]$ ] , where @xmath39 and @xmath40 are the endpoints of @xmath35 . then the canonical semi - ring @xmath41 is _ not _ finitely generated as a graded semi - ring over @xmath13 . [ cor : main:2 ] 1 . let @xmath5 be a hyperelliptic @xmath33-tropical curve of genus at least @xmath42 . then @xmath6 is _ not _ finitely generated as a graded semi - ring over @xmath13 . 2 . let @xmath43 be a complete graph on vertices at least @xmath44 , and let @xmath5 be the tropical curve associated to @xmath43 , where each edge of @xmath43 is assigned the same positive integer as length . then @xmath6 is _ not _ finitely generated as a graded semi - ring over @xmath13 . for theorem [ thm : main:2 ] , we give , in fact , a sufficient condition for non - finite generation of the graded semi - ring @xmath45 over @xmath13 for any @xmath33-divisor @xmath19 of degree at least @xmath42 on a @xmath33-tropical curve @xmath5 ( theorem [ criterion ] ) . it seems likely that , for _ any _ tropical curve of genus @xmath15 , the canonical semi - ring @xmath46 will not be finitely generated as a graded semi - ring over @xmath13 , which we pose as a question . for the proof of theorem [ thm : main:2 ] , we use the notion of _ extremals _ of @xmath31 introduced by haase , musiker and yu @xcite . then theorem [ thm : main:1](b ) is deduced as a certain discrete version of theorem [ thm : main:2 ] . theorem [ thm : main:1](a ) is shown by using gordan s lemma ( see ( * ? ? ? * , proposition 1 ) ) . in this section , we prove theorem [ thm : main:2 ] and corollary [ cor : main:2 ] . in this section , we first put together necessary definitions and results on the theory of divisors on tropical curves , which will be used later . our basic references are @xcite . in this article , all finite graphs are assumed to be connected and allowed to have loops and multiple edges . for a finite graph @xmath3 , let @xmath47 and @xmath48 denote the set of vertices and the set of edges , respectively . bridge _ is an edge of @xmath3 which makes @xmath3 disconnected . a metric space @xmath5 is called _ a metric graph _ if there exist a finite graph @xmath3 and a function @xmath49 ( called the edge - length function ) such that @xmath5 is obtained by gluing the intervals @xmath50 $ ] for @xmath51 at their endpoints so as to keep the combinatorial data of @xmath3 . the pair @xmath52 is called a _ model _ for @xmath5 . _ in this article , we assume that a metric space @xmath5 is not homeomorphic to the circle @xmath53_. for a point @xmath54 of @xmath5 , we define the _ valence _ @xmath55 of @xmath54 to be the number of connected components in @xmath56 for any sufficiently small neighborhood @xmath57 of @xmath54 . let @xmath58 be a finite subset of @xmath5 which includes all points of valence different from @xmath42 , and @xmath59 be a finite graph whose vertices are the points in @xmath58 and whose edges correspond to the connected components of @xmath60 . if we define a function @xmath61 such that @xmath62 is equal to the length of the corresponding component for each edge @xmath35 , then @xmath63 is a model for @xmath5 . the model @xmath63 is called the _ canonical model _ for @xmath5 if we take the set of all points of valence different from @xmath42 as the finite subset @xmath58 . a metric graph @xmath5 with the canonical model @xmath52 is called a _ @xmath33-metric graph _ if @xmath62 is an integer for each edge @xmath35 of @xmath3 . in this case , the points of @xmath5 with integer distance to the vertices of @xmath3 are called _ @xmath33-points _ , and we denote the set of @xmath33-points by @xmath64 . tropical curves are defined in a similar way as metric graphs . a metric space @xmath5 is called a _ tropical curve _ if there exist a finite graph @xmath3 and a function @xmath65 such that @xmath5 is obtained by gluing the intervals @xmath50 $ ] for @xmath51 at their endpoints so as to keep the combinatorial data of @xmath3 , where the only edges adjacent to a one - valent vertex may have length @xmath66 . the pair @xmath52 is called a _ model _ for @xmath5 . we define the _ canonical model _ of a tropical curve in the same way as that of a metric graph . a tropical curve @xmath5 with the canonical model @xmath52 is called a _ @xmath33-tropical curve _ if @xmath62 is either an integer or equal to @xmath66 for each edge @xmath35 of @xmath3 . in this case , the points of @xmath5 with integer distance to the vertices of @xmath3 are called _ @xmath33-points_. a _ divisor _ on a tropical curve @xmath5 is a finite formal sum of points of @xmath5 , and a _ @xmath33-divisor _ on a @xmath33-tropical curve @xmath5 is a finite formal sum of @xmath33-points of @xmath5 . we denote the set of all divisors on @xmath5 by @xmath67 . if @xmath19 is a divisor on @xmath5 , we write it as @xmath68 , $ ] where @xmath69 is an integer and @xmath70 $ ] is merely a symbol . for a divisor @xmath19 , we define the _ degree _ @xmath71 to be the integer @xmath72 and the _ support _ @xmath73 to be the set of all points of @xmath5 occurring in @xmath19 with a non - zero coefficient . a divisor @xmath19 is called _ effective _ , and we write @xmath74 , if @xmath69 is a non - negative integer for all @xmath75 . on a @xmath33-tropical curve , a divisor @xmath19 is called a _ @xmath33-divisor _ if @xmath73 is a subset of @xmath76 . the _ canonical divisor _ on a tropical curve @xmath5 is defined to be @xmath77 . $ ] a _ rational function _ @xmath21 on a tropical curve @xmath5 is a continuous function @xmath78 that is piecewise linear with finitely many pieces and integer slopes , and may take on values @xmath79 only at the one - valent points . the set of all rational functions on @xmath5 is denoted by @xmath80 . for a rational function @xmath21 and a vertex @xmath54 , we define the _ order _ @xmath81 of @xmath21 at @xmath54 as the sum of outgoing slopes at @xmath54 . the _ principal divisor _ associated to @xmath21 is defined to be @xmath82 .\ ] ] we say that two divisors @xmath19 and @xmath83 are _ linearly equivalent _ , and we write @xmath84 , if there exists a rational function @xmath21 such that @xmath85 . now , we define the most important objects in this article . let @xmath19 be a divisor on a tropical curve @xmath5 . we set @xmath86 for @xmath21 , @xmath87 and @xmath88 , we define the tropical sum @xmath89 and the tropical @xmath13-action @xmath90 as follows : @xmath91 an _ extremal _ of @xmath31 is an element such that @xmath92 implies @xmath93 or @xmath94 for any @xmath95 , @xmath96 . a subset @xmath97 is called a _ subgraph _ if @xmath98 is a compact subset with a finite number of connected components . for a subgraph @xmath98 and a positive real number @xmath99 , we define the rational function _ chip firing move _ @xmath100 as @xmath101 we say that a subgraph @xmath98 can _ fire _ on a divisor @xmath19 if the divisor @xmath102 is effective for a sufficiently small positive real number @xmath99 . here , by a sufficiently small positive real number , we mean that @xmath99 is chosen to be small enough so that the chips " do not pass through each other or pass through points of valence @xmath42 . [ props ] 1 . @xmath31 is a semi - module over @xmath13 . the set of extremals of @xmath31 is finite modulo @xmath13-action . @xmath31 is generated by the extremals . the following lemma is useful for finding extremals : [ extremalcriterion ] a rational function @xmath21 is an extremal of @xmath31 if and only if there are not two proper subgraphs @xmath103 and @xmath104 covering @xmath5 such that each can fire on @xmath22 . let @xmath5 be a tropical curve , and let @xmath105 be the canonical divisor . the direct sum @xmath106 is called the _ canonical semi - ring _ of @xmath5 , and denoted by @xmath6 . for @xmath107 and @xmath108 , we define the tropical product @xmath109 as @xmath110 we show that @xmath6 has indeed a graded semi - ring structure over @xmath13 . [ semiring ] let @xmath5 be a tropical curve . then the canonical semi - ring @xmath6 has naturally a graded semi - ring structure over @xmath13 . for any divisor @xmath19 on @xmath5 , in general , the direct sum @xmath45 has naturally a graded semi - ring structure over @xmath13 . _ proof._we prove only the general case . let @xmath21 and @xmath95 be elements of @xmath111 and @xmath112 , respectively . since the order of a rational function at a point is defined as the sum of outgoing slopes and the tropical product is defined as the ordinary sum , it follows that @xmath113 . therefore @xmath114 . here both @xmath115 and @xmath116 are effective , so @xmath117 is also effective . this means that the tropical product @xmath109 is an element of @xmath118 . together proposition [ props](a ) , we obtain the assertion . 500 since we have @xmath119 , the semi - ring @xmath45 can be seen as a semi - ring over the @xmath120-th part @xmath121 . [ criterion ] let @xmath5 be a @xmath33-tropical curve of genus @xmath15 , and let @xmath19 be a @xmath33-divisor of degree @xmath122 . assume that there exist an edge @xmath35 of the canonical model of @xmath5 and a positive integer @xmath36 such that @xmath35 is not a bridge and @xmath123 is linearly equivalent to @xmath124 + \frac{n d}{2}[q]$ ] , where @xmath39 and @xmath40 are the endpoints of @xmath35 . then @xmath45 is not finitely generated as a graded semi - ring over @xmath13 . _ proof._let @xmath125 be the length of @xmath35 . note that , if @xmath35 is a loop , then @xmath126 . we begin by showing the following lemma . [ mainlemma ] let @xmath19 , @xmath39 , @xmath40 be as in theorem [ criterion ] . if there exists a positive integer @xmath127 such that @xmath128 is linearly equivalent to @xmath129 + \frac{s d}{2}[q]$ ] , then there exists an extremal of @xmath130 which is not generated by elements of @xmath131 over @xmath13 . _ proof._put @xmath132 . since @xmath128 is linearly equivalent to @xmath133 + \frac{n}{2}[q]$ ] , it follows that @xmath134 is linearly equivalent to @xmath135 + l n[q]$ ] . identify the edge @xmath35 with an interval @xmath136 $ ] such that @xmath39 and @xmath40 are identified with @xmath120 and @xmath125 , respectively . let @xmath137 be the point identified with the point @xmath138 of the interval . by definition , @xmath137 is not a @xmath33-point . first , we show two claims . the divisor @xmath139 is linearly equivalent to @xmath140 + ( 2 l n - 1)[r]$ ] . _ proof._let @xmath141 be the rational function which takes on value @xmath120 on @xmath142 , and value @xmath143 at @xmath137 , and is extended linearly to @xmath144 . then the orders of @xmath141 at @xmath39 , @xmath40 , and @xmath137 are @xmath145 moreover , the order of @xmath141 at any point of @xmath146 is equal to @xmath120 by construction . from these values we conclude that @xmath147 + l n [ q ] + { \mathop{\mathrm{div}}}(\tilde{f})$ ] is equal to @xmath140 + ( 2 l n - 1)[r]$ ] . therefore @xmath134 is linearly equivalent to @xmath140 + ( 2l n - 1)[r]$ ] . 500 let @xmath21 be the rational function such that @xmath148 + ( 2l n - 1)[r]$ ] . then @xmath21 is an extremal of @xmath149 . _ proof._since @xmath39 is an endpoint of @xmath35 and @xmath35 is an edge of the canonical model , we have @xmath150 . moreover , by the assumption that @xmath35 is not a bridge , we have @xmath151 . suppose that @xmath103 is a subgraph of @xmath5 that can fire on @xmath148 + ( 2l n - 1)[r]$ ] . then the boundary set @xmath152 of @xmath103 in @xmath5 is contained in @xmath153 . since @xmath151 , we have @xmath154 or @xmath155 . ( here @xmath156 denotes the open interval in @xmath35 connecting @xmath39 to @xmath137 . ) from lemma [ extremalcriterion ] we conclude that @xmath21 is an extremal of @xmath157 . 500 we prove that @xmath21 is not generated by elements of @xmath158 over @xmath13 by contradiction . suppose that @xmath21 is generated by elements of @xmath158 over @xmath13 . then we have @xmath159 where @xmath160 is an element of @xmath161 and the sum @xmath162 is equal to @xmath163 for each @xmath164 . note that there are at least two terms in @xmath165 for each @xmath164 . by lemma [ semiring ] we can take @xmath166 and @xmath167 and @xmath168 such that @xmath169 and @xmath170 . by proposition [ props](b ) we may assume that @xmath171 and @xmath172 are the extremals of @xmath173 and @xmath174 , respectively . then we have @xmath175 where each @xmath171 and @xmath172 is an extremal of @xmath173 and @xmath174 , respectively . since @xmath21 is an extremal , it follows that @xmath21 is equal to @xmath176 after changing indices if necessary . put @xmath177 , @xmath178 , @xmath179 , and @xmath180 . recall that @xmath181 . now , we have @xmath182 + ( 2l n - 1)[r ] = 2s l d + { \mathop{\mathrm{div}}}(f ) = l d + { \mathop{\mathrm{div}}}(g ) + k d + { \mathop{\mathrm{div}}}(h ) .\ ] ] since both @xmath183 and @xmath184 are effective , we may assume that @xmath185 + ( l d-1)[r ] , \\ k d + { \mathop{\mathrm{div}}}(h ) & = k d[r ] , \end{aligned}\ ] ] after changing the role of @xmath95 and @xmath186 if necessary . in this setting , we deduce a contradiction by studying the property of the rational function @xmath186 . since @xmath187 - k d$ ] , all the zeros and poles of @xmath186 lie in @xmath188 . let @xmath189 be the points of @xmath190 in this order , where @xmath191 is @xmath39 and @xmath192 is @xmath40 . moreover , let @xmath193 be the segment which connects @xmath194 to @xmath195 , and @xmath196 be the segment which connects @xmath197 to @xmath198 , where we set @xmath199 and @xmath200 . we denote the each length of @xmath201 and @xmath202 by @xmath203 and @xmath204 , respectively . the sum of outgoing slopes of @xmath186 at @xmath54 as a rational function on @xmath201 and @xmath205 are denoted by @xmath206 and @xmath207 , respectively . , scaledwidth=60.0% ] since @xmath186 is continuous , we have @xmath208 now , by the equality @xmath187 -k d$ ] , we have @xmath209 for @xmath210 , and for @xmath211 , and we have @xmath212 for @xmath213 , and for @xmath214 . from these relations , we deduce that @xmath215 similarly , we deduce that @xmath216 since @xmath19 is a @xmath33-divisor and @xmath137 is not a @xmath33-point , we have @xmath217 . thus @xmath218 it follows that where we use ( [ eq:1 ] ) , ( [ eq:2 ] ) in the second equality , and ( [ eq:3 ] ) in the last equality . by construction , we have @xmath219 hence we have @xmath220 the value @xmath221 is an integer . _ proof._first , we claim that the value @xmath222 is an integer . indeed , since @xmath35 is not a bridge , there exists a path @xmath223 in @xmath142 such that @xmath39 and @xmath40 are the endpoints of @xmath223 . since @xmath186 is a piecewise linear function with integer slopes along @xmath223 and both zeros and poles of @xmath186 on @xmath142 are @xmath33-points , it follows that the difference @xmath224 is an integer . since @xmath5 is a @xmath33-tropical curve and @xmath19 is a @xmath33-divisor , it follows that @xmath203 , @xmath204 , @xmath225 , @xmath226 @xmath227 , @xmath125 are integers . moreover , by definition , @xmath228 is an integer . thus @xmath221 is an integer . 500 since @xmath229 and @xmath230 , and @xmath231 and @xmath232 are relatively prime , respectively , it follows that there exists a positive integer @xmath233 such that @xmath234 . then we have @xmath235 since @xmath236 , it follows that @xmath237 and @xmath238 , but this contradicts @xmath122 . thus @xmath21 is not generated by elements of @xmath239 over @xmath13 . 500 now , we return to the proof of theorem [ criterion ] . we prove that @xmath45 is not finitely generated as a graded semi - ring over @xmath13 by contradiction . suppose that @xmath45 is finitely generated as a graded semi - ring over @xmath13 . since elements of @xmath240 is generated by the extremals over @xmath13 , we may assume that all the generators of @xmath45 is an extremal of @xmath240 for some positive integer @xmath241 . let @xmath233 be the maximal number among such numbers . fix a positive integer @xmath242 such that @xmath243 is bigger than @xmath233 , and put @xmath244 . then we have @xmath245 + \frac{kn d}{2}[q ] = \frac{s d}{2}[p ] + \frac{s d}{2}[q].\ ] ] applying lemma [ mainlemma ] , we get an extremal of @xmath130 which is not generated by the elements of @xmath246 over @xmath13 , but this contradicts the maximality of @xmath233 . thus @xmath45 is not finitely generated as a graded semi - ring over @xmath13 . 500 [ sufficientcondition ] let @xmath5 be a @xmath33-tropical curve of genus @xmath15 . assume that there exist an edge @xmath35 of the canonical model of @xmath5 and a positive integer @xmath36 such that @xmath35 is not a bridge and @xmath37 is linearly equivalent to @xmath38 + n(g-1)[q]$ ] , where @xmath39 and @xmath40 are the endpoints of @xmath35 . then the canonical semi - ring @xmath6 is not finitely generated as a graded semi - ring over @xmath13 . a finite graph is called a _ complete graph _ on @xmath36 vertices if it is a finite graph with @xmath36 vertices in which every pair of distinct vertices is connected by a unique edge . [ corollary ] 1 . let @xmath5 be a hyperelliptic @xmath33-tropical curve of genus at least @xmath42 . then @xmath6 is not finitely generated as a graded semi - ring over @xmath13 . 2 . let @xmath36 be an integer at least @xmath44 , let @xmath43 be a complete graph on @xmath36 vertices , and let @xmath5 be the tropical curve associated to @xmath43 , where each edge of @xmath43 is assigned the same positive integer as length . then @xmath6 is not finitely generated as a graded semi - ring over @xmath13 . _ proof._for ( a ) , let @xmath52 be the canonical model of @xmath5 . by chan s theorem ( * ? ? ? * theorem 3.12 ) , there exists an edge @xmath35 of @xmath3 such that @xmath247 + [ q ] ) = 1 $ ] , where @xmath39 and @xmath40 are the endpoints of @xmath35 . by the riemann - roch formula , it follows that @xmath248 + ( g-1)[q]$ ] is linearly equivalent to @xmath34 . applying theorem [ criterion ] ( with @xmath249 ) , the statement ( a ) follows . for ( b ) , there are two cases , one is that @xmath36 is an odd number , and the other is that @xmath36 is an even number . fix any two vertices @xmath250 and @xmath251 of @xmath43 , and let @xmath35 be the unique edge which connects these vertices . since the genus of @xmath5 is equal to @xmath252 , the degree of the canonical divisor is equal to @xmath253 . if @xmath36 is odd , then @xmath34 is equivalent to @xmath254 + \frac{n(n-3)}{2}[w]$ ] . if @xmath36 is even , then @xmath255 is equivalent to @xmath256 + n(n-3)[w]$ ] . therefore , in both cases , we can apply theorem [ criterion ] , and the statement ( b ) follows . 500 in this section , we prove theorem [ thm : main:1 ] . we denote the valence of a vertex @xmath54 by @xmath55 . a _ divisor _ on a finite graph @xmath3 is a finite formal sum of vertices and we denote the set of all divisors on @xmath3 by @xmath257 . if @xmath19 is a divisor on @xmath3 , we write it as @xmath258 , $ ] where @xmath69 is an integer and @xmath70 $ ] is merely a symbol . the _ degree _ of a divisor and an _ effective _ divisor are defined in the same way as in [ tcdiv ] . the _ canonical divisor _ on a finite graph @xmath3 is defined to be @xmath259 . $ ] a _ rational function _ @xmath21 on a finite graph @xmath3 is a @xmath33-valued function on vertices @xmath47 . the set of all rational functions on @xmath3 is denoted by @xmath260 . for a rational function @xmath21 and a vertex @xmath54 , we define the _ order _ @xmath81 of @xmath21 at @xmath54 as the sum of differences between the value at @xmath54 and at each vertex adjacent to @xmath54 , that is , we define it to be the integer @xmath261 where @xmath262 means that @xmath54 and @xmath263 are the endpoints of @xmath51 . the _ principal divisor _ associated to @xmath21 is defined to be @xmath264 .\ ] ] an _ extremal _ of @xmath269 is an element such that @xmath92 implies @xmath93 or @xmath94 for any @xmath95 , @xmath270 . for a subset @xmath271 of vertices @xmath47 , we define the rational function @xmath272 on @xmath3 as @xmath273 we say that a subset @xmath271 of vertices @xmath47 can _ fire _ on a divisor @xmath19 if the divisor @xmath274 is effective . _ proof._for ( a ) , it is clear that @xmath90 is an element of @xmath269 for @xmath267 and @xmath275 . so it is sufficient to show that @xmath89 is an element of @xmath269 for any @xmath21 , @xmath266 . if we have @xmath276 for a fixed vertex @xmath54 , then it follows that @xmath277 similarly , we have @xmath278 if @xmath279 at @xmath54 . since both @xmath22 and @xmath280 are effective , it follows that @xmath281 is effective . then the statement ( a ) follows . for ( b ) , we identify @xmath20 with the lattice points of a polyhedron in an euclidean space , which is a finite set . since the way of identification will be described in the proof of theorem [ finitea ] , we omit the detail now . then the statement ( b ) follows . [ semiringforfg ] let @xmath3 be a finite graph . then the canonical semi - ring @xmath4 has naturally a graded semi - ring structure over @xmath24 . for any divisor @xmath19 on @xmath3 , in general , the direct sum @xmath287 has naturally a graded semi - ring structure over @xmath24 . let @xmath3 be the finite graph with two vertices and three edges each of which connects the vertices . let @xmath28 be the finite graph obtained by replacing each edge of @xmath3 with a segment which consists of @xmath289 edges . note that @xmath28 has @xmath290 vertices and @xmath291 edges . let @xmath137 be the @xmath292-th vertex counted from @xmath39 on a segment , where @xmath39 is a vertex of valence different from @xmath42 . the canonical divisor @xmath293 is equal to that of @xmath3 by definition , and the divisor @xmath294 is linearly equivalent to @xmath140 + ( 2n - 1)[r]$ ] . it follows that there exists an extremal @xmath21 of @xmath295 such that @xmath296 is equal to @xmath140 + ( 2n - 1)[r]$ ] . suppose that @xmath21 is generated by the elements of @xmath297 over @xmath24 . then we may assume that @xmath21 is equal to @xmath298 , where @xmath95 and @xmath186 are the extremal of @xmath299 and @xmath300 , respectively , and @xmath301 is equal to @xmath36 . it follows that @xmath302 + ( 2l - 1)[r ] , \\ k k_{g_n } + { \mathop{\mathrm{div}}}(h ) & = 2k [ r ] .\end{aligned}\ ] ] let @xmath303 and @xmath251 be the second vertex counted from @xmath39 on each segment different from @xmath35 , where @xmath35 is the segment on which @xmath137 is a vertex . after some calculations , we get @xmath304 then we have @xmath305 and @xmath306 is an integer . _ proof._let the vertices @xmath310 , and let @xmath311 be the graph laplacian , that is , the @xmath312 symmetric matrix such that each entry @xmath313 is equal to the number of edges which connect @xmath195 to @xmath314 if @xmath315 , and the value @xmath316 if @xmath317 . using the laplacian , we can describe @xmath318 as @xmath319 we identify @xmath318 with the lattice points of a polyhedron @xmath320 in @xmath321 by a map @xmath322 which maps @xmath21 to @xmath323 , where @xmath320 is a polyhedron of the form @xmath324 by the fundamental theorem of polyhedra , it follows that @xmath320 is a convex hull of finitely many vectors . _ proof._by the above identification , it is sufficient to show that each lattice point of @xmath0 whose @xmath292-th coordinate is equal to @xmath241 corresponds to a lattice point of @xmath328 . let @xmath329 be a lattice point of @xmath0 . by definition , @xmath330 is an element of @xmath325 . since @xmath331 it follows that @xmath332 hence , @xmath303 is a lattice point of @xmath328 . conversely , let @xmath303 be a lattice point of @xmath328 and let @xmath333 be the vector whose i - th coordinate @xmath334 is equal to @xmath335 . then @xmath251 is an element of @xmath325 in the same way as above . in particular , @xmath336 is a lattice point of @xmath0 . moreover , the sum of the lattice points of @xmath0 corresponds to the product of the elements of @xmath337 for some @xmath241 . this follows from the same reason as the above correspondence , so we omit the detail . by this correspondence and the gordan s lemma ( see ( * ? ? ? * , proposition 1 ) ) , all the elements of @xmath327 for any @xmath241 is generated by the elements of @xmath338 for finitely many @xmath36 over @xmath24 . | for a projective curve @xmath0 and the canonical divisor @xmath1 on @xmath0 , it is classically known that the canonical ring @xmath2 is finitely generated in degree at most three . in this article
, we study whether analogous statements hold for finite graphs and tropical curves . for any finite graph @xmath3
, we show that the canonical semi - ring @xmath4 is finitely generated but that the degree of generators are not bounded by a universal constant . for any hyperelliptic tropical curve @xmath5 with integer edge - length ,
we show that the canonical semi - ring @xmath6 is not finitely generated , and , for tropical curves with integer edge - length in general , we give a sufficient condition for non - finite generation .
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quantum information is an important branch of quantum physics . it includes mainly quantum communication and quantum computation @xcite . by far , many interesting quantum systems have been presented for quantum information processing , such as nuclear magnetic resonance @xcite , quantum dots @xcite , diamond nitrogen vacancy ( nv ) centers @xcite , photonic systems @xcite , circuit quantum electrodynamics ( qed ) @xcite , and so on . due to the good scalability @xcite and convenient operation on superconducting qubits , circuit qed has attracted much attention in recent years . composed of the superconducting circuit and the superconducting 1d resonator , circuit qed @xcite has some good characters for completing quantum information processing . the superconducting circuit can act as a qubit perfectly . the energy - level structure of the qubit can be divided into @xmath0 , @xmath1 , @xmath2 , and @xmath3 types @xcite which can not be found in atom systems . a relative long life time of a superconducting qubit has been realized to reach @xmath4 ms @xcite . the strong coupling strength between a superconducting qubit and a superconducting resonator @xcite has been demonstrated in the experiment . all these characters make circuit qed as a good platform for the quantum computation based on superconducting qubits . in 2009 , dicarlo _ et al . _ demonstrated a two - qubit algorithms with a superconducting quantum processor @xcite . in 2012 , reed _ _ realized a three - qubit quantum error correction with superconducting circuits @xcite , and in the same year , lucero _ et al . _ computed the prime factors with a josephson phase qubit quantum processor @xcite in which they integrated five superconducting resonators and four superconducting qubits in a quantum processor . a superconducting resonator can act as a cavity and a quantum bus , which can be coupled to the distant qubits . the quality of the resonator can be reached to @xmath5 and even @xmath6 @xcite , that is , the superconducting resonators can also afford a powerful platform for quantum information processing . in 2007 , _ resolved the photon number states in a superconducting circuit @xcite . in 2010 , johnson _ realized a quantum non - demolition detection of single microwave photons in a circuit @xcite , and in the same year , strauch _ et al . _ presented a method to synthesize an arbitrary quantum state of two superconducting resonators @xcite . in 2012 , strauch proposed an all - resonant control of superconducting resonators with a drive field @xcite . in 2013 , we proposed a selective - resonance scheme to perform a fast quantum entangling operation for quantum logic gates on superconducting qubits @xcite , assisted by one or two superconducting resonators . by combination of the selective resonance and the tunable period relation between a wanted quantum rabi oscillation and an unwanted one besides the positive influence from the non - computational third levels of the superconducting qubits , these universal quantum gates are significantly faster than previous proposals and do not require any kind of drive fields . recently , the generation of the noon state @xcite on two resonators attracted much more attention . in 2010 , strauch , jacobs , and simmonds @xcite proposed a scheme for completing the generation of the noon state on two resonators without using the third non - computational excited energy level . the superconducting qubit was operated with a selective rotation by using a drive field whose amplitude should much smaller than the photon - number - dependent stark shifts on the qubit . that is , the operation time of the qubit should be extended a little longer . in 2010 , merkel and wilhelm @xcite proposed a theoretic scheme for generating noon states on two resonators by using two superconducting qubits and three superconducting resonators . in 2011 , wang _ et al . _ @xcite demonstrated merkel - wilhelm scheme in experiment . in ref.@xcite , a novel method was proposed to generate the noon state on two resonators by using a complicated classical microwave pulse and an all - resonant manipulation . it can get a very high - fidelity noon state within a much shorter time without using the non - computational excited energy level . in 2013 , su _ et al . _ @xcite proposed an interesting scheme for the generation of the noon state on two resonators with the resonant operation between the transmon qubit and the superconducting resonator , assisted by the single - qubit rotation . the scheme can be completed with @xmath7 steps , and in the first @xmath8 steps , the qubit should be maintained in the third - excited state corresponding to the case that the photon number in each resonator is zero . in this paper , we proposed a scheme to produce the noon state on two resonators in a quantum processor composed of two tunable superconducting resonators coupled to a tunable @xmath9-type three - energy - level superconducting qutrit . our scheme requires two kind of quantum operations . one is the resonant operation on the superconducting qutrit and the resonators . the other is the single - qubit manipulation which can be completed by applying a drive field on the qutrit . our scheme can be used to produce the noon state on two resonators effectively in a simple and fast way , compared with merkel - wilhelm scheme . moreover , it does not require us to remain the qutrit in the third - excited state all the time , which relaxes largely the requirements of its implementation in experiment , compared with the previous work in ref . and @xmath10 are the two microwave - photon resonators . @xmath11 ( @xmath12 ) is the coupling strength between the resonator @xmath13 and the superconducting qutrit in the transition between the states @xmath14 and @xmath15 ( @xmath15 and @xmath16 ) . , width=309 ] let us consider a quantum system composed of two superconducting resonators coupled to a superconducting qutrit , shown in fig . [ fig1 ] ( a ) . the energy - level structure of the qutrit is the @xmath9 type , which can be found in a superconducting charge qubit , shown in fig . [ fig1 ] ( b ) . in order to construct the noon state on the two resonators @xmath17 and @xmath10 , we exploit the lowest three energy levels of the qutrit , denoted by @xmath18 , @xmath19 , and @xmath20 with the energy @xmath21 . the hamiltonian of the system composed of the two resonators and the qutrit is ( under the rotating - wave approximation , and we choose @xmath22 below ) @xmath23 . \label{h}\end{aligned}\ ] ] here , @xmath24 and @xmath25 are the transition frequency and the creation operator of the resonator @xmath13 , respectively . @xmath26 and @xmath27 are the creation operators of the two transitions @xmath28 and @xmath29 of the qutrit , respectively . @xmath30 is the coupling strength between the resonator @xmath13 and the qutrit in the two transitions @xmath31 and @xmath32 , and @xmath33 is the coupling strengths between the resonator @xmath13 and the qutrit in the two transitions @xmath15 and @xmath34 . in order to turn on or off the interaction between the resonators and the qutrit , on one hand , one can tune the transition frequency of the qutrit by using the external magnetic flux , or tune the transition frequency of the resonator to make them resonate or largely detune with each other . on the other hand , one can tune the coupling strength between the qutrit and the resonator . it worth noticing that a tunable resonator @xcite and a tunable coupling qubit @xcite have been demonstrated in experiment . the principle of our scheme for generating the noon state on two microwave - photon resonators efficiently is shown in fig . [ fig1](a ) . suppose the initial state of the system is @xmath35 here the subscripts @xmath36 and @xmath37 represent the two resonators @xmath17 and @xmath10 , respectively . that is , the qutrit is in the state @xmath38 , and the resonators are in the state @xmath39 . here and below , @xmath40 is the fock state of the resonator @xmath13 , which means there are @xmath41 microwave photons in the resonator @xmath13 ( @xmath42 ) . to generate the noon state @xcite @xmath43 on @xmath17 and @xmath10 ( @xmath44 is a special situation of the noon state ) , our scheme needs @xmath45 steps . the first @xmath8 steps are described as follows . step @xmath36 : by making both @xmath17 and @xmath10 detune largely with the qutrit , one can use a drive field with the frequency equivalent to the transition frequency @xmath46 of the qutrit to pump the state of the qutrit from @xmath47 to @xmath48 . the amplitude of the drive field is chosen with a proper value for avoiding to pump the state from @xmath49 to @xmath47 . here after the operation time @xmath51 ( @xmath52 is the proper amplitude of the drive field for pumping the qutrit from @xmath53 to @xmath48 ) , the state of the system evolves into @xmath54 subsequently , one can tune the transition frequencies of the qutrit and the two resonators to make @xmath17 resonate with the qutrit in the transition @xmath55 . if the coupling strength between @xmath17 and the qutrit is tuned with a proper value before the resonance , one can neglect the interaction between @xmath17 and the qutrit in the transition @xmath56 . meanwhile , @xmath10 and the qutrit detune largely with each other . after the interaction time @xmath57 , the state of the system becomes @xmath58 step @xmath59 ( @xmath60 ) : by repeating the operation of the step 1 for @xmath61 times and maintaining @xmath10 detuning largely with @xmath17 and the qutrit all the time , the state of the system is changed to be @xmath62 the whole operation time is @xmath63 here , @xmath64 is the rotated - operation time of the qutrit and @xmath65 is the resonated - operation time between the qutrit and the @xmath17 . the details of the first @xmath8 steps have been described above . the next @xmath66 steps are described as follows . step @xmath67 : by making both @xmath17 and @xmath10 , detune largely with the qutrit , one can apply a drive field with the frequency equivalent to the transition frequency @xmath68 of the qutrit to rotate the states of the qutrit with @xmath56 . by choosing the proper amplitude of the drive field , one can avoid to flip the qutrit with @xmath55 . after the operation time @xmath69 , the state of the system evolves from eq.([n1 ] ) to @xmath70 applying a drive field with the frequency equivalent to the transition frequency @xmath46 of the qutrit , one can pump the state of the qutrit from @xmath19 to @xmath20 . the amplitude of the drive field is chosen with a proper value for avoiding to pump the state from @xmath18 to @xmath19 . after the operation time @xmath71 , the state of the system evolves into @xmath72 subsequently , one can tune the transition frequencies of the qutrit and the two resonators to make @xmath10 resonate with the qutrit in the transition @xmath29 . if the coupling strength between t @xmath17 and the qutrit is tuned with a proper value before the resonance , one can neglect the interaction between @xmath10 and the qutrit in the transition @xmath73 . meanwhile , @xmath17 and the qutrit detune largely with each other . after the interaction time @xmath74 , the state of the system becomes @xmath75 step @xmath76 @xmath77 : by repeating the operation of the step @xmath67 for @xmath78 times , and maintaining @xmath17 detuning largely with the qutrit all the time , the state of the system is changed to be @xmath79 the final step : applying a single - qubit operation to complete the rotations of the states @xmath80 and @xmath81 , the state of the system evolves into @xmath82 by resonating @xmath10 and the qutrit in the transition @xmath56 , and making @xmath17 detune largely with the qutrit , the state shown in eq.([m+14 ] ) is changed to be @xmath83 here , we have generated the noon state on two microwave - photon resonators efficiently . the operation time of the second @xmath66 steps is @xmath84 @xmath85 is the rotation - operation time of the qutrit and @xmath86 is the resonance - operation time between the qutrit and @xmath10 . in which , we neglect the operation time of the single - qubit operation in the final step for generating the noon state with large number of the @xmath8 and @xmath66 . we have described the process of our scheme for generating the noon state on two superconducting resonators which are coupled to a @xmath9-type - energy - level structure superconducting charge qutrit . it includes two kinds of quantum operations . the first one is the resonant operation on the qutrit and the resonators . the second one is the single - qubit operation on the qutrit . they are the high - fidelity , high - efficiency , and simple quantum operations in experiment in circuit qed systems . the whole operation time of our scheme for generating the noon state @xmath87 is @xmath88 in the calculation for the operation time in our scheme , we neglect the time for changing the transition frequencies of the superconducting qutrit and the superconducting resonator , and the operation time of the single - qubit operation in the final step . compared with the one in ref.@xcite , our scheme for generating the noon state on superconducting resonators is much faster as it is composed of the resonant controls . compared with the one in refs.@xcite , both the number of the resonators and that of the qutrits required in our scheme are much smaller as there are three superconducting resonators and two superconducting qutrits in the scheme in refs.@xcite , but only two superconducting resonators and a superconducting qutrit used in our scheme . moreover , the single - qubit operation required in our scheme can be achieved with the simple classical drive field , and it is simpler than the one used in ref.@xcite as the amplitude of the drive field should be designed with a complex type and it is difficult to be realized in experiment in the latter . in ref . @xcite , a similar method is used to generate the noon state on two resonators . in their work , the transmon qutrit should be maintained in the first @xmath8 steps in the third excited state when there is no microwave photons in each resonators . it worth noticing that the higher excited states lead to a lower fidelity operation @xcite . luckily , our scheme does not require us to maintain the qutrit in its third excited state all the time , which relaxes the requirements of its implementation in experiment , compared with the one in ref.@xcite . compared with a transmon qutrit , the level anharmonicity of a charge qutrit is larger and it is better for us to tune the different transitions of the charge qutrit resonant to the resonator @xcite . in summary , we have proposed an efficient scheme to generate the noon states on two superconducting resonators , assisted by a superconducting qutrit . it requires some high - fidelity quantum operations , that is , the resonant operation on the qutrit and the resonator and the single - qubit operation on the qutrit . our scheme is a fast and simple one . moreover , it does not require to maintain the qutrit in the third excited state with a long time , which relaxes the requirements of its implementation in experiment . this work is supported by the national natural science foundation of china under grant no . 11174039 and nect-11 - 0031 . hu c y , young a , obrien j l , et al . giant optical faraday rotation induced by a single - electron spin in a quantum dot : applications to entangling remote spins via a single photon . phys rev b , 2008 , 78 : 085307 yang w l , yin z q , xu z y , et al . one - step implementation of multiqubit conditional phase gating with nitrogen - vacancy centers coupled to a high - q silica microsphere cavity . appl phys lett , 2010 , 96 : 241113 qian y , zhang y q , xu j b. amplifying stationary quantum discord and entanglement between a superconducting qubit and a data bus by time - dependent electromagnetic field . chin sci bulletin , 2012 , 57 : 1637 - 1642 | we present an efficient scheme for the generation of noon states of photons in circuit qed assisted by a superconducting charge qutrit .
it is completed with two kinds of manipulations , that is , the resonant operation on the qutrit and the resonator , and the single - qubit operation on the qutrit , and they both are high - fidelity operations . compared with the one by a superconducting transmon qutrit proposed by su et al .
( sci .
rep . * 4 * , 3898 ( 2014 ) ) , our scheme does not require to maintain the qutrit in the third excited state with a long time , which relaxes the difficulty of its implementation in experiment . moreover , the level anharmonicity of a charge qutrit is larger and it is better for us to tune the different transitions of the charge qutrit resonant to the resonator , which makes our scheme faster than others .
+ * key words : * entanglement production , noon states , microwave - photon resonators , superconducting charge qutrit , circuit qed |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the hubbard model @xcite is one of the generic models in many particle physics . due to difficulties with the analytic solution of the hubbard model in two dimensions , this model is intensively studied with various numerical algorithms , e.g. exact diagonalization @xcite , @xcite , @xcite @xcite , stochastic diagonalization @xcite , @xcite and quantum monte carlo algorithms @xcite , @xcite , @xcite , @xcite . the single band hubbard model with additional next nearest neighbor hopping is given in real the space by : @xmath2 the sum @xmath3 is over the nearest neighbors and @xmath4 is the sum over the next nearest neighbors . @xmath5 is the creation operator for an electron with spin @xmath6 on site @xmath7 and @xmath8 is the corresponding number operator . throughout this article we take @xmath9 as energy unit . in the momentum space this hamiltonian reads as : @xmath10 with @xmath11 the usual hubbard - model ( @xmath12 ) has a van hove singularity in the density of states at half filling in the noninteracting case ( @xmath13 ) . it is possible to move this van hove singularity to any electron filling by extending the hubbard model by an additional next nearest neighbor hopping @xmath1-term in the kinetic energy . the @xmath0-hubbard model @xcite shows superconductivity for repulsive interactions @xmath14 with d@xmath15-symmetry @xcite and for attractive interaction with on site s - symmetry @xcite , @xcite , @xcite . as a measure for the superconductivity we calculate the reduced two particle density matrix according to the concept of yang @xcite . from this two particle density matrix we calculate the two particle correlation functions for different symmetries and additionally we calculate the vertex correlation function @xcite . the van hove scenario predicts an increase of @xmath16 for fillings close to a van hove singularity @xcite . we study the influence of the van hove singularity on the superconducting correlation functions by modifying the @xmath1-hopping parameter for fixed fillings . here we use the lanczos - algorithm @xcite , @xcite as exact diagonalization technique to determine the ground state and from there ground state properties of the @xmath0-hubbard model . the basic limitations on the calculations of large system sizes with the lanczos - algorithm is the huge memory consumption of this method . but to our surprise we found that even for hubbard systems of size @xmath17 , which the lanczos method is capable of handling , the cpu consumption of the simulations was substantial and made a detailed scan of the parameter space given by interaction strength @xmath14 , filling @xmath18 and next nearest neighbor hopping @xmath1 almost impossible . we therefore implemented the lanczos - method on ibm sp2 parallel computer with mpi ( message passing interface ) for the communication between the processes to speed up the calculations . before turning our attention to the parallel techniques we outline the basic concept of the lanczos method @xcite . in the case of the exact diagonalization the hamiltonian @xmath19 of the system is written in matrix or heisenberg representation . one chooses an orthonormal single particle basis , to represent the many - particle states . for the @xmath0-hubbard model we use the momentum - space representation . each many particle basis state @xmath20 is a product of the spin up @xmath21 and the spin down @xmath22 component : @xmath23 and spin @xmath6 . @xmath24 is the vacuum state . each many particle state @xmath25 can be represented by means of the basis states @xmath26 and coefficients @xmath27 : @xmath28 the hilbert space of a system consisting of a lattice of @xmath29 sites and @xmath30 electrons with spin up and @xmath31 electrons with spin down has the dimension @xmath32 as the size of the hilbert space also determines the size of the computer memory , that is used in the calculation , one tries to reduce the hilbert space . the usual way to restrict the size of the hilbert space @xmath33 is to apply symmetries of the lattice and the hamiltonian . the translation invariance is a symmetry easily implemented when solving the @xmath0-hubbard model in momentum space representation with the lanczos algorithm . the hilbert space decomposes into subspaces @xmath34 containing only basis states @xmath20 , which have the same total momentum @xmath35 : @xmath36 where @xmath37 and @xmath38 are the momenta of the creation operators @xmath39 resp . @xmath40 in eq . [ eqphi ] . we denote the number of states of particles with the same spin with the total momentum @xmath41 as @xmath42 . in a @xmath17 system with @xmath43 the number of states @xmath44 varies for different momentum between @xmath45 and @xmath46 . the states @xmath47 must have the total momentum @xmath48 , so that the product state latexmath:[$|\phi_{i_\uparrow,\uparrow } \rangle \otimes @xmath34 . altogether the subspace @xmath34 has the size @xmath50 for various fillings @xmath51 table [ tabstates ] gives an overview . when storing the coefficients @xmath27 with 8 byte floating point numbers the memory demand for one state is @xmath52 mbyte for a lattice with 16 sites and @xmath53 electrons . even if one uses the translation symmetry it remains a demand of @xmath54 mbyte for each state . in the standard lanczos algorithm @xcite , @xcite it is necessary to store three many particle states @xmath55 to calculate the ground state and the ground state energy . we implemented the lanczos - iteration @xcite : @xmath56 with the coefficients @xmath57 ( diagonal ) and @xmath58 ( offdiagonal ) of the tridiagonal matrix @xmath59 and the lanczos - vectors @xmath60 . from this tridiagonal matrix @xmath59 we calculate the eigenvalues of @xmath19 . in the lanczos - scheme it is not necessary to transform the matrix @xmath19 . therefore it is even not necessary to store the matrix elements @xmath61 ; they are only calculated , when they are needed for the further evaluation of the lanczos - iteration ( eq . [ eqlanczos ] ) . in this section we concentrate our effort on how to speed up the simulations with the lanczos method . the determination of the ground state properties of a hamiltonian with the lanczos algorithm consists of two main parts concerning the consumption of cpu - time . first the ground state energy @xmath62 and the ground state @xmath63 are calculated . the second main part is the determination of the two particle density matrix @xcite : @xmath64 as the main observable of interest . to handle an arbitrary state @xmath55 it is necessary to know for each basis state @xmath26 the indices @xmath37 and @xmath65 of the creation operators in eq . [ eqphi ] and the weights @xmath27 . a simple possibility for such an algorithm is the bitcoding or bitrepresentation of the basis states . here the momenta @xmath37 are labeled from @xmath66 to @xmath29 and the momenta @xmath65 from @xmath67 to @xmath68 . then the states @xmath26 are expressed by an one dimensional array of bits , which are one for occupied sites , and otherwise the bits are zero . if one interprets this array as binary representation of an integer number one has an algorithm to assign each basis state @xmath69 an index @xmath70 . as example we take a lattice with 4 sites and each two electrons with spin up and down : @xmath71 for any many particle state only the coefficients @xmath72 need to be stored . but the bitrepresentation has the great disadvantage , that a huge amount of memory is wasted , because the bitrepresentation of many integer numbers @xmath70 does not correspond to a valid basis state . for a system with @xmath29 latticepoints in an array of the length @xmath73 one only stores @xmath74 numbers . for the above example @xmath75 and @xmath53 there is @xmath76 . therefore it is desirable to use another algorithm ( @xmath77 ) . one example is the hashing algorithm @xcite . we developed a new algorithm . though we only present results for the momentum space representation , this algorithm can also be implemented very efficiently for the exact diagonalization in real space @xcite . first we are numbering the momenta from 1 to @xmath29 ( @xmath78 ) . second we define @xmath79 to fix the sign . the state with the number @xmath66 is @xmath80 in @xmath81 the electron @xmath82 moves from @xmath83 to @xmath84 . in @xmath85 this `` last '' electron has reached the latticepoint @xmath29 . next in @xmath86 the two creation operators with the highest index are increased by one , @xmath87 then the `` last '' creation operator moves . these are the states with the numbers @xmath88 to @xmath89 . next the `` last '' two electrons go to the sites @xmath84 and @xmath90 . if both electrons have reached the final two lattice sites ( @xmath91 , @xmath29 ) three electrons move in the same manner through the lattice . one gets the number @xmath92 of the state @xmath93 with @xmath94 where @xmath95 is the position of the electron @xmath96 in the lattice . using the translational invariance of the hamiltonian in the k - space representation means , that we keep the total momentum @xmath35 of the basis states @xmath97 fixed . in this case one can only choose the `` spin - up '' part of the basis state free and take such a `` spin - down '' state , that eq . [ eqk ] is full filled . this means , we must use another convention to label the states . we generate all states @xmath98 in the sequence as described above and calculate the momentum . for each momentum we count independently the indices . the coefficients @xmath27 are now labeled in following way : first we combine with state number 1 , momentum 1 and spin up with all possible states with spin down for a given @xmath35 . then we do the same with state number 2 , momentum 1 and spin up . next we switch to state 1 with momentum 2 and spin up and so on . in the parallel algorithm each of the @xmath99 processes stores the coefficients @xmath27 of @xmath100 basis states . @xmath27 is stored on process @xmath101 . in the lanczos method it is necessary to perform a matrix - vector multiplication between the matrix @xmath19 and a lanczos - vector @xmath102 . in the parallel algorithm this is carried out in the following way : for @xmath103 : * each process @xmath104 calculates the index numbers @xmath70 of the basis states @xmath105 , the multiplication results for the @xmath7-th state and the process , on which the states @xmath70 are stored . * each process exchanges the multiplication results and the numbers @xmath70 of the basis states , which are not stored on the process , with process @xmath106 . to calculate the two - particle density matrix @xmath107 in an efficient way , one only calculates the elements @xmath108 which are nonzero . that means one takes a basis state @xmath109 , one of the @xmath110 possible combinations of @xmath111 , @xmath112 , @xmath113 and @xmath114 and applies @xmath115 to @xmath116 , afterwards one calculates the index number of this transformed basis state @xmath117 . each process stores the complete matrix @xmath107 . this matrix takes for example in the @xmath17 system @xmath118 byte or @xmath119 kbyte of memory . in this algorithm one has to calculate @xmath120 expectation values , compared to @xmath121 expectation values that would be calculated if one took all combinations @xmath97 and @xmath122 into account . in a @xmath123 system with @xmath53 electrons @xmath124 and the amount of saved cpu - time is significant . but in a @xmath17 system with only @xmath125 electrons @xmath126 and this algorithm is slower . to calculate @xmath107 we use a similar way for the exchange of the weights and numbers as for matrix - vector multiplication ( see section [ sectionmatmult ] ) . at the end the values of the arrays @xmath107 of all processes are summed on one process . first we take a look on the dependence of the cpu time of one lanczos iteration ( eq.[eqlanczos ] ) for a different number of processes . as example we use a @xmath123 lattice with three different numbers of electrons ( fig . [ cpuiter ] ) . the small decay from 1 to 2 processes is due to communication between processes which is only necessary for more than one process . then one sees as expected a decrease of computation time . as expected this decrease vanishes for an increasing number of processes , since the communication is growing with the number of processes . in figure [ cpucorr ] we examine the cpu - consumption for the two - particle density matrix for the same system size and fillings as in figure [ cpuiter ] . here the gain of time is much larger than for the calculation of the energy . in this case more computation is performed for the determination of one matrix element . for more than 2 processes the dependence of nodes and cpu - time is nearly linear . this can be understood , if we look on the numbering of the states . most of the states @xmath127 are stored on the same node and only for a fraction of these states the weights must be interchanged with an other process . summarizing the algorithm for this parallel implementation of the lanczos - iteration and the determination of the two - particle density matrix is a coarse grained algorithm and therefore achieves a good speed - up on a parallel computer like the ibm sp2 , with only some , but very powerful , processors . for very many processors the communication grows dramatically and no further speed - up is reachable . now we want to turn our attention from the technical points of view to physical properties of the @xmath0-hubbard model . especially we study the influence of the next nearest hopping parameter @xmath1 on the ground state energy and superconducting correlation functions in the ground state of a @xmath17 cluster . we focus to @xmath128 electrons which corresponds to a filling @xmath129 . this is a so called closed shell situation , which can be also handled with the projector quantum monte carlo method @xcite . first we study the ground state energy @xmath62 in the attractive @xmath0-hubbard model ( fig . [ emin ] ) . for small and intermediate interaction strength ( @xmath130 ) there is a visible difference between the energy with @xmath12 and @xmath131 . this difference results from the changes in the structure of @xmath132 ( eq.[eqepsilon ] ) with @xmath1 . but for large interaction strength @xmath133 the influence of the kinetic part is vanishing . in this interaction regime the ground state energy is approximately linear with @xmath14 and approaches slowly the energy @xmath134 of the system without hopping . next we turn our attention to the superconducting correlation functions . in the concept of off diagonal long range order @xcite the largest eigenvalue and eigenvector of @xmath107 is calculated . the quantum monte carlo algorithms handle system sizes , where it is impossible to calculate the complete two - particle density matrix due to the memory consumptions @xcite . in order to compare the exact diagonalization results with the quantum monte carlo data we study two - particle correlation functions for certain symmetries @xcite , e.g. @xmath135 where the index s denotes the on site s - wave symmetry and d the @xmath136-wave symmetry . the factor @xmath137 gives the signs of the d - wave . ( @xmath138 in x- and @xmath139 in y - direction ) these full correlation functions have nonzero values even for a system with no interaction . responsible for this are the one - particle correlation functions @xmath140 which decay to zero with @xmath141 and thus do not really contribute to the long range behavior that signals superconductivity . to exclude the contribution of the one - particle correlation functions we define the vertex correlation function as @xmath142 in figure [ corrvgl ] we show @xmath143 and @xmath144 in dependence of the distance @xmath145 . for @xmath146 there is a visible difference only for @xmath147 . for @xmath148 the one - particle contributions are dominant . in this case it is important to study the vertex correlation function to get the `` superconducting '' correlations . but already for @xmath149 and @xmath150 the difference between full and vertex correlation function is less than 30% and it is less important to take @xmath151 in account . as a measure for the superconductivity in a system we show in fig . [ neguvertex ] the average of the vertex correlation function @xmath152 for a small interaction strength @xmath133 the increase of the correlation functions is small . between @xmath149 and @xmath153 there is a strong increase . finally at @xmath154 the curves flatten . for correlation functions the influence of the additional hopping @xmath1 remains important even if there is nearly no difference in the energies ( cp . [ emin ] and [ neguvertex ] , @xmath155 ) . therefore we study the influence of @xmath1 for the interaction @xmath156 ( fig . [ negutprime ] ) . the correlation functions have a broad maximum around @xmath157 . as it is commonly accepted @xcite the finite size gap has an influence on the superconducting correlation functions . the finite size gap @xmath158 in the energy dispersion @xmath132 ( eq . [ eqepsilon ] ) of the free system ( @xmath13 ) between the highest occupied state and the lowest unoccupied state is given in this case by : @xmath159 this means the finite size gap is becoming smaller with increasing @xmath160 . according to @xcite this will lead to an increase in the superconducting correlations . figure [ negutprime ] confirms this for @xmath161 . but at @xmath162 the correlations decrease again . therefore also the structure of the energy dispersion @xmath132 has an influence on the correlation functions . in figure [ negutprime ] the maximum of the correlation functions is not at @xmath163 , where the noninteracting system has a van hove singularity in the thermodynamic limet . there a maximum of @xmath16 is predicted by the van hove scenario @xcite . but for small system sizes one can not really speak of a van hove singularity . therefore the results of figure [ negutprime ] are not in contradiction to the van hove scenario . yet , the question remains , whether the observed increase of the correlation function results only from a vanishing finite size gap or is related to changes of the fermi surface . to clarify this point it will be necessary to calculate larger systems . next we study the influence of the @xmath1 hopping for repulsive interaction @xmath164 . quantum monte carlo calculations show a plateau for the d@xmath15 correlation function in the @xmath0-hubbard model @xcite , @xcite . figure [ posutprime ] shows the average correlation functions with @xmath136 symmetry . here the full correlation function is nearly independent of @xmath1 for @xmath165 . the vertex correlation function is much smaller than in the attractive case ( cp . [ negutprime ] and fig . [ posutprime ] ) . the increase of @xmath166 by a factor of about 10 between @xmath12 and @xmath167 is much larger than the increase of @xmath168 and @xmath169 in the attractive @xmath0-hubbard model and of @xmath170 in the repulsive model . in figure [ posulong ] the vertex correlation function with @xmath136 symmetry is plotted against the distance @xmath145 of the `` cooper pairs '' . for @xmath171 the vertex correlation function has , in contrast to larger @xmath1 , no longer a negative value at @xmath172 and is positive for all distances @xmath145 . this negative value at @xmath172 is also seen in the quantum monte carlo results for larger systems @xcite , @xcite . in the case of @xmath157 , where the gap @xmath158 gets zero , @xmath173 changes its shape completely and most values are negative and also the average is negative . in contrast to the attractive @xmath0-hubbard model , where the plateau is decreasing gradually , in the repulsive case one observes a complete break down of the plateau for @xmath174 . this means again , that the vertex correlation function depends strongly on the energy dispersion @xmath132 ( eq.[eqepsilon ] ) , which is transformed due to the @xmath1-hopping . as in the attractive @xmath0-hubbard model the maximum of the superconducting correlations is not at @xmath163 , where a van - hove singularity is in the non interacting infinite system . to clarify the origin of this behavior it is necessary to study larger systems . for a possible connection with the van - hove - scenario it is also necessary to calculate the density of states for this parameter regime we have presented an effective algorithm for the implementation of the exact diagonalization of the @xmath0-hubbard model in momentum - space respresentation . as method for the exact diagonalization we use the lanczos algorithm . we showed a detailed description of the parallel algorithm . the speed - up of the code is almost linear for the correlation functions and is increasing with increasing size of the hilbert space . the key point in our algorithm is a new method of labeling the states that is compact and in contrast to previous methods @xcite , @xcite also gives a consecutive order without any interruption . this makes the distribution on the different processes for parallelization a straightforward task . with the access to powerful modern parallel computers we were able to scan the parameter space of the @xmath0-hubbard model in more detail . the influence of the @xmath1-hopping to the ground state energy is vanishing with increasing interaction strength in the attractive @xmath0-hubbard model . this is in contradiction to the correlation functions with on site s - wave symmetry , where the influence of @xmath1 is remaining for all studied attractive interactions . the average full and vertex on site s - wave correlation functions have a broad maximum at @xmath175 , where the gap @xmath158 is vanishing , in the attractive @xmath0-hubbard model . in the repulsive @xmath0-hubbard model only the d@xmath15 vertex correlation functions show a strong increase with decreasing @xmath1 and gap @xmath158 . at @xmath175 and @xmath176 @xmath177 has a break down to a negative value . the origin of this behavior and a possible connection with the van - hove scenario for high @xmath16 superconductors is yet not clear . simulations for larger systems are necessary . we are grateful for the leibnitz rechenzentrum mnchen ( lrz ) for providing us a generous amount of cpu - time on their ibm sp2 parallel computer . werner fettes wants to thank the `` deutsche forschungs gemeinschaft '' ( dfg ) for the financial support . | we present a new parallel algorithm for the exact diagonalization of the @xmath0-hubbard model with the lanczos - method . by invoking a new scheme of labeling the states we were able to obtain a speedup of up to four on 16 nodes of an ibm sp2 for the calculation of the ground state energy and an almost linear speedup for the calculation of the correlation functions . using this algorithm we performed an extensive study of the influence of the next - nearest hopping parameter @xmath1 in the @xmath0-hubbard model on ground state energy and the superconducting correlation functions for both attractive and repulsive interaction . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
_ iscsi _ is a protocol designed to transport scsi commands over a tcp / ip network . + _ iscsi _ can be used as a building block for network storage using existing ip infrastructure in a lan / wan environment . it can connect different types of block - oriented storage devices to servers . + _ iscsi _ was initially standardized by ansi t10 and further developed by the ip storage working group of the ietf @xcite , which will publish soon an rfc . many vendors in the storage industry as well as research projects are currently working on the implementation of the iscsi protocol . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ " the small computer systems interface ( scsi ) is a popular family of protocols for communicating with i / o devices , especially storage devices . scsi is a client - server architecture . clients of a scsi interface are called " initiators " . initiators issue scsi " commands " to request services from components , logical units , of a server known as a " target " . a " scsi transport " maps the client - server scsi protocol to a specific interconnect . initiators are one endpoint of a scsi transport and targets are the other endpoint . the iscsi protocol describes a means of transporting of the scsi packets over tcp / ip , providing for an interoperable solution which can take advantage of existing internet infrastructure , internet management facilities and address distance limitations . " draft - ietf - ips - iscsi-20 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hyperscsi _ is a protocol that sends scsi commands using raw ethernet packets instead of the tcp / ip packets used for _ iscsi_. thus , it bypasses the tcp / ip stack of the os and does not suffer from the shortcomings of tcp / ip . _ hyperscsi _ focuses on turning ethernet into a usable storage infrastructure by adding missing components such as flow control , segmentation , reassembly , encryption , access control lists and security . it can be used to connect different type of storage , such as scsi , ide and usb devices . _ hyperscsi _ is developed by the _ modular connected storage architecture _ group in the network storage technology division of the data storage institute from the agency for science , technology and research of singapore @xcite . enbd is a linux kernel module coupled with a user space daemon that sends block requests from a linux client to a linux server using a tcp / ip connection . it uses multichannel communications and implements internal failover and automatic balancing between the channels . it supports encryption and authentication . + this block access technology is only useful with a linux kernel because of the linux specific block request format . + it is developed by the linux community @xcite under a gpl license . the following hardware was used to perform the tests : * _ test2 _ : + dual pentium 3 - 1 ghz + 3com gigabit ethernet card based on broadcom bcm 5700 chipset + 1 western digital wd1800jb 180 gbytes + 3ware raid controller 7000-series * _ test11 _ : + dual pentium 4 - 2.4 ghz ( hyperthreading enabled ) + 6 western digital wd1800jb 180 gbytes + 3ware raid controllers 7000-series or promise ultra133 ide controllers + 3com gigabit ethernet card based on broadcom bcm 5700 chipset * _ test13 _ : + dual amd mp 2200 + + 6 western digital wd1800jb 180 gbytes + 3ware raid controllers 7000-series or promise ultra133 ide controllers + 3com gigabit ethernet card based on broadcom bcm 5700 chipset * iscsi server : eurologic elantra ics2100 ip - san storage appliance - v1.0 @xcite + 3 scsi drives all the machines have a redhat 7.3 based distribution , with kernel 2.4.19 or 2.4.20 . + the following optimizations were made to improve the performance : sysctl -w vm.min-readahead=127 sysctl -w vm.max-readahead=256 sysctl -w vm.bdflush = 2 500 0 0 500 1000 60 20 0 elvtune -r 512 -w 1024 /dev / hd\{a , c , e , g , i , k } two benchmarks were used to measure the io bandwidth and the cpu load on the machines : * _ bonnie++ _ : v 1.03 @xcite + this benchmark measures the performance of harddrives and filesystems . it aims at simulating a database like access pattern . + we are interested in two results : _sequential output block_ and _sequential input block_. + bonnie++ uses a filesize of 9gbytes with a chunksize of 8kbytes . bonnie++ reports the cpu load for each test . however , we found that the reported cpu load war incorrect . so we used a standard monitoring tool ( vmstat ) to measure the cpu load during bonnie++ runs instead . * _ seqent_random_io64 : _ + in this benchmark , we were interested in three results : * * _ write _ performance : bandwidth measured for writing a file of 5 gbytes , with a blocksize of 1.5 mbytes . * * _ sequential reading _ performance : bandwidth measured for sequential reading of a file of 5 gbytes with a blocksize of 1.5 mbytes . * * _ random reading _ performance : bandwidth measured for random reads within a file of 5 gbytes with a blocksize of 1.5 mbytes . + this benchmark is a custom program used at cern to evaluate the performance of disk servers . it simulates an access pattern used by cern applications . _ vmstat _ has been used to monitor the cpu load on each machine . the server was the eurologic ics2100 ip - san storage appliance @xcite . the client was _ test13 _ , with kernel 2.4.19smp . two software initiators were used to connect to the iscsi server : ibmiscsi @xcite and linux - iscsi @xcite . we used two versions of linux - iscsi : 2.1.2.9 , implementing version 0.8 of the iscsi draft , and 3.1.0.6 , implementing version 0.16 of the iscsi draft . the results are given in the table below : [ cols="^,^,^,^,^,^ " , ] _ comments : _ softwareraid delivers more bandwidth than hardwareraid , but at a higher cpu cost . i would like to thank all the people from the it / adc group at cern for helping me in this study and particularly markus schulz , arie van praag , remi tordeux , oscar ponce cruz , jan iven , peter kelemen and emanuele leonardi for their support , comments and ideas . 1 ietf ip storage working group + http://www.ietf.org/html.charters/ips-charter.html mcsa hyperscsi + http://nst.dsi.a-star.edu.sg/mcsa/hyperscsi/index.html enhanced network block device + http://www.it.uc3m.es/~ptb/nbd/ eurologic elantra ics2100 + http://www.eurologic.com/products_elantra.htm bonnie++ + http://www.coker.com.au/bonnie++/ ibmiscsi + http://www-124.ibm.com/developerworks/projects/naslib/ linux - iscsi + http://linux-iscsi.sourceforge.net/ | we report on our investigations on some technologies that can be used to build disk servers and networks of disk servers using commodity hardware and software solutions .
it focuses on the performance that can be achieved by these systems and gives measured figures for different configurations .
it is divided into two parts : iscsi and other technologies and hardware and software raid solutions .
the first part studies different technologies that can be used by clients to access disk servers using a gigabit ethernet network .
it covers block access technologies ( iscsi , hyperscsi , enbd ) .
experimental figures are given for different numbers of clients and servers .
the second part compares a system based on 3ware hardware raid controllers , a system using linux software raid and ide cards and a system mixing both hardware raid and software raid .
performance measurements for reading and writing are given for different raid levels . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
ultracold atoms provide an excellent forum to study complex quantum mechanical behavior . an example is the superfluid to mott insulator transition @xcite , where experimental efforts have imaged this as a quantum phase transition at the single atom level @xcite . a quantum phase transition is a fundamental change in the ground state as a parameter is altered , in the superfluid - mott insulator example the parameter is lattice depth . near this quantum phase transition the temperature dependence of superfluidity and quantum criticality has been studied @xcite . another system in ultracold atoms , which exhibits a quantum phase transition , is the 2 component bose einstein condensate ( bec ) , where by tuning the interactions between the components the gas can change from a miscible to an immiscible phase , as has been experimentally demonstrated in rb @xcite . theoretical studies showed that if @xmath0 is greater ( less ) than zero , the gas is miscible ( immiscible ) @xcite . here @xmath1 is the coupling strength in the mean field treatment between the @xmath2 and @xmath3 component . interestingly , 2 component becs have been used to create vortices @xcite and study non - equilibrium dynamics @xcite . in this work , we look at the character of the excitations across the quantum phase transition in the 2 component bec system . for a miscible system , the modes are collective in nature , and the condensates move either in - phase or out - of - phase with each other . for the immiscible system , the excitations are either collective or interface excitations . this work offers a new perspective into the nature the miscible - immiscible transition with characterization of the excitations . there has been previous work on excitation spectra for trapped 2 component bec @xcite . that work studied symmetry breaking as a function of particle number and found that a mode goes to zero when the system becomes immiscible , but they did not study the nature of the quasiparticles . ref . @xcite studied the ground state and characterized the mode that goes soft ( energy goes to zero ) , here we extend the analysis to many low - lying modes of the system . to obtain the excitation spectrum , we solve the bogoliubov de gennes equations @xcite for a trapped gas with contact interactions . first , we must solve the gross pitaevskii equation for the 2 condensates ( @xmath4 ) : @xmath5 @xmath6 is the kinetic energy and trapping potential and @xmath7 is the chemical potential for the @xmath2 component . we normalize @xmath4 so that @xmath8=1 . now we consider the ground state and its excitations to be of the form : @xmath9 . substituting this into the time dependent version of eq . ( [ gpe ] ) and collecting powers of @xmath10 and linear terms in @xmath11 , we find the excitations are given by : @xmath12 we have assumed @xmath4 is real , and that the different components have equal number and mass . the second term contains both the exchange and anomalous term , which couples @xmath13 to @xmath14 and @xmath13 to @xmath15 , respectively . these terms would be non - local if the interaction was finite range . these excitations are normalized in the standard way : @xmath16-@xmath17 @xcite . to perform this work , we focus on the quasi2d case where high resolution experimental imaging is possible @xcite . for clarity , we will focus on two examples : one is miscible and the other immiscible , both far away from the transition so the character of the excitations is clear . we consider @xmath18100 and @xmath19=1.01 g ; for the miscible example , we pick @xmath20=0.5 g and an immiscible example , we pick @xmath20=2 g , where @xmath21 is the strength of the contact interaction , @xmath22 is the 3d s - wave the scattering length ( @xmath23 ) , @xmath24 is the axial harmonic oscillator length and @xmath25 is the trapping frequency in the tightly confined direction . we only consider @xmath26 and equal masses . we rescale the equations into trap units , so the energy scale is @xmath27 and the length scale is @xmath28 where @xmath29 is the trapping frequency in the x - y plane . we can loosely relate this to experiments , for g=100 if we pick @xmath30=1000 , @xmath25/@xmath29=100 , and @xmath22=100 @xmath31 , then this example corresponds to radial trapping frequencies , of @xmath3238 hz for k and @xmath3211 hz for cs . it is worth mentioning that the chemical potentials for each component are about equal ( @xmath33 ) . more importantly , for the immiscible system , @xmath34 is @xmath35 and this gives a healing length of @xmath36 ( for the miscible system @xmath37 ) . further details of how we solve these equations ( [ gpe],[bdg ] ) appear in ref . @xcite . only slightly less than @xmath38 , and in ( b ) we show the quantum gas just after quantum phase transition when the ground state breaks rotational symmetry and @xmath20 only slightly greater than @xmath38 . for ( c ) we have shown the standard example of an immiscible gas with @xmath39 . the bec labeled 1 ( 2 ) is shown as blue ( red ) . ] in fig . [ ground ] we see the ground state changes character as we vary @xmath20 . the component with the smaller @xmath40 is more dense in the middle of the trap . we define @xmath38 as the value of @xmath20 when the ground state changes character to a broken symmetry state which begins the immiscible regime . when @xmath41 , the ground state of the system has azimuthal symmetry and the 2 becs overlap , ( a ) . however as @xmath20 is increased to @xmath38 , the ground state suddenly changes , and the azimuthal symmetry is broken , see ( b ) . as @xmath20 is further increased , the 2 becs separate further and decrease their overlap as it becomes energetically costly , ( c ) . a similar evolution of the ground state as a function of @xmath20 was reported in ref . @xcite . it is challenging to find the ground state for all @xmath20 . to do so , we use the conjugate gradient method . we have found the best initial guess is one with a slightly broken symmetry and poor overlap with the final group state . we seed the noise so that the interface would be along the y axis . if the overlap between the initial guess and the ground state is too large then it is easy to get stuck in a local energy minimum . when the conjugate gradient method fails to find the true ground state , the solutions to the bogoliubov de gennes equations have complex eigenvalues when we vary @xmath20 , the previous solution is thrown out . in this way we reliably find the excitation spectrum of the 2 component bec with only real eigenvalues . is at 1.04 g for trapped example with @xmath42=100 . note the presence of strong avoided crossings and broken degeneracies after rotational symmetry has been broken , @xmath43 . for the miscible system as @xmath42 is increased , the out - of - phase collective excitations dramatically lower in energy . some of the excitations shown in fig . [ quasi ] ( [ quasi2 ] ) are labeled on the left ( right ) of this figure . ] in fig . [ energytrans ] we show bogoliubov excitation energies ( @xmath44 ) as a function of @xmath20 . this shows the transition from miscible to immiscible at @xmath45 , where a mode goes soft . homogeneous theory predicts this transition at @xmath46 @xcite . the discrepancy is explained by the trap and finite size of the gas @xcite . in fact , if we were to increase g ( keep @xmath47 fixed ) ; @xmath48 decreases toward 1 . for example if g=400 ( @xmath49 ) then @xmath50 . a recent study explored how the trapping and the kinetic energy contributions impact the criteria for immisciblity @xcite . our findings are consistent with their results . to further understand fig . [ energytrans ] , we start with @xmath41 where the systems is miscible . the quasi - particles are readily classified based on their azimuthal symmetry and the relative motion of the 2 condensates . as @xmath20 increases towards @xmath38 , many modes decrease in energy . then at @xmath38 two modes go soft or their energies go to zero . for @xmath43 , many degeneracies are broken , and there are many avoided crossing as @xmath20 is further increased . this is where the excitations mix and change character . for the miscible side of spectrum , there are energy crossings , but they are protected symmetry and do not couple . to further understand this transition , we look at the mode which goes soft at @xmath38 . with @xmath51 . energy is shown for each quasi - particle in trap units . ] in fig . [ quasi ] we show density perturbations associated with 2 low - energy excitations for the miscible system ( far left of fig . [ energytrans ] with @xmath51 ) . density perturbations from bogoliubov de gennes theory ( t=0 ) are given by @xmath52 for the mode @xmath53 . in fig . [ quasi ] we show both ( a ) the out - of - phase and ( b ) the in - phase slosh modes . the condensates for this example are co - spatial and nearly identical . they look similar to those in fig . [ ground ] ( a ) . in this figure , we separated the two components for clarity . the density moves from the regions define by the dashed lines to the regimes defined by the solid lines . the color of the perturbations matches the color for the associated condensate . we have drawn arrows to illustrate the motion of the density perturbations . the contours are shown for 0.25 , 0.5 and 0.75 of the maximum value of the perturbation , and condensate density is shown in the background . the energy of the mode is reported on the figure in trap units . in fig . [ quasi ] ( a ) we show a slosh mode , but the motions of the 2 becs are out - of - phase with each other . the solid lines coincide with the dashed lines for the other condensate s motion . for example , the blue condensate , ( i ) , sloshes from @xmath54 to @xmath55 while the red condensate , ( ii ) , sloshes from @xmath55 to @xmath54 . there is no center - of - mass motion in this case . in contrast , ( b ) shows a mode with center - of - mass motion which is a kohn mode of energy 1@xmath27 . the motion of each condensate coincides with the other ; both the blue and red condensate slosh from @xmath55 to @xmath54 in phase with each other . it is important to note that the out - of - phase slosh mode has the lowest energy , and goes soft at the quantum phase transition . for @xmath41 , modes with @xmath56 , where @xmath57 is the azimuthal quantum number , have a degenerate twin . in our case with real quasi - particle modes , degenerate modes are related by a rotation of @xmath58 . for example in [ quasi ] ( a ) and ( b ) there are degenerate twins are just rotated by @xmath59 . referring back to the energy spectra in fig . [ energytrans ] , we see that as @xmath20 increases towards @xmath38 the energy of the mode in fig [ quasi ] ( a ) decreases ( while ( b ) stays at @xmath60 ) . more generally , all out - of - phase modes significantly lower in energy as @xmath42 is increased to @xmath38 . in fact , the out - of - phase modes with @xmath61 , @xmath62 , @xmath63 , and @xmath64 are all below @xmath60 at @xmath38 . then at @xmath38 the energy of the out - of - phase slosh goes to zero and one mode stays zero for @xmath43 . the other mode ( rotated by @xmath59 ) shoots up in energy as @xmath20 is further increased beyond @xmath38 . the ground state spontaneously breaks rotational symmetry - which the hamiltonian has - and this leads to an extra zero energy goldstone mode in the excitations spectrum @xcite . there are already two goldstone modes associated with broken phase symmetry of each condensate , and they are : @xmath65 and @xmath66 . if one looks more closely at the third mode with @xmath67 , one finds it is a rotation of the interface ( @xmath68 ) . this extra zero energy mode has already been observed in the 2 component becs @xcite . goldstone modes have been discussed in more detail for spinor becs @xcite , and ref . @xcite found similar behavior for a goldstone mode in an attractive condensate with a bogoliubov de gennes treatment . as we have said for the miscible system , the classification of the modes is simple : we use azimuthal symmetry and relative motion of the two condensates . but for the immiscible system , there is no rotational symmetry , so the characterization of the excitations must be different . to study this in more detail , we look at several quasi - particles . . the x and y axes are in trap units . ( a , b ) are slosh modes , ( c , d ) are quadrupole modes ( e , f ) are breathing modes , and ( g , h ) are interface modes . the energy is reported in trap units and for this example @xmath69 . ] in fig . [ quasi2 ] we show the density perturbations associated with quasi - particle excitations for the immiscible system ( @xmath70 , @xmath71 , @xmath72 , and g=100 ) . there are two types of quasi - particle modes : first , bulk excitations which look like those from a standard condensate , and second , interface excitations where the excitations are localized to the interface between the 2 condensates . since there is no azimuthal symmetry , to classify the bulk modes we need to access how their motion is oriented relative to the interface . in fig . [ quasi2 ] , to depict the motion of the density perturbations , we show arrows in a few examples . the density moves from the dashed regions to the solid regions . we also show the energy of the excitation in trap units . the contours are shown for 0.25 , 0.5 and 0.75 of the maximum value of the perturbation and condensate density shown in the background . the collective modes look like those in a standard bec , however the two becs now act collectively to retain the excitation character . first , we look at the slosh modes of the systems . they are shown in fig . [ quasi2 ] ( a , b ) . in ( a ) the slosh mode with the center - of - mass displacement parallel to the interface ( kohn mode ) is shown and in ( b ) a slosh mode and center - of - mass displacement is perpendicular to interface ( also a kohn mode ) is shown . the arrows show that in ( a ) both the blue and red condensate sloshes from @xmath55 to @xmath54 . for example ( b ) both the blue and red condensate sloshes from @xmath73 to @xmath74 . next we show the quadrupole modes in fig . [ quasi2 ] ( c , d ) . ( c ) shows a quadrupole mode with a nodal line along interface , and ( d ) shows a quadrupole mode where the density increases at interface . these excitations are typical of a single component bec where the excitations are related by a @xmath58 rotation . but in this case , the 2 becs collude to make the excitation . additionally , these two excitations are very similar in energy . we show two breathing modes in fig . [ quasi2 ] ( e , f ) . they can be classified as in - phase and out - of - phase motion of the 2 condensates . in ( e ) we show an in - phase breathing mode , where both becs inhale at once , or they both move into or out of the center of the trap in unison . in ( f ) we show an out - of - phase breathing mode , where one condensate inhales and the other exhales . the energies of these modes are notably different : 2 and 2.73 @xmath27 . there is another class of excitation in the immiscible 2 component bec : interface excitations . two examples are shown in fig . [ quasi2 ] ( g , h ) . these excitations are localized along the interface , and in general they are out - of - phase excitations , i.e. the density of one moves to where the other is leaving . note these are low energy excitations , in fact ( g ) is the lowest energy excitations , for the system at 0.58 @xmath27 and ( h ) is only slightly higher than the two kohn modes at 1.08 @xmath27 . if @xmath42 is increased the mode in ( h ) will decrease below 1@xmath27 . so if the chemical potential is increased , then the interface modes become lower in energy . to illustrate this , we look at how the excitation energies change as a function of @xmath42 while keeping the ratio of the interactions fixed ( @xmath75 ) . this is shown in fig [ energyn ] ( a ) . for reference , on the far right where @xmath42=600 and @xmath76 , we have marked the interface modes with red @xmath77 s , there are 16 interface modes with energy under 3.5@xmath27 . in contrast , on the far left where @xmath51 , we have marked the interface modes with red @xmath78 s and there are only 7 modes under 3.5@xmath27 . as one moves up in excitation energy , each new excitation simply adds another bend to the interface . one more important point of fig . [ energyn ] ( a ) , is that as @xmath42 increases the energies of interface modes decrease . ) . the quasiparticles from fig . [ quasi2 ] are on the far left side of spectrum . the interface excitations are marked by @xmath77 or @xmath78 on either side of the figure . the energies of the quasiparticle shown in fig . [ quasi2 ] ( g , h ) are labeled . in ( b ) and ( c ) the density perturbations for the interface excitations when @xmath76 or @xmath42=600 , they are labeled on the right side of ( a ) . their energy is reported in trap units.,title="fig : " ] + ) . the quasiparticles from fig . [ quasi2 ] are on the far left side of spectrum . the interface excitations are marked by @xmath77 or @xmath78 on either side of the figure . the energies of the quasiparticle shown in fig . [ quasi2 ] ( g , h ) are labeled . in ( b ) and ( c ) the density perturbations for the interface excitations when @xmath76 or @xmath42=600 , they are labeled on the right side of ( a ) . their energy is reported in trap units.,title="fig : " ] ) . the quasiparticles from fig . [ quasi2 ] are on the far left side of spectrum . the interface excitations are marked by @xmath77 or @xmath78 on either side of the figure . the energies of the quasiparticle shown in fig . [ quasi2 ] ( g , h ) are labeled . in ( b ) and ( c ) the density perturbations for the interface excitations when @xmath76 or @xmath42=600 , they are labeled on the right side of ( a ) . their energy is reported in trap units.,title="fig : " ] fig . [ energyn ] ( b ) and ( c ) show two examples of the higher energy interface modes . first , fig . [ energyn ] ( b ) is a mode with 6 bends in the interface . an interesting point about ( c ) is that it is a hybrid mode , it also has some density perturbations near the edge of the gas , away from the interface . the mode shown in ( c ) is in a region where several modes are crossing and their character is changing . if we further increase @xmath42 , the interface mode lowers in energy and loses it collective nature and looks more like mode in ( b ) . related excitations have been studied in non - equilibrium simulations of 2 component becs where rayleigh - taylor instabilities have been predicted @xcite . in these studies , the value of @xmath79 is changed and this drives one bec into the other , the interface then becomes unstable and a rayleigh - taylor instability forms . in conclusion , we have characterized the excitations of a 2 component bec within the bogoliubov de gennes framework . we found that as @xmath20 is increased from a miscible regime to an immiscible regime , fig . [ energytrans ] , generally , all of the out - of - phase excitations lower in energy . the energy of the out - of - phase slosh mode goes to zero , fig . [ quasi ] ( a ) . this mode becomes new goldstone modes when the rotational symmetry is spontaneously broken in the immiscible system . we looked at the excitations of the immiscible systems when @xmath80 in fig . [ quasi2 ] . we found that there are bulk modes which look similar to the excitations of a single trapped bec . there are also excitations localized at the boundary between the condensates . one of these interface modes is lowest energy mode when the becs are immiscible , and there are many other low energy interface modes , see the red @xmath78 s or x s in fig . [ energyn ] ( a ) . furthermore , if one goes to a more strongly interacting bec regime ( while still immiscible ) , the interface modes lower in energy . future work will seek to understand the relationship between bogoliubov excitations across the miscible - immiscible transition and critical phenomena . the effect of temperature on this system will be studied within the hartree fock bogoliubov framework , where bogoliubov excitations are thermally occupied . additionally , we will look at how this collective excitation changes with non - local dipolar interactions @xcite . the author gratefully acknowledges support through a ldrd ecr grant , lanl which is operated by lans , llc for the nnsa of the u.s . doe under contract no . de - ac52 - 06na25396 . this research was supported in part by the national science foundation under grant no . nsf phy11 - 25915 . the author also gratefully acknowledges conversations with e. timmermans , l. a. collins , and r. m. wilson . m. greiner _ et al . _ , nature * 415 * , 39 ( 2002 ) . w. s. bakr _ nature * 462 * , 74 ( 2009 ) . s. trotzky _ et al . _ , nat * 6 * , 998 ( 2010 ) . _ , science . * 335 * , 1070 ( 2012 ) . s. b. papp , j. m. pino , and c. e. wieman , phys . * 101 * , 040402 ( 2008 ) . h. pu and n. bigelow , phys . lett . * 80 * , 1130 ( 1998 ) ; _ ibid . _ * 80 * , 1134 ( 1998 ) . e. timmermans , phys . lett . * 81 * , 5718 ( 1998 ) . m. r. matthews _ et al . _ , lett . * 83 * , 2498 ( 1999 ) . k. m. mertes _ et al . _ , lett . * 99 * , 190402 ( 2007 ) . r. p. anderson , c. ticknor , a. i. sidorov , and b. v. hall phys . a * 80 * , 023603 ( 2009 ) . e. nicklas _ et al . * 107 * , 193001 ( 2011 ) . d. gordon and c. m. savage , phys . a * 58 * 1440 ( 1998 ) . j. g. kim and e. k lee , phys . e * 65 * , 066201 ( 2002 ) . a. l. fetter and j. d. waleck , _ quantum theory of many - particle systems _ , ( dover , new york 2003 ) . . hung , x. zhang , n. gemelke , and c. chin , nature * 470 * , 236 ( 2011 ) . c. ticknor , phys . a * 85 * , 033629 ( 2012 ) ; phys . a * 86 * , 053602 ( 2012 ) . r. navarro , r. carretero - gonzlez , and p. g. kevrekidis , phys . a * 80 * , 023613 ( 2009 ) . l. wen _ et al . _ phys . rev . a * 85 * , 043602 ( 2012 ) . for a review see : t. brauner , symmetry * 2 * , 609 ( 2010 ) . t .- l . ho and v. b. shenoy , phys . * 77 * , 3276 ( 1996 ) . s. uchino , m. kobayashi , and m. udea , phys . a * 81 * , 063623 ( 2010 ) ; s. uchino , m. kobayashi , m. nitta , and m. ueda phys . rev . lett . * 105 * , 230406 ( 2010 ) . r. kanamoto , h. saito , and m. ueda , phys . rev . lett . 94 , 090404 ( 2005 ) . k. sasaki , n. suzuki , d. akamatsu , and h. saito , phys . a * 80 * , 063611 ( 2009 ) ; n. suzuki , h. takeuchi , k. kasamatsu , m. tsubota , and h. saito , phys . a * 82 * , 063604 ( 2010 ) ; h. takeuchi , n. suzuki , k. kasamatsu , h. saito , and m. tsubota , phys . b * 81 * , 094517 ( 2010 ) ; t. kadokura , t. aioi , k. sasaki , t. kishimoto , and h. saito , phys . rev . a * 85 * , 013602 ( 2012 ) . s. gautam and d. angom , phys . a * 81 * , 053616 ( 2010 ) . a. bezett , v. bychkov , e. lundh , d. kobyakov , and m. marklund , phys . a * 82 * , 043608 ( 2010 ) ; d. kobyakov , v. bychkov , e. lundh , a. bezett , and m. marklund , phys . a * 86 * , 023614 ( 2012 ) . r. m. wilson , c. ticknor , j. l. bohn , and eddy timmermans , phys . a * 86 * , 033606 ( 2012 ) . | we present analysis of the excitation spectrum for a 2 component quasi2d bose einstein condensate .
we study how excitations change character across the miscible to immiscible phase transition .
we find that the bulk excitations are typical of a single - component bec with the addition of interface bending excitations .
we study how these excitations change as a function of the interaction strength . |
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the recent market turbulence caused by the credit crunch has exposed in a drastic way the consequences of overconfidence in financial modelling assumptions . typically , a financial model , such as the famous black - scholes model , will assume that the price of an asset follows a given stochastic process whose parameters need to be calibrated to market prices . if a model becomes an accepted standard and most market participants adopt it , problems can occur when assumptions that hold under normal market conditions are also expected to hold under abnormal ones . an example is the stock market crash of @xmath2 , where the volatilities used for pricing at - the - money options were also used for pricing far out - of - the - money put options . as the market headed downwards , it turned out that the true hedging cost for somebody who had sold such puts was far greater than the received premium . another good example is described in the recent paper @xcite , where the authors demonstrate for cdos and cdo@xmath3s what can happen to asset prices when model parameters that are hard to observe or estimate with sufficient accuracy are put to a true stress test . however , they write : `` the good news is that this mistake can be fixed . for example , a bayesian approach that explicitly acknowledges that parameters are uncertain would go a long way towards solving this problem . '' @xcite another well - established way to obtain estimates for such parameters from observable data , which we will follow here , is via maximum entropy methods ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . such an estimate `` is the least biased estimate possible on the given information , i.e. , it is maximally noncommittal with regard to missing information . '' @xcite for example , the probability distribution over the interval @xmath4 $ ] which maximises entropy is the uniform distribution . there is no entropy maximiser for distributions over @xmath5 . however , when the mean and variance are specified , the gauss distribution with these parameters maximises entropy . we concentrate on the distribution of an asset price at a given time in the future , for which there are some option data . we develop a highly robust technique to find a maximum entropy distribution ( med ) for the asset in case we have call and digital option prices . the density is obtained by partitioning the range of possible stock prices into buckets , i.e. the intervals between adjacent strikes given by the option data , but , in contrast to the black - scholes model , making no a priori assumption about the asset s distribution . instead , we maximise the boltzmann - shannon entropy to obtain a distribution that respects only the given option prices and is otherwise unbiased . the density can in turn be used to interpolate implied volatilities and , by repeating this operation for a range of maturities , obtain a volatility surface . the results agree surprisingly well with observed volatility surfaces from the markets . buchen and kelly ( @xcite ) have proposed a similar entropy maximisation method to infer the probability distribution for an asset from call prices . this maximisation problem corresponds to finding a set of lagrange multipliers . in @xcite the authors write : `` there is a problem with this type of calculation , '' meaning that the formal lagrange multipliers approach is not mathematically rigorous . using convex programming arguments they legitimise those calculations . like @xcite we legitimise the results found in @xcite . however , we follow a simpler approach by applying a result of csiszr s @xcite . both @xcite and @xcite present numerical methods to find the lagrange multipliers by solving an @xmath6-dimensional non - linear problem ( where @xmath6 is the number of constraints given by call prices ) . as mentioned in @xcite , in the case of close strikes the problem can be poorly conditioned . in the present work , we add @xmath6 digital prices to our constraints and the resulting numerical problem is highly simplified : instead of an @xmath6-dimensional equation , we need to solve a one - dimensional problem @xmath7 for @xmath6 different values of @xmath8 ( the same @xmath9 though ) , allowing for easy parallelisation and avoiding any ill - conditioned problem . additionally , @xmath9 is a strictly monotonic function whose derivative is known analytically . therefore , the newton - raphson method can be used for excellent speed of convergence and stability . alternatively to iterative methods , one may try to find an analytical approximation for @xmath10 . in a nutshell , the advantage we obtain is the localisation of the maximum entropy density into asset price buckets , in which the functional form is a simple exponential function . but of course there is a price to pay for this localisation technique , and the price here is the necessity of an additional constraint for each call option used . this extra constraint is the price of a digital option at the same strike . the density in our case differs slightly from the one given by the method in @xcite . we therefore investigate the differences between them . in both approaches one can also use information from a so - called prior density , if available , leading to the concept of relative entropy ( also called @xmath1-divergence and kullback - leibler information number ) , and we compare the densities obtained by this method . after finding the med we give the expressions for the cumulative distribution function and its inverse . these formulas involve only arithmetic operations and exponential- and logarithm - functions . they are therefore very easy to implement and fast to compute . this is a highly useful feature for fast monte carlo simulations . furthermore , we obtain an analytical formula for the price of a call at a given strike . by calculating several such prices at different strikes , one can recover the implied volatility smile . we also include a section in which we calibrate to real market data . digital options on the s&p 500 index ( spx ) and the cboe volatility index ( vix ) are traded on the chicago board option exchange ( cboe ) , where they are called _ binary options_. they are specified such that `` expiration dates and settlement values are the same as for traditional options '' @xcite , which is just what we need in our setup . we show results for two cases : the first , in which we have cboe quotes for call _ and _ digital options on the spx and calibrate to them , and the second , for a different maturity , in which we only have quotes for call options and therefore have to estimate digital prices from call spreads . the method we propose here is found to work very well in both cases , and we compare our results to those obtained by calibrating only to call prices as in @xcite . we are given a fixed maturity @xmath11 , strictly increasing strikes @xmath12 @xmath13 , and undiscounted call and digital prices @xmath14 at these strikes . the payoffs of the call and digital options are given in equations and . @xmath15 denotes the discount factor . throughout we make the convention @xmath16 assuming risk neutral pricing , we will determine a density @xmath17 for the underlying asset price @xmath18 which maximises entropy @xmath19 under the constraints @xmath20 } = \tilde{c}_i , \quad i.e. \quad \int_{k_i}^\infty ( x - k_i ) g(x ) dx = \tilde{c}_i\ ] ] and @xmath21 } = \tilde{d}_i , \quad i.e. \quad \int_{k_i}^\infty g(x ) dx = \tilde{d}_i\ ] ] for all @xmath22 . in particular , these two constraints for @xmath23 mean that @xmath17 is a density , since @xmath24 , and that the martingale condition @xmath25 } = \int_0^\infty x g(x ) dx = \tilde{c}_0\ ] ] is satisfied , since @xmath26 is the forward price of @xmath27 for time @xmath11 . from the second constraint it immediately follows that @xmath28 looking at a call spread with strikes @xmath29 raised to level @xmath30 , i.e. a derivative that pays @xmath18 if @xmath31 and zero otherwise , we obtain the condition @xmath32 we now calculate the density @xmath17 under the constraints given above . the purpose of theorem [ localglobalmaximiser ] is to show that the local constraints and are equivalent to the global constraints and . moreover , @xmath33 and , thus , we only need to maximise @xmath34 subject to and over each bucket . let @xmath35 be the set of positive borel - measurable functions defined on @xmath36 define @xmath37 and , for all @xmath22 , @xmath38 [ intersectionx ] @xmath39 it is straightforward to show this using , , and . @xmath40 for @xmath41 we define @xmath42 [ localglobalmaximiser ] if @xmath17 is a maximiser of @xmath43 on @xmath44 , then @xmath17 is a maximiser of @xmath45 on @xmath46 . conversely , if @xmath17 is a maximiser of @xmath45 on @xmath46 for all @xmath47 then @xmath17 is a maximiser of @xmath43 on @xmath44 . let @xmath17 be a maximiser of @xmath43 on @xmath44 , and let @xmath48 . define @xmath49 since @xmath50 on @xmath51 , we have @xmath52 . moreover , for @xmath53 , we have @xmath54 on @xmath55 , and thus @xmath56 . it follows from proposition [ intersectionx ] that @xmath57 . hence , from the maximality of @xmath58 , we get @xmath59 . a simple computation gives @xmath60 , and therefore @xmath61 . it follows that @xmath17 maximises @xmath45 on @xmath46 . conversely , suppose that @xmath17 is a maximiser of @xmath45 on @xmath46 for all @xmath62 let @xmath63 . we have @xmath64 which means that @xmath17 is a maximiser of @xmath43 on @xmath44 . @xmath40 we now give a heuristic way of determining the entropy maximiser , but in the next subsection we also give a rigorous proof that this is indeed the correct result . formally applying the lagrange multipliers theorem , we conclude that the maximiser has the form @xmath65 to see this , define the functionals @xmath66 and solve the equation @xmath67 for the frchet derivatives . it follows that @xmath68 therefore , on the interval @xmath51 , we must have @xmath69 and , introducing @xmath70 and @xmath71 , we obtain . using the explicit form of @xmath17 just found in and gives @xmath72 for all @xmath22 . for @xmath73 , solving for @xmath74 and then for @xmath75 , using integration by parts , gives @xmath76 define @xmath77 it follows that @xmath78 and @xmath79 ( here @xmath80 means derivative with respect to @xmath81 ) . ] figure [ fig:1 ] shows the graphs of @xmath82 and @xmath83 for @xmath84 and @xmath85 . it suggests that equation has a unique solution if the quantity on the right hand side is in @xmath86k_i , k_{i+1}[$ ] . this turns out to be the case , as we show with the following proposition . let @xmath87 . if there is no arbitrage opportunity implied by @xmath88 , then there is a unique solution @xmath89 for equations and . define @xmath90 we first show that we must have @xmath91 . this can be seen by comparing the prices of three derivatives : they pay , respectively , @xmath30 , @xmath18 and @xmath92 if @xmath31 and zero otherwise . under the assumption that there is no arbitrage opportunity , it follows immediately that the second derivative is more expensive than the first one and cheaper that the third one . it is also clear that they can be replicated by portfolios of calls and digitals and their prices , in increasing order , are @xmath93 from the definition of @xmath94 the middle quantity above is @xmath95 and the result follows . next we show that if @xmath96 , then there is a unique solution @xmath89 for equations and . we begin with the case @xmath73 . as we have just seen , and are then equivalent to and . without loss of generality , we may assume @xmath97 and @xmath98 . indeed , it is straightforward to see that the change of variables @xmath99 transforms the equation @xmath100 into @xmath101 , with @xmath1020 , 1[$ ] . using lhpital s rule we obtain that the function @xmath103 given by @xmath104 is a continuous extension of @xmath105 . it is easy to see that @xmath106 and @xmath107 . hence the equation @xmath108 has a solution . to prove that the solution is unique , we shall now show that @xmath9 is strictly increasing . again by lhpital s rule , we obtain that @xmath9 is differentiable at @xmath109 and @xmath110 ( this is particularly useful because @xmath109 is an ideal starting point for the newton - raphson method ) . for @xmath111 we have @xmath112 recall that @xmath113 for all @xmath114 . hence @xmath115 we conclude that @xmath116 for all @xmath117 . therefore @xmath9 is strictly increasing . finally , we consider the case @xmath118 . equations and then become @xmath119 the first equation implies that @xmath120 . solving it for @xmath121 and the second equation for @xmath122 gives @xmath123 @xmath40 note that we have shown that @xmath9 is itself a continuously differentiable probability distribution function . for such a function there might already exist an inversion - algorithm . like others , we have formally derived the expression for the entropy maximiser using the lagrange multipliers method . however , as pointed out in @xcite `` there is a problem with this type of calculation . '' recall that the lagrange multipliers theorem requires continuous differentiability for objective and constraint functionals in a neighbourhood of the maximiser . however , the boltzmann - shannon entropy functional is finite only for densities in @xmath124 which has empty interior on @xmath125 . therefore a maximiser is not an interior point of @xmath126 . even worse , the entropy is far from being continuously differentiable since it is nowhere continuous . in @xcite , convex programming arguments are considered to circumvent this problem . here we present a new approach based on a result by csiszr @xcite . when no prior density is given we are interested in the ( non - relative ) entropy of @xmath17 @xmath127 where @xmath128 is an interval . however , csiszr s results deal with relative entropy of a probability density @xmath17 with respect to a probability measure @xmath129 on @xmath1 @xmath130 roughly speaking , we are interested in the `` relative entropy '' with respect to the lebesgue measure which , in general , is not a probability measure . for @xmath131 , @xmath132 is bounded . in that case , the problem can easily fit in csiszr s framework by considering the normalised lebesgue probability measure @xmath133 . however , it is impossible to use this trick for the global problem , @xmath134 , and for the last bucket , @xmath135 , since there is no normalisation constant which turns the lebesgue measure into a probability measure on these intervals . nevertheless , it is possible to turn the two problems over unbounded intervals into equivalent ones that do fit in csiszr s framework . this is the subject of the next proposition . moreover , the same arguments also apply to bounded intervals . therefore , in contrast to @xcite , we do not need to make any distinction between bounded and unbounded intervals . for the sake of simplicity , the statement in the following proposition considers only two _ main _ constraints , namely , the total mass and the mean . this includes the bucket problems and excludes the global problem ( where additional constraints are given ) . however , the proof works even for an infinite number of constraints provided the two main ones are among them . [ bijection ] let @xmath136 be an interval . define @xmath137 for all @xmath138 , where @xmath139 is a normalisation constant such that @xmath140 is a probability measure on @xmath1 . let @xmath141 . then the mapping @xmath142 is a bijection from @xmath143 onto @xmath144 moreover , @xmath17 is a maximiser of @xmath145 on @xmath146 if and only if @xmath147 is a maximiser of @xmath148 on @xmath149 . define @xmath150 by @xmath151 . since @xmath152 is strictly positive , it follows immediately that @xmath153 is a well defined bijection . we shall show that @xmath153 preserves some linear functionals . let @xmath154 and @xmath155 . then we have @xmath156 in particular , applying this result to @xmath157 and to @xmath158 , it follows immediately that @xmath153 maps @xmath146 onto @xmath149 . to complete the proof it suffices to show that if @xmath159 , then @xmath160 in fact , we shall show a stronger result , namely , that the two differences above are equal this is equivalent to showing that @xmath161 does not depend on @xmath162 . we have @xmath163 @xmath40 later we will restate and apply a partial version of a theorem by cziszr . but before we do so , let us say a few words about it . it is very natural to apply the lagrange multipliers theorem for maximisation problems under constraints . however , there are many cases where other techniques are used - for instance in the proof of the existence of projection on a convex set of a hilbert space . in that case , geometric arguments , including the parallelogram identity , are used . many texts suggest thinking of the relative entropy of one probability measure with respect to another as a quantity measuring how much they differ . moreover , they present some similarities between relative entropy and a metric . unfortunately , they say , this analogy does not go too far . csiszr s paper pushes these similarities a bit further , showing a relation analogous to the parallelogram identity . furthermore , he proves the existence of an entropy minimiser under convex constraints by similar arguments that show the existence of projection on convex subsets of hilbert spaces . we restate here a partial version of his theorem 3.1 sufficient for our purposes . [ csiszr ] let @xmath129 be a probability on a measurable space @xmath164 . let @xmath165 be an arbitrary set of real - valued @xmath166-measurable functions on @xmath167 and @xmath168 be real constants . let @xmath169 be the set of all those probabilities @xmath170 on @xmath171 for which the integrals @xmath172 exist and equal @xmath173 @xmath174 . then , if there exists @xmath175 such that @xmath176 and its radon - nikodym derivative has the form @xmath177 where @xmath178 and @xmath179 belongs to the linear space spanned by the @xmath180 s , then @xmath181 for all @xmath182 such that @xmath183 . now we prove that @xmath17 , given by , and , is indeed an entropy maximiser . let @xmath184 , @xmath132 . let @xmath74 and @xmath75 be defined by equations and . then @xmath185 given by @xmath186 maximises @xmath45 on @xmath46 . set @xmath187 and @xmath188 . let @xmath152 , @xmath129 , @xmath146 and @xmath149 be as in proposition [ bijection ] . note that for this choice of @xmath1 , @xmath189 and @xmath190 , we have @xmath191 and @xmath192 . let @xmath193 , @xmath194 be the @xmath195-algebra of lebesgue measurable subsets of @xmath1 , @xmath196 , @xmath197 ( @xmath198 ) and @xmath169 as in csiszr s theorem . given @xmath199 , define the measure @xmath200 by @xmath201 . from the definition of @xmath149 , it follows that @xmath202 . conversely , if @xmath182 and @xmath183 , then @xmath203 . then a simple computation yields @xmath204 by definition of @xmath74 and @xmath75 we have @xmath205 . proposition [ bijection ] yields @xmath206 . moreover , @xmath207 for all @xmath138 . let @xmath208 . it follows that @xmath175 , and its radon - nikodym derivative with respect to @xmath129 ( which is @xmath209 ) has the form with @xmath210 and @xmath211 . therefore , csiszr s theorem gives @xmath181 for all @xmath182 such that @xmath183 . in particular , for all @xmath199 , from we obtain @xmath212 we conclude that @xmath213 maximises @xmath148 on @xmath149 and , again by proposition [ bijection ] , that @xmath17 is a maximiser of @xmath45 on @xmath46 . @xmath40 we have the explicit form of the density given by equation . this allows us to give formulas in several important cases . to do this , we first state two useful results for the following proofs . for @xmath214 , we have @xmath215 \nonumber \\ & = & \frac{\alpha_i}{\beta_i}(ke^{\beta_ik } - k_ie^{\beta_ik_i } ) - \frac{\alpha_i}{\beta_i^2}(e^{\beta_ik } - e^{\beta_ik_i } ) . \label{k_i - k - mean}\end{aligned}\ ] ] it is straightforward to integrate the density @xmath17 and obtain an explicit form of the probability distribution @xmath216 its inverse can also be expressed analytically , which is a useful feature for monte carlo simulations . the results are stated in the following proposition . [ distributionfunction ] suppose @xmath217 . then @xmath218 given @xmath219 , find @xmath220 such that @xmath221\tilde d_{i+1 } , \tilde d_i]$ ] . then @xmath222 we treat only the case @xmath223 . the simpler case @xmath224 is left to the reader . first , notice that @xmath225 . then , using , we get @xmath226 since @xmath227 , we have @xmath228 . therefore solving for @xmath229 concludes the proof . @xmath40 it is also straightforward to express the prices of call and digital options analytically . [ prices ] given a strike @xmath230 , find @xmath231 such that @xmath214 . if @xmath223 , then @xmath232 if @xmath224 , then @xmath233 again we prove only the case @xmath223 . from we obtain @xmath234 for the ( undiscounted ) call price we have @xmath235 now putting and into leads to the stated result . @xmath40 finally , using euler s relationship for homogeneous functions , we can also give an explicit formula for spot - delta . given a strike @xmath230 , find @xmath231 such that @xmath214 . let @xmath27 be today s underlying spot price and @xmath236 be the spot - delta of a call with strike @xmath237 maturing at @xmath11 . if @xmath223 , then @xmath238 if @xmath224 , then @xmath239 again we consider only the case @xmath223 and leave the simpler case @xmath224 for the reader . let @xmath240 and @xmath241 be , respectively , the discounted prices of call and digital options with strike @xmath237 maturing at @xmath11 , i.e. @xmath242 and @xmath243 , where @xmath244 and @xmath245 are as in proposition [ prices ] . since @xmath240 is a positively homogeneous function of degree @xmath246 in @xmath247 , from euler s theorem we have @xmath248 recalling that @xmath249 , we can rewrite the last relation as @xmath250 now using and gives the result . @xmath40 note that the analogous statement for the forward - delta can be obtained by replacing the spot - price with the forward - price in the corollary and proof above . if we hold a prior belief about the distribution , we can maximise relative entropy instead in order to stay as `` close '' as possible to the prior distribution . suppose @xmath251 is a probability density for this prior distribution . for @xmath252 , define relative entropy @xmath253 ( the kullback - leibler information number or @xmath1-divergence is given by @xmath254 . this can be thought of as a measure of distance between two distributions . for example , @xmath255 , and @xmath256 if and only if @xmath257 . ) we have @xmath258 and essentially the same argument as the one given above shows that the maximum relative entropy density ( mred ) @xmath259 is given by @xmath260 therefore the resulting density @xmath261 is now given by the product of a piecewise exponential density and the prior density . even in the simple case where the prior density @xmath262 is just log - normal , we no longer have explicit formulas for call and digital prices . since we can not separate the two constraints @xmath263 for each @xmath47 as in equations and , we must solve them simultaneously using numerical integration and a two - dimensional root - finder . however , if the prior density @xmath262 is already given by an med , then @xmath264 and we can solve everything analytically as before . we also recover explicit formulas for call and digital prices . buchen and kelly @xcite propose a similar method to find an entropy - maximising density @xmath265 under constraints given by european payoffs . the case of most interest is where these are the payoff - functions of call options at different strikes @xmath266 and the actual constraints are given by ( undiscounted ) call option prices @xmath267 such that @xmath268 } = \tilde{c}_i\ ] ] must hold for all @xmath269 . the density @xmath265 must therefore satisfy the conditions @xmath270 and @xmath271 to find @xmath265 , they construct the functional @xmath272 where @xmath273 are the lagrange multipliers , and then solve the equation @xmath274 the solution is given by @xmath275 where @xmath276 is a normalising constant . buchen and kelly show that numerically , finding the parameters @xmath277 is an m - dimensional root - finding problem that can be tackled with the multi - dimensional newton algorithm . they show how to compute the jacobian , and that it is invertible , by expressing it as a covariance matrix . if a call option with strike @xmath278 , i.e. the forward , is among the input data , the mean of the distribution is given . since the total mass , @xmath246 , is also known , we have the two main constraints needed to apply the arguments from subsection [ rigorousproof ] and can therefore also rigorously find the entropy maximiser when only call options are given as input . of course , the forward should be known in most situations , so that this is certainly the most important case . similarly , if a prior distribution @xmath262 is given , the distribution maximising relative entropy under the same constraints is given by @xmath279 where @xmath280 is again the normalising constant . in this section we give some numerical examples for the entropy maximisers described so far . we suppose that the market data is given by @xmath281 we assume a flat volatility and make no skew correction when calculating the digital prices in this scenario . we calculate three densities using strikes * @xmath282 * @xmath283 * @xmath284 lllllllllll + strike & 0.00 & 20.00 & 40.00 & 60.00 & 80.00 & 100.00 & 120.00 & 140.00 & 160.00 & 180.00 + call & 100.0000 & 80.0000 & 60.0005 & 40.1454 & 22.2656 & 9.9477 & 3.7059 & 1.2139 & 0.3659 & 0.1049 + digital & 1.0000 & 1.0000 & 0.9998 & 0.9725 & 0.7786 & 0.4503 & 0.1965 & 0.0707 & 0.0225 & 0.0066 + + entropy & 4.6714 + @xmath285 & 1.3582e-04 & n / a & n / a & n / a & n / a & 1.8835 & n / a & n / a & n / a & n / a + @xmath81 & 0.0539 & n / a & n / a & n / a & n / a & -0.0453 & n / a & n / a & n / a & n / a + call & 100.0000 & 80.0402 & 60.2562 & 40.9886 & 23.2384 & 9.9477 & 4.0232 & 1.6271 & 0.6581 & 0.2661 + implied vol . & n / a & 62.13% & 46.26% & 36.17% & 28.88% & 25.00% & 25.95% & 27.04% & 27.84% & 28.41% + digital & 1.0000 & 0.9951 & 0.9808 & 0.9386 & 0.8146 & 0.4503 & 0.1821 & 0.0736 & 0.0298 & 0.0120 + + entropy & 4.6143 + @xmath285 & 6.0682e-08 & n / a & n / a & 0.0016 & n / a & 0.5397 & n / a & 14.2333 & n / a & n / a + @xmath81 & 0.1894 & n / a & n / a & 0.0255 & n / a & -0.0343 & n / a & -0.0582 & n / a & n / a + call & 100.0000 & 80.0001 & 60.0033 & 40.1454 & 22.4905 & 9.9477 & 3.7539 & 1.2139 & 0.3790 & 0.1183 + implied vol . & n / a & 38.76% & 28.60% & 25.00% & 25.93% & 25.00% & 25.14% & 25.00% & 25.15% & 25.38% + digital & 1.0000 & 1.0000 & 0.9994 & 0.9725 & 0.7765 & 0.4503 & 0.1978 & 0.0707 & 0.0221 & 0.0069 + + entropy & 4.6076 + @xmath285 & 6.0682e-08 & n / a & n / a & 1.5393e-04 & 0.0129 & 0.2389 & 1.6987 & 14.2333 & n / a & n / a + @xmath81 & 0.1894 & n / a & n / a & 0.0584 & 0.0027 & -0.0268 & -0.0433 & -0.0582 & n / a & n / a + call & 100.0000 & 80.0001 & 60.0033 & 40.1454 & 22.2656 & 9.9477 & 3.7059 & 1.2139 & 0.3790 & 0.1183 + implied vol . & n / a & 38.76% & 28.60% & 25.00% & 25.00% & 25.00% & 25.00% & 25.00% & 25.15% & 25.38% + digital & 1.0000 & 1.0000 & 0.9994 & 0.9725 & 0.7765 & 0.4503 & 0.1978 & 0.0707 & 0.0221 & 0.0069 + table [ tab:1 ] gives the ( undiscounted ) option prices we used and the parameters describing the density . and three maximum entropy densities obtained by calibrating to @xmath246 , @xmath286 and @xmath287 strikes . ] figure [ fig:2 ] shows the three densities and the actual log - normal density . it can be seen that already with @xmath287 strikes and the forward , the fit of the piecewise - exponential distribution to the log - normal distribution is very good . in practice , however , implied volatilities are not flat as in the example above , i.e. they are not the same for different strikes and maturities . this is discussed in detail in gatheral s book @xcite , in particular in the section `` the spx implied volatility surface '' in chapter 3 . we show that our method has by its nature a tendency to give good fits to observed volatility surfaces . to do this , we will assume that we now only have at - the - money ( atm ) option prices , and that already with this minimal amount of market data our method generates a very realistic looking volatility surface . we show in figure [ fig:3 ] the implied volatility surface obtained by using just the atm strike . more precisely , if we assume @xmath288 and @xmath289 again , then the atm strike is @xmath290 . moreover , we consider a constant atm volatility @xmath291 ( it could of course be time dependent ) . for each maturity @xmath292 , we compute the black - scholes prices of the atm call @xmath293 and atm digital @xmath294 . then applying the maximum entropy approach to just one strike @xmath295 , we compute @xmath74 and @xmath75 for @xmath296 . in other terms , we recover the maximum entropy density of @xmath18 compatible with @xmath297 , @xmath293 and @xmath294 . we emphasise that no other strike , call or digital is used in this calibration . according to the med , the price of a call with strike @xmath237 and maturity @xmath11 is given by @xmath298 , where @xmath299 is given in proposition [ prices ] . from @xmath300 we recover the implied volatility @xmath301 with a bisection root - finder from the black - scholes formula . readers interested in a more robust method can consult the one proposed in @xcite . as expected , as a consequence of calibration , @xmath302 . what is surprising is the fact that the curve @xmath303 has a profile very similar to smile curves typically seen in equity markets . now , by varying @xmath11 one constructs a volatility surface which , again , is qualitatively very similar to those observed in equity markets . the maximum entropy method seems to be able to transform just one volatility number from a flat black - scholes ( atm ) world into a very realistic looking volatility surface , with important features such as a strongly pronounced smile at the short end that decays as the maturity increases . different strikes and different numbers of option prices can of course be used at different maturities , so that any arbitrage - free option data can easily be converted into an implied volatility surface . the density @xmath17 is usually discontinuous at the @xmath30 s . the distribution function is of course continuous . many monte carlo models work by drawing a random uniform variable and inverting the distribution . in black - scholes type models , for example , a normal distribution has to be inverted at some stage . in our case , only one logarithm needs to be taken , a circumstance which accelerates a simulation . let the prior distribution be a log - normal distribution with fixed volatility parameter @xmath195 @xmath304 we still have an explicit form of the density , namely @xmath261 , where @xmath17 is a piecewise exponential density , although the parameters @xmath305 are of course different from the parameters @xmath306 used for the med of the previous subsection . since we are now unable to express call prices analytically , we calculate them via numerical integration . lllllll strike & 0.00 & 60.00 & 80.00 & 100.00 & 120.00 & 140.00 + + @xmath307 & 12.2600 & n / a & n / a & 0.0833 & n / a & n / a + @xmath308 & -0.0298 & n / a & n / a & 0.0206 & n / a & n / a + + @xmath307 & 11.2900 & 7.2379 & n / a & 0.2930 & n / a & 0.3267 + @xmath308 & -0.0194 & -0.0237 & n / a & 0.0098 & n / a & 0.0116 + + @xmath307 & 11.2900 & 5.9910 & 2.0430 & 0.5970 & 0.5815 & 0.3267 + @xmath308 & -0.0194 & -0.0210 & -0.0097 & 0.0031 & 0.0047 & 0.0116 + table [ tab:2 ] gives the parameters describing the density . of course , should a prior density already meet the constraints , we will have @xmath309 and @xmath310 for all @xmath311 . , actual log - normal density with @xmath312 and three maximum relative entropy densities obtained by calibrating to @xmath246 , @xmath286 and @xmath287 strikes . ] figure [ fig:4 ] shows the three maximum relative entropy densities and the prior log - normal density ( @xmath313 ) . the density for the forward and one call is already much closer to the actual one than in the previous case , so that convergence is not as pronounced as before when the number of strikes is increased . we see that @xmath17 has the effect of pushing the prior density downwards and widening it as to be closer to the actual density . the explicit form of the density given by equation allows one to obtain analytic expressions for call and digital prices like those in proposition [ prices ] . as an example , using just the forward and an at - the - money call , i.e. @xmath314 , we obtained @xmath315 on our computer . this leads to a very similar volatility smile as the one given at @xmath316 in subsection [ med - cd ] above . we refer to @xcite for graphs and numerical data regarding this distribution . as in subsection [ relmed - cd ] , in general there will be no analytic expressions for call or digital prices . if the chosen prior distribution is continuous , then the resulting relative entropy maximiser will also be continuous . again , we advise the reader to look at @xcite for graphs and numerical data regarding this distribution . digital options are traded on the chicago board option exchange ( cboe ) . they are called _ binary options _ there . we quote the following paragraph from the `` binaries '' product description @xcite : `` cboe offers binary options on the s&p 500 index ( spx ) and the cboe volatility index ( vix ) . the ticker symbols for these binary contracts is bsz and bvz respectively . expiration dates and settlement values are the same as for traditional options . '' the specification that digital option strikes and maturities are the same as those of call options is exactly what we need for our setup . we calibrate to cboe option prices from 10 april 2010 for two different maturities . the first maturity is 18 september 2010 . we have digital option bid and ask quotes for strikes @xmath237 from @xmath317 to @xmath318 usd , usually in steps of @xmath319 usd . we also have call option bid and ask quotes for these same strikes . we calibrate to the `` mid '' prices , i.e. the average of the bid and ask quotes , at the ten strikes from @xmath317 to @xmath318 in steps of @xmath320 usd . rrrrrrr & & & + strike & digital & call & digital & call & digital & call + 950 & 0.9400 & 246.30 & 0.9259 & 246.30 & 0.9400 & 246.30 + 975 & 0.9150 & 223.20 & 0.9171 & 223.25 & 0.9153 & 223.12 + 1000 & 0.8950 & 200.50 & 0.9014 & 200.50 & 0.8950 & 200.50 + 1025 & 0.8750 & 178.15 & 0.8787 & 178.24 & 0.8795 & 178.30 + 1050 & 0.8550 & 156.60 & 0.8516 & 156.60 & 0.8550 & 156.60 + 1075 & 0.8150 & 135.70 & 0.8191 & 135.70 & 0.8195 & 135.65 + 1100 & 0.7750 & 115.70 & 0.7802 & 115.70 & 0.7750 & 115.70 + 1125 & 0.7250 & 96.75 & 0.7336 & 96.76 & 0.7367 & 96.76 + 1150 & 0.6700 & 79.10 & 0.6776 & 79.10 & 0.6700 & 79.10 + 1175 & 0.6050 & 63.00 & 0.6117 & 62.97 & 0.6137 & 63.01 + 1200 & 0.5350 & 48.60 & 0.5357 & 48.60 & 0.5350 & 48.60 + 1225 & 0.4550 & 36.25 & 0.4541 & 36.23 & 0.4585 & 36.13 + 1250 & 0.3550 & 25.90 & 0.3720 & 25.90 & 0.3550 & 25.90 + 1300 & 0.1850 & 11.35 & 0.2112 & 11.35 & 0.1850 & 11.35 + 1350 & 0.0700 & 4.10 & 0.0896 & 4.10 & 0.0700 & 4.10 + 1400 & 0.0450 & 1.33 & 0.0307 & 1.33 & 0.0450 & 1.33 + table [ tab:3 ] shows cboe prices for call and digital options on the spx from 10 april 2010 in columns 2 and 3 . columns 4 and 5 show option prices obtained by calibrating an med to call prices at strikes @xmath317 , @xmath321 , @xmath322 , @xmath323 , @xmath318 . columns 6 and 7 show option prices obtained by calibrating an med to call and digital prices at the same strikes . note that the second med matches market call _ and _ digital prices at the strikes calibrated to exactly , whereas the first med matches only the call prices . the second maturity is 31 december 2010 . we have call option bid and ask quotes for strikes @xmath237 from @xmath324 to @xmath325 usd in steps of @xmath320 usd . we do not have any digital option quotes for this maturity . as a substitute , we calculate symmetric call spread prices @xmath326 using mid call prices . we calibrate to this data at the three strikes @xmath327 . in the previous example , we showed that our method can be used to calibrate to quotes at many ( @xmath328 ) strikes , and that this leads to a very good fit . in this example , we calibrate to only a small number ( @xmath286 ) of strikes in order to show that our method has an excellent natural tendency to fit a market smile . the call spread prices @xmath329 needed for are obtained by using the call quotes at @xmath330 . [ tab:4 ] rrrrrrrrr & & & & + & & & & + strike & call & implied vol . & call & implied vol . & call & implied vol . & call & implied vol . + 500 & 681.15 & 62.90% & 678.87 & 59.45% & 677.05 & 56.22% & 679.66 & 60.70% + 550 & 631.75 & 57.19% & 630.27 & 55.25% & 628.62 & 52.84% & 630.92 & 56.12% + 600 & 582.35 & 51.94% & 581.72 & 51.22% & 580.41 & 49.63% & 582.18 & 51.75% + 650 & 533.45 & 47.55% & 533.22 & 47.32% & 532.44 & 46.53% & 533.45 & 47.55% + 700 & 484.75 & 43.52% & 484.75 & 43.52% & 484.75 & 43.52% & 484.75 & 43.52% + 750 & 436.55 & 39.99% & 436.54 & 39.98% & 437.36 & 40.59% & 436.55 & 39.99% + 800 & 388.85 & 36.77% & 388.90 & 36.81% & 390.39 & 37.75% & 388.97 & 36.85% + 850 & 341.85 & 33.85% & 342.02 & 33.95% & 343.97 & 35.01% & 342.11 & 34.00% + 900 & 295.95 & 31.28% & 296.11 & 31.36% & 298.32 & 32.39% & 296.20 & 31.40% + 950 & 251.25 & 28.89% & 251.49 & 28.99% & 253.72 & 29.88% & 251.55 & 29.01% + 1000 & 208.25 & 26.70% & 208.54 & 26.80% & 210.56 & 27.50% & 208.54 & 26.80% + 1050 & 167.50 & 24.68% & 167.76 & 24.76% & 169.36 & 25.25% & 167.72 & 24.75% + 1100 & 129.70 & 22.82% & 129.84 & 22.86% & 130.87 & 23.15% & 129.77 & 22.84% + 1150 & 95.60 & 21.09% & 95.64 & 21.10% & 96.06 & 21.21% & 95.60 & 21.09% + 1200 & 66.30 & 19.52% & 66.30 & 19.52% & 66.30 & 19.52% & 66.30 & 19.52% + 1250 & 42.70 & 18.13% & 43.09 & 18.24% & 42.93 & 18.20% & 42.70 & 18.13% + 1300 & 25.10 & 16.91% & 25.83 & 17.13% & 25.61 & 17.06% & 25.19 & 16.93% + 1350 & 13.35 & 15.87% & 13.79 & 16.05% & 13.63 & 15.99% & 13.35 & 15.87% + 1400 & 6.35 & 15.01% & 6.35 & 15.01% & 6.35 & 15.01% & 6.35 & 15.01% + 1450 & 2.68 & 14.27% & 2.74 & 14.34% & 2.83 & 14.43% & 2.68 & 14.27% + 1500 & 1.13 & 13.91% & 1.18 & 14.02% & 1.26 & 14.16% & 1.08 & 13.82% + 1550 & 0.43 & 13.57% & 0.51 & 13.88% & 0.56 & 14.05% & 0.43 & 13.59% + 1600 & 0.20 & 13.69% & 0.22 & 13.83% & 0.25 & 14.03% & 0.17 & 13.49% + table [ tab:4 ] shows cboe prices for call options on the spx from 10 april 2010 and their implied volatilities in columns 2 and 3 . columns 4 and 5 show call option prices and implied volatilities obtained by calibrating an med to call prices at strikes @xmath331 , @xmath332 , @xmath318 . columns 6 and 7 show call option prices and implied volatilities obtained by calibrating an med to call and digital prices at the same strikes , using call spread prices at @xmath333 , @xmath334 , @xmath335 as substitutes for the digital prices . columns 8 and 9 show call option prices and implied volatilities obtained by calibrating an med to call and digital prices at all nine strikes . of course the med obtained using calls and digitals at three strikes uses more `` information '' than the med obtained using just calls at three strikes . it is therefore not surprising that it leads to a better fit . however , to show that this fit is already very good , we also report the med obtained from call prices at all nine strikes used in this example . figure [ fig:5 ] illustrates graphically that these last two med s are indeed very close to each other . . in most situations the information observed in the market regarding an asset consists of option prices at a discrete set of strikes . using this to extrapolate the second derivative of a function everywhere , as suggested by breeden and litzenberger @xcite or the volatility approach @xcite , @xcite , relies on additional assumptions about the distribution of returns , the sde the asset follows and/or the choice of an interpolation method . even when there are strong reasons for such assumptions , we believe that it is important to know the shape of the distribution function given by the principle of maximum entropy ( pme ) in case these assumptions turn out to be flawed . in the local volatility model call prices for all strikes in @xmath336 are needed , and , additionally , it assumes that the smile volatility is twice continuously differentiable . hence this approach requires an infinity of non - quoted prices together with a strong regularity . `` since the market provides call prices at only a small number of strike prices , the second derivative must be estimated by interpolation . this method is not very robust as the results are very sensitive to the interpolation scheme used . '' @xcite for the method proposed here , if there are no observable digital quotes in the market , the `` artificial '' data required consists only of digital prices for a finite , usually small , set of strikes @xmath337 . if one assumes ( and our approach does not ) that the volatility smile is differentiable with respect to the strike at the points of @xmath338 , then prescribing digital prices there is indeed equivalent to prescribing the value of the smile derivative at those points . this is still a much weaker requirement than that of the local volatility model . moreover , the example in section [ spx - calibration ] shows that centered call spread prices are very good estimators for digital prices . entropy has been one of the main concepts in information theory @xcite , and since market participants react to information when taking their positions , we believe entropy is a very natural tool for use in finance . entropy can be seen as a measure of how unbiased a probability distribution is . hence , by maximising entropy , what we propose is to find the most unbiased probability distribution which agrees with information provided from the market . we then show how this hypothesis leads to a piecewise exponential density . the method we propose can be used reliably and efficiently in practice . on the one hand , we have seen that it produces a remarkably realistic volatility surface from just one volatility number as in the original black - scholes model , with a steep skew for short maturities that decays with increasing maturity . on the other hand , if the actual distribution is known , then with option prices given at five or more strikes , the fit to it is very close . in particular , it can be used as a robust interpolation method for volatility curves . if additionally there is knowledge of a prior distribution , the principle of maximum relative entropy can be applied to find a density that takes this into account and also meets the new constraints . we give an example of such a scenario with two log - normal distributions , and show that the convergence to the actual distribution is particularly quick . buchen and kelly have proposed a similar method of finding a probability density that maximises entropy when the market data consists only of call options . the density they obtain is continuous . however , to find its parameters they must solve a multi - dimensional root - finding problem with the newton - raphson algorithm . one criticism often raised in this application of the pme is that the method of finding the form of the density uses lagrange multipliers and is not rigorous . indeed , this technique works well in practice and leads to the correct form , but we also give a complete mathematical proof that avoids them . relative entropy has often been compared to a metric for probability distributions . our proof uses results by csiszr that give additional insights into `` distances '' between distributions and establish remarkable `` geometric '' results . since we have an explicit form of the density , we are able to give analytical formulas for the distribution , inverse distribution , and call and digital option prices . using euler s relation for homogeneous functions , we give formulas for spot- and forward - deltas . we also include two examples in which we calibrate to real market data from the cboe . we show that our method performs very well in both cases and compare our results to those obtained by using buchen and kelly s approach . cassio neri currently works as quantitative analyst for lloyds banking group in london . lorenz schneider is assistant professor in quantitative finance at emlyon business school in lyon . they have ph.d.s in mathematics from universities paris ix and vi , respectively . | we obtain the maximum entropy distribution for an asset from call and digital option prices . a rigorous mathematical proof of its existence and exponential form is given , which can also be applied to legitimise a formal derivation by buchen and kelly @xcite .
we give a simple and robust algorithm for our method and compare our results to theirs .
we present numerical results which show that our approach implies very realistic volatility surfaces even when calibrating only to at - the - money options .
finally , we apply our approach to options on the s&p 500 index .
* keywords : * entropy @xmath0 information theory @xmath0 @xmath1-divergence @xmath0 asset distribution @xmath0 option pricing @xmath0 volatility smile * mathematics subject classification ( 2000 ) : * 91b24 @xmath0 91b28 @xmath0 91b70 @xmath0 94a17 * jel : * c16 @xmath0 c63 @xmath0 g13 |
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in a standard imaging system , light scattered from an object forms a diffraction pattern which encodes information about the object fourier components . a lens recombines the scattered rays so that they interfere correctly to form an ) . , title="fig:",scaledwidth=20.0% ] , title="fig:",scaledwidth=20.0% ] these ideas , along with the development of powerful light sources , producing collimated beams of coherent x - rays , enabled the development of coherent x - ray diffraction microscopy @xcite ( aka lensless or diffractive imaging ) . this technique aims at imaging , through coherent illumination , fourier amplitude measurements and adequate sampling , macroscopic objects such as entire cellular organisms @xcite , or nanoporous aerogel structures @xcite . see @xcite for a review . diffraction microscopy solves the phase problem using increasingly sophisticated algorithms based on the support constraint , which assumes adequate sampling . the object being imaged is limited within a support region @xmath0 : @xmath1 the sampling conditions required to benefit from the support constraint have limited the adoption of projection algorithms to other experimental geometries that allow only for sub - nyquist sampling , most notably bragg sampling from periodic crystalline structures . modern sampling theory however , tells us that that nyquist sampling conditions dictated by the support are the worst case scenario for an arbitrary object . in other words , shannon was a pessimist : he did not account for the signal structure . compressive sensing theory tells us that the number of measurements are dictated by the signal structure rather than it s length . by structured we mean that the signal has only a few non - zero coefficients when represented in terms of some basis , or can be well approximated well by a few non zero coefficients : they can be described in terms of a few atoms , a few stars , a few wavelet coefficients , or possibly a few protein folds . in other words , an object of interest is often sparse or concentrated in a small number of non - zero coefficients in a well chosen basis , i.e. it can be compressed with no or almost no loss of information . the meanings of `` well - chosen '' and `` of interest '' are slightly circular : a basis is well - chosen if it succinctly describes a signal of interest ; likewise , a signal is of interest if it can be described with just a handful of basis elements . since we do not know where these few terms are located , conventional wisdom would indicate that one has to first measure the full sample at the desired resolution , since overlooking an important component of a signal seems almost inevitable if the whole haystack is nt thoroughly searched over . there would seem to be no alternative to processing each signal in its entirety before we can compress it and store only the desired information ( such as the location of the atoms in a molecule ) . but a new theory of `` compressive sampling '' has shown how an image of interest or structured signals generally can be reconstructed , exactly , from a surprisingly small set of direct measurements . cands and colleagues @xcite have defined a notion of `` uniform uncertainty '' that guarantees , with arbitrarily high probability , an exact solution when the signal is sparse and a good approximation when it is compressible or noisy . their uniform uncertainty condition is satisfied , among others , by fourier measurements of a sparse real space object . the question of whether modern sensing theory is applicable to fourier amplitude measurements was first raised by moravec , romberg and baraniuk @xcite who provide an upper bound sampling condition for the successful retrieval of a sparse signal autocorrelation , and discuss other conjectures with far reaching consequences for low - resolution undersampled phase retrieval . since the theory is relatively new and not widely known to the phase retrieval community , modern sampling theory is briefly reviewed . the notion that a diffraction pattern from a sparse object can be reconstructed at sub - nyquist sampling is not entirely new . the so called `` direct methods '' are routinely used for atomic resolution imaging of increasing complex molecular structures . direct methods enforce the condition that the resulting molecule is composed of a finite number of atoms . the conditions for successful ab - initio phase retrieval using these methods are strict : it requires ( 1 ) atomic resolution and ( 2 ) about 5 strong peaks per atom . condition ( 2 ) means that the algorithms do not scale well with a large number of atoms since the number of strong reflections decreases rapidly with the number of atoms . suppose that one collects an incomplete set of frequency samples ( amplitude and phase ) of a discrete signal @xmath2 of length @xmath3 . the goal is to reconstruct the full signal @xmath4 given only @xmath5 samples in the fourier domain where the `` visible frequencies '' are a subset @xmath6 ( of size @xmath5 ) of the set of all frequencies @xmath7 . at first glance , solving the underdetermined system of equations appears hopeless , as it is easy to make up examples for which it clearly can not be done . but suppose now that the signal @xmath4 is compressible , meaning that it essentially depends on a number of degrees of freedom which is smaller than @xmath3 . then in fact , accurate and sometimes exact recovery is possible by solving a simple convex optimization problem . ( candes romberg and tao @xcite ) : assume that @xmath4 is @xmath8 -sparse , ( e.g. @xmath8 atomic charges in real space with @xmath3 resolution elements ) , and that we are given @xmath5 fourier coefficients with frequencies selected uniformly at random . suppose that the number of observations obeys @xmath9 . then minimizing @xmath10 reconstructs @xmath4 exactly with overwhelming probability . in particular , writing @xmath11 , then the probability of success exceeds @xmath12 . the theorem shows that a simple convex minimization will find the exact solution without any knowledge about the support , the number of nonzero coordinates of @xmath13 , their locations , and their amplitudes which we assume are all completely unknown a priori . following @xcite we formulate this more explicitly . the algorithm that optimizes the @xmath10 norm : latexmath:[\ ] ] although this result describes only local convergence , it shows how a few wrong peaks can be recovered easily . 27 l. pauling and m. d. shappell , zeits . * 75 * , 128 ( 1930 ) . bernal , i. fankuchen , m. f. perutz , `` an x - ray study of chymotrypsin and haemoglobin . '' nature 141 , 523 - 524 ( 1938 ) . d. sayre , `` on the implication of a theorem due to shannon '' , acta cryst . * 5 * , ( 1952 ) 843 . y. m. bruck and l. g. sodin . `` on the ambiguity of the image reconstruction problem . '' optics communications , * 30*(3):304 - 308 , ( 1979 ) . `` fourier phase problems are uniquely solvable in more than one dimension . i : underlying theory '' optik , * 61*(3):247 - 262 , 1982 . m. h. hayes , `` the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform , '' ieee trans . 30*(2 ) , 140 - 154 , ( 1982 ) . m. h. hayes and j. h. mcclellan , `` reducible polynomials in more than one variable , '' proc . 70*(2 ) , 197 - 198 , ( 1982 ) . j. miao , p. charalambous , j. kirz , d. sayre , `` extending the methodology of x - ray crystallography to allow imaging of micrometre - sized non - crystalline specimens , '' nature * 400 * , 342 - 344 ( 1999 ) . d. shapiro , p. thibault , t. beetz , v. elser , m. howells , c. jacobsen , j. kirz , e. lima , h. miao , a. neiman , d. sayre , `` biological imaging by soft x - ray diffraction microscopy , '' proc . * 102 * , 1543 - 1546 ( 2005 ) . nature * 442 * , 63 - 67 ( 2006 ) . a. barty , s. marchesini , h. n. chapman , c. cui , m. r. howells , d. a. shapiro , a. m. minor , j. c. h. spence , u. weierstall , j. ilavsky , a. noy , s. p. hau - riege , a. b. artyukhin , t. baumann , t. willey , j. stolken , t. van buuren , j. h. kinney , `` three - dimensional coherent x - ray diffraction imaging of a ceramic nanofoam : determination of structural deformation mechanisms , '' * 101 * , 055501 ( 2008 ) , [ arxiv:0708.4035 ] . p. w. hawkes & j. c. h. spence ( eds . ) , _ science of microscopy _ ( springer , 2007 ) . e. j. cands , j. romberg and t. tao , `` robust uncertainty principles : exact signal reconstruction from highly incomplete frequency information , '' ieee trans . inform . theory , * 52 * , 489 - 509 ( 2006 ) [ arxiv : math/0409186 ] . m. l. moravec , j. k. romberg , r. g. baraniuk , richard , compressive phase retrieval , wavelets xii . spie * 6701 * , 670120 ( 2007 ) . e. van den berg and m. p. friedlander , spgl1 : a solver for large - scale sparse reconstruction , http://www.cs.ubc.ca/labs/scl/index.php/main/spgl1 e. van den berg and m. p. friedlander , `` probing the pareto frontier for basis pursuit solutions '' , ubc computer science technical report tr-2008 - 01 , january 2008 . available at http://www.optimization-online.org/db_f w. h. richardson , bayesian - based iterative method of image restoration " . * 62 * , 55 - 59 ( 1972 ) . l. b. lucy , an iterative technique for the rectification of observed distributions " . astronomical journal textbf79 745 - 754 ( 1974 ) . fienup , `` phase retrieval for undersampled broadband images , '' , * 16 * , 1831 - 1839 ( 1999 ) . s. marchesini , h. he , h. n. chapman , s. p. hau - riege , a. noy , m. r. howells , u. weierstall , j.c.h . spence , `` x - ray image reconstruction from a diffraction pattern alone , '' * 68 * , 140101(r ) 1 - 4 , ( 2003 ) , [ arxiv : physics/0306174 ] . l. d. marks , w. sinkler and e. landree , `` a feasible set approach to the crystallographic phase problem '' , acta cryst . * a55 * , 601 - 612 ( 1999 ) . j. r. fienup , `` reconstruction of an object from the modulus of its fourier transform , '' * 3 * , 27 - 29 ( 1978 ) . j. r. fienup , `` phase retrieval algorithms : a comparison '' , * 21 * , 2758 - 2769 ( 1982 ) . j. p. abrahams , a. w. g. leslie , acta cryst . * 52 * , 30 - 42 ( 1996 ) . v. elser , `` phase retrieval by iterated projections , '' * 20 * , 40 - 55 ( 2003 ) . h. h. bauschke , p. l. combettes , and d. r. luke , `` hybrid projection reflection method for phase retrieval , '' * 20 * , 1025 - 1034 ( 2003 ) . d. r. luke , `` relaxed averaged alternating reflections for diffraction imaging , '' inverse problems * 21 * , 37 - 50 ( 2005 ) . , ( arxiv : math.oc/0405208 ) . l. combettes , `` the convex feasibility problem in image recovery , in advances in imaging and electron physics , '' ( p. hawkes , ed . ) , vol . 155 - 270 . ( academic press , new york 1996 ) . s. marchesini , `` a unified evaluation of iterative projection algorithms for phase retrieval , '' rev . inst . * 78 * , 011301 1 - 10 ( 2007 ) , [ arxiv : physics/0603201 ] . g. oszlnyi and a. sto , `` ab initio structure solution by charge flipping , '' acta cryst . * a60 * , 134 - 141 ( 2004 ) [ arxiv : cond - mat/0308129 ] . j. wu , k. leinenweber , j. c. h. spence , `` ab initio phasing of x - ray powder diffraction patterns by charge flipping , '' nature materials * 5 * , 647 - 652 ( 2006 ) . e. j. cands , d. l. donoho , `` new tight frames of curvelets and optimal representations of objects with piecewise c@xmath94 singularities . '' pure appl . math . * 57 * , 219 - 266 ( 2004 ) . s. marchesini , 24,32890 - 3296 ( 2007 ) , [ arxiv : physics/0611233 ] . g. oszlnyi and a. sto , `` ab initio structure solution by charge flipping . ii . use of weak reflections , '' acta crystallogr . * a61 * , 147 - 152 ( 2005 ) . i. daubechies , m. defrise , and c. d. mol , `` an iterative thresholding algorithm for linear inverse problems with a sparsity constraint , '' comm . pure appl . math . * 57*(11 ) , 1413 - 1457 , ( 2004 ) . | any object on earth has two fundamental properties : it is finite , and it is made of atoms . structural information about an object can be obtained from diffraction amplitude measurements that account for either one of these traits .
nyquist - sampling of the fourier amplitudes is sufficient to image single particles of finite size at any resolution .
atomic resolution data is routinely used to image molecules replicated in a crystal structure . here
we report an algorithm that requires neither information , but uses the fact that an image of a natural object is compressible .
intended applications include tomographic diffractive imaging , crystallography , powder diffraction , small angle x - ray scattering and random fourier amplitude measurements . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the power spectrum of the cosmic microwave radiation ( cmbr ) carries much cosmological information about primordial density fluctuations in the early universe . as photons leave the last scattering surface and travel across the universe , however , these brightness fluctuations are modified by intervening structures , causing secondary fluctuations , which we would expect to become more important on small angular scales . the power spectrum of the cmbr alone can be used to determine cosmological parameters . recently it has been shown , however , that a geometrical degeneracy effect prevents some combinations of cosmological parameters from being disentangled by the power spectrum alone ( @xcite ; @xcite ; @xcite ) . the primordial density fluctuations and matter content determine the positions and magnitudes of the doppler peaks at the last scattering surface . these fluctuations are transferred to apparent angular scales determined by their angular diameter distance . as a result , cold dark matter ( cdm ) models with the same primordial density fluctuations , matter content , and angular diameter distance can not be distinguished . the models are `` effectively degenerate '' in the sense that their power spectrum is degenerate for parameter determination on intermediate and small scales . although the observed power spectrum also depends on the time variation of the metric , via the integrated sachs - wolfe effect , this breaks the degeneracy only at large angular scales . unfortunately observations of the power spectrum do not provide strong constraints on models at large scales , since this is where the statistics of the data are dominated by the cosmic variance due to the fact that we have only one realization of our cosmological model , the universe itself , and we encounter a sampling problem ) . therefore we can determine , for example , only combinations such as @xmath2 and @xmath3 ( where @xmath4 and @xmath5 are the @xmath6 matter and baryon density parameters and @xmath7 is the dimensionless hubble parameter ) . it has been noticed that gravitational lensing can break the geometric degeneracy at small angular scales , @xmath8 , in such a way that the cosmological parameters can be determined separately ( @xcite ; @xcite ) . the effect of gravitational lensing on the cmbr was studied by several authors ( see for example @xcite ; @xcite ; @xcite ) . static gravitational lenses do not change a smooth cmbr , but the fluctuations get distorted by lensing . as a result , power from the acoustic peaks is transferred to small angular scales , conserving the variance of the spectrum . the amount of power transferred depends on the cosmological model , thus , in principle , we can determine separately @xmath4 , @xmath5 , and @xmath7 . this method makes use of the small angular scale part of the power spectrum , where the amplitude of primordial fluctuations is declining and secondary fluctuations are becoming more important . the question naturally arises : how do contributions to the power spectrum from secondary fluctuations influence parameter determination based on the small scale cmbr power spectrum ? the most important secondary fluctuations are the thermal static and kinematic sunyaev - zeldovich ( ssz and ksz ) effects ( @xcite ) , the rees - sciama ( rs ) effect ( @xcite ) , the moving cluster of galaxies ( mcg ) effect ( @xcite ; @xcite ; @xcite ) , point sources ( @xcite ) , and , if the universe was re - ionized at some early stage , the ostriker - vishniac effect ( @xcite ; @xcite ) . in this paper we concentrate on secondary effects caused by clusters of galaxies . since detailed reviews are available on the sz and the mcg effects ( @xcite ; @xcite ) , here we just summarize their major features . the thermal sz effect is a change in the cmbr via inverse compton scattering by electrons in the hot atmosphere of an intervening cluster of galaxies . we use the terms kinematic or static sz effect depending on whether or not the intracluster gas possesses bulk motion . to date only the static thermal sz effect has been detected ( @xcite ) . the mcg effect is a special type of rs effect , due to the time - varying gravitational field of a cluster of galaxies as it moves relative to the rest frame of the cmbr . unlike the original rs effect , the mcg effect is not caused by intrinsic variation of the gravitational field , so that in the rest frame of the cluster , the photons fall into and climb out of the same gravitational field . however , in the rest frame of the cluster the cmbr is not isotropic , but has a dipole pattern , being brighter in the direction of the cluster peculiar velocity vector . photons passing the cluster are deflected towards its center . thus in the direction of the cluster peculiar velocity vector ( ahead of the cluster ) one can see a cooler part of the dipole pattern . towards the tail of the cluster , one can see a brighter part of the dipole ( ahead of the cluster ) . when transferring back to the rest frame of the cmbr , we transfer the dipole out , but the fluctuations remain , showing a bipolar pattern of positive and negative peaks . at cluster center there is no deflection , thus there is no effect . the amplitude of the mcg effect is proportional to the product of the gravitational deflection angle and the peculiar velocity of the cluster . the most important characteristics of the ssz , ksz , and mcg effects in the context of cosmology is that their amplitudes do not depend on the redshift of the clusters causing the effect . using thermodynamic temperature units , their maximum amplitude are about 500 @xmath9k , 20 @xmath9k , and 10 @xmath9k , respectively . the ssz and ksz effects have the same spatial dependence as the line of sight optical depth , the mcg effect has a unique bipolar pattern . assuming a king approximation for the total mass and an isothermal beta model for the intracluster gas , the full width at half maximum ( fwhm ) of the ssz and ksz effects @xmath10 , where @xmath11 is the core radius , depending on @xmath12 , which is typically between 2/3 and 1 . the mcg effect has a much larger spatial extent , with fwhm for each part of the bipolar distribution @xmath13 . the spectra of the effects are also important : the ssz effect has a unique spectrum which changes sign from negative to positive at about 218 ghz . the ksz and mcg effects have the same frequency dependence as the primordial fluctuations . the most important difference between the sz effect and the mcg effect is that the sz effect is caused by intracluster gas , the mcg effect is caused by gravitational lensing by the total mass regardless the physical nature of that mass . therefore the sz effect only arises from clusters with intracluster gas . clusters can produce significant mcg effects even if devoid of intracluster gas . the effects of clusters of galaxies on the cmbr in a given cosmology have been a subject of intensive research since the late 1980s . there are several different ways of extracting information from these effects . source counts of the ssz effect were estimated by using the press - schechter mass function ( psmf ) and scaling relations ( @xcite ; @xcite ; 1994 ; @xcite ; @xcite , de luca , desert & puget 1995 , @xcite ; 1997 , @xcite ) . the importance of the ssz effect was demonstrated and it was shown that thousands of detections are expected with the next generation of satellites . contributions to the cmbr from the rs and the ksz effects were derived by tuluie , laguna & anninos ( 1996 ) and seljak ( 1996 ) for cdm models with zero cosmological constant . tuluie et al . used n - body simulations and a ray - tracing technique , seljak used n - body simulations and second order perturbation theory . contributions from the ssz and ksz effects originating from large scale mass concentrations ( superclusters ) were studied by persi et al . bersanelli et al . ( 1996 ) , in their extensive study of the cmbr for the planck mission , estimated the contribution to the power spectrum from the ssz and ksz effects . aghanim et al . ( 1998 ) estimated the effects of the ksz and mcg effects on the cmbr including their contributions to the cmbr power spectrum . aghanim et al . simulated @xmath14 maps with pixel size of @xmath15 . they used the psmf normalized to x - ray data ( assuming an x - ray luminosity - mass relation ) . the total mass was assumed to have a navarro - frenk - white profile ( @xcite ) , and the intracluster gas was assumed to follow an isothermal beta model distribution . the time evolution of the electron temperature and the core radius were assumed to follow models of bartlett & silk ( 1994 ) , which are based on self - similar models of kaiser ( 1986 ) . according to aghanim et al . ( 1998 ) s results , the ksz effect is many orders of magnitude stronger than the primordial cmbr on small angular scales , and therefore the effect would prevent the use of the power spectrum to break the geometric degeneracy . atrio - barandela & mucket ( 1998 ) estimated the power spectra of the ssz effect in a standard dark matter dominated model with different lower mass cut - offs . contributions to the power from the ostriker - vishniac effect in cdm models were estimated by jaffe & kamionkowski ( 1998 ) . in this paper we estimate the contributions to the cmbr power spectrum from the ssz , ksz , and mcg effects on small angular scales adopting cold dark matter dominated models . in our models we assumed scale invariant primordial fluctuations with a processed spectrum having a power law form on cluster scales with a power law index of @xmath16 ( bahcall & fan 1998 ) . this maybe used as a first approximation as long as contributions from very low and/or very high mass clusters are small ( cf . our discussion about mass cut offs at section [ s : psmf ] ) . we use three representative models in our study : _ model 1 _ , open cdm ( ocdm ) model : a low density open model with @xmath17 , @xmath18 , @xmath19 ; _ model 2 _ , flat lambda cdm ( cdm ) model : a low density flat model with @xmath17 , @xmath20 , @xmath21 ; _ model 3 _ , standard cdm ( scdm ) model : a flat model with @xmath22 , @xmath18 , and @xmath23 . in section [ s : method ] we outline our method of estimating the power spectra of secondary fluctuations caused by clusters of galaxies and discuss our normalization method for the psmf . sections [ s : psmf ] and [ s : physpar ] describe how we used the psmf and the scaling relations to obtain masses and other physical parameters of clusters . in section [ s : powerspectr ] we present the spherical harmonic expansion of the ssz , ksz and mcg effects , and our method of estimating their power spectra . section [ s : simulation ] describes our simulations to evaluate the integrals over clusters . sections [ s : results ] and [ s : discussion ] present our results and discuss the differences from previous work . we used the press - schechter mass function ( psmf , @xcite ) as a distribution function for cluster masses . we used @xmath24 as indicated by observations ( bahcall and fan 1998 , and references therein ) . we used observationally determined cluster abundances as a constraint on the psmf . where necessary , we altered the model parameters resulting from the usual top hat spherical collapse model since that model is only an approximation . in our scdm model we changed only the overall normalization of the psmf by multiplying it by 0.23 ( a similar normalization was used by @xcite ) . this procedure is inconsistent with the interpretation of the psmf as a probability distribution ( it does not integrate to unity ) , but we use results for the scdm model only as a comparison to the other two models . in our lambda - cdm model we multiplied the critical density threshold , @xmath25 , obtained from the spherical collapse model ( equation [ [ e : delta_c ] ] ) , by 1.23 ( which is equivalent to changing the @xmath26 normalization ) and made no other changes . our ocdm model needed no adjustments . with these changes all three models agree well with the present day ( @xmath27 ) observed mass spectrum ( figure [ f : psmf_obs ] ; bahcall & cen 1993 ) . for high masses , the first two models ( ocdm , and cdm ) also agree with the observed @xmath28 dependence of the high mass cumulative mass function ( figure [ f : psmf_z_obs ] ; bahcall , fan & cen 1997 ) . the cdm and the scdm models agree with cmbr constraints , while the ocdm model is rejected by these constraints ( @xcite ) . as a cautionary note , it is useful to keep it in mind that taking _ all _ data into account none of these models are acceptable . we assumed that the total mass distribution follows a truncated king profile ( @xcite ) . for the intra - cluster gas we assumed an isothermal beta model ( cavaliere & fusco - femiano 1976 ) . isothermal beta model fits to x - ray images of clusters give @xmath29 ( @xcite ) . determinations of the @xmath12 parameter based on spectroscopy suggest @xmath30 ( @xcite ; @xcite ) . numerical simulations imply a range for @xmath12 from 1 to about 1.3 ( @xmath31 : @xcite ) ; @xmath32 : @xcite ; @xcite suggest @xmath33 ) . we follow the precepts of @xcite and adopt @xmath34 . this choice provides a mass temperature function which is in a good agreement with the observed function ( @xcite ) . the fitted x - ray spatial distribution is highly dependent on the x - ray structure of the core , and may be expected to be less reliable in the outer regions of clusters . the sz effect is more sensitive to the outer regions ( the sz effect is proportional to the electron density as opposed to thermal bremsstrahlung , which is proportional to density squared ) . choosing the spectroscopically derived @xmath34 gives a smaller sz effect , and so our choice of @xmath12 should provide a conservative estimate of the contribution of the sz effect to the power spectrum . a slightly larger @xmath12 would not change our results significantly . although the beta model describes the inner intra - cluster gas well , for more accurate sz work an improved cluster model , which fits the outer regions better , will be needed . the other physical parameters were determined using the virial theorem , a spherical collapse model , and models of the intra - cluster gas by colafrancesco & vittorio ( 1994 ) . we assumed a maxwellian distribution for the peculiar velocities , @xmath35 , and used results of n - body simulations by gramann et al . ( 1995 ) to normalize the distribution . we took velocity bias into account , and assumed that the peculiar velocities are isotropically distributed . we used analytical approximations to calculate the contributions of the ssz , ksz and mcg effects to the cmbr power spectrum . these contributions are important only at small angular scales , where we can neglect the overlap between cluster images and ignore the weak cluster - cluster correlation , and therefore we can approximate the resulting power spectrum by summing the contributions from individual clusters ( similar methods were used by @xcite ; @xcite ; and @xcite ) . we expanded the ssz , ksz , and mcg effects as laplace series ( i.e. , series of spherical harmonics ) , then determined the individual cluster contributions and summed over the clusters ( for a detailed description see @xcite ) . our approximation breaks down at large angular scales , but at these scales primordial fluctuations dominate the cmbr , and the cluster contribution is only a minor perturbation , so that only a rough indication of the cluster effect is needed . according to the press - schechter method , the co - moving number density of clusters of total mass @xmath36 at redshift @xmath28 ( the psmf ) is given by @xmath37 where @xmath4 is the matter density today in units of the critical density , @xmath38 is the current critical density of the universe ( we adopt a dimensionless hubble parameter @xmath39 in our work ) ; @xmath40 , where the present mass variance for a power law power spectrum , @xmath41 , is @xmath42 @xmath43 , @xmath44 is the mass within an @xmath45 mpc sphere , and @xmath26 is the normalization ( lacey & cole 1993 , press & schechter 1974 ) . the over - density threshold linearly extrapolated to the present may be expressed as ( @xcite ; @xcite ) @xmath46 & $ \omega_0 < 1 $ , $ \lambda = 0 $ \cr 0.15 ( 12 \pi)^{2/3 } \omega_m^{0.0055 } d_\lambda(\omega_0 , 0)/d_\lambda(\omega_0 , z ) & $ \omega_0 < 1 $ , $ \lambda = 1 - \omega_0 $ \cr } , \ ] ] where the conformal time for open models is @xmath47 .\ ] ] for open models with cosmological constant @xmath18 the linear growth factor is ( peebles 1980 ) @xmath48 where @xmath49 , and the density parameter @xmath50 is @xmath51 for models with a non - zero cosmological constant the integral can not be done analytically , thus we use an approximation ( lahav et al 1991 ; caroll , press & turner 1992 ) to obtain the equivalent expression to ( [ e : dz_lam0 ] ) , @xmath52 \biggl [ 1 + { \omega_\lambda(z ) \over 70 } \biggr ] \biggr\}^{-1 } .\ ] ] the density parameters , @xmath53 and @xmath54 , for spatially flat universes with @xmath55 , are @xmath56 the normalization of these growth functions is chosen so that at high redshifts they approximately match the time variation of density contrast in an einstein - de sitter ( @xmath22 ) universe , which is a good approximation to the early universe whatever its density parameter today . the total number of clusters at redshift @xmath28 ( in a redshift interval of @xmath57 ) is @xmath58 where @xmath59 and @xmath60 are the lower and upper mass cut offs for clusters . we used @xmath61 for the ssz and ksz effects , @xmath62 for the mcg effect , and @xmath63 for all effects . the lower cut off , @xmath59 , for sz effects signifies the lowest cluster mass for which we expect a well - developed intracluster atmosphere . in the case of the mcg effect , @xmath59 is the mass limit from which we consider a mass concentration as a cluster ( `` formation '' ) . we found that low mass clusters do not contribute substantially to the power spectrum , thus the lower cut off , @xmath59 , has little effect on our results . the upper cut off , @xmath60 , has no effect on our results ( as long as it is large enough ) : the probability of getting such a large cluster is negligible , so that the contribution from more massive clusters is negligible . we assumed a truncated king profile for the total mass distribution @xmath64 where @xmath65 is the core radius , @xmath66 is the cut off , and an isothermal @xmath12 model for the intra - cluster gas @xmath67 where @xmath68 and @xmath69 are the electron number density at radius @xmath70 and at the center of the cluster ( cavaliere & fusco - femiano 1976 ) . analytical studies and numerical simulations show that the gas density profile scales with the total density , and that the gas central electron density may be expressed as @xmath71 where @xmath72 is the average hydrogen mass fraction , @xmath73 is the intra - cluster gas mass fraction . @xmath74 , the central mass density , is determined from the total mass by integrating equation ( [ e : king_ro ] ) . little is known about the total mass and redshift dependence of the intra - cluster gas from observations . here , we adopt colafrancesco & vittorio ( 1994 ) s model which assumes that changes in the intra - cluster gas are driven by entropy variation and/or shock compression and heating . according to their model , the gas mass fraction may be expressed as @xmath75 where the normalization , @xmath76 , is based on local rich clusters , and we used @xmath77 and @xmath78 , which are consistent with available data . using the virial radius to express the core radius , @xmath79 , and assuming spherical collapse , we obtain @xmath80^{1/3 } { 1 \over 1 + z } , \ ] ] where @xmath81 is the overdensity of the cluster relative to the background ( @xcite ) . for @xmath18 models ( our ocdm and scdm models ) the over density may be expressed as @xmath82 , \ ] ] where @xmath83 ( oukbir & blanchard 1997 ) . for spatially flat models with finite cosmological constant ( our cdm model ) we have @xmath84 , \ ] ] where we used the approximation of kitayama & suto ( 1996 ) . numerical models of cluster formation show that cluster temperature scales with total mass . using the virial theorem and assuming spherical collapse with a recent - formation approximation in a standard cdm model , the electron temperature , @xmath85 , becomes @xmath86 where @xmath87 is the density contrast of a spherical top - hat perturbation relative to the background density just after virialization ( cf . for example eke , cole & frenk 1996 ) . the recent - formation approximation , however is valid only for @xmath22 . for our low matter density open model ( _ model 1 _ , ocdm ) , we use a model which takes into account accretion during the evolution of clusters , and leads to the following scaling : @xmath88 \ , { \rm kev } .\ ] ] this was derived for open models with zero cosmological constant ( @xcite ) , but since structure formation evolves similarly in a low density model with the same matter density and a zero cosmological constant , we use it for our _ model 2 _ ( cdm ) as an approximation . we assumed a maxwellian for the cluster peculiar velocity distribution , @xmath35 , as expected from a gaussian initial density field : @xmath89 where @xmath90 is the maxwellian width of the peculiar velocity distribution . the rms peculiar velocity from linear theory , smoothed with a top - hat window function of radius @xmath91 , @xmath92 , is given by @xmath93 where @xmath94 is the scale factor , and the moments , @xmath95 , are defined as @xmath96 where @xmath97 is the fourier transform of the power spectrum and equation ( [ e : pecvel1 ] ) uses the moment @xmath98 ( @xcite ) . the velocity factor , @xmath99 , can be approximated as ( @xcite ; 1984 ) @xmath100 & $ \lambda = 1 - \omega_0 $ \cr } .\ ] ] the cluster peculiar velocity rms differs from this since we assume that clusters form at the peaks of the density distribution , and with this bias may be expressed as @xmath101\ ] ] ( bardeen et al 1986 ) . colberg et al . ( 1998 ) calculated the velocity bias in a series of cdm models using a top - hat filter and processed cdm power spectra . the correction factor has a weak dependence on @xmath4 : it is about 0.8 for low density and flat cdm models . we obtain the maxwellian width in equation ( [ e : p_v_pec ] ) from the rms peculiar velocity from averaging a maxwellian : @xmath102 we expressed @xmath103 as @xmath104/ [ h(0 ) a(0 ) f(0 ) ] $ ] . the normalization at @xmath6 was determined by using results on the peculiar velocity distribution from numerical simulations ( @xcite ) . thus we obtain the following expression for the maxwellian width of the peculiar velocities , @xmath105 , with velocity bias for models with no cosmological constant ( ocdm , scdm , @xmath18 ) : @xmath106 for our cdm model ( @xmath107 ) we obtain @xmath108 , \\ & & \nonumber\end{aligned}\ ] ] this normalization is significantly larger than some recent measurements suggest ( bahcall & oh 1996 ) , but it is a good match to others ( gramann 1998 ) . this uncertainty should be remembered when interpreting our final results . ignoring the correlation between clusters , the power spectrum becomes @xmath109 where @xmath110 is the contribution from clusters with total mass @xmath36 at redshift @xmath28 , and @xmath111 denotes the ssz , ksz or mcg effects . @xmath112 is the differential volume element ( assumed isotropy ) @xmath113^{-1/2 } , \ ] ] where the effective distance @xmath114 is @xmath115 & $ \lambda = 0$\cr { c \over h_0 } \int_0^z dx \bigl[\omega_0 ( 1 + x)^3 + 1 - \omega_0 \bigr]^{-1/2 } & $ \lambda = 1 - \omega_0$}\ ] ] ( peebles 1993 ) . in general , the coefficients @xmath116 may be determined by calculating the spherical harmonic expansion of the cluster image by averaging out the azimuthal parameter , @xmath117 , @xmath118 our task is to determine the @xmath119 coefficients . the ssz and ksz effects are cylindrically symmetric for spherical clusters , therefore we may describe them using only one coordinate , the angular distance from the cluster center . we separate the effects into amplitudes and geometrical form factors which carry their spatial dependence . the ssz and ksz effects in thermodynamic temperature units may be expressed as @xmath120 where the central effects for the ssz and ksz effects are @xmath121 \ , \theta \ , \tau_0 , \ ] ] and @xmath122 \ , \beta^2 - \bigl[1 + \theta c_1(x ) + \theta^2 c_2(x ) \bigr]\ , \beta \ , p_1(\alpha ) \\ & + & \bigl [ d_0(x ) + \theta d_1(x ) \bigr ] \ , \beta^2 \ , p_2(\alpha ) \biggr\}\ , \tau_0 \nonumber .\end{aligned}\ ] ] in these expressions @xmath123 is the legendre polynomial of order of @xmath124 , @xmath125 is the dimensionless frequency , @xmath126 is the dimensionless temperature , @xmath127 is the angle between the cluster s peculiar velocity vector and its position vector , @xmath7 , @xmath128 , @xmath129 and @xmath130 are the planck constant , frequency , boltzmann constant , and temperature of the cmbr , @xmath131 k ( @xcite ) , and the lengthy expressions for the spectral functions @xmath132 , @xmath133 and @xmath134 may be found in nozawa , itoh & kohyama ( 1998 ) . these functions arise from an expansion of the boltzmann equation and although they are inaccurate for high temperature clusters , their precision is sufficient for our purposes here ( for a discussion see @xcite and references therein ) . the optical depth through the cluster center for gas model ( [ e : n_el ] ) is @xmath135 and the geometrical form factor is @xmath136 where the function @xmath137 is defined as @xmath138 @xmath139 holds in the small angle approximation , and the integral , @xmath140 , is @xmath141 where @xmath142 is the gamma function , @xmath143 is gauss hyper - geometric function , and @xmath127 must be greater than 1/2 ( @xcite ) . the geometrical form factor is normalized to one at the cluster center ( @xmath144 ) . we may expand the ssz and ksz effects in legendre series as @xmath145 where @xmath146 are legendre coefficients of @xmath147 , and @xmath148 refers to the ssz or the ksz effect . we determine the legendre coefficients using a small angle approximation , as @xmath149 \ , \theta \ , d \theta , \ ] ] where we used the approximation @xmath150 , \ ] ] where @xmath151 is a bessel function of the first kind and zero order ( @xcite ) . we can convert legendre coefficients to laplace coefficients by expressing the laplace series of such a function as @xmath152 where @xmath153 , and @xmath154 and @xmath155 are the spherical harmonics and legendre polynomials . therefore the conversion can be done as @xmath156 thus the laplace series of the ssz and ksz effects become @xmath157 using equation ( [ e : delta_sz ] ) , the contribution of one cluster to the power spectrum of the ssz and ksz effects becomes @xmath158 where @xmath159 and @xmath146 are given by equations ( [ e : delta_i_s_r ] ) , ( [ e : delta_i_k_r ] ) , and ( [ e : zeta_ell ] ) . similarly , the mcg effect may be expressed as @xmath160 where @xmath161 is the geometrical form factor . @xmath162 is the angle of the line of sight relative to the center of the cluster . the azimuthal angle , @xmath163 , is measured in the plane of the sky from the direction of the tangential component of the peculiar velocity . the maximum of the mcg effect is @xmath164 where @xmath165 is the maximum deflection angle , @xmath127 is the angle between the cluster s peculiar velocity vector , @xmath166 , and its position vector , and @xmath167 is the speed of light in vacuum . for a spherically symmetric thin lens the deflection angle is given by @xmath168 where @xmath169 is the impact parameter at the source , and @xmath170 is the mass enclosed by a cylindrical volume with axis parallel to the line of sight and radius equal to the impact parameter @xmath169 ( cf . for example schneider , ehlers & falco 1992 ) . using the king approximation for the density distribution ( equation [ [ e : king_ro ] ] ) , the total mass in cylindrical coordinates , @xmath171 , becomes @xmath172 where @xmath173 in the small angle approximation . a straightforward integration and equation ( [ e : delta_mcg0 ] ) lead to @xmath174 where the function @xmath175 is @xmath176 .\ ] ] thus the geometrical form factor in our case becomes @xmath177 where @xmath178 is the maximum value of the function @xmath179 . the mcg effect depends only on @xmath180 , therefore we need to determine only the @xmath181 terms in the spherical harmonic expansion . expressing the spherical harmonics by associated legendre polynomials , equation ( [ e : xi_g ] ) expands as @xmath182 , \ ] ] where we used the identity @xmath183 in order to obtain a real function , the imaginary terms must vanish , therefore we must have @xmath184 using orthogonality , expressing the spherical harmonics in terms of associated legendre polynomials and using equations ( [ e : xi_ell ] ) and ( [ e : xi_thetaphi ] ) , we obtain @xmath185 where @xmath186 the @xmath163 integral can be performed since @xmath175 and @xmath187 do not depend on @xmath163 giving @xmath188 from which we find @xmath189 \ , d \theta .\ ] ] here we used the small angle approximation for the associated legendre polynomials : @xmath190 , \ ] ] where @xmath191 is the bessel function of the first kind and order 1 ( for a derivation see appendix ) . thus the laplace series of the mcg effect is @xmath192 where @xmath193 is given by equation ( [ e : xi_ell_1_j1 ] ) . for the power spectrum of the mcg effect , using equation ( [ e : delta_mcg_y ] ) , we obtain @xmath194 the observed effects are calculated by convolving the theoretical fluctuation pattern with the telescope s point spread function ( psf ) . one advantage of using the spherical harmonic coefficients is that this convolution is just a multiplication in spherical harmonic space . assuming an axially symmetric psf , @xmath91 , its legendre polynomial expansion may be expressed as @xmath195 where the unit vectors , @xmath196 and @xmath197 , point to an arbitrary direction ( where we want to evaluate the expansion ) and to the center of the psf . assuming a non - axially symmetric effect , its spherical harmonic expansion can be written as @xmath198 where @xmath199 runs from zero to infinity and @xmath117 runs from -@xmath199 to @xmath199 . using the addition theorem for spherical harmonics and their orthogonality , the convolution of these two functions , @xmath200 , becomes @xmath201 we used monte carlo simulations to generate an ensemble of clusters of galaxies with masses sampled from the psmf ( equation [ e : psmf ] ) with parameters those of our ocdm , cdm and scdm models . we obtained the central electron number density and temperature , and the cluster core radius from scaling relations ( equations [ e : n_el_0 ] , [ e : t_eom1 ] , [ e : t_eopen ] , and [ e : r_cor ] ) . we choose to sample the psmf using a rejection method . the magnitude of the peculiar velocity may be sampled using an inversion method on the maxwellian ( [ e : p_v_pec ] ) , and yields @xmath202 where @xmath203 is the inverse of the incomplete gamma function and @xmath103 can be determined by using equations ( [ e : sigm_v_pec ] ) , ( [ e : sigma_p_l0 ] ) and ( [ e : sigma_p_lam ] ) . @xmath204 is a uniformly distributed random number in ( 0,1 ) . we assumed an isotropic distribution in space for the directions of the peculiar velocity vectors , and ignored correlations between cluster peculiar velocities . the tangential and radial peculiar velocities are distributed as projections of equation ( [ e : v_p_rn ] ) accordingly . as an illustration , in figure [ f : mcg1 ] we show results from one simulation using our scdm model projected on a grid . the observational mass function ( figure [ f : psmf_obs ] ) is specified by @xmath205 , the mass contained within co - moving radius of 1.5 mpc . to convert @xmath205 to the virial mass , @xmath206 , which we use in the psmf , we assume that the mass profile near 1.5 mpc can be approximated with @xmath207 . we obtain @xmath208 or , substituting the numerical values , @xmath209 we used q = 0.64 ( @xcite ) to obtain curves shown in figure [ f : psmf_z_obs ] . the power spectrum for an ensemble of clusters may be determined by summing the individual contributions of the simulated clusters ( equation [ [ e : c_ell_int ] ] ) . we binned clusters by their apparent core radii , @xmath210 , then we summed the amplitudes in each bin . the numerical evaluation of integral ( [ e : c_ell_int ] ) , in this case , can be performed as @xmath211 and @xmath212 where the index @xmath213 runs over clusters whose core radii fall within the @xmath214 bin . our results for the power spectra ( more exactly the dimensionless @xmath215 ) from our _ model 1 _ ( ocdm ) are shown on figure [ f : powerspocdm ] . figures [ f : powersplcdm ] and [ f : powerspscdm ] show our results for _ model 2 _ , ( cdm ) and _ model 3 _ , ( scdm ) respectively . as a comparison , in each figure , we plot the corresponding primordial cmbr power spectrum ( solid line ) with @xmath216 normalization including the effects of gravitational lensing calculated by using a new version of cmbfast ( @xcite ; @xcite ) . on large angular scales ( up to about @xmath217 ) the cosmic variance dominates ( not shown ) . on small angular scales the shape of the power spectra depends on the apparent angular sizes of the clusters and the amplitudes of the effects . the apparent angular size depends on how the core radius and the angular diameter distance change with redshift , while the amplitude is sensitive to the gas content , gas temperature and total mass as a function of redshift . figures [ f : powerspocdm]-[f : powerspscdm ] demonstrate that for small angular scales ( @xmath0 ) the contribution to the power spectrum from the ssz effect exceeds that of the primordial cmbr in all our models . the contributions of the ksz and mcg effects become important only on small scales , but , at those scales , they may dominate over the lensed primordial fluctuations . due to the early structure formation , there are more clusters at high redshift in our ocdm and cdm than in our scdm simulations . therefore the contribution to the power spectrum from clusters in a low matter density model is substantially larger than in a scdm model . also , most clusters are closer to us in a scdm model , thus the contribution from clusters to the power spectrum peaks at higher angular scales ( lower @xmath199 ) than in low matter density models . the ksz and mcg effects have their coherence length ( peak contributions ) at @xmath218 . the coherence length of the ssz effect is about @xmath219 for our scdm model and at about @xmath220 for our ocdm and cdm models . in general , the contributions to the power spectrum from the ssz effect are about 2 and 3 orders of magnitude greater than those from the ksz and mcg effects . at very small angular scales , @xmath221 , the contribution to the power spectrum from the mcg effect might exceed that of the ssz or ksz effects , and even the primordial fluctuations in the cmbr , but this depends on the details of the evolution of cluster atmospheres . our simulations give somewhat different results for the ksz and mcg effects than those of aghanim et al . ( 1998 ) . in our simulations the amplitudes of the ksz and mcg effects for low matter density models are about an order of magnitude greater than those for our scdm model , and have a coherence length of about @xmath222 , while rising monotonically at smaller @xmath199 . according to aghanim et al.s simulations , with similar cut off to ours , @xmath223 , contributions from the ksz effect in all models constantly grow and show no signs of leveling off , and their amplitude has a very weak dependence on cosmological model . contributions from the mcg effect on the other hand show a plateau in all aghanim et al.s models for @xmath224 , and for the scdm model , the mcg effect is larger than for the other two models . quantitatively , our models show cluster - related effects that are weaker by a factor of 10 for the mcg effect in our scdm model and a factor of 100 for the ksz effect for all models . we attribute these differences mostly to the different evolution models for the intracluster gas . the ratio between the overall amplitudes of the ksz and mcg effects in our calculations is about the same as in aghanim et al . ( 1998 ) s results . our results show that the power spectra of the ksz and mcg effects do not exceed the gravitationally lensed primordial power spectrum up to @xmath225 , while the power spectrum of the ssz effect becomes dominant at @xmath226 in all our models . contributions to the power spectrum from the ssz and ksz effects based on aghanim et al.s model would exceed those from the cmbr at @xmath227 even if one takes gravitational lensing of the primordial cmbr into account . our results are comparable to those obtained by tuluie et al . ( 1996 ) and seljak ( 1996 ) . persi et al . ( 1995 ) s results for contributions to the power spectrum from the ssz ( ksz ) effect are about the same as ( an order of magnitude higher than ) our results suggest . an observed power spectrum is made up from the sum of all astrophysical effects and noise . we rely on the different frequency and/or power spectra of the secondary effects to separate these foregrounds from the primordial cmbr signal ( @xcite ) . of the effects discussed here , it should be easy to separate the ssz effect by using multi - frequency measurements of its unique spectrum . the separation of primordial fluctuations in the cmbr and fluctuations caused by the ksz and mcg effects is more difficult since their frequency spectra are the same . optimal filters have been designed to separate the ksz effect ( @xcite ; @xcite ) : here it helps to know the ssz effect for the same cluster , since that would give us a position and even an estimate for the expected amplitude of the effect . aghanim et al . ( 1998 ) discussed methods to separate the mcg effect : this is facilitated by its unique dipole pattern with sharp peaks ( figure [ f : mcg1 ] ) . primordial fluctuations are usually assumed to be gaussian , where the probability of getting such a strongly peaked bipolar pattern is small , and we would expect the strong small angular scale gradient near a known cluster of galaxies to be a definite indication of the presence of the mcg effect . also , knowing the position of clusters helps to find the effect . however , contributions from other effects , such as early ionization and discrete radio sources causes further confusion , and may be expected to make it difficult to determine the power from the sz or mcg effects . we analyzed the contributions to the power spectrum from the ssz , ksz and mcg effects to check their impact on the determination of cosmological parameters , especially at large @xmath199 where gravitational lensing may break the geometric degeneracy . in figure [ f : lens01 ] we show the small scale lensed primordial fluctuation power spectra of our three models ( ocdm , scdm , cdm ; solid lines ) with power spectra resulting from the sum of fluctuations due to the lensed primordial cmbr and the ssz effect ( long dashed lines ) , and from the sum of the lensed primordial cmbr , the ksz and the mcg effects ( short dashed lines ) . according to our models , if the fluctuations due to the ssz effect are fully separated , the ksz and mcg effects do not prevent the use of this part of the power spectrum to break the geometric degeneracy and distinguish between different cdm models . note that normalization at the first doppler peak , rather than the usual _ cobe _ normalization , would lower the contributions to the power spectrum from primordial fluctuations in a cdm model relative to those from a scdm model , and thus secondary effects would become more important relative to the primordial cmbr fluctuations . our simulations also show that the power spectrum of the ssz effect may itself be used to break the geometric degeneracy . since the separation of the ssz effect from other secondary effects should be straightforward , we should be able to determine the power spectrum of the ssz effect alone . as can be seen from figure [ f : powersp3 ] , this power spectrum depends on @xmath2 and @xmath3 , providing an additional constraint on these parameters . we note , however , that the amplitude of the ssz effect is model dependent . since the contributions to the power spectrum from the sz and mcg effects are model dependent , to evaluate fully their power spectra we need a better observationally - supported model for the intracluster gas . sensitive , high - resolution all - sky , x - ray observations could map the emission from intracluster gas up to high redshift providing strong constraints on gas formation and evolution and thus a good basis for modeling the ssz and ksz effects ( @xcite ) . number counts of clusters based on their ssz effect can also be used to constrain cosmological models ( @xcite ) . there are many possibilities of using observations to break the geometric degeneracy . for example measurements of the cmbr polarization , the hubble constant , or light curves of type ia supernovae have been discussed ( zaldarriaga et al . 1997 ; @xcite ; @xcite ) . also , combination of measurements of the ssz effect and thermal bremsstrahlung ( x - ray ) emission from clusters can be used to determine the hubble constant for a large number of clusters , providing a statistical sample which might enable us to determine the hubble constant , and perhaps the acceleration parameter , to good accuracy ( @xcite ) . secondary fluctuations introduce non - gaussianity into the primordial spectrum at small scales . this non - gaussianity should be taken into account when estimating cmbr non - gaussianity at these scales . winitzky ( 1998 ) estimated the effect of lensing and concluded that planck may observe non - gaussianity due to lensing near the angular scale of maximum effect , @xmath228 . other processes , including the ssz , ksz , and especially the mcg effect , introduce a highly non - gaussian signal as is easily seen for the mcg effect on figure [ f : mcg1 ] . a similar non - gaussian pattern arises from moving cosmic strings ( the kaiser - stebbins effect , compare our figure [ f : mcg1 ] to figure 6a of @xcite ) . our results indicate that at @xmath229 the mcg effect might be comparable in strength to the primordial fluctuations . evidence for non - gaussianity has been reported by ferreira , magueijo & gorski ( 1998 ) and gaztanaga , fosalba , & elizalde ( 1997 ) at angular scales @xmath230 and @xmath231 . they do not exclude the possibility that this non - gaussianity has been introduced by foregrounds , but our results show that clusters can not introduce detectable non - gaussianity on such scales ( figure [ f : powersp3 ] ) . we convolved our theoretical results ( figures [ f : powerspocdm ] - [ f : powerspscdm ] ) with the expected point spread functions ( psfs ) of instruments on the map and planck missions to estimate the level of the secondary fluctuations caused by clusters of galaxies on the observable power spectrum . the observed @xmath232 values become in general , contributions from unresolved cluster static effects add to provide a cumulative contribution to the cmbr power spectrum . contributions from the ksz and mcg effects from unresolved sources tend to cancel . in the case of the mcg effect this is because each unresolved source contribution would be zero owing to the dipole spatial pattern of the effect . for small - scale ksz effects there are several sources in the field of view of the telescope , and different sources have positive or negative contributions depending on the sign of their line of sight peculiar velocity , and therefore they tend to cancel each other . the larger the beam size , the more effective is the cancellation of the mcg and ksz effects . note that the spatial extension of the mcg effect is much larger than that of the ksz effect , so many clusters may be unresolved in their ksz and resolved in their mcg effect . the mcg effect might be relatively more important at high redshifts , since it does not require a well - developed cluster atmosphere . in figures [ f : map94 ] and [ f : planck353 ] we show the contributions to the power spectrum from primordial fluctuations , and the ssz , ksz and mcg effects , convolved with the psf of the planned @xmath238 ghz receiver on map , and the planned @xmath239 ghz bolometer on planck . the amplitude of the fluctuations from the the ssz effect is negative at @xmath238 ghz and positive at @xmath239 ghz , but only the the absolute value of the effect contributes to the power spectrum . the different maximum @xmath199 values for the map and planck systems ( @xmath240 1000 and 2000 , respectively ) can clearly be seen on figures [ f : map94 ] and [ f : planck353 ] . because of these cutoffs , the observable power spectrum is dominated by primordial fluctuations at all @xmath199 for these missions . according to our results , the ssz effect may cause a 1% enhancement in the amplitude of the doppler peaks , which is at the limit of the sensitivity of the map and planck missions . from the analysis of the power spectrum , this would lead to an overestimation of the parameter @xmath2 by about 1% , . the shift in the position of peaks as a function of @xmath199 caused by the ssz effect is less important since the spectrum of the ssz effect has only a weak dependence on @xmath199 . in table [ t : rms_all ] we show the @xmath241 values of the contributions to the cmbr from the ssz , ksz , and mcg effects convolved with the the 94 ghz map and 353 ghz planck receivers for our three models ( ocdm ; cdm ; scdm ) . as a comparison , we display the corresponding rms values of the primordial fluctuations . the rms values of all these secondary effects are an order of magnitude smaller than rms values from primordial fluctuations . the most important contribution is that of the ssz effect at these frequencies . the ksz and mcg effects give similar contributions with the ksz effect being about a factor of two stronger . aghanim et al . ( 1998 ) s results for the rms values of the mcg effect is a factor of 10 ( scdm ) or a factor of 3 ( ocdm and cdm ) larger than our results . note , however , that rms values give only a crude estimate of the magnitude of the effects . at large angular scales the primordial fluctuations are about 100 ( for the ssz effect ) or @xmath242 ( ksz , mcg effects ) times stronger than the secondary fluctuations . an ideal observation to measure the contribution to the power spectrum from the ssz effect would use high angular resolution ( @xmath226 ) and high frequency ( @xmath243 ghz ) . suzie probes the power spectrum at angular scale @xmath244 at 140 ghz . the 2@xmath245 upper limit on the power at this scale from suzie is @xmath246 ( @xcite ) . unfortunately our models suggest that at this frequency the primordial contribution to the power spectrum is about 10 times stronger than that from the ssz effect . a promising experiment is scuba , which probes the anisotropies at angular scale @xmath225 and frequency 348.4 ghz . their preliminary 2@xmath245 upper limit on the power spectrum at this scale is @xmath247 . much further work is planned , and should lower this limit by a factor of 3 - 10 ( @xcite ) . smm is grateful to bristol university for a full scholarship , where most of this work was done . this work was finished while smm held a national research council research associateship at nasa goddard space flight center . we thank n. aghanim for comments on an earlier version of the manuscript , and our referee , dr bartlett , for his detailed comments and for helping us to clarify some aspects of our approximations . we thank u. seljak and m. zaldarriaga for making the cmbfast code available . we derive a small angle approximation for the associated legendre polynomial @xmath187 . we express the associated legendre polynomial @xmath187 in terms of the legendre polynomial @xmath155 ( see for example @xcite ) as cccccccccc + model & & map & 94 ghz & & & & planck & 353 ghz & + + + & cmbr & ssz & pksz & mcg & & cmbr & ssz & pksz & mcg + & @xmath9k & @xmath9k & @xmath9k & @xmath9k & & @xmath9k & @xmath9k & @xmath9k & @xmath9k + + + 1 . ocdm & 86 & 7.0 & 0.54 & 0.20 & & 100 & 14 & 0.81 & 0.33 + 2 . lcdm & 119 & 5.9 & 0.49 & 0.20 & & 130 & 11 & 0.71 & 0.32 + 3 . scdm & 93 & 3.2 & 0.36 & 0.12 & & 101 & 5.7 & 0.50 & 0.17 + | we estimate the contributions to the cosmic microwave background radiation ( cmbr ) power spectrum from the static and kinematic sunyaev - zeldovich ( sz ) effects , and from the moving cluster of galaxies ( mcg ) effect .
we conclude , in agreement with other studies , that at sufficiently small scales secondary fluctuations caused by clusters provide important contributions to the cmbr . at @xmath0 ,
these secondary fluctuations become important relative to lensed primordial fluctuations .
gravitational lensing at small angular scales has been proposed as a way to break the `` geometric degeneracy '' in determining fundamental cosmological parameters .
we show that this method requires the separation of the static sz effect , but the kinematic sz effect and the mcg effect are less important .
the power spectrum of secondary fluctuations caused by clusters of galaxies , if separated from the spectrum of lensed primordial fluctuations , might provide an independent constraint on several important cosmological parameters .
@xmath1 subject headings : cosmic microwave background galaxies : clusters : general methods : numerical |
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over the last fifteen years , the dynamics of glassforming liquids under nanoscale confinement has attracted much attention . a great variety of molecular systems under many different types of confinement has been subjected to virtually all available experimental techniques , while at the same time extensive studies of simple model systems using molecular dynamics simulations were undertaken @xcite . the main reason for this rapidly growing interest is that confinement is considered as a potential tool to investigate the concept of cooperativity , a key ingredient of many glass transition theories @xcite . indeed , there are now many evidences that the dynamics of deeply supercooled liquids is inhomogeneous and that dynamically correlated groups of molecules play a crucial role in the slowing - down of the dynamics when the temperature is decreased . but , up to now , many questions pertaining , for instance , to the shape , size or temperature evolution of these dynamical heterogeneities remain essentially unanswered ( for recent progress , see , however , ref . @xcite ) . in confinement , geometric constraints associated with the pore shape are imposed to the adsorbed fluid and new characteristic length scales , like the pore size , come into play . thus , by looking for alterations in the dynamics under confinement compared to the bulk , one can hope to pinpoint some of the elusive characteristic features of the dynamical heterogeneities . for instance , in the simplest scenarios , deviations from the bulk behavior are expected to occur due to finite size effects , when the typical size of the dynamical heterogeneities in the bulk would become larger than the pore size . confinement effects would thus provide a ruler to measure dynamical heterogeneities . the situation actually turns out more complex . indeed , a direct comparison between the bulk and confined fluids is only meaningful if the physical phenomena which are specific to confinement have a weak impact on the properties of the imbibed fluid or at least if their influence is sufficiently well understood that it can be corrected for . this is usually not so and strong confinement effects are often observed , for instance , the formation of structured layers of almost immobile molecules at the fluid - solid interface . so , in fact , dynamics in confinement should be addressed as a problem of its own , not necessarily with reference to the bulk . the variety of the systems to consider is immense . porous media can differ in the size , shape and topology of their pore space . they can be made of various materials or receive different surface treatments , leading to a wide range of fluid - solid interactions which adds to the already great variability of the intermolecular interactions met with usual glassformers . thus , owing to the complexity of the field , a reasonable microscopic theory , able to catch at least some of these aspects , could be very helpful . indeed , applied to various models , it would allow to explore thoroughly the phenomenology of confined glassforming liquids and maybe to disentangle the contributions of the different physical phenomena which interplay in these systems . many porous media , like controlled porous glasses and aerogels , are disordered . in the past few years , a very useful and quite successful model to deal with this kind of systems has been the so - called `` quenched - annealed '' ( qa ) binary mixture , first introduced by madden and glandt @xcite . in this model , sketched in fig . [ figsketch ] , the fluid molecules ( the annealed component ) equilibrate in a matrix of particles frozen in a disordered configuration sampled from a given probability distribution ( the quenched component ) . the matrix is assumed to be statistically homogeneous , so that , while for any single realization , the system lacks translational and rotational invariance , all expectation values computed with the matrix probability distribution will have the same properties as in a truly translationally and rotationally invariant system . a common , but not unique , prescription is to take the equilibrium distribution of some simple fluid system , so that the various samples of the matrix can be thought of as the results of instantaneous thermal quenches of this original equilibrium system , hence the denomination `` quenched '' for the matrix component . thanks to the assumption of statistical homogeneity , as far as the computation of matrix averaged quantities is concerned , qa mixtures can be studied with great ease using simple extensions of standard liquid state theoretical methods . this has been put to good use to derive equations describing the structure and the thermodynamics of these systems , either via diagrammatic techniques @xcite or by application of the replica trick @xcite . sketch of a qa system . in black , the immobile matrix particles . in white , with arrows symbolizing their movement , the fluid particles . ] the aim of the present work is now to develop a dynamical theory for qa mixtures , able to deal with the problem of the glass transition in confinement . this will take the form of an extension to qa systems of the ideal mode - coupling theory ( mct ) for the liquid - glass transition @xcite . the mct occupies a central place in the study of the dynamics of supercooled liquids in the bulk , as one of the very few available microscopic theories in this field . it has well known deficiencies , in particular the fact that its predicted sharp ergodicity breaking transition , the so - called ideal glass transition , is always located in the regime of weak to mild supercooling , rather far from the calorimetric glass transition point . but , quite remarkably , it has been able to correctly predict novel nontrivial relaxation patterns which develop in systems like colloidal suspensions with short - ranged attractions @xcite or fluids of symmetric dumbbells @xcite , when the parameters of these models are varied . it seems thus sensible to turn to this theoretical framework for a systematic investigation of confined glassforming liquids , aiming at understanding , at least qualitatively , how the microscopic details of the fluid - solid system impact its dynamics . moreover , it is encouraging that recent computer simulation data for confined fluids could be interpreted with the universal predictions of the mct @xcite , showing that the mode - coupling glass transition scenario might indeed be of some relevance in confinement as well . note that ref . @xcite precisely deals with a qa system , as do refs . @xcite , where other simulation studies are reported . theories related to the one to be derived have already been discussed in the literature . the mode - coupling approach to the diffusion - localization transition in the classical random lorentz gas @xcite is of particular relevance , since this system is actually a qa mixture taken in the limit of a vanishing density of the annealed component . besides , the present theory will borrow some ideas from this approach . other works have dealt with the conductor - insulator transition of quantum fluids in random potentials , originally neglecting the fluid - fluid interactions @xcite which were later reintroduced by thakur and neilson @xcite . they appear as special cases of the present theory , with additional implicit and uncontrolled approximations in the treatment of the effect of randomness on the static fluid correlations @xcite . finally , a similar nonlinear feedback mechanism has been derived for the freezing of a polymer chain in a quenched random medium using the self - consistent hartree approximation for the langevin dynamics of the system @xcite . the paper is organized as follows . in sec . ii , the mct equations for the collective dynamics of a qa mixture are derived and discussed , while sec . iii explores the relation between the dynamics in a self - induced glassy phase and in a quenched random environment . in sec . iv , dynamical phase diagrams for two simple models are computed , which show two basic types of ideal liquid - glass transitions discussed in more details in sec . v. section vi is devoted to concluding remarks . preliminary reports on the present work can be found in refs . @xcite and @xcite . in this section , the mode - coupling equations for the qa binary mixture are derived and discussed . but , before proceeding with the dynamical theory , a few static quantities have to be defined . as mentioned in the introduction , in a qa system , the disordered porous medium is represented by a collection of @xmath0 rigorously immobile point particles , randomly placed in a volume @xmath1 at positions denoted by @xmath2 , according to a given probability distribution @xmath3 @xcite . its overall density is @xmath4 and the fourier components of its frozen microscopic density , or , in short , its frozen density fluctuations , are given by @xmath5 where @xmath6 denotes the wavevector . their disorder - averaged correlation functions , which , because of the assumed statistical homogeneity , are diagonal in @xmath6 and only depend on its modulus @xmath7 , define the matrix structure factor @xmath8 where @xmath9 denotes an average over the matrix realizations . the fluid component consists of @xmath10 point particles ( density @xmath11 ) of mass @xmath12 , which equilibrate at a temperature @xmath13 in the random potential energy landscape created by the frozen matrix particles . as in the bulk , for the present theory , we will be interested in the dynamics of the fluid density fluctuations @xmath14 where @xmath15 is the position of the fluid particle @xmath16 at time @xmath17 . at equal times , they allow to define the fluid structure factor @xmath18 and the fluid - matrix structure factor @xmath19 where @xmath20 denotes a thermal average _ taken for a given realization of the matrix _ , the disorder average @xmath9 being performed _ subsequently_. for an ordinary binary mixture , the knowledge of the above three structure factors would be enough to fully characterize the structure at the pair level . this is not the case for a qa mixture . indeed , because the matrix component is quenched , for any single realization , the system lacks translational and rotational invariance . it results that , at variance with a bulk fluid , non - zero average density fluctuations exist at equilibrium , i.e. , @xmath21 . it is only after averaging over disorder that the symmetry is restored , so that @xmath22 . thus , one is led to consider relaxing and non - relaxing fluid density fluctuations , corresponding to @xmath23 and @xmath24 , respectively , and to define the connected fluid structure factor @xmath25 and the disconnected or blocked fluid structure factor @xmath26 such that @xmath27 . this splitting of the fluid pair correlations is well known from the replica theory of qa systems , where it leads to the peculiar structure of the so - called replica ornstein - zernike ( oz ) equations @xcite , which are given in appendix [ app.oz ] for reference . non - zero average fluid density fluctuations at equilibrium mean that , even without any dynamical ergodicity breaking , they will have time - persistent correlations . indeed , defining the normalized total density fluctuation autocorrelation function @xmath28 one expects using standard arguments that @xmath29 this is a general consequence of the fact that the fluid evolves in an inhomogeneous environment and we stress that this is a true static phenomenon . usually , mode - coupling theories are derived assuming that the statics of the problem is solved . for this reason , since the calculation of the contribution from the blocked correlations is actually a static problem , the theory has been developed using the relaxing part of the fluid density fluctuations as the central dynamical variable . attempts to derive a dynamical theory starting from the full density fluctuations have resulted in complicated equations , which appeared unfaithful to the statics of the problem , and thus were abandoned . the theory is derived using standard projection operator methods , as shown in ref . @xcite for bulk systems . as usual , in a first step , one obtains a generalized langevin equation for the time evolution of the normalized connected autocorrelation function of the density fluctuations @xmath30 it is formally the same as for the bulk , i.e. , @xmath31 with initial conditions @xmath32 , @xmath33 , and @xmath34 the second step involves the calculation of the slow decaying portion of the memory kernel @xmath35 with a mode - coupling approach . we will assume that the slow dynamics is dominated by three types of quadratic variables , which can be separated in two classes . in the first class , we have variables quadratic in the relaxing density fluctuations , @xmath36 , in close analogy with bulk mct @xcite . in the second class , inspired by previous studies on the lorentz gas @xcite , we consider variables expressing couplings of the relaxing density fluctuations to the two frozen density fluctuations present in the problem , @xmath37 and @xmath38 . the last variable was omitted in all previous works , including a recent account of the present theory @xcite , with consequences to be discussed below . the calculation is outlined in appendix [ app.derivation ] and we only quote the result here . it reads @xmath39 , where @xmath40 is a friction coefficient associated with fast dynamical processes , and @xmath41,\ ] ] with [ vertices ] @xmath42 ^ 2 s^{c}_k s^{c}_{|\mathbf{q - k}|},\ ] ] and @xmath43 ^ 2 s^{c}_k s^{b}_{|\mathbf{q - k}|},\ ] ] where @xmath44 , the fourier transform of the connected direct correlation function , has been introduced . based on these equations , a few general comments are in order . first of all , the derived expressions retain the mathematical structure of the typical mode - coupling equations which have been extensively studied in ref . @xcite . thus , all known properties of the solutions of mct equations , in particular near the transition , apply in the case of the qa mixture . a significant addition compared to the bulk is however the presence of a linear term in the memory function , which opens the possibility of continuous ideal glass transitions . like in the theory for the bulk , which is recovered in the limit @xmath45 , the slow dynamics is fully determined by smoothly varying static quantities . interestingly , no explicit reference to the matrix is visible in the equations . indeed , the only required information is @xmath46 , @xmath47 , @xmath48 , and @xmath44 , i.e. , quantities characterizing the fluid component of the qa system . these functions and the relations between them are generically meaningful for the description of fluids evolving in statistically homogeneous random environments and they are by no means restricted to the model of the qa mixture ( for the case of non - particle - based random fields , see refs . therefore , one can expect that the present dynamical theory shares the same degree of generality and is applicable in its present form out of the strict context of the qa binary mixture . in fact , this shows up nicely in the process of deriving the equations , since one finds that the contributions resulting from the coupling of the random forces to @xmath37 are identically zero ( see appendix [ app.derivation ] ) . thus , it is enough to consider only two types of quadratic variables , @xmath49 and @xmath50 , precisely those which are not specific to qa systems , to obtain the same dynamical equations . further evidence of the generic character of the present equations is given in appendix [ app.pspin ] , where the dynamics of a mean - field spin - glass model in a random magnetic field is shown to obey equations with exactly the same structure . along the same line , it is noteworthy that the total fluid structure factor @xmath51 does not appear in the derived dynamical equations . only @xmath47 and @xmath48 do . it results that , in the framework of the mct , the global fluid correlations are of limited relevance to discuss the relation between the statics and the dynamics . a much more important aspect is rather the balance between the connected and disconnected contributions to these correlations . this result is very welcome , since similar differences in the roles played by these three types of correlations are known to be crucial in the physics of qa systems . this is best illustrated by the compressibility sum rule @xcite , which precisely involves @xmath47 and not @xmath51 , as one would naively expect from the relation for the bulk . it is thus reassuring that this feature has been preserved despite the uncontrolled approximations involved in the derivation of the theory . moreover , this finding allows one to clarify the issues raised in refs . @xcite about the possibility of a description of the dynamics in confinement with theories based on structural quantities only . indeed , it was observed there that systems with identical global fluid correlations could have significantly different dynamics , a result which appeared as a challenge to such approaches . the present mct shows that this is not necessarily so and that one should also consider the connected and blocked correlations before a conclusion can be reached . about the form of the vertices , one can note the familiar expression of @xmath52 , which measures the coupling of the random forces to @xmath49 . it is the same as in a bulk system , with connected quantities simply replacing the fluid structure factor and direct correlation function . this is true for @xmath53 , given by eq . , as well . thus , we find that the relaxing part of the density fluctuations in a qa system just behaves dynamically like the corresponding degree of freedom in the bulk . this is not really surprising , since a similar correspondence is already visible in the statics , for instance in the oz equation or in the convolution approximation , which are both used in the calculation of @xmath52 . such a correspondence was missing in all our attempts to derive a mct starting from the total density fluctuations and this is one of the reasons for which the resulting equations were considered unsatisfactory . @xmath54 is less obvious and combines features of @xmath52 and of the vertex for the tagged particle dynamics in the bulk @xcite . in particular , like the latter , it diverges when @xmath55 . finally , the above equations differ slightly from those reported in ref . @xcite , where a first account of the theory was given . there , @xmath56 in eq . is replaced by @xmath57 , where @xmath58 is the fourier transform of the blocked direct correlation function . the only difference between the two approaches is that @xmath59 was not included as a slow variable in the earlier mode - coupling scheme . a priori , this is an unjustified approximation . indeed , as shown in appendix [ app.derivation ] , the coupling to this variable is actually so strong , that , when both @xmath60 and @xmath50 are considered , the contribution of the former becomes identically zero , while the effect of the random environment is integrally transferred to the latter . there are nevertheless circumstances in which the absence of @xmath59 looks perfectly well motivated , for instance when a single particle is moving in the porous matrix , corresponding to the limit @xmath61 , or when the adsorbed fluid is an ideal gas @xcite . in both cases , the present mct reduces to @xmath62 and @xmath63 ^ 2 \hat{h}^{b}_{|\mathbf{q - k}| } \phi_{k}(t),\ ] ] where @xmath64 is the fourier transform of the blocked total pair correlation function . but , since the only forces exerted on the fluid are those due to the random matrix , on physical grounds , the only mode - coupling contributions to the relaxation kernel are expected to come from @xmath65 . all previous theories of the lorentz gas have been based on this insight , which leads to the same equations as above , except that @xmath66 in eq . is replaced with @xmath67 @xcite . note that , for an adsorbed ideal gas , both theories correctly predict that the dynamics is independent of @xmath46 . it seems thus that there is a subtle interplay between the approximations involved in the derivation of the mct and the peculiar structure of the static correlations in the qa mixture . at present , it is not clear what to conclude from this observation , but it is interesting that the difference between the two mct schemes involves the blocked direct correlation function @xcite . indeed , this is a rather delicate object , not easily captured by simple approximations . this is best understood in the replica framework , where @xmath68 is obtained as the zero replica limit of the direct correlation function between two non - interacting fluid replicas only correlated through their common interaction with the matrix @xcite . it results that @xmath68 has a highly non - additive character and that , for instance , most simple closures of the replica oz equations fail to provide expressions for this function which do not vanish identically . thus , it is a rather nontrivial finding that the mct approximation scheme is actually sensitive to the existence of this function . from the previous discussion , it is clear that the separation of the fluid density fluctuations into relaxing and frozen parts plays a crucial role in the derivation of the mct for qa systems . in that case , the freezing is of static origin , due to the random external field generated by the quenched matrix . but , as it is well known from the mct for bulk fluids , freezing of the density fluctuations may also occur dynamically , when the system enters in the ideal glassy state . it seems thus interesting to compare both situations . this can be seen as a case of self - induced versus quenched disorder , similar to what has been discussed many times in the literature @xcite . the residual relaxation of a bulk fluid in its ideal glassy state has been studied in ref . it is described by mode - coupling equations of the same form as above , except that the characteristic frequency is given by @xmath69 and the vertices are [ verticesresidual ] @xmath70 ^ 2 \left\{(1-f_k ) s_k\right\ } \left\{(1-f_{|\mathbf{q - k}| } ) s_{|\mathbf{q - k}|}\right\}\ ] ] and @xmath71 ^ 2 \left\{(1-f_k ) s_k \right\ } \left\ { f_{|\mathbf{q - k}| } s_{|\mathbf{q - k}| } \right\},\ ] ] where @xmath46 is the density of the fluid , @xmath72 its structure factor , @xmath73 the fourier transform of its direct correlation function , and @xmath74 the debye - waller factor of the glass . accordingly , @xmath75 corresponds to the dynamically frozen part of the density fluctuations , while @xmath76 corresponds to their relaxing part . the analogies between these equations and eqs . and are striking . there is almost a perfect correspondence between @xmath76 and @xmath77 on the one hand , and @xmath75 and @xmath78 on the other hand . thus , we find that , irrespective of the mechanism of freezing of the fluid density fluctuations , the resulting relaxing and frozen contributions essentially play the same role in both systems . but there remains one significant difference between both sets of equations . indeed , one of the fourier transformed direct correlation functions in eq . , the one which carries the same wavevector as the frozen part of the structure factor , is replaced by a simple constant factor @xmath79 in eq . . the origin of this constant term should probably be traced back to the asymmetric nature of the qa system , where the fluid reacts to the matrix but not the other way around , resulting in a non - equilibrium character of its static and dynamical correlations , in the sense that the matrix is not equilibrated with the fluid . it would then reflect the lack of dynamical self - consistency between the glassy dynamics in a qa system and the frozen background on top of which it develops . it is tempting to try and obtain the mct for the qa binary mixture as a limiting case of the mct for the ordinary binary mixture @xcite , where one component would become the quenched matrix . the above discussion shows that this is not possible . indeed , based on heuristic considerations , one can get close to eqs . and , for instance by canceling all terms which would result in a time dependence of the fluid - matrix and matrix - matrix correlations , but the non - equilibrium density factor identified above seems impossible to generate from the theory for the fully annealed system . thus , as far as the development of theoretical approaches is concerned , and this statement is probably not restricted to the mct framework , the dynamics of a qa system should definitely be considered from the start as different from an infinite mass ( for newtonian systems ) or a zero bare diffusivity ( for brownian dynamics ) limit of the dynamics of a fully annealed mixture . this is actually not so unexpected since , already in the statics , similar difficulties were met in early attempts to derive the equations valid for the qa mixture starting from those describing fully equilibrated systems ( see the discussion of refs . @xcite in ref . @xcite ) . thus , the present theory is not a special case of earlier mode - coupling studies of the dynamics of particles moving in a glassy matrix @xcite . there , the mct for binary mixtures was used in a regime where one component , made of big particles , was completely glassy , i.e. , both the collective and tagged particle correlators did not relax to zero , while the other component , made of small spheres , was not necessarily localized , i.e. , the tagged particle correlators could relax to zero . in fact , both theories are complementary . in the early approach , the glassy matrix has to be the product of the self - consistent mode - coupling dynamics , but the model incorporates thermal fluctuations of the solid and aspects of its response to the presence of the fluid , while in the present approach , one has much more freedom to choose the structure of the confining medium , including realistic models of porous solids @xcite , but the disordered matrix is rigorously inert . there are anyway qualitative analogies between both models . for instance , there is a clear link between the fact that the collective motion of the small particles in a binary mixture necessarily becomes non ergodic at the same point as that of the big particles and the unavoidable existence of static blocked correlations for a fluid in a random matrix . we now move to the quantitative predictions of the theory , which require numerical solutions of the mct equations . in this section , we report dynamical phase diagrams , obtained by mapping , in the parameter space of a given model , the domain where @xmath80 , corresponding to the ergodic fluid phase , and the one where @xmath81 , corresponding to the non - ergodic ideal glassy state . the interface between the two domains forms the ideal liquid - glass transition manifold ; in the present work , it will always be a line , since only systems with a two - dimensional parameter space are considered . details of the dynamical changes when crossing this line are discussed in the next section . as already mentioned in sec . ii , a significant difference between the mct equations for qa systems and those for bulk glassformers is the presence of a linear term in the memory kernel . it results that the discontinuous or type b transition scenario known from the bulk , where the infinite time limit of @xmath82 jumps discontinuously from zero to a finite value when going from the liquid to the glass , is not the only possibility anymore . continuous or type a transition scenarios , where @xmath74 grows continuously from zero when entering the glassy phase , indeed become possible . this is best understood by reference to the so - called @xmath83 ( @xmath84 ) schematic models @xcite , in which the time evolution of a single correlation function @xmath85 is ruled by a two parameter memory kernel of the form @xmath86 it can be readily shown that , when the second term dominates ( @xmath87 small ) , these models have a line of type b transitions starting at @xmath88 , and , when the first term dominates ( @xmath89 small ) , they have a line of type a transitions starting at @xmath90 . the study of the @xmath83 models also gives us information on the possible topologies of the dynamical phase diagrams , which depend on the way the transition lines meet . two simple cases are found @xcite . if @xmath91 , the two lines join smoothly at a common endpoint , where a topologically stable degenerate @xmath92 singularity is located . if @xmath93 , the two lines intersect and , in the glassy domain , only the extension of the type b transition line subsists beyond the intersection , forming a glass - glass transition line terminated by an ordinary @xmath92 singularity . it turns out that these two prototypical shapes of phase diagrams are obtained with two very simple , closely related qa mixture models , to which the present study is restricted . in both , the fluid - fluid and fluid - matrix interactions are pure hard core repulsions of the same diameter @xmath94 . the only difference lies in the matrix correlations . in model i , the matrix configurations are assumed to be quenched from an equilibrium fluid of hard spheres of diameter @xmath94 , so that the matrix particles do not overlap , while in model ii the matrix particles are completely uncorrelated and overlap freely . in the following , both systems will be parametrized by the two dimensionless densities @xmath95 and @xmath96 , and the percus - yevick ( py ) approximation @xcite will be used to compute the required structural quantities . note that , in this approximation , @xmath97 , so that the difficulties mentioned at the end of sec . ii are irrelevant . the non - ergodicity parameter @xmath74 is a solution of the nonlinear set of equations @xmath98,\ ] ] which has to be solved numerically in order to locate the liquid and glassy phases when @xmath99 and @xmath100 are varied . all computations in the present work have been achieved using the method of ref . @xcite , to which the interested reader is referred for technical details , and we only provide the quantitative information needed to reproduce the present results , i.e. , that the wavevector integrals have been discretized to points on a grid of @xmath101 equally spaced values with step size @xmath102 , starting at @xmath103 @xcite . we have checked by two means that this discretization is not too coarse , in particular with respect to the small @xmath7 divergence of the memory kernel which is cut off . first , test calculations at various fluid and matrix densities have been performed on a finer @xmath7 grid , hence with a smaller @xmath104 . second , in the @xmath61 limit , we have compared our prediction for the diffusion - localization point of model i with leutheusser s analytic result obtained within an additional hydrodynamic approximation which allows to integrate exactly over the full @xmath7 range @xcite . in all cases , only modest quantitative differences of a few percents on the location of the transition points were found . when dealing with complex transition scenarios , including higher - order singularities and glass - glass transition lines , a useful quantity to consider is the largest eigenvalue @xmath105 of the stability matrix of the set of equations . indeed , there holds @xmath106 , and @xmath105 goes to @xmath107 when a transition is approached from the strong coupling side @xcite . thus , @xmath108 can be used as a convergence criterion for the determination of the transition points @xcite . in this work , @xmath109 was assured for ordinary transition points and @xmath110 was required in the regions where the transition lines of types a and b meet . the dynamical phase diagrams of models i and ii are reported in fig . [ figtrans ] . in the left panel , they are plotted in the @xmath111 plane . this is the obvious parameter space of the problem , but , with this choice of variables , no account is given of the differences in structure between the two matrix models . this clearly limits the possibilities of a meaningful comparison between the two systems . thus , in the right panel , both qa mixtures have been tentatively parametrized by the same physical constant , their henry constant @xmath112 . for systems with hard core interactions , @xmath112 is equal to the fraction of the total volume accessible to the center of an adsorbate particle in a matrix of density @xmath100 and is probably the most simple and generic scalar quantity characterizing the confining effect of a solid matrix on an adsorbed fluid . it is given by @xmath113 for model ii @xcite , and by @xmath114 $ ] for model i , where @xmath115 is the excess chemical potential of the equilibrium hard sphere fluid with volume fraction @xmath100 @xcite . the latter has been estimated following the compressibility route within the py approximation @xcite . as it can be readily seen in fig . [ figtrans ] , both phase diagrams essentially have the same overall shape , especially when they are plotted as functions of @xmath112 . their main qualitative difference at this global scale , which is the difference in concavity of their upper parts visible in fig . [ figtrans](a ) , turns out to be rather insignificant , as it is the simple and direct consequence of the fact that the strength of confinement increases more slowly with @xmath100 in model ii than in model i , because of the overlapping matrix particles . it completely disappears in fig . [ figtrans](b ) . the nature of the ideal glass transitions met in the phase diagrams is just as expected from the study of schematic models . starting from the type b liquid - glass transition point for the bulk ( at @xmath116 ) , where the memory kernel is purely quadratic , a line of type b transitions develops when increasing @xmath100 , and , starting from the type a diffusion - localization point ( at @xmath117 ) , where the memory kernel is purely linear , a line of type a transitions emerges when increasing @xmath99 . the same is true of the way the two lines meet in the phase diagrams . on the one hand , for model i , the type a and b lines have a common endpoint , denoted by e , where they smoothly join and form a degenerate @xmath92 singularity , like in the @xmath118 model . on the other hand , for model ii , the two lines intersect at a crossing point c and the extension of the type b transition line in the glassy domain becomes a glass - glass transition line ending with an ordinary @xmath92 singularity , denoted by e as well . this is the scenario obtained with the @xmath83 models , for @xmath93 . note that , to our knowledge , this is the first time that these widely studied one equation toy models find physical realizations as fluid systems . the glass - glass transition line of model ii , located between points c and e , is barely visible at the scale of fig . [ figtrans ] . it is short and not well separated from the type a liquid - glass transition line . taking into account the idealized nature of the predictions of the mct , it would probably be impossible to detect it unambiguously in computer simulations of this system . only the very specific features of the dynamics in the vicinity of a higher - order singularity , be it degenerate or not , should be visible , like logarithmic decay laws and subdiffusive behaviors @xcite . but , at least , the present calculation shows that this scenario can actually be realized and that this does not require any exotic physical ingredient . now that a suitable starting point is available , by playing with the parameters of the model , one can try and obtain a system for which this glass - glass transition line would be extended enough for its signatures to be observable in simulations . a final requirement for a complete characterization of the transitions studied in this work is the knowledge of the so - called exponent parameter @xmath119 @xcite . it determines many aspects of the dynamics near a transition ( see the next section ) and , for this reason , plays a crucial role in the theory . one has @xmath120 and @xmath121 for type a and b transitions , respectively . also , @xmath119 reaches @xmath107 , its maximum value , at endpoint singularities and jumps discontinuously at crossing points . this is thus a useful parameter to follow during the computation of the phase diagrams . it is plotted in fig . [ figlambda ] as a function of @xmath99 at the transition . for model ii , points c@xmath122 and c@xmath123 mark the discontinuity associated with the crossing point c in fig . [ figtrans ] and the existence of glass - glass transitions is clearly visible , with values of @xmath119 given by the line between points c@xmath122 and e. for the same model , a nonmonotonic variation of @xmath119 along the type b line can be noted as well . exponent parameter @xmath119 along the transition lines of models i and ii . the lowest part , where @xmath119 goes to zero with @xmath99 , has been omitted for readability . for both models , e denotes the @xmath92 singularity , with @xmath124 . c@xmath122 and c@xmath123 delimit the discontinuity associated with the crossing point between the type a and b transition lines in model ii . ] in addition to these transition scenarios , and formally not related to them , another remarkable prediction of the present theory is a reentry phenomenon for matrix densities higher than the localization threshold ( obtained for @xmath117 ) . indeed , as shown in fig . [ figtrans](a ) , for a given , not too high @xmath100 in this domain , ergodicity can be broken either by an increase or a decrease of the fluid density . in hard core systems , freezing by an increase of the fluid density can be qualitatively understood from simple free volume arguments , which provide a direct explanation for the decrease of @xmath99 at the transition as a function of @xmath100 in the upper part of the phase diagrams . because of the volume excluded by the matrix particles , the larger the matrix density is , the smaller the fluid density has to be for structural arrest to occur . from the results for model i , we might further note that this decrease is such that the total compacity @xmath125 at the transition is a decreasing function of @xmath100 as well , reflecting the fact that the inclusion of immobile matrix particles in the system slows down the dynamics more efficiently than the inclusion of the same amount of mobile fluid particles . such a behavior , which is hardly surprising , has already been observed in molecular dynamics simulations @xcite . the possibility of an ergodicity breaking transition by a decrease of the fluid density , which is reflected in the bottom part of the phase diagrams by the increase with @xmath100 of the transition @xmath99 , is more unexpected . we interpret this prediction as the signature of a delocalization phenomenon induced by fluid - fluid interactions . more precisely , we propose that the occasional collisions between the fluid particles at low @xmath99 can destroy the dynamical correlations responsible for the localization of individual particles in dense enough matrices . for this process to provide an efficient ergodicity restoring relaxation channel , a reasonable criterion is that the localization domains should overlap to allow the fluid particles to interact . thus , the localization length computed at @xmath117 , which decreases when @xmath100 increases , should be comparable to the average distance between two fluid particles , which decreases when @xmath99 increases . it is then immediate that , starting in the localized state , the larger @xmath100 is , the higher @xmath99 has to be in order to restore ergodicity . the physical implications of this result will be discussed in more details in the last section . dynamical scenarios involving reentrant glass transition lines , higher - order singularities , and glass - glass transition lines , have already been found for colloidal suspensions with short - ranged attractions @xcite . by analyzing various contributions to the memory kernel , these features were shown to result from the interplay of two well defined phenomena , cage effect and bond formation , driven by the hard core and attractive parts of the interaction , respectively . it seems thus interesting to attempt such an analysis for the present problem . guided by the results of sec . iii , we propose that a contribution represented by a memory kernel @xmath126 derived from @xmath127 by replacing @xmath79 in eq . with @xmath128 should be isolated . indeed , the resulting expression then coincides with the one describing the residual dynamics of a bulk ideal glass . therefore , one can reasonably expect that , for a qa system , @xmath126 will provide a fair representation of the mechanism of caging by fluid particles in the presence of permanent density fluctuations , which is precisely the one at work in bulk glassy systems . the remaining linear kernel @xmath129 can then be attributed to the confinement - specific phenomena , i.e. , the localization effect of the matrix combined with the unusual decorrelation mechanism due to fluid - fluid collisions discussed above . in principle , one could try to separate these two processes , using the fact that , for an ideal gas , only the localization effect is present and leads to a matrix density at the transition which is independent of the fluid density . however , already with the present limited separation , the somewhat artificial character of the procedure shows up in the form of negative values of @xmath130 , which restrict the density domain where stable solutions of the mct equations can be found , so no further decomposition of @xmath130 was attempted . the hypothetical phase diagrams computed with the partial kernels @xmath126 and @xmath130 for model i are reported in fig . [ figfictitious ] , where they are compared to the one obtained with the full @xmath127 . clearly , the reentry phenomenon observed in the complete phase diagram can be explained by the interplay of the two contributions discussed above . but , at variance with the colloidal systems , the higher - order singularity is not located in the domain where they cross over . the interpretation of this finding is ambiguous . on the one hand , since a higher - order singularity is found on the transition line computed with @xmath126 as well , one might argue that the singularity should be considered as an integral part of the scenario of caging by fluid particles . then , in the relevant domain , confinement would simply appear as a modifier of the cage structure , progressively changing the nature of the ideal jamming transition from discontinuous to continuous . on the other hand , the bifurcation analysis of the mct scenario shows that a higher - order singularity is necessarily formed when two lines of type a and b transitions meet . then , since the type a and b lines arise from points representative of systems ruled by confinement and bulk caging , respectively , one might consider that the singularity is the product of the interplay of these two phenomena . in favor of the latter interpretation , it has been recently suggested that higher - order singularities generically result from a competition between different arrest mechanisms @xcite . note however that the conclusions of ref . @xcite are drawn from the consideration of bulk systems only , which necessarily enter into the glassy state through type b transitions and for which there is no mathematical constraint in the theory imposing a priori the existence of a singularity . it is not immediate that the proposed statement is valid or even needed for systems where type a and b liquid - glass transitions coexist . real and hypothetical dynamical phase diagrams of model i , computed with the total and partial memory kernels @xmath127 , @xmath126 , and @xmath130 . the curves are labeled accordingly and the degenerate @xmath92 singularities are denoted by e. ] it results from the previous section that , if one stays away from the higher - order singularities , the crossing points , and the glass - glass transitions , which require fine tuning of the parameters of the models , the present mct for qa mixtures predicts two generic liquid - glass transition scenarios . one is discontinuous or type b , the other is continuous or type a. in this section , we discuss the different features of these transitions which are relevant for comparisons of experimental or simulation data with the predictions of the theory . the analytic results will be quoted without their proofs , which can be found in ref . they will be illustrated with detailed computations for model i at two matrix densities , @xmath131 and @xmath132 . for the former value , a type b transition occurs at @xmath133 , with @xmath134 . for the latter , a type a transition is found at @xmath135 , with @xmath136 . for this matrix density , we consider the type a transition on the upper branch of the phase diagram , as it is of greater relevance for the problem of the glass transition in confinement . in both cases , keeping @xmath137 allows one to consider that the transitions are far enough from the higher - order singularity . here , a comment on the method of solution of the mode - coupling equations is in order . indeed , as mentioned in the previous section , it involves a cutoff of the low @xmath7 divergence of the memory kernel . since it has been demonstrated that this divergence can change the qualitative properties of the solutions in the asymptotic regime near the transition @xcite , this approximation and the use of the results of ref . @xcite , which have been obtained under the assumption of nonsingular vertices , might look problematic . in fact , this is not the case for physical reasons . indeed , in the lorentz gas limit , the low @xmath7 singularity of the memory function has been shown to be an ill feature of the mode - coupling approximation @xcite . thus , from a physical point of view , the discretized equations considered in the present calculation are actually more satisfactory than the continuous ones and the results reviewed in ref . @xcite can legitimately be applied . we first consider the density dependence of the non - ergodicity parameter @xmath74 , shown in fig . [ fignonergvsphif ] . as it should , @xmath74 takes a finite value at the ideal glass transition point of type b , while it grows continuously from zero at the type a transition . in the glassy state , the bifurcation analysis of eq . to leading order yields two universal power law behaviors , @xmath138 for a type b scenario , and @xmath139 near a type a transition accordingly , in fig . [ fignonergvsphif ] , the curves corresponding to the type b and a transitions start with infinite and finite slopes , respectively . the wavevector dependence of @xmath74 , though not universal , is an important prediction of the mct as well . it is reported in fig . [ fignonergvsq ] . in fig . [ fignonergvsq](a ) , @xmath74 is shown at the type b transition , while in fig . [ fignonergvsq](b ) , since @xmath140 at a type a transition , the results at @xmath141 are plotted . for reference , the two relevant structure factors @xmath77 and @xmath78 at @xmath142 are also given . note that in both cases , @xmath78 shows maxima both where @xmath77 has maxima or minima . at the type b transition , except for the peak at @xmath143 where @xmath74 reaches 1 as a consequence of the diverging kernel , the non - ergodicity parameter is very similar to the one found for a bulk hard sphere fluid and oscillates with @xmath77 , which precisely represents bulk - like correlations . the overall amplitude is smaller than in the bulk , reflecting the fact that , when @xmath100 increases , the system evolves towards a continuous transition scenario . in comparison , the non - ergodicity parameter near the type a transition appears rather featureless . @xmath74 simply decreases from @xmath107 at @xmath143 , with , as @xmath99 is increased , a small shoulder developing in the wavevector regime where @xmath77 has its main peak and @xmath78 its second peak . with the present models where the fluid and matrix particles have the same size , it is not clear which changes in the static correlations are actually responsible for the growth of this contribution . beside these results for the infinite time limit of the density correlation functions , the full dynamics is of great interest as well . for this computation , which uses the algorithm described in ref . @xcite , we follow ref . @xcite and reduce the generalized langevin equation to its form valid for brownian dynamics , @xmath144 with @xmath145 and the initial condition @xmath32 . this simplification affects the short time transient part of the dynamics , but not its long time properties . in the following , the unit of time shall be chosen such that @xmath146 . the time evolution of the density correlation function @xmath147 at @xmath148 , corresponding to the main peak of @xmath77 , is reported in fig . [ figcorrel ] for state points in the vicinity of the two transitions discussed above . the curves for other values of @xmath7 are qualitatively similar . in fig . [ figcorrel](a ) , the two step dynamics typical of the discontinuous ideal glass transition scenario and well known from the study of bulk systems is easily recognized in the curves corresponding to the liquid state . the second step , associated to the decay from the plateau where @xmath149 , obeys the so - called superposition principle , which states that , in this regime and for a given @xmath7 , the shape of the dynamics is independent of the state point . the relaxation functions only differ through the characteristic time scale @xmath150 , which displays a power law divergence , @xmath151 when the transition is approached . one shows that @xmath152 where the exponents @xmath153 and @xmath154 ( @xmath155 , @xmath156 ) are related to @xmath119 through @xmath157 @xmath158 denoting euler s gamma function . in the glassy state , only the first relaxation step remains and , when @xmath17 goes to infinity , @xmath147 reaches @xmath74 , which increases with @xmath99 as shown in fig . [ fignonergvsphif](a ) . the dynamics near the type a transition visible in fig . [ figcorrel](b ) looks significantly different , with a single step relaxation scenario , both in the liquid and glassy phases . the slowing - down of the dynamics manifests itself through a weak long time tail which extends to longer times when @xmath99 is increased and turns above @xmath142 into a finite asymptote which grows as shown in fig . [ fignonergvsphif](b ) . there are nevertheless strong similarities between the two dynamics , provided one concentrates on the time domain where @xmath159 ( i.e. , where @xmath147 is small for a type a transition ) . for type b dynamics , this corresponds to the so - called fast @xmath160 relaxation regime and we first specialize the discussion to this case . then , close enough to the transition , a reduction theorem holds , according to which the wavevector and time dependencies of @xmath147 factorize , yielding @xmath161 at the critical point , the critical decay law @xmath162 is obtained , where @xmath153 is given by eq . and @xmath163 is a time scale obtained by matching the short and long time dynamics . for finite values of @xmath164 , one finds the scaling laws @xmath165 where the master functions @xmath166 are the solutions of @xmath167 and the scaling variables obey @xmath168 for large @xmath169 , in the non - ergodic phase , @xmath170 goes to a constant and eq . results from the expression of @xmath171 . in the same time regime , in the ergodic phase , another power law behavior sets in for @xmath172 , yielding the so - called von schweidler decay law , @xmath173 where @xmath154 is given by eq . and @xmath174 is a positive constant which can be determined by matching eq . with eq . for @xmath175 . by combining the power law behaviors of @xmath171 and @xmath176 in the resulting expression of @xmath177 , one recovers the divergence of @xmath150 , eq . . moving now to the type a transitions , one finds that , both in the ergodic and non - ergodic states , @xmath147 essentially behaves near zero as it does when it approaches @xmath178 in the glassy phase in a type b scenario . in particular , the critical decay law remains valid at the transition . for finite values of @xmath164 and independently of its sign , eq . has to be modified in order to properly define @xmath177 ( see ref . @xcite for details ) , then one finds that a scaling law holds , of the form @xmath179 where @xmath170 is the same function as above . however , as in eq . , the exponents characterizing the scaling variables @xmath180 and @xmath181 are twice those for a type b transition , i.e. , @xmath182 , where the time evolution of @xmath183 ( simply @xmath82 for the type a transition ) at @xmath148 is plotted in a log - log scale in order to evidence the power laws and . in this graphs , an interesting consequence of the scaling laws for small @xmath169 is clearly visible , which is the symmetric departure from the critical decay law at long times for points located in the liquid and glassy phases at the same distance from the transition . this is a particularly important feature of the dynamics in the type a scenario . the above results do not apply in the vicinity of higher - order singularities or crossing points , where a refined mathematical analysis is required @xcite . a detailed discussion is beyond the scope of the present overview of the theory and we shall only mention that such liquid - glass transition points signal themselves in the dynamics through the appearance of logarithmic decay laws @xcite . finally , already for bulk systems , it is often quite difficult to unambiguously demonstrate that the above features are actually present in some experimental or simulation data , since , as mentioned in the introduction , the mct offers an idealized picture of the glass transition phenomenon and there are always significant alterations to the theoretical scenario . an additional difficulty can be anticipated in the case of qa systems . indeed , one usually has access to the total density correlation function @xmath184 and not to the connected function @xmath82 . both are related through @xmath185 thus , the glassy dynamics is modulated by static factors and in particular develops itself on top of a state dependent static background . the separation in the long time behavior of @xmath184 of the evolutions which are of purely static origin from those which characterize the glassy dynamics might then be delicate . this could be especially critical for type a transitions , where the signatures of the continuous transition have to be identified just on top of the slowly drifting static contribution . figures demonstrating the problem can be found in ref . in this paper , a mode - coupling theory for the slow dynamics of fluids adsorbed in disordered porous solids made of spherical particles frozen in random positions has been developed . derived by properly taking into account the peculiar structure of the correlations in these systems and by including a contribution which had been forgotten in a previous work @xcite , its equations are found to display many appealing features . for instance , they show universality , in the sense that they do not contain any explicit reference to the precise nature of the random environment in which the fluid evolves . also , they compare favorably with previous mode - coupling equations derived in other contexts , for the residual dynamics in the glassy phase of a bulk fluid ( sec . iii ) or for the equilibrium dynamics of a mean - field spin - glass in a random magnetic field ( appendix [ app.pspin ] ) . thus , from a formal point of view , the theory appears rather satisfactory . nevertheless , a few difficulties remain . first , there is the fact that , in the limit of vanishing fluid - fluid interactions , the present theory does not coincide with the mct which can be derived by assuming from the start that there are no such interactions . second , there is the divergence of the memory kernel for small wavevectors and the resulting spurious long time anomalies @xcite . we do not believe that these issues are really harmful , even if their handling requires ad hoc approximations , but they are worth stressing , since their solutions would probably teach us something on the nature of the approximations underlying the mct scheme and on possible extensions of the theory . for instance , it has been argued by leutheusser that the inclusion of vertex corrections within a kinetic theory approach would solve the second problem , but no operational scheme was proposed for this calculation @xcite . the numerical solution of the mct equations for two simple fluid - matrix models leads to a variety of transition scenarios , which are either discontinuous for dilute matrices or continuous for dense matrices . depending on the model , in the intermediate region where the nature of the transition changes , degenerate or genuine higher - order singularities and glass - glass transition lines are found . another remarkable prediction of the theory is the possibility of a reentry phenomenon for high matrix densities above the localization threshold , which has been interpreted as the signature of a decorrelation process induced by fluid - fluid collisions . before going further , one should note that , strictly speaking , this prediction of the theory can not be correct in the case of hard core fluid - matrix interactions . indeed , as recently confirmed by extensive computer simulations of the lorentz gas @xcite , the localization transition is driven by the percolation transition of the matrix void space , i.e. , localization occurs because , above a certain critical matrix density @xmath186 , the void space only consists of finite disconnected domains . in such a scenario , it is obvious that , whatever the fluid density , it is impossible to have an ergodic system above @xmath186 , since fluid - fluid interactions will never change the geometry of the matrix . this contradiction between the percolation theory and the mct clearly raises the issue of the relation between the two approaches , for which we propose the following simple argument . the mct applied to the problem of the diffusion - localization transition attempts to capture the onset of percolation in an indirect way , by giving an account of the increasingly correlated nature of the fluid - matrix collision events as the threshold is approached . in this respect , one should note that none of the static structure functions on which it is based does show a sensitivity to the phenomenon of percolation . following leutheusser , the theory works at the level of a self - consistent treatment of ring collision processes @xcite . this turns out to be enough to predict a diffusion - localization transition , but , clearly , the infinite sequences of correlated collisions which would really reflect the permanent trapping of the fluid particle in a finite domain above the percolation threshold are missing . from this incomplete characterization of the dynamical processes escorting the percolation phenomenon , it results that the mct diffusion - localization transition is actually an ideal version of the true percolation transition , in the usual sense that the mct predicts ideal glass transitions , and that the theory is not able to detect that , in an exact treatement , the percolation threshold fixes an absolute limit to diffusive behavior . when fluid - fluid collisions come into play at finite fluid densities , this leaves room for the prediction of a reentry phenomenon in contradiction with percolation theory . we believe that it is for the same reason that the mct also misses the fact that , at any matrix density , there is always a non - vanishing probability that particles will be trapped in a finite domain disconnected from the rest of the void space , so that the exploration of the available void space is never completely ergodic @xcite . it is thus clear that the prediction of a reentrant behavior of the ergodicity breaking transition line in the low fluid density domain should not be taken too literally . in fact , a reasonable expectation based on this finding is that , below the localization threshold , but in the regime where transient trapping effects are important , there might be an acceleration of the dynamics due to fluid - fluid collisions . interestingly , such a behavior has already been observed in a computer simulation study of a two dimensional lattice gas model with fixed randomly placed scatterers @xcite . indeed , it was found that , starting from the zero fluid density limit , the diffusion coefficient of a tagged particle first increases with the fluid density . an interpretation in terms of a decorrelation process similar to the one discussed in sec . iv was then proposed and validated by varying the parameters of the model . as a possible source of the difficulties of the theory , one might blame the fact that it works at the level of disorder averaged quantities . indeed , the procedure of averaging over disorder is equivalent to an averaging over the volume of a macroscopic system , an operation in which many microscopic details of the statics and dynamics become blurred . this might confer a mean - field character to the theory , where the contribution of the matrix would actually be taken into account at the level of a diffuse effective localizing potential , with a possible loss of important local constraints . unfortunately , this is a necessary step in order to develop a theory which is comparable in complexity with the one for bulk systems , since it allows one to consider the system as homogeneous . some progress has recently been made on the application of the mct to inhomogeneous situations @xcite , so it should be possible to relax the condition of homogeneity in order to study the dynamics of fluids confined in pores of simple geometry which are often preferred in simulation works , but there is no doubt that the more complex wavevector dependence of the resulting theory will make it harder to obtain numerical solutions of the equations . moreover , beyond this purely technical aspect , the present formulation in terms of disorder averaged quantities has a practical interest as well . indeed , many real porous media are disordered and most experimental techniques measure quantities which are averaged over the volume of a macroscopic sample and thus equivalent to the disorder averages considered by the mct . so , the current theoretical setup seems well suited for direct comparisons with experimental results . for molecular dynamics simulations , however , since rather small systems are usually considered , it might be necessary to explicitly perform the disorder average over a representative sample of matrix configurations before a comparison with the theory can be done . altogether , in spite of the above merely technical issues , we believe that the present mode - coupling theory represents a valuable step towards a better understanding of the slow dynamics of confined glassforming liquids . indeed , it remains rather simple and , since it is a microscopic approach , it allows one to study in detail the effect on the dynamics of changes in the different ingredients of a fluid - matrix model ( fluid - fluid and fluid - matrix interactions , structure of the matrix ) . thus , this provides us with a tool to efficiently and thoroughly explore the phenomenology of dynamics in confinement . this is clearly illustrated by our findings for two very simple systems with pure hard core interactions , which already display new and nontrivial glass transition scenarios . then remains the question of the validation of the theoretical predictions . because the model of the qa mixture is quite simple and the theory makes detailed predictions , molecular dynamics studies should be able to give clear - cut answers . the presently available results look rather encouraging , but more simulation work is definitely needed . it is a pleasure to thank g. tarjus , w. gtze , and w. kob , for useful comments , and f. hfling for a valuable discussion and the communication of unpublished results . for reference , we quote the replica oz equations relating the various pair correlation functions in qa binary mixtures @xcite . they read , in fourier space , @xmath187 with @xmath188 and @xmath189 . as usual , @xmath190 and @xmath180 denote total and direct correlation functions , respectively . @xmath191 denotes the fourier transform of @xmath192 and the superscripts have the same meaning as for the structure factors ( see sec . ii ) . using the relations @xmath193 ( remember that @xmath194 , hence @xmath195 ) the oz equations can be formally solved for the structure factors , leading to @xmath196 \frac{1}{(1 - n_f \hat{c}^{c}_q)^2}.\end{aligned}\ ] ] in this appendix , the derivation of the mode - coupling part of the memory kernel is outlined . this calculation is a direct extension of the one for bulk systems which is described in its most minute details in ref . @xcite . the memory function in eq . is defined as @xmath197 where @xmath198 i \mathcal{q}_1 \mathcal{l } g^f_\mathbf{q}$ ] is the projected random force obtained from the longitudinal momentum density fluctuation @xmath199 @xcite . @xmath200 is the liouville operator of the system and @xmath201 is the complementary operator of the projector @xmath202 which projects any dynamical variable onto the subspace spanned by @xmath203 and @xmath204 . the calculation of the mode - coupling part of the kernel amounts to replacing @xmath205 in eq . by its projection @xmath206 onto the subspace spanned by @xmath207 , @xmath208 , and @xmath209 , where we introduce the projection operator @xmath210 such that @xmath211 in this expression , we have anticipated that , within the mode - coupling approximation , @xmath210 is diagonal in @xmath212 and the subspaces spanned by the @xmath174s and @xmath213s are orthogonal . the prime in the first sum indicates that , to avoid double - counting , the wavevectors are assumed to be ordered somehow and the sum is restricted to @xmath214 . @xmath215 and @xmath216 are normalization matrices insuring that @xmath217 and @xmath218 . in the course of this calculation , four - point density correlation functions are generated . thus , in order to eventually obtain closed dynamical equations , a factorization approximation is needed to express them as products of two - point density correlation functions . we follow the usual mode - coupling prescription and find that , within this approximation , the only non - vanishing four - point functions are given by @xmath219 specializing to @xmath220 and computing @xmath221 and @xmath222 , it results that @xmath223^{-1 } , \\ % h^{(11)}_{\mathbf{q},\mathbf{k } } = & \left [ n_f n_m s^{c}_k \right]^{-1 } \frac{s^{b}_{|\mathbf{q - k}|}}{s^{mm}_{|\mathbf{q - k}| } s^{b}_{|\mathbf{q - k}| } - ( s^{fm}_{|\mathbf{q - k}|})^2 } , \\ % h^{(22)}_{\mathbf{q},\mathbf{k } } = & \left [ n_f^2 s^{c}_k \right]^{-1 } \frac{s^{mm}_{|\mathbf{q - k}|}}{s^{mm}_{|\mathbf{q - k}| } s^{b}_{|\mathbf{q - k}| } - ( s^{fm}_{|\mathbf{q - k}|})^2 } , \\ % h^{(12)}_{\mathbf{q},\mathbf{k } } = & - \left [ n_f \sqrt{n_f n_m } s^{c}_k \right]^{-1 } \frac{s^{fm}_{|\mathbf{q - k}|}}{s^{mm}_{|\mathbf{q - k}| } s^{b}_{|\mathbf{q - k}| } - ( s^{fm}_{|\mathbf{q - k}|})^2}.\end{aligned}\ ] ] it remains to express @xmath224 and @xmath225 . first , using yvon s theorem , one finds @xmath226,\\ % \overline{\langle i\mathcal{l}g^f_\mathbf{q } c^{(1)}_{\mathbf{-q ,- k}}\rangle } & = i q k_b t \sqrt{n_f n_m } \frac{\mathbf{q}\cdot \mathbf{k}}{q^2 } s^{fm}_{|\mathbf{q - k}|},\\ % \overline{\langle i\mathcal{l}g^f_\mathbf{q } c^{(2)}_{\mathbf{-q ,- k}}\rangle } & = i q k_b t n_f \frac{\mathbf{q}\cdot \mathbf{k}}{q^2 } s^{b}_{|\mathbf{q - k}|}.\end{aligned}\ ] ] then , with the help of the extension of the convolution approximation @xcite to qa systems , @xmath227 it comes @xmath228 the desired results are obtained by substracting the matching equations . we might now complete the explicit calculation of @xmath206 . indeed , if we define @xmath229 such that @xmath230 the above results immediately lead to @xmath231,\\ % w^{(1)}_\mathbf{q , k } & = 0 , \\ % w^{(2)}_\mathbf{q , k } & = \frac{i q k_b t}{n_f } \left [ \frac{\mathbf{q}\cdot \mathbf{k}}{q^2 } \frac{1}{s^{c}_k } - 1 \right].\end{aligned}\ ] ] note the vanishing of the fluid - matrix contribution . eventually , injecting the resulting expression of @xmath206 into @xmath232 one obtains eqs . and after a few elementary steps . exactly the same calculation can be done with subsets of the above three quadratic variables . one then easily demonstrates that , if only @xmath233 and @xmath234 are considered , the same equations are obtained , while , if one works with @xmath233 and @xmath235 , the equations of ref . @xcite are reproduced , which reduce to those of refs . @xcite in the zero fluid density limit . a fruitful source of new theoretical developments on the physics of glassy systems has been the finding that the mct provides an exact description of the equilibrium dynamics of a certain class of mean - field spin - glass models with multispin interactions @xcite . in this appendix , based on ref . @xcite ( the interested reader is referred to this paper for details ) , we show that the equations describing the dynamics of such a mean - field spin - glass model in a random magnetic field reproduce the structure of those of the mct for a fluid in a random environment . thus , the correspondence between the two approaches still holds in the presence of an external source of disorder . we consider the fully connected mean - field spherical spin - glass model with three spin interactions in a random magnetic field , whose hamiltonian is @xmath236 = - \sum_{1\le i < j < k\le n}j^{(3)}_{ijk } s_i s_j s_k - \sum_{1\le i\le n } j^{(1)}_{i } s_i,\ ] ] where the spins @xmath237 are @xmath238 real variables subject to the constraint @xmath239 , and the random coupling constants @xmath240 and fields @xmath241 are uncorrelated zero mean gaussian variables with variances @xmath242 and @xmath243 , respectively . in zero field ( @xmath244 ) , this model is known to generate the quadratic memory kernel introduced in refs . @xcite and @xcite as a one wavevector approximation to the bulk mct functional . thus , it is especially suitable for our purpose , since it displays nonlinearities of the same degree as in the mct . in the presence of a random magnetic field , just like qa mixtures are characterized by static frozen density fluctuations , the present spin - glass model is characterized by static frozen local magnetizations @xmath245 which vanish when the average over disorder ( both on couplings and fields ) is performed . accordingly , we might define @xmath246 and introduce three correlation functions , which are the analogues of @xmath48 , @xmath47 , and @xmath82 . they are , respectively , the two overlap functions @xmath247 and @xmath248 which obey @xmath249 , and the normalized connected spin correlation function , @xmath250 the exact solution of the langevin dynamics of the model using standard methods @xcite provides us with an equation for the time evolution of @xmath85 which reads , for a high enough temperature @xmath13 , a direct comparison of these equations with those of sec . ii immediately shows that both sets of equations exactly have the same formal structure . in particular , beyond the simple fact that the memory functions are similar polynomials of the dynamical correlation functions , the same parametrization in terms of connected and disconnected static correlations is found . we conclude with two short remarks . first , as for qa mixtures , we note that the external random field does not enter explicitly in the above equations ( no @xmath253 is present ) . because the model is exactly soluble , it is easy to understand how this happens . in fact , the details of the random field are only needed for the computation of the static correlations , through the equality @xmath254 once this calculation is done , they can be forgotten . the same is probably true for fluids in random environments and this supports the assumption that the mct equations for qa systems actually have a wider domain of applicability , provided they are expressed in terms of fluid quantities only . second , it can be shown that eq . also describes the residual relaxation of the present spin - glass model in zero field , when the system is equilibrated in its low temperature phase and the total spin correlation function has a nonvanishing infinite time limit @xmath7 @xcite . then , @xmath255 and @xmath256 simply have to be replaced by @xmath7 and @xmath257 . thus , in the present model , the difference discussed in sec . iii between self - induced glassiness and the effect of a quenched random field does not seem to exist . this could have been anticipated , since by construction , fully connected models are unable to capture phenomena which only show up in the wavevector dependence of the coupling constants in fluid systems . l. fabbian , w. gtze , f. sciortino , p. tartaglia , and f. thiery , phys . e * 59 * , r1347 ( 1999 ) ; * 60 * , 2430 ( 1999 ) ; j. bergenholtz and m. fuchs , * 59 * , 5706 ( 1999 ) ; k. dawson , g. foffi , m. fuchs , w. gtze , f. sciortino , m. sperl , p. tartaglia , t. voigtmann , and e. zaccarelli , phys . e * 63 * , 011401 ( 2001 ) . similar discrepancies appear generically when different routes are followed to derive the mct equations for the tagged particle dynamics in a qa system . they will be discussed in more details , with their quantitative consequences , in a forthcoming paper ( v. krakoviack , in preparation ) . the possible values of @xmath261 are limited by the method of solution of the oz / py equations on a real space grid of equally spaced points with a step size commensurate to the hard core diameter @xmath94 . for our calculation , the present choice of @xmath261 offers the closest match with the value @xmath262 used in ref . @xcite , where analytic expressions for the structure functions were readily available . | we derive a mode - coupling theory for the slow dynamics of fluids confined in disordered porous media represented by spherical particles randomly placed in space .
its equations display the usual nonlinear structure met in this theoretical framework , except for a linear contribution to the memory kernel which adds to the usual quadratic term .
the coupling coefficients involve structural quantities which are specific of fluids evolving in random environments and have expressions which are consistent with those found in related problems .
numerical solutions for two simple models with pure hard core interactions lead to the prediction of a variety of glass transition scenarios , which are either continuous or discontinuous and include the possibility of higher - order singularities and glass - glass transitions .
the main features of the dynamics in the two most generic cases are reviewed and illustrated with detailed computations .
moreover , a reentry phenomenon is predicted in the low fluid - high matrix density regime and is interpreted as the signature of a decorrelation mechanism by fluid - fluid collisions competing with the localization effect of the solid matrix . |
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there is gap in the scientific analysis of the fraction ply methods one of the best methods of search in computer chess and other strategy games . as hans berliner pointed out about the scheme of partial depths , ... the success of these micros ( micro - processor based programs ) attests to the efficacy of the procedure . unfortunately , little has been published on this . this research has the objective of developing a theoretical model of the partial depth scheme based on information theory , implementing it and providing experimental evidence for the method and for the model . an introduction to games theory and information theory is given in the background section . a model based on the principles of information theory is outlined and then the formula for partial depths scheme is calculated . search experiments are performed and then the results are interpreted . in the appendix can be found an introduction to some concepts in chess , and to the axioms of information theory . an important mathematical branch for modeling chess is games theory , the study of strategic interactions . assuming the game is described by a tree , a finite game is a game with a finite number of nodes in its game tree . it has been proven that chess is a finite game . the rule of draw at three repetitions and the 50 moves rule ensures that chess is a finite game . sequential games are games where players have some knowledge about earlier actions . a game is of perfect information if all players know the moves previously made by all players . zermelo proved that in chess either player @xmath0 has a winning pure strategy , player @xmath1 has a winning pure strategy , or either player can force a draw . a zero sum game is a game where what one player looses the other wins . chess is a two - player , zero - sum , perfect information game , a classical model of many strategic interactions . by convention , w is the white player in chess because it moves first while b is the black player because it moves second . let m(x ) be the set of moves possible after the path x in the game has been undertaken . w choses his first move @xmath2 in the set m of moves available . b chooses his move @xmath3 in the set m(@xmath2 ) : @xmath3 @xmath4 m(@xmath2 ) then w chooses his second move @xmath5 , in the set m(@xmath2,@xmath3 ) : @xmath5 @xmath4 m(@xmath2,@xmath3 ) then b chooses his his second move @xmath6 in the set m(@xmath7,@xmath3,@xmath5 ) : @xmath6 @xmath4 m(@xmath2,@xmath3,@xmath5 ) at the end , w chooses his last move @xmath8 in the set m(@xmath2 , @xmath3 , ... , @xmath9 , @xmath10 ) . in consequence @xmath8 @xmath4 m(@xmath2 , @xmath3 , ... , @xmath9 , @xmath10 ) let n be a finite integer and m , m(@xmath2 ) , m(@xmath2,@xmath3), ... ,m(@xmath2 , @xmath3 , ... , @xmath9 , @xmath10,@xmath8 ) be any successively defined sets for the moves @xmath2,@xmath3, ... ,@xmath8,@xmath11 satisfying the relations : @xmath12 and @xmath13 a realization of the game is any 2n - tuple ( @xmath2 , @xmath3 , ... , @xmath9 , @xmath10,@xmath8,@xmath11 ) satisfying the relations ( 1 ) and ( 2 ) a realization is called variation in the game of chess . let r be the set of realizations ( variations ) , of the chess game . consider a partition of r in three sets @xmath14 , @xmath15 and @xmath16 so that for any realization in @xmath14 , player1 ( white in chess ) wins the game , for any realization in @xmath15 , player2 ( black in chess ) wins the game and for any realization in @xmath16 , there is no winner ( it is a draw in chess ) . then r can be partitioned in 3 subsets so that @xmath17 w has a winning strategy if @xmath18 @xmath2 @xmath4 m , @xmath19 @xmath3 @xmath4 @xmath20 , @xmath18 @xmath5 @xmath4 m(@xmath2,@xmath3 ) , @xmath19 @xmath6 @xmath4 m(@xmath21 , @xmath22 , @xmath23 ) ... @xmath18 @xmath8 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10 ) , @xmath19 @xmath11 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10,@xmath8 ) , where the variation @xmath24 w has a non - loosing strategy if @xmath18 @xmath2 @xmath4 m , @xmath19 @xmath3 @xmath4 @xmath20 , @xmath18 @xmath5 @xmath4 m(@xmath2,@xmath3 ) , @xmath19 @xmath6 @xmath4 m(@xmath7 , @xmath3 , @xmath5 ) ... @xmath18 @xmath8 @xmath4 m(@xmath3,@xmath2, ... ,@xmath9,@xmath10 ) , @xmath19 @xmath11 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10,@xmath8 ) , where the variation @xmath25 b has a winning strategy if @xmath18 @xmath3 @xmath4 m , @xmath19 @xmath2 @xmath4 @xmath20 , @xmath18 @xmath6 @xmath4 m(@xmath2,@xmath3,@xmath5 ) , @xmath19 @xmath5 @xmath4 m(@xmath7 , @xmath3 ) ... @xmath18 @xmath11 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10,@xmath8 ) , @xmath19 @xmath8 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10 ) , where the variation @xmath26 b has a non - loosing strategy if @xmath18 @xmath3 @xmath4 m , @xmath19 @xmath2 @xmath4 @xmath20 , @xmath18 @xmath5 @xmath4 m(@xmath2,@xmath3 ) , @xmath19 @xmath5 @xmath4 m(@xmath7 , @xmath3 ) ... @xmath18 @xmath11 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10,@xmath8 ) , @xmath19 @xmath8 @xmath4 m(@xmath2,@xmath3, ... ,@xmath9,@xmath10 ) , where the variation @xmath27 considering a game obeying the conditions stated above , then each of the next three statements are true : ( 1 ) . w has a winning strategy or b has a non - losing strategy . b has a winning strategy or w has a non - losing strategy . + ( 3 ) . if @xmath16 = @xmath28 , then w has a winning strategy or b has a winning strategy . + if @xmath16 is @xmath28 , one of the players will win and if @xmath16 is identical with r the outcome of the game will result in a draw at perfect play from both sides . it is not know yet the outcome of the game of chess at perfect play . + the previous theorem proves the existence of winning and non - losing strategies , but gives no method to find these strategies . a method would be to transform the game model into a computational problem and solve it by computational means . because the state space of the problem is very big , the players will not have in general , full control over the game and often will not know precisely the outcome of the strategies chosen . the amount of information gained in the search over the state space will be the information used to take the decision . the quality of the decision must be a function of the information gained as it is the case in economics and as it is expected from intuition . of critical importance in the model described is the information theory . it is proper to make a short outline of information theory concepts used in the information theoretic model of strategy games and in particular chess and computer chess . a discrete random variable @xmath29 is completely defined by the finite set of values it can take s , and the probability distribution @xmath30 . the value @xmath31 is the probability that the random variable @xmath29 takes the value x. the probability distribution @xmath32 : s @xmath33 [ 0,1 ] is a non - negative function that satisfies the normalization condition @xmath34 the expected value of f(x ) may be defined as @xmath35 this definition of entropy may be seen as a consequence of the axioms of information theory . it may also be defined independently @xcite . as a place in science and in engineering , entropy has a very important role . entropy is a fundamental concept of the mathematical theory of communication , of the foundations of thermodynamics , of quantum physics and quantum computing . the entropy @xmath36 of a discrete random variable @xmath29 with probability distribution p(x ) may be defined as @xmath37 entropy is a relatively new concept , yet it is already used as the foundation for many scientific fields . this article creates the foundation for the use of information in computer chess and in computer strategy games in general . however the concept of entropy must be fundamental to any search process where decisions are taken . some of the properties of entropy used to measure the information content in many systems are the following : [ [ non - negativity - of - entropy ] ] non - negativity of entropy + + + + + + + + + + + + + + + + + + + + + + + + + + + @xmath38 uncertainty is always equal or greater than 0.if the entropy , h is 0 , the uncertainty is 0 and the random variable x takes a certain value with probability @xmath39 = 1 consider all probability distributions on a set s with m elements . h is maximum if all events x have the same probability , @xmath40 = @xmath41 if x and y are two independent random variables , then @xmath42 the entropy of a pair of variable x and y is @xmath43 for a pair of random variables one has in general @xmath44 additivity of composite events the average information associated with the choice of an event x is additive , being the sum of the information associated to the choice of subset and the information associated with the choice of the event inside the subset , weighted by the probability of the subset the entropy rate of a sequence @xmath45 = @xmath46 , t @xmath4 n @xmath47 mutual information is a way to measure the correlation of two variables @xmath48 all the equations and definitions presented have a very important role in the model proposed as will be seen later in the article . @xmath49 @xmath50 if any only if x and y are independent variables . a necessary condition for a truly selective search given by hans berliner is the following : the search follows the areas with highest information in the tree @xcite `` it must be able to focus the search on the place where the greatest information can be gained toward terminating the search '' . berliner describes the essential role played by information in chess , however he does not formalize the concept of information in chess as an information theoretic concept . from the perspective of the depth in understanding the decision process in chess the article @xcite is exceptional but it does not formulate his insight in a mathematical frame . it contains great chess and computer chess analysis but it does not define the method in mathematical definitions , concepts and equations . mark winands in @xcite outlines a method based on fractional depth where the fractional ply fp of a move with a category c is given by @xmath51 his approach is experimental and based on data mining as the method presented previously . in the article @xcite david levy , david broughton , mark taylor describe the selective extension algorithm . the method is based on assigning an appropriate additive measure for the interestingness of the terminal node of a path . consider a path in a search tree consisting of the moves @xmath52 , @xmath53 , @xmath54 and the resulting position being a terminal node . the probability that a terminal node in that path is in the principal continuation is @xmath55 the measure of the interestingness of a node in this method is @xmath56 + lg [ p ( m_{ij } ] + lg [ p ( m_{ijk } ) ] \label{eq}\ ] ] the problem is to describe the mathematical meaning of information in computer chess , develop the principles and formulas that can be used to control the search and provide experimental evidence for the search heuristic as well as for the role of information gain in obtaining good results at an acceptable cost . the contributions of this research are the creation of the information theoretical model for search in computer chess , the description of the information gain in computer chess and a scientific explanation of the partial depth scheme . the paths explored are the areas of the search tree with the highest amount of information gain . other contributions are , the calculation of information gain for important moves , the calculation of a formula describing the size of the ply added for various moves , the experimental evidence given for the effect of information gain on search for chess problems . [ [ search - on - informed - game - trees ] ] search on informed game trees + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in @xcite it is introduced the use of heuristic information in the sense of upper and lower bound but no reference to any information theoretic concept is given . actually the information theoretic model would consider a distribution not only an interval as in @xcite . wim pijls and arie de bruin presented a interpretation of heuristic information based on lower and upper estimates for a node and integrated it in alpha beta , proving in the same time the correctness of the method under the following specifications . consider the specifications of the procedure alpha - beta . if the input parameters are the following : ( 1 ) n , a node in the game tree , ( 2 ) alpha and beta , two real numbers and ( 3 ) f , a real number , the output parameter , and the conditions : ( 1)pre : alpha @xmath57 beta ( 2)post : alpha @xmath57 f @xmath57 beta @xmath58 , f @xmath59 alpha @xmath60 f(n ) @xmath59 f @xmath59 alpha f @xmath61 beta @xmath60 f(n ) @xmath61 f @xmath61 beta then the procedure alpha - beta ( defined with heuristic information , but not quantified as in information theory ) meets the specification . @xcite considering the representation given by @xcite , assume for some game trees , heuristic information on the minimax value f(n ) is available for any node . the information may be represented as a pair h = ( u , l ) , where u and l map nodes of the tree into real numbers . u is a heuristic function representing the upper bound on the node . l is a heuristic function representing the lower bound on the node . for every internal node , n the condition u(n ) @xmath61 f(n ) @xmath61 l(n ) must be satisfied . for any terminal node n the condition u(n ) = f(n ) = l(n ) must be satisfied . this may even be considered as a condition for a leaf . a heuristic pair h = ( u , l ) is consistent if + u(c ) @xmath59 u(n ) for every child c of a given max node n and + l(c ) @xmath61 l(n ) for every child c of a given min node n the following theorem published and proven by @xcite relates the information of alpha - beta and the set of nodes visited . let @xmath62 = ( @xmath63,@xmath64 ) and @xmath65 = ( @xmath66,@xmath67 ) denote heuristic pairs on a tree g , such that @xmath63(n ) @xmath59 @xmath66(n ) and @xmath64(n ) @xmath61 @xmath67(n ) for any node n. let @xmath68 and @xmath69 denote the set of nodes , that are visited during execution of the alpha - beta procedure on g with @xmath62 and @xmath65 respectively , then @xmath68 @xmath70 @xmath69 . in the light of the new description it is possible to reformulate the search problem in a strategy game . the problem is to plan the search process minimizing the entropy on the value of the starting position considering limits in costs . the best case is when entropy , or uncertainty in the value of a position becomes 0 with an acceptable cost in search . this is feasible in chess and it happens every time when a successful combination is executed and results in mate or significant advantage . it is possible to formulate the problem of search in computer chess and in other games as a problem of entropy minimization . @xmath71 subject to a limit in the number of position that can be explored . the entropy of a position can be approximated by the sum of entropy rates of the pieces minus the entropy reduction due the strategical configurations . this can be expressed as : @xmath72 where @xmath73 represents the entropy of a piece and @xmath74 represents the entropy of a structure with possible strategic importance . this gives also a more general perspective on the meaning of a game piece . a game piece can be seen as a stochastic function having the state of the board as entrance and generating possible trajectories and the associated probabilities . these probabilities form a distribution having an uncertainty associated . the entropy of a positional pattern , strategic or tactic may be considered a form of joint entropy of the set of variables represented by pieces positions and their trajectory . the pieces forming a strategic or tactic pattern have correlated trajectories which may be considered as forming a plan . @xmath75 \label{eq}\ ] ] @xmath76 where @xmath77 is a subset of pieces involved in a strategic pattern and the probabilities @xmath78 represent the probability of realization of such strategic or tactical pattern . the reduction of entropy caused by strategic and tactical patterns such as double attacks , pins , is determined by both the frequency of such structures and by a significant increase in the probability that one of the sides will win after this position is realized . we may consider the pieces undertaking a common plan as a form of correlated subsystems with mutual information i(piece1,piece2 , ... ) . it results that undertaking a plan may result in a decrease in entropy and a decrease in the need to calculate each variation . it is known from practice that planning decreases the need to calculate each variation and this gives an experimental indication for the practical importance of the concept of entropy as it is defined here in the context of chess . each of the tactical procedures , pinning , forks , double attack , discovered attack and so on , can be understood formally in this way . a big reduction in the uncertainty in regard to the outcome of the game occurs , as the odds are often that such a structure will result in a decisive gain for a player . when such a structure appears as a choice it is likely that a rational player will chose it with high probability . the entropy of these structures may be calculated with a data mining approach to determine how likely they appear in games . an approximation if we do not consider the strategic structures would be : @xmath79 [ [ assumption - analysis ] ] assumption analysis : + + + + + + + + + + + + + + + + + + + + the entropy of the position is smaller in general than the sum of the entropies of pieces because there are certain positional patterns such as openings , end - games , various pawn configurations in a chess position which result in a smaller number of combinations , results in order and a smaller entropy . closer to reality would be such a statement : @xmath80 it is possible to define the information gain during the search process based on the reduction in uncertainty in the following way : @xmath81 where h represents the uncertainty in the value of the position and @xmath82 @xmath83 represents the variation of uncertainty in the current position after a move is made . it is the information gained after making a move . in the case when @xmath84 we speak of information gain , if @xmath85 we understand information lost through approximate evaluation or other operation . it is possible to describe the information gain of the search process by defining the heuristic efficiency @xmath86 when @xmath87 @xmath88 1 the information gain results after a move is @xmath89 this concept may be considered similar to the the concept of information gain for decision trees , the kullback - leibler divergence . we may see the same principle also here , the higher the difference between entropies , the higher the information gain , which makes very much sense also intuitively and it provides a new theoretical justification for the empirical heuristics of chess and computer chess . the partial depths method is a generalization of the classic alpha beta in that it offers a greater importance to moves considered significant for the search . if all moves have the same importance then , the partial depth scheme can be reduced to the ordinary alpha - beta scheme . it can be described also as an importance sampling search . the partial depth scheme has been used by various authors . as hans berliner observed , few has been published about this method @xcite.the contribution of this article goes in this direction . it is possible to define a function returning the depth : @xmath90 this is a generalization of the classic alpha - beta because in classic alpha - beta @xmath91 depth @xmath92 constant ; if the decision to add a certain depth to the path is dependent only on the current move and position , then if @xmath91 path @xmath93 1 the decision depends only on the current position . the increase in depth is dependent on the path in this method , where the path is composed of moves @xmath94 , @xmath95 , @xmath96 , .... . in the classic alpha - beta the depth increase is constant regardless of the type of move . the principle behind a theory of optimal search should be the allocation of search resources based on the optimality of information gain per cost . it results that the fraction of a search ply added to the depth of the path with a move should be in inverse proportion to the quality of the move . the standard approach gives equal importance to all moves , the fraction ply method gives more importance to significant moves . therefore it must be described a quantitative measure for the quality of a move . the reduction from the normal depth of 1 ply should be proportional to the quantitative measure of the quality of a move . the fraction ply fd must decrease with the quality of the move relative to optimal . the fraction ply added would be equal in this system to the decrease of a full play with the approximate entropy reduction achieved by that move compared to a move having the highest entropy reduction . for instance for a capture of a rock the entropy reduction is @xmath97 an axiom of efficient search in chess , in computer chess and of efficient search in general should be that the probability of executing a move must be equal to the heuristic efficiency of that move which is equal to the information efficiency of expanding the node resulted after the move . the same principle can be considered in general for trajectories . by notation , let the heuristic efficiency be he and @xmath98 be the probability of a move in category @xmath99 to be executed . the heuristic efficiency is a fundamental measure of the ability of a search procedure to gain information from the state space . the heuristic efficiency depends in this analysis on the categories of moves and trajectories defined . the examples are for moves with individual tactical values , however the analysis can be extended also to tactical plans generated by pins , forks and other tactical patterns . because such analysis would require some readers to look for the meaning of these structures in chess books and also because space considerations the moves generating such configurations would not be presented as examples . no additional theoretical difficulties would emerge from the introduction of these move categories . the same applies to strategical elements . following the principles outlined , a formula for the fraction ply can be derived . @xmath100 considering that @xmath101 and @xmath102 it means @xmath103 for k = 1 , @xmath104 from this , @xmath105 of course a different value than 1 can be given to the constant k and this will propagate without changing the meaning of the equations . the constant k would increase the flexibility of implementations actually , offering more freedom in this direction . now consider the same equation for the move category with the best information gain . it means @xmath106 assuming the moves from the best category , the most informational efficient will always be executed in the search , the following condition must be satisfied : @xmath107 then @xmath108 so @xmath109 the cost for execution of any of the two moves is the same . equating this cost , it results @xmath110 it means @xmath111 which is a very intuitive result . in general , for a @xmath112 , the probability of a trajectory to be explored should be in this system @xmath113 let @xmath114 be the probability that a move is executed and one more ply is added to the search . the size of the ply added should be function of this probability . it is logically to consider the size of the play as a quantity increasing with the probability of the move not being executed . the probability of the move not being executed is @xmath115 therefore assuming an abstraction , a linear relation of the form : size of ply = k * ( probability of a move not being executed ) then the relation between the size of the ply and the probability of the move to be chosen would be for k = 1 @xmath116 this may be considered even a theorem describing the size of the fraction ply in computer chess and even for other exptime problems under the above assumptions and resulting from the above calculations . starting from the previous equation , it is possible to use the relative entropies of pieces and positional patterns to implement the previous formula . consider the check as the move with the ultimate decrease in entropy because its forceful nature and because it has a higher frequency in the vicinity of the objective , the mate than any other move . then all the other moves may be rated as function of the check move . let such value be @xmath117 . here can be used a constant reflecting the above mentioned properties of such move . it must be noted that not all checks are equally significant . several categories of checks can be introduced instead of a single check category . also in the application , not all checks are equally important , check and capture for example gains a better priority but in this example does not have a smaller depth . as a consequence , if the normal increase in search depth is counted as 1 for moves without significance the fractional ply for a check is : @xmath118 then d = 0 in this system because the best move should be always executed and then the depth added should be 0 . for a capture of queen the entropy rate of the system decreases with @xmath119 . then the fractional ply for a queen capture is @xmath120 after calculations , d = 0.02 for a capture of rock the entropy rate of the system decreases with @xmath97 . then the fractional ply for a rock capture is @xmath121 after calculations , d = 1 - 0.776 = 0.223 instead of using the entropy rates for calculating the size of the fractional depth it is possible to use the value of pieces which is strongly correlated for most of the systems with the entropy rate of the pieces.as it can be seen from the calculation above , the higher the differences in entropy between consecutive positions in a variation , the higher the information gain . this can be understood as a divergence between distributions of consecutive moves . the more they diverge the higher the information gain after a move . as a test case it is used a combination which gives us the possibility to define the quality of the response to a position in a precise way . the meaning of the columns is the following : + column 1:experiment number - represents the number of the search experiment + column 2:nodes searched - represents the number of nodes searched in the experiment + column 3:term dividing the reduction in ply - represents the number dividing the term decreasing the size of the normal ply added to the current depth + column 4 : max depth attained - the maximum depth in standard plies attained , here it is added 1 for each ply + column 5 : max uniform depth - the maximum allowed depth in the partial depth scheme considering a step of 6 decreased with a value depending to the quality of the move + column 6 : solved or not - 1 if the case has been solved with the parameters from the other columns + column 7 : step size - the number added to the partial depth for each new level of search in case of moves without importance + the following is the table with the results of the search experiments : [ cols="<,<,<,<,<,<,<,<",options="header " , ] at first a step representing a fraction of 1 has been used . however , better results have been obtained by using a step bigger than 1 for not so interesting moves . the cause is the decrease in the sensitivity of the output and of other search dependent parameters in regard to the variations of other parameters and of the positional configurations . the detection of the variation leading to the objective early decreases the number of nodes searched very much . the fact that the mate has been found at 13 plies depth after only 20000 nodes searched shows the line to mate has been one of the first lines tried at each level , even without using knowledge . as it can be seen from the table if the mate is detected relatively fast the number of nodes searched is more than 10 times smaller . the next plot shows this . the maximums in the number of nodes represents the configurations ( a set of parameters ) for which the mate has not been fast detected . the ox represents the number dividing the factor giving importance to some significant moves and on oy it is represented the number of nodes searched . plot of the increase in number of nodes when the importance given to moves with high information gain is decreased on ox it is represented the virtual depth . on oy it is represented the number of nodes . as it can be seen , even a deeper search that detects the decisive line will explore less nodes than a shallower search that does not find the decisive line . for this heuristic and for most of the combinations , when the mate or a strongly dominant line is found fast , the drop in the number of nodes searched is as high as 10 times , even if the uniform search is parametrized for a higher depth . the number of nodes to be searched increases very much with the decrease of importance given to important moves and to lines of high informational value . the following plot , based on data from the previous table shows the increase in the number of nodes explored with the decrease in the importance given to information gain when the solution is found . the less importance to the information gaining moves and lines is given , the greater the need for a higher amount of nodes to be searched in order to find the solution . on ox it is represented the term dividing the reduction in ply which represents the number dividing the term decreasing the size of the normal ply added to the current depth . on oy it is represented the number of nodes . the plot shows the explosion of nodes required to find a solution when the importance given to high information lines is decreased . as the importance given to high information lines is decreased the number of nodes searched has to be increased . the importance given to information is decreased so the depth of search must be increased to find the solution . the following plot has the same significance but for the case when the solution is not found . the plot of nodes searched vs depth when the solution is detected fast shows a far less pronounced combinatorial explosion then when the solution is not found . the plot shows the explosion of nodes required to find a solution when the importance given to high information lines is decreased . as the importance given to high information lines is decreased the number of nodes searched has to be increased . it increases even faster when the decisive line is not detected . for a high depth of search , the search cost registers an explosion when no decisive move is found reasonably fast . when less importance is given to high information gain moves the number of plies has to be increased to compensate this and the number of nodes explodes with the number of plies . the plot shows the necessary increase of depth when the importance of high information gain moves is decreased . for the case when the problem is solved the plot is : now we can analyze the data for the cases when the solution is not achieved . for the case when the position is not solved is a similar plot but the search at the respective depth has been realized at a far greater cost than when the solution has been found fast : the maximum depth achieved decreases with a decrease in the importance given to areas of the tree with high information . maximum depth vs importance given to information gain . if less importance is given to moves with high information gain more resources are needed for attaining a maximum given depth . this is the case for solving some combinations . as it can be seen from the previous plot , the maximal depth has been achieved also when the solution has not been found but as it can be observed from the above table and plots , at an ever increased cost . for the search experiments when the solution has not been found the highest depth remains the same but this time the cost of resources needed to sustain that depth increased very fast , faster than in the previous plot when the solution has been found . the search detects the mate even if the maximum length is just one ply deeper than the length of the combination . even if we keep the maximum depth constant at far greater cost the searches are less likely to find the decisive lines as it can be seen from plots . as the search has been changed and less importance has been given to interesting moves , the range in the length of the variations became smaller as less energy has been allocated for the most informative search lines than previously and more energy to the less informative lines . after shifting ever more resources from the informative line to other lines , the objective , the solution of the combination , has not been attained any more by the best lines who did not have the energy this time to penetrate deep enough . the best variations did not have any more the critical energy to penetrate the depth of the state space and solve the problem . the weaker lines were not feasible as a path for finding any acceptable solution . from this we can understand the fundamental effect of resource allocation . and how marginal shifts in resources can lead in this context to completely different result . if somebody used the same depth increase for each move , therefore allocating the resources uniformly to the variations only a supercomputer can go as deep as it is needed for finding the solution to this combination which is not among the deepest . with the introduction of knowledge and heuristics much greater performance would be possible . the experiment concentrates on one heuristic and its effect on the search is highly significant . in order to solve deep combinations where some responses are not forced a program must have chess specialized knowledge ( or an extension of the information theoretical model of computer chess to all chess theory ) in order to give importance to variations without active moves but with significant tactical maneuvering between forceful moves such as checks and captures . [ [ stochastic - modeling - in - computer - chess ] ] stochastic modeling in computer chess + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in the context of game theory , chess is a deterministic game . the practical side of decision in chess and computer chess has many probabilistic elements . the decision is deterministic , but the system that takes the decision is not deterministic , it is a stochastic system . the human decision - making system and its features such as perception and brain processes are known to be stochastic systems . in the case of computer chess many of the search processes are also stochastic , as it has been seen from the previous examples . the implications of the information theoretic model in terms of heuristic development are discussed in this paper . the extension of the model for more elements of computer chess are left for a different research . the limitations of the model are given by the ability to detect the information gain resulted from different moves and to quantify the information gain resulted from these moves . the objective for future research is to explain also other methods from computer chess using the information theoretic model . applications also in the case of other strategy games are also a future objective . the model starts from the axiomatic framework of information theory and describes in a formal way the role of information in the efficiency and effectiveness of the heuristics used in computer chess and other strategy board games . the model proposed considers information in its formal information theoretical meaning as the objective of exploration and the essential factor in the quality of decision in chess and computer chess as well as in other similar games . the method of partial depths scheme , well known in practice has been described mathematically by observing the fundamental fact that information gain is the criteria that determines the decrease in the uncertainty of the position . the uncertainty of the position is described in a mathematical way through the concept of entropy . the information gain describes in a information theoretic way the decrease in uncertainty resulted from making a move . in this way , a quantification of search information is realized . this refers to entropy as it is understood in information theory but it is possible to build parallels also with thermodynamics . previous approaches relied on intuitive formulas and descriptions of the best moves in terms of interestingness or in terms of chess theory or using knowledge extracted from the games of strong players . the approach of the method proposed here is different in that it explains an important method such as the fraction ply method using mathematical methods and formulas that can be derived from the axioms of information theory and determines important coefficients such as the fraction ply associated with moves . the problem of nxn chess is a generalization of the 8x8 chess . it can be expected that the general approach proposed would give a general method for the nxn problem where specialized knowledge is not known and would also provide a method to analyze other exptime - complete problems which can be transformed in the nxn chess . the method provides a new understanding of chess , a game analyzed scientifically before by scientists such as norbert wiener , john von newumann , allan turing , claude shannon , richard bellman and other famous scientists . the method proposed generalizes previous approaches and grounds them on information theory a field with a strong theoretical axiomatic system . it can be expected that the method can provide an example on how to quantify search for difficult problems from classes with high complexity and connect search in computer science also to physics through the common concept of entropy . [ [ acknowledgment - the - author - acknowledges - with - thanks - his - discussions - with - alberto - giovanni - busetto - and - prof .- j .- buhmann ] ] _ acknowledgment : _ the author acknowledges with thanks his discussions with alberto giovanni busetto and prof . j. buhmann + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 100 t.a . marshland , computer chess methods johnathan schaeffer , the history heuristic and alpha - beta search enhancements in practice claude shannon , programming a computer for playing chess glenn strong , the minimax algorithm t.a . marshland , m. campbell , parallel search of strongly ordered game trees r.f . chess the easy way kotov , think like a grandmaster the wikipedia page of games theory as of november 2011 the wikipedia page of computer chess as of november 2011 the wikipedia page of computer go as of november 2011 the wikipedia page of complexity of games as of november 2011 the wikipedia page of thermodynamic entropy as of november 2011 the wikipedia page of information theory as of november 2011 the wikipedia page of thermodynamic entropy as of november 2011 the wikipedia page of thermodynamic entropy and it entropy as of november 2011 alexandru godescu , thesis for the degree of engineer in computer science , pub 2006 alexandru godescu , case study presentation eth zurich 2011 martin j. osborne , ariel rubinstein , a course in game theory theory of games and economic behavior , commemorative edition , ( princeton classic edition ) john von neumann , oskar morgenstern , harold william kuhn and ariel rubinstein s. fraenkel and d. lichtenstein , computing a perfect strategy for n*n chess requires time exponential in n , proc . automata , languages , and programming , springer lncs 115 ( 1981 ) 278 - 293 and j. comb . a 31 ( 1981 ) 199 - 214 d.s . nau , quality of decision versus depth of search on game trees . in : ( 2nd ext . ed . ) , ph.d . thesis , duke university , durham , nc ( 1979 ) . nau , pathology on game trees : a summary of results . in : proceedings aaai-80 ( 1980 ) , pp . 102 - 104 . nau , an investigation of the causes of pathology in games . artificial intelligence 19 3 ( 1980 ) , pp . 257 - 278 d.s . nau , decision quality as a function of search depth in decision trees , j. acm 30 4 ( 1983 ) , pp . 687 - 708 d.s . nau , pathology on game trees revisited , and an alternative to mini - maxing . artificial intelligence 21 1 - 2 ( 1983 ) , pp . 687 - 708 m.m . botvinnik , computers in chess : solving inexact search problems m.m . botvinnik , computers , chess and long range planning h. j. berliner , a chronology of computer chess and its literature h. j. berliner , the b * tree search algorithm : a best - first proof procedure , artificial intelligence 1978 r. keeny , the chess combination from philidor to karpov , learn tactics from champions b. brgmann , monte carlo go , max - plank - institute of physics h.m . markovitz ( march 1952 ) `` portfolio selection '' . the journal of finance 7 ( 1 ) : 77 - 91 t. cover , j. thomas , elements of information theory , second edition marc mezard , andrea montanari , information , physics , and computation wim pijls , arie de bruin , searching informed game trees eliot slater , statistics for the chess computer and the factor of mobility f. etiene de vylder , advanced risk theory , a self contained introduction d e knuth an analysis of alpha - beta pruning , artificial intelligence 1975 g.m . adelson - velsky , v.l . arlazarov , m.v . donskoy , some methods of controlling the tree search in chess programs , artificial intelligence 1975 the wikipedia page of the relative entropy as of november 2011 the wikipedia page of information gain in decision trees as of november 2011 the wikiepdia page of information gain renato renner , lecture notes , quantum information theory , august 16 , 2011 judea pearl , on the nature of pathology in game searching mark winands , enhanced realization probability search david levy , david broughton , mark taylor ( 1989 ) , the sex algorithm in computer chess icga journal , vol . 3 game - tree search algorithm based on realization probability , icga journal , vol . 25 , no . 3 ray solomonoff , the universal distribution and machine learning , the kolmogorov lecture , feb 27 , 2003 , royal holloway , univ . of london . the computer journal , vol 46 , no . 6 , 2003 . jonathan schaeffer , andreas junghanns , search versus knowledge in game - playing programs revisited , aleksander sadikov , ivan bratko , igor konenko , search vs knowledge : empirical study of minimax on krk endgame dana s. nau , mitja lustrek , austin parker , ivan bratko , matjaz gams , when is better not to look ahead ? mitja lustrek , matjaz gams , ivan bratko , is real - valued minimax pathological ? mitja lustrek , matjaz gams , ivan bratko , why minimax works : an alternative explanation aleksander sadikov , ivan bratko , igor konenko , bias and pathology in minimax search .... double minimax(double alfa , double beta , int depth , int k , int type , move mv , double previousval , double virtualdepth ) { move * listnewmoves = ( move * ) new move [ 100 ] ; move mr ; double value = 0 , temp = 0 , ev = 0 ; int c , number ; if ( ( virtualdepth > = maxdepth || depth > = maxextension ) ) { return evaluation(type , mv ) ; } else { if ( tip = = 1 ) { value = -10000 ; } else { value = 10000 ; } generator(mv , listnewmoves , number ) ; for(int i=1 ; i < = number ; i++ ) { listnewmoves[i].eval = fabs ( evaluation(tip , listnewmoves[i ] ) - previousval ) ; double b = -1 ; if ( ischeck ( listnewmoves[i ] ) ) listnewmoves[i].eval + = 10000 ; } if ( number = = 0 ) { if ( tip = = 1 ) if ( ! is_legal_w(mv ) ) return inf_plus ; else return 0 ; } else { if ( ! is_legal_n(mv ) ) return inf_neg ; else return 0 ; } } else for(int k1=1;k1 < = number;k1++ ) { double max = -1 ; int ic = -1 ; for(int c = 1 ; c < = number ; c++ ) { double comp = listnewmoves[c].eval ; if ( comp > max ) { max = listnewmoves[c].eval ; ic = c ; } } mr.eval = listnewmoves[ic].eval ; double evalposition = listnewmoves[ic].eval ; lista_pozitii_urm[ic].eval = -2 ; copy(mr.move , listnewmoves.move ) ; copy ( mr.tabla , listnewmove[ic].tabla ) ; mr.turn = lista_pozitii_urm[ic].turn ; double nextv = evaluation(tip , mr ) ; if ( evalposition > 2000 ) value = - minimax ( -beta , -alfa , depth + 1 , ic , -tip , mr , nextv , virtualdepth ) ; else { double add = log(fabs(0.1 + ( evalposition/100)))/(log(10.0 ) ) + 5.0/log(number + 2 ) ; value = - minimax ( -beta , -alfa , depth + 1 , ic , -tip , mr , nextv , virtualdepth + 6 - add ) ; } if ( value > = alfa ) alfa = value ; if ( alfa > = beta ) { cutoff++ ; break ; } } return alfa ; } } .... [ [ the - reason - for - presenting - some - concepts - of - chess - theory . ] ] the reason for presenting some concepts of chess theory . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + some of the concepts of chess are useful in understanding the ideas of the paper . regardless of the level of knowledge and skill in mathematics without a minimal understanding of important concepts in chess it may be difficult to follow the arguments . it is not essential in what follows vast knowledge of chess or a very high level of chess calculation skills . however , some understanding of the decision process in human chess , how masters decide for a move is important for understanding the theory of chess and computer chess presented here . the theory presented here describes also the chess knowledge in a new perspective assuming that decision in human chess is also based on information gained during positional analysis . an account of the method used by chess grandmasters when deciding for a move is given in a very well regarded chess book . @xcite . [ [ combination ] ] combination + + + + + + + + + + + + a combination is in chess a tree of variations , containing only or mostly tactical and forceful moves , at least a sacrifice and resulting in a material or positional advantage or even in check mate and the adversary can not prevent its outcome . the following is the starting position of a combination . the problem is to find the solution , the moves leading to the objective of the game , the mate . [ [ the - objective - of - the - game . ] ] the objective of the game . + + + + + + + + + + + + + + + + + + + + + + + + + + the objective of the game is to achieve a position where the adversary does not have any legal move and his king is under attack . for example a mate position resulting from the previous positions is : [ [ the - concept - of - variation ] ] the concept of variation + + + + + + + + + + + + + + + + + + + + + + + + + a variation in chess is a string of consecutive moves from the current position . the problem is to find the variation from the start position to mate . in order to make impossible for the adversary to escape the fate , the mate , it is desirable to find a variation that prevents him from doing so , restricting as much as possible his range of options with the threat of decisive moves . [ [ forceful - variation ] ] forceful variation + + + + + + + + + + + + + + + + + + + a forceful variation is a variation where each move of one player gives a limited number of legal option or feasible options to the adversary , forcing the adversary to react to an immediate threat . the solution to the problem , which represents also one of the test cases is the following : \1 . q - n6 ch ! ; pxq 2 . bxqnpch ; k - b1 3 . r - qb7ch ; k - q1 4 . r - b7 ch ; k - b1 5 . rxrch ; q - k1 6 . rxqch ; k - q2 7 . r - q8 mate [ [ attack - on - a - piece ] ] attack on a piece + + + + + + + + + + + + + + + + + + in chess , an attack on a piece is a move that threatens to capture the attacked piece at the very next move . for example after the first move , a surprising move the most valuable piece of white is under attack by the blacks pawn . [ [ the - concept - of - sacrifice - in - chess ] ] the concept of sacrifice in chess + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a sacrifice in chess represents a capture or a move with a piece , considering that the player who performs the chess sacrifice knows that the piece could be captured at the next turn . if the player loses a piece without realizing the piece could be lost then it is a blunder , not a sacrifice . the sacrifice of a piece in chess considers the player is aware the piece may be captured but has a plan that assumes after its realization it would place the initiator in advantage or may even win the game . for example the reply of the black in the forceful variation shown is to capture the queen . while this is not the only option possible , all other options lead to defeat faster for the defending side . the solution requires 7 double moves or 13 plies of search in depth . the entropy as an information theoretic concept may be defined in a precise axiomatic way . @xcite . let a sequence of symmetric functions @xmath122 satisfying the following properties : + ( 1 ) normalization : @xmath123 ( 2 ) continuity:@xmath124 is a continuous function of p + ( 3 ) @xmath125 it results @xmath126 must be of the form @xmath127 | the article describes a model of chess based on information theory . a mathematical model of the partial depth scheme
is outlined and a formula for the partial depth added for each ply is calculated from the principles of the model . an implementation of alpha - beta with partial depth is given .
the method is tested using an experimental strategy having as objective to show the effect of allocation of a higher amount of search resources on areas of the search tree with higher information .
the search proceeds in the direction of lines with higher information gain . the effects on search performance of allocating higher search resources on lines with higher information gain
are tested experimentaly and conclusive results are obtained . in order to isolate the effects of the partial depth scheme
no other heuristic is used .
= 1 |
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in their book @xcite , m. kulenovi and g. ladas initiated a systematic study of the difference equation @xmath11 for nonnegative real numbers @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 such that @xmath6 and @xmath7 , and for nonnegative or positive initial conditions @xmath12 , @xmath13 . under these conditions , ( [ eq : 3 - 3 orig . ] ) has a unique positive equilibrium . one of their main ideas in this undertaking was to make the task more manageable by considering separate cases when one or more of the parameters in ( [ eq : 3 - 3 orig . ] ) is zero . the need for this strategy is made apparent by cases such as the well known _ lyness equation _ @xcite , @xcite , @xcite . @xmath14 whose dynamics differ significantly from other equations in this class . there are a total of 42 cases that arise from ( [ eq : 3 - 3 orig . ] ) in the manner just discussed , under the hypotheses @xmath6 and @xmath7 . the recent publications @xcite , @xcite give a detailed account of the progress up to 2007 in the study of dynamics of the class of equations ( [ eq : 3 - 3 orig . ] ) . after a sustained effort by many researchers ( for extensive references , see @xcite , @xcite ) , there are some cases that have resisted a complete analysis . we list them below in normalized form , as presented in @xcite , @xcite . @xmath15 the dynamics of equation ( [ eq : ladas 2 - 3 ] ) has been settled recently in @xcite , @xcite . global attractivity of the positive equilibrium of equation ( [ eq : ladas y2k ] ) has been proved recently in @xcite . since eq.([eq : kul ] ) can be reduced to eq.([eq : ladas y2k ] ) through a change of variables @xcite , global behavior of solutions to ( [ eq : kul ] ) is also settled . equation ( [ eq : kul 1 ] ) is another equation that can be reduced to ( [ eq : ladas y2k ] ) , through the change of variables @xmath16 @xcite . ladas and co - workers @xcite , @xcite , @xcite , have posed a series of conjectures on these equations . one of them is the following . * conjecture [ ladas et al . ] * _ for equations ( [ eq : ladas 3 - 2 ] ) and ( [ eq : ladas 3 - 3 ] ) , every solution converges to the positive equilibrium or to a prime period - two solution . _ in this article , we prove this conjecture . our main results are the following . [ th : 3 - 3 ] for every choice of positive parameters @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 , all solutions to the difference equation @xmath17 converge to the positive equilibrium or to a prime period - two solution . [ th : 3 - 2 orig . ] for every choice of positive parameters @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 , all solutions to the difference equation @xmath18 converge to the positive equilibrium or to a prime period - two solution . a reduction of the number of parameters of eq.([eq : 3 - 2 orig . ] ) is obtained with the change of variables @xmath19 , which yields the equation @xmath20 where @xmath21 , @xmath22 , and @xmath23 . the number of parameters of eq.(3 - 3 ) can also be reduced , which we proceed to do next . consider the following affine change of variables which is helpful to reduce number of parameters and simplify calculations : @xmath24 with ( [ change of coordinates ] ) , eqn.(3 - 3 ) may now be rewritten as @xmath25 where @xmath26 theorems [ th : 3 - 3 ] and [ th : 3 - 2 orig . ] can be reformulated in terms of the parameters @xmath27 , @xmath28 and @xmath29 as follows . [ th : 3 - 2-l ] let @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 be positive numbers , and let @xmath27 , @xmath28 , @xmath29 and @xmath30 be given by relations ( [ eq : pqr ] ) . then every solution to eqn.(3 - 2-l ) converges to the unique equilibrium or to a prime period - two solution . [ th : 3 - 2 ] let @xmath27 , @xmath28 , @xmath29 be positive numbers . then every solution to eqn.(3 - 2 ) converges to the unique equilibrium or to a prime period - two solution . in this paper we prove theorems [ th : 3 - 2-l ] and [ th : 3 - 2 ] ; theorems [ th : 3 - 3 ] and [ th : 3 - 2 orig . ] follow as an immediate corollary . the two main differences between eq.(3 - 2-l ) and eq.(3 - 2 ) are the set of initial conditions , and the possibility of having a negative value of @xmath29 in eq.(3 - 2-l ) , while only positive values of @xmath29 are allowed in eq.(3 - 2 ) . nevertheless , for both eq.(3 - 2-l ) and eq.(3 - 2 ) the unique equilibrium has the formula : @xmath31 although it is not possible to prove theorem [ th : 3 - 3 ] as a simple corollary to theorem [ th : 3 - 2 orig . ] , the changes of variables leading to theorems [ th : 3 - 2-l ] and [ th : 3 - 2 ] will result in proofs to the former theorems that are greatly simplified . our main results theorem [ th : 3 - 3 ] and theorem [ th : 3 - 2 orig . ] imply that when prime period - two solutions to eq.(3 - 3 ) or eq.(3 - 2 ) do not exist , then the unique equilibrium is a global attractor . we have not treated here certain questions about the global dynamics of eq.(3 - 3 ) and eq.(3 - 2 ) , such as the character of the prime period - two solutions to either equation , or even for more general rational second order equations , when such solutions exist . this matter will be treated in an upcoming article of the authors @xcite . this work is organized as follows . the main results are stated in section [ sec : introduction ] . results from the literature which are used here are given in section [ sec : results from the literature ] for convenience . in section [ section : existence ] , it is shown that either every solution to eq.(3 - 2-l ) converges to the equilibrium , or there exists an invariant and attracting interval @xmath32 with the property that the function @xmath33 associated with the difference equation is coordinate - wise strictly - monotonic on @xmath34 . in section [ sec : the equation ( 3 - 2 ) ] , a global convergence result is obtained for eq.(3 - 2 ) over a specific range of parameters and for initial conditions in an invariant compact interval . theorem [ th : 3 - 2-l ] is proved in section [ sec : proof 3 - 2-l ] , and the proof of theorem [ th : 3 - 2 ] is given in section [ sec : proof 3 - 2 ] . section [ appendix : computer 1 ] includes computer algebra system code for performing certain calculations that involve polynomials with a large number of terms ( over 365,000 in one case ) . these computer calculations are used to support certain statements in section [ sec : the equation ( 3 - 2 ) ] . finally , we refer the reader to @xcite for terminology and definitions that concern difference equations . the results in this subsection are from the literature , and they are given here for easy reference . the first result is a reformulation of theorems ( 1.4.5 ) ( 1.4.8 ) in @xcite . [ th : mm first ] suppose a continuous function @xmath35 ^ 2 \rightarrow [ a , b]$ ] satisfies one of i.iv . : * @xmath33 is nondecreasing in @xmath36 , @xmath37 , and @xmath38 ^ 2 , \quad ( \ , f(m , m)=m\ \&\ f(m , m)=m\ , ) \implies m = m\ ] ] * @xmath33 is nonincreasing in @xmath36 , @xmath37 , and @xmath38 ^ 2 , \quad ( \ , f(m , m)=m\ \&\ f(m , m)=m\ , ) \implies m = m\ ] ] * @xmath33 is nonincreasing in @xmath36 and nondecreasing in @xmath37 , and @xmath38 ^ 2 , \quad ( \ , f(m , m)=m\ \&\ f(m , m)=m\ , ) \implies m = m\ ] ] * @xmath33 is nondecreasing in @xmath36 and nonincreasing in @xmath37 , and @xmath38 ^ 2 , \quad ( \ , f(m , m)=m\ \&\ f(m , m)=m\ , ) \implies m = m\ ] ] then @xmath39 has a unique equilibrium in @xmath40 $ ] , and every solution with initial values in @xmath41 $ ] converges to the equilibrium . the following result is theorem a.0.8 in @xcite . [ th : mm 3 - 3 ] suppose a continuous function @xmath35 ^ 3 \rightarrow [ a , b]$ ] is nonincreasing in all variables , and @xmath38 ^ 3 , \quad ( \ , f(m , m , m)=m\ \&\ f(m , m , m)=m\ , ) \implies m = m\ ] ] then @xmath42 has a unique equilibrium in @xmath40 $ ] , and every solution with initial values in @xmath41 $ ] converges to the equilibrium . [ th : camouzis - ladas ] let @xmath32 be a set of real numbers and let @xmath43 be a function @xmath44 which decreases in @xmath45 and increases in @xmath46 . then for every solution @xmath47 of the equation @xmath48 the subsequences @xmath49 and @xmath50 of even and odd terms do exactly one of the following : * they are both monotonically increasing . * they are both monotonically decreasing . * eventually , one of them is monotonically increasing and the other is monotonically decreasing . theorem [ th : camouzis - ladas ] has this corollary . if @xmath32 is a compact interval , then every solution of eq.([eq : camouzis - ladas ] ) converges to an equilibrium or to a prime period - two solution . [ th : kocic - ladas ] assume the following conditions hold : * @xmath51 $ ] . * @xmath52 is decreasing in @xmath36 and strictly decreasing in @xmath37 . * @xmath53 is strictly increasing in @xmath36 . * the equation @xmath54 has a unique positive equilibrium @xmath55 . then @xmath55 is a global attractor of all positive solutions of eq.([eq : kocic - ladas ] ) . in this section we prove a proposition which is key for later developments . we will need the function @xmath56 associated to eq.(3 - 2-l ) . [ prop : invariant and attracting ] at least one of the following statements is true : * every solution to ( 3 - 2-l ) converges to the equilibrium . * there exist @xmath57 , @xmath58 with @xmath59 s.t . * * @xmath60 $ ] is an invariant interval for eq.(3 - 2-l ) , i.e. , @xmath61\times [ m^{*},m^ { * } ] ) \subset [ m^{*},m^{*}]$ ] . * * every solution to eq.(3 - 2-l ) eventually enters @xmath60 $ ] . * * @xmath33 is coordinate - wise strictly monotonic on @xmath60^{2}$ ] . the next lemma states that the function @xmath62 associated to eq.(3 - 2-l ) is bounded . [ lemma : f is bounded ] there exist _ positive _ constants @xmath63 and @xmath64 such that @xmath65 and @xmath66 in particular , @xmath67 is invariant for f } f\left([\mathcal{l},\mathcal{u}]\times[\mathcal{l},\mathcal{u}]\right ) \subset [ \mathcal{l},\mathcal{u}]\ ] ] the function @xmath68 associated to eq.(3 - 3 ) is bounded : @xmath69 set @xmath70 and @xmath71 . the affine change of coordinates ( [ change of coordinates ] ) maps the rectangular region @xmath72 ^ 2 $ ] onto a rectangular region @xmath73 ^ 2 $ ] which satisfies ( [ ineq : bounds for f ] ) and ( [ eq : [ u , l ] is invariant for f ] ) . [ lemma : p = q ] if @xmath74 , then every solution to eq.(3 - 2-l ) converges to the unique equilibrium . if @xmath74 then @xmath75 and @xmath76 . thus , depending on the sign of @xmath29 , the function @xmath33 is either nondecreasing in both coordinates , or nonincreasing in both coordinates on @xmath77 . by lemma [ lemma : f is bounded ] , all solutions @xmath78 satisfy @xmath79 $ ] for @xmath80 . a direct algebraic calculation may be used to show that all solutions @xmath81 $ ] of either one of the systems of equations @xmath82 necessarily satisfy @xmath83 . in either case , the hypotheses ( i ) or ( ii ) of theorem [ th : mm first ] are satisfied , and the conclusion of the lemma follows . we will need the following elementary result , which is given here without proof . [ lemma : signs of partials ] suppose @xmath84 . the function @xmath33 has continuous partial derivatives on @xmath85 , and * @xmath86 if and only if @xmath87 , and @xmath88 if and only if @xmath89 . * @xmath90 if and only if @xmath91 , and @xmath92 if and only if @xmath93 . we will need to refer to the values @xmath94 and @xmath95 where the partial derivatives of @xmath33 change sign . if @xmath96 , set @xmath97 for @xmath98 , let @xmath99 ^ 2\ } \quad \mbox{and } \quad \phi(m , m ) : = \max \ { f(x , y ) : ( x , y ) \in [ m , m]^2\}\ ] ] [ lemma : m < phi ] suppose @xmath96 . if @xmath100\subset [ \mathcal{l},\mathcal{u}]$ ] is an invariant interval for eq.(3 - 2-l ) with @xmath101 or @xmath102 , then @xmath103 or @xmath104 or @xmath105 . by definition of @xmath106 and @xmath107 , @xmath108 and @xmath109 . suppose @xmath110 the proof will be complete when it is shown that @xmath83 . there are a total of four cases to consider : ( a ) @xmath111 and @xmath112 , ( b ) @xmath113 and @xmath114 , ( c ) @xmath111 and @xmath114 , and ( d ) @xmath113 and @xmath112 . we present the proof of case ( a ) only , as the proof of the other cases is similar . if @xmath111 and @xmath112 , then @xmath115 $ ] and @xmath116 $ ] . note that @xmath117\times[m , m ] = [ m , m]\times [ m , k_1]\ \bigcup \ [ m , m]\times [ k_1,m].\ ] ] by lemma [ lemma : signs of partials ] , the signs of the partial derivatives of @xmath33 are constant on the interior of each of the sets @xmath100\times [ m , k_1 ] $ ] and @xmath100\times [ k_1,m]$ ] , as shown in the diagram . ( 2.5,2.5 ) ( 0.5,2.5 ) ( 2.5,1.4 ) ( 0.5,0.5 ) ( -0.8,.6)@xmath118 ( 2.8,1.4)@xmath119 ( -0.8,2.4)@xmath120 ( 0.6 , 0.6)(2,2 ) ( 1,0.9)@xmath121 ( 0.6,1.5)(1,0)2 ( 1,1.9)@xmath122 [ fig : diagram 1 ] since @xmath33 is nonincreasing in both @xmath36 and @xmath37 on @xmath123\times[m , k_{1}]$ ] , @xmath124\times[m , k_{1}].\ ] ] similarly , @xmath33 is nondecreasing in @xmath36 and nonincreasing in @xmath37 on @xmath123\times[k_{1 } , m]$ ] , hence @xmath125\times[k_{1 } , m].\ ] ] from ( [ eq : ineq1-new1 ] ) and ( [ eq : ineq2-new1 ] ) one has @xmath126 combine ( [ eq : phi = f(m , m ) ] ) with relation ( [ eq : m = phi ] ) to obtain the system of equations @xmath127 eliminating @xmath128 from system ( [ eq : sys phi phi ] ) gives the cubic in @xmath129 @xmath130 which has the roots @xmath131 only one root in the list ( [ eq : the m roots ] ) is positive , namely @xmath132 substituting into one of the equations of system ( [ eq : sys phi phi ] ) one also obtains @xmath133 , which gives the desired relation @xmath105 . let @xmath134 , @xmath135 , and for @xmath136 let @xmath137 , @xmath138 . by the definitions of @xmath139 , @xmath140 , @xmath141 and @xmath142 , we have that @xmath143 \subset [ m_\ell , m_\ell]$ ] for @xmath144 . thus the sequence @xmath145 is nondecreasing and @xmath146 is nonincreasing . let @xmath147 and @xmath148 . suppose @xmath96 . either there exists @xmath149 such that @xmath150 = \emptyset$ ] , or @xmath151 . arguing by contradiction , suppose @xmath152 and for all @xmath153 , @xmath154 \not = \emptyset$ ] . since the intervals @xmath155 $ ] are nested and @xmath156 = [ m^*,m^*]$ ] , it follows that @xmath157 \not = \emptyset$ ] . by lemma [ lemma : m < phi ] , we have @xmath158 continuity of the functions @xmath106 and @xmath107 implies @xmath159 statements ( [ ineq : mstar1 ] ) and ( [ ineq : mstar2 ] ) give a contradiction . * proof of proposition [ prop : invariant and attracting]*. suppose statement ( a ) is not true . by lemma [ lemma : p = q ] , one must have @xmath160 . note that if @xmath161 is a solution to eq.(3 - 2-l ) , then @xmath162 $ ] for @xmath136 . if @xmath163 , since @xmath164 and @xmath165 we have @xmath166 , but this is statement ( a ) which we are negating . thus @xmath152 , and by lemma [ lemma : m < phi ] there exists @xmath149 such that @xmath150=\emptyset$ ] , so @xmath33 is coordinate - wise monotonic on @xmath167 $ ] . the set @xmath167 $ ] is invariant , and every solution enters @xmath167 $ ] starting at least with the term with subindex @xmath168 . we have shown that if statement ( a ) is not true , then statement ( b ) is necessarily true . this completes the proof of the proposition . @xmath169 in this section we restrict our attention to the equation @xmath172 where @xmath173 for @xmath174 , @xmath175 , and @xmath170 , eq.(3 - 2 ) has a unique _ positive _ equilibrium @xmath176 we note that if @xmath177 is an invariant compact interval , then necessarily @xmath178 . the goal in this section is to prove the following proposition , which will provide an important part of the proofs of theorems [ th : 3 - 3 ] and [ th : 3 - 2 ] . [ prop : f up down r > 0 on i ] let @xmath27 , @xmath28 and @xmath29 be real numbers such that @xmath179 and let @xmath180\subset ( \frac{q\,r}{p - q } , \frac{p}{q})$ ] be a compact invariant interval for eq.(3 - 2 ) . then every solution to eq.(3 - 2 ) with @xmath181 $ ] converges to the equilibrium . proposition [ prop : f up down r > 0 on i ] follows from lemmas [ lemma : f up down ybar p<1 q>1 r>0 ] , [ lemma : f up down ybar sufficient condition ] and [ lemma : ybar is ga when suff cond 2 ] , which are stated and proved next . [ lemma : f up down ybar p<1 q>1 r>0 ] assume the hypotheses to proposition [ prop : f up down r > 0 on i ] . if either @xmath182 or @xmath183 , then every solution to eq.(3 - 2 ) with @xmath181 $ ] converges to the equilibrium . we verify that hypothesis ( iv ) of theorem [ th : mm first ] is true . since @xmath184 for @xmath185 $ ] , the function @xmath33 is increasing in @xmath36 and decreasing in @xmath37 for @xmath186 ^ 2 $ ] by lemma [ lemma : signs of partials ] . let @xmath187 $ ] be such that @xmath188 and @xmath189 we show first that system ( [ eq : mm for p<1 ] ) has no solutions if either @xmath182 or @xmath183 . by eliminating denominators in both equations in ( [ eq : mm for p<1 ] ) , @xmath190 and by subtracting terms in ( [ eq : mm for p<1 b ] ) one obtains @xmath191 since @xmath192 , we have @xmath193 , which implies that for @xmath194 there are no solutions to system ( [ eq : mm for p<1 ] ) which have both coordinates positive . now assume @xmath195 ; from ( [ eq : mm for p<1 2 ] ) , @xmath196 , and substitute the latter into ( [ eq : mm for p<1 ] ) to see that @xmath197 is a solution to the quadratic equation @xmath198 by a symmetry argument , one has that @xmath199 is also a solution to ( [ eq : mm quadratic p>1 ] ) . by inspection of the coefficients of the polynomial in the left - hand - side of ( [ eq : mm quadratic p>1 ] ) one sees that two positive solutions are possible only when @xmath200 . to get the conclusion of the lemma , note that the fact that ( [ eq : mm for p<1 ] ) has no solutions with @xmath192 is just hypothesis ( iv ) of theorem [ th : mm first ] . [ lemma : f up down ybar sufficient condition ] assume the hypotheses to proposition [ prop : f up down r > 0 on i ] . if @xmath201 then every solution to eq.(3 - 2 ) with @xmath181 $ ] converges to the equilibrium . by substituting @xmath202 into @xmath203 we obtain @xmath204 that is , @xmath205 where the @xmath36 has been kept in @xmath206 for bookkeeping purposes . thus @xmath206 is constant in @xmath36 . we claim @xmath206 is decreasing in both @xmath37 and @xmath207 . to see that the partial derivative @xmath208 is negative just use @xmath112 and the inequality @xmath209 , which is true by lemma [ lemma : signs of partials ] . the remaining partial derivative is @xmath210 where @xmath211 we have , @xmath212 since @xmath213 , @xmath214\ ] ] thus we conclude that @xmath215 for @xmath216 $ ] . to complete the proof we verify the hypotheses of theorem [ th : mm 3 - 3 ] . we claim that the system of equations @xmath217 has no solutions @xmath120 with @xmath192 whenever hypothesis ( [ sufficient condition ] ) holds . by eliminating denominators in both equations in ( [ eq : mm for embed ] ) one obtains @xmath218 and by subtracting terms in ( [ eq : mm for p<1 b embed ] ) one obtains @xmath219 since @xmath192 , we may use the second factor in the left - hand - side term of ( [ eq : mm for p<1 2 embed ] ) to solve for @xmath128 in terms of @xmath129 , which upon substitution into @xmath220 and simplification yields the equation @xmath221 where @xmath222 by hypothesis ( [ sufficient condition ] ) we have @xmath223 , hence @xmath224 , which implies @xmath225 . by direct inspection one can see that @xmath226 and @xmath227 . thus ( [ eq : no solutions ] ) has no positive solutions , and we conclude that ( [ eq : mm for embed ] ) has no solutions @xmath228 $ ] with @xmath192 . we have verified the hypotheses of theorem [ th : mm 3 - 3 ] , and the conclusion of the lemma follows . [ lemma : las ] let @xmath174 , @xmath175 and @xmath170 . if the positive equilibrium @xmath229 of eq.(3 - 2 ) satisfies @xmath230 , then @xmath229 is locally asymptotically stable ( l.a.s . ) . solving for @xmath29 in @xmath231 gives @xmath232 then a calculation shows @xmath233 set @xmath234 and @xmath235 . the equilibrium @xmath229 is locally asymptotically stable if the roots of the characteristic polynomial @xmath236 have modulus less than one @xcite . by the schur - cohn theorem , @xmath229 is l.a.s . if and only if @xmath237 . it can be easily verified that @xmath238 if and only if @xmath239 which is true regardless of the allowable parameter values . since @xmath240 by the hypothesis , we have @xmath241 , hence some algebra gives @xmath242 if and only if @xmath243 but ( [ eq : schur conn ] ) is a true statement by formula ( [ eq : ybar formula ] ) . we conclude @xmath229 is l.a.s . [ lemma : ybar is ga when suff cond 2 ] assume the hypotheses to proposition [ prop : f up down r > 0 on i ] . if @xmath244 then every solution to eq.(3 - 2 ) with @xmath181 $ ] converges to the equilibrium . the proof begins with a change of variable in eq.(3 - 2 ) to produce a transformed equation with normalized coefficients analogous to those in the standard _ normalized lyness equation _ @xcite , @xcite , @xcite @xmath245 we seek to use an argument of proof similar to the one used in @xcite , in which one takes advantage of the existence of _ invariant curves _ of lyness equation to produce a lyapunov - like function for eq.(3 - 2 ) . set @xmath246 in eq.(3 - 2 ) to obain the equation @xmath247 where @xmath248 we shall denote with @xmath249 the unique equilibrium of eq.([3 - 2-z ] ) . note that @xmath250 it is convenient to parametrize eq.([3 - 2-z ] ) in terms of the equilibrium . we will use the symbol @xmath45 to represent the equilibrium @xmath249 of eq.([3 - 2-z ] ) . by direct substitution of the equilibrium @xmath251 into eq.([3 - 2-z ] ) we obtain @xmath252 by ( [ eq : alpha ] ) , @xmath111 iff @xmath253 . using ( [ eq : alpha ] ) to eliminate @xmath29 from eq.([3 - 2-z ] ) gives the following equation for @xmath254 , @xmath255 and @xmath256 , equivalent to eq.([3 - 2-z ] ) : @xmath257 therefore it suffices to prove that all solutions of eq.([eq : y2k2u ] ) converge to the equilibrium @xmath45 . the following statement is crucial for the proof of the proposition . [ claim : ybar > 1 ] @xmath258 if and only if @xmath259 . since @xmath260 , we have @xmath261 if and only if @xmath262 , which holds if and only if @xmath263 after an elementary simplification , the latter inequality can be rewritten as @xmath259 . by the hypotheses of the lemma , by claim [ claim : ybar > 1 ] , and by ( [ eq : new parameters ] ) and ( [ eq : alpha ] ) we have @xmath264 we now introduce a function which is the invariant function for ( [ eq : lyness ] ) with constant @xmath265 ( in this case the the equilibrium of ( [ eq : lyness ] ) is @xmath45 ) : @xmath266 note that @xmath267 for all @xmath268 whenever @xmath261 . by using elementary calculus , one can show that the function @xmath269 has a strict global minimum at @xmath270 @xcite , @xcite , i.e. , @xmath271 we need some elementary properties of the sublevel sets @xmath272 we denote with @xmath273 , @xmath274 the four regions @xmath275 let @xmath276 be the map associated to eq . ( [ eq : y2k2u ] ) ( see @xcite ) . [ claim : q2q4 ] if @xmath277 , then @xmath278 . set @xmath279 a calculation yields @xmath280 where @xmath281 by ( [ eq : ineqs u , g , b ] ) , for @xmath282 we have @xmath283 and @xmath284 with @xmath285 , therefore @xmath286 , @xmath287 and @xmath288 . consequently @xmath289 for @xmath282 . to see that @xmath289 for @xmath290 as well , rewrite @xmath291 and @xmath292 as follows : @xmath293 for @xmath290 we have @xmath294 and @xmath285 . thus @xmath295 , @xmath296 , and @xmath297 , which imply @xmath298 . [ claim : q1q3 , g > b ] suppose @xmath299 . if @xmath300 , then @xmath301 . this proof requires extensive use of a computer algebra system to verify certain inequalities involving rational expressions . here we give an outline of the steps , and refer the reader to section [ appendix : computer 1 ] for the details . since @xmath302 , and @xmath303 we may write @xmath304 the expression @xmath305 may be written as a single ratio of polynomials , @xmath306 with @xmath307 . the next step is to show @xmath308 for @xmath309 . points @xmath310 in @xmath311 may be written in the form @xmath312 , @xmath313 , where @xmath314 . substituting @xmath36 , @xmath37 , @xmath45 and @xmath315 in terms of @xmath46 , @xmath316 , @xmath317 and @xmath318 into the expression for @xmath319 one obtains a rational expression @xmath320 with positive denominator . the numerator @xmath321 has some negative coefficients . at this points two cases are considered , @xmath322 , and @xmath323 . these can be written as @xmath324 and @xmath325 for nonnegative @xmath326 . substitution of each one of the latter expressions in @xmath321 gives a polynomial with positive coefficients . this proves @xmath327 for @xmath328 . if now we assume @xmath329 with @xmath285 , we may write @xmath330 the rest of the proof is as in the first case already discussed . details can be found in section [ appendix : computer 1 ] . [ claim : q1q3 , g < b ] suppose @xmath331 and @xmath261 . if @xmath300 , then @xmath332 . the proof is analogous to the proof of claim [ claim : q1q3 , g > b ] . we provide an outline . more details can be found in section [ appendix : computer 1 ] . since @xmath333 , we may write @xmath334 with @xmath335 . also , @xmath336 implies @xmath337 , and @xmath338 for @xmath339 . since @xmath340 we may write @xmath341 for @xmath342 . the expression @xmath343 may be written as a single ratio of polynomials , @xmath306 with @xmath307 . the next step is to show @xmath308 for @xmath309 . this is done in a way similar to the procedure described in in claim [ claim : q1q3 , g > b ] . to complete the proof of the lemma , let @xmath344 . let @xmath345 be the solution to ( [ eq : y2k2u ] ) with initial condition @xmath346 , and let @xmath347 be the corresponding orbit of @xmath348 . the following argument is essentially the same as the one found in @xcite ; we provided here for convenience . define @xmath349 note that @xmath350 , which can be shown by applying claims [ claim : q2q4 ] , [ claim : q1q3 , g > b ] and [ claim : q1q3 , g < b ] repeatedly as needed to obtain a nonincreasing subsequence of @xmath351 that is bounded below by @xmath352 . let @xmath353 be a subsequence convergent to @xmath354 . therefore there exists @xmath355 such that @xmath356 that is , @xmath357 the set @xmath358 is closed by continuity of @xmath269 . boundedness of @xmath358 follows from @xmath359 thus @xmath358 is compact , and there exists a convergent subsequence @xmath360 with limit @xmath361 . note that @xmath362 we claim that @xmath363 . if not , then by claims [ claim : q2q4 ] [ claim : q1q3 , g > b ] and [ claim : q1q3 , g < b ] , @xmath364 let @xmath365 denotes the euclidean norm . by ( [ eq : cont ineq ] ) and continuity , there exists @xmath366 such that @xmath367 choose @xmath368 large enough so that @xmath369 but then ( [ eq : by continuity ] ) and ( [ eq : choose l ] ) imply @xmath370 which contradicts the definition ( [ eq : hatc def ] ) of @xmath354 . we conclude @xmath363 . from this and the definition of convergence of sequences we have that for every @xmath371 there exists @xmath372 such that @xmath373 . finally , since @xmath374 we have that for every @xmath371 there exists @xmath372 such that @xmath375 and @xmath376 . since @xmath45 is a locally asymptotically stable equilibrium for eq.([eq : y2k2u ] ) by lemma [ lemma : las ] , it follows that @xmath377 . this completes the proof of the lemma . to prove theorem [ th : 3 - 2-l ] it is enough to assume statement ( b ) of proposition [ prop : invariant and attracting ] . also by lemma [ lemma : p = q ] we may assume @xmath96 without loss of generality . thus we make the following standing assumption , valid throughout the rest of this section for eq.(3 - 2-l ) . * standing assumption ( sa ) * _ assume @xmath160 and that there exist @xmath57 , @xmath58 with @xmath378 such that for eq.(3 - 2-l ) and its associated function @xmath33 , _ * @xmath60 $ ] is an invariant interval . * every solution eventually enters @xmath60 $ ] . * @xmath33 is coordinate - wise strictly monotonic on @xmath60^{2}$ ] . the function @xmath33 is assumed to be coordinate - wise monotonic on @xmath379 $ ] , and there are four possible cases in which this can happen : ( a ) @xmath33 is increasing in both variables , ( b ) @xmath33 is decreasing in both variables , ( c ) @xmath33 is decreasing in @xmath36 and increasing in @xmath37 , and ( d ) @xmath33 is increasing in @xmath36 and decreasing in @xmath37 . we present several lemmas before completing the proof of theorem [ th : 3 - 2-l ] . by considering the restriction of the map @xmath348 of eq.(3 - 2-l ) to @xmath379 ^ 2 $ ] , an application of the schauder fixed point theorem @xcite gives that @xmath379 ^ 2 $ ] contains the fixed point of @xmath348 , namely @xmath380 . thus we have the following result . [ lemma : ybar in invariant interval ] @xmath381 $ ] . [ lemma : f(up , up ) and f(down , down ) ] neither one of the systems of equations @xmath382 have solutions @xmath383 ^ 2 $ ] with @xmath384 . since @xmath385 is the only solution to @xmath386 , it is clear that only @xmath380 satisfies ( s@xmath387 ) . now let @xmath120 be a solution to ( s@xmath388 ) . from straightforward algebra applied to @xmath389 one arrives at @xmath390 , which implies @xmath83 . [ lemma : 0<q < p ] suppose @xmath33 is increasing in @xmath36 and decreasing in @xmath37 for @xmath391 $ ] . then @xmath392 . by the standing assumption ( sa ) , @xmath96 . by lemma [ lemma : signs of partials ] , the coordinate - wise monotonicity hypothesis , and the fact @xmath381 $ ] from lemma [ lemma : ybar in invariant interval ] , we have @xmath393 the inequalities in ( [ eq : p - q sign ] ) can not hold simultaneously unless @xmath394 . [ lemma : f up down lemma bounds ] if @xmath33 is increasing in @xmath36 and decreasing in @xmath37 for @xmath391 $ ] , then @xmath395 ^ 2 ) \subset ( 1,\frac{p}{q})$ ] . for @xmath391 $ ] , the function @xmath396 is well defined and is componentwise strictly monotonic on the set @xmath397 . then , @xmath398 [ lemma : f up down iff ] let @xmath174 , @xmath175 and @xmath170 . if @xmath33 is increasing in @xmath36 and decreasing in @xmath37 on @xmath379 $ ] , then @xmath399 since @xmath381 $ ] by lemma [ lemma : ybar in invariant interval ] , we have @xmath400 and @xmath401 . by lemma [ lemma : 0<q < p ] , @xmath112 , and by lemma [ lemma : signs of partials ] , @xmath402 then , @xmath403 in addition , by lemma [ lemma : signs of partials ] , @xmath404 [ lemma : p - q+r>0 ] suppose @xmath33 is increasing in @xmath36 and decreasing in @xmath37 for @xmath391 $ ] . if @xmath113 , then @xmath405 . since @xmath406 for @xmath391 $ ] , and by lemma [ lemma : signs of partials ] , lemma [ lemma : ybar in invariant interval ] and by lemma [ lemma : 0<q < p ] , we have @xmath407 , that is , @xmath408 if the right - hand - side of inequality ( [ eq : ybar bigger than ] ) is nonnegative , then , after squaring both sides of ( [ eq : ybar bigger than ] ) we have @xmath409 further simplification of ( [ eq : ybar bigger than 2 ] ) and the hypothesis @xmath113 yield @xmath410 which , after some elementary algebra , implies @xmath405 . now assume the right - hand - side of inequality ( [ eq : ybar bigger than ] ) is negative , relation that we may rewrite as @xmath411 if @xmath412 , then @xmath413,which gives the conclusion @xmath414 . if @xmath415 , that is , @xmath416 , then @xmath417 therefore if @xmath418 the conclusion of the lemma follows from this and from ( [ eq : p - q+r > q+r+1 ] ) . assume @xmath419 from relations ( [ eq : pqr ] ) we have @xmath420 hence assumption ( [ eq : assumption q+r+1<0 ] ) and relation ( [ eq : q+r+1<0 ] ) imply @xmath421 further algebra gives @xmath422 since @xmath423 by ( [ eq : long term ] ) , from inequality ( [ eq : further algebra ] ) we have @xmath424 finally , from ( [ eq : pqr ] ) we have @xmath425 combining ( [ eq : one more ] ) with ( [ eq : yet one more ] ) we obtain @xmath426 . [ lemma : f up down ybar is ga ] if @xmath113 and @xmath33 is increasing in @xmath36 and decreasing in @xmath37 for @xmath391 $ ] , then every solution converges to the equilibrium . since @xmath414 by lemma [ lemma : p - q+r>0 ] , we have @xmath427 , which together with lemma [ lemma : f up down lemma bounds ] implies that @xmath428 $ ] is an invariant , attracting compact interval such that @xmath33 is increasing in @xmath36 and decreasing in @xmath37 on @xmath428 ^ 2 $ ] . since @xmath429 ^ 2 ) \subset ( 1,\frac{p}{q})$ ] , we see that every solution to eq.(3 - 2-l ) eventually enters the invariant interval @xmath430 . the change of variables @xmath431 transforms the equation @xmath432 into the equivalent equation @xmath433 where @xmath434 we claim that for @xmath435 , ( a ) @xmath436 is increasing in @xmath316 , ( b ) @xmath437 is decreasing in @xmath316 , and ( c ) @xmath437 is decreasing in @xmath46 . indeed , since @xmath112 , @xmath113 , @xmath414 , and @xmath438 we have @xmath439 @xmath440 @xmath441 also , note that eq.([eq : new kl ] ) has a unique equilibrium @xmath249 . therefore hypotheses ( 1)(4 ) of theorem [ th : kocic - ladas ] are satisfied , so every solution @xmath442 to eq.([eq : new kl ] ) converges to @xmath249 . by reversing the change of variables , one can conclude that every solution to eq.([eq : old kl ] ) converges to the equilibrium . * proof of theorem [ th : 3 - 2-l ] . * the four parts of the proof are : * _ @xmath33 is increasing in both @xmath36 and @xmath37 on @xmath379 ^ 2 $ ] : _ by lemma [ lemma : f(up , up ) and f(down , down ) ] the hypotheses of theorem [ th : mm first ] part ( i ) . is satisfied , hence every solution converges to the equilibrium @xmath229 . * _ @xmath33 is decreasing in both @xmath36 and @xmath37 on @xmath379 ^ 2 $ ] : _ by lemma [ lemma : f(up , up ) and f(down , down ) ] the hypotheses of theorem [ th : mm first ] part ( ii ) . is satisfied , hence every solution converges to the equilibrium @xmath229 . * _ @xmath33 is decreasing in @xmath36 and increasing in @xmath37 on @xmath379 ^ 2 $ ] : _ by the corollary to theorem [ th : camouzis - ladas ] we conclude every solution converges to the unique equilibrium or to a prime period - two solution . * _ @xmath33 is increasing in @xmath36 and decreasing in @xmath37 on @xmath379 ^ 2 $ ] : _ by lemmas [ lemma : signs of partials ] , [ lemma : 0<q < p ] , and [ lemma : f up down lemma bounds ] , there is no loss of generality in assuming @xmath379 \subset ( k,\frac{p}{q})$ ] , where @xmath443 , which we do . we consider two subcases . if @xmath111 , then lemma [ lemma : 0<q < p ] , lemma [ lemma : f up down iff ] and proposition [ prop : f up down r > 0 on i ] imply that every solution converges to the unique equilibrium . if @xmath113 , then lemma [ lemma : f up down ybar is ga ] implies that every solution converges to the unique equilibrium . this completes the proof of theorem [ th : 3 - 2-l ] . since theorem [ th : 3 - 2-l ] is just a version of theorem [ th : 3 - 3 ] obtained by an affine change of coordinates , we have also proved theorem [ th : 3 - 3 ] as well . @xmath169 the first lemma guarantees solutions to eq.(3 - 2 ) to be bounded . [ lemma : 3 - 2 is bounded ] let @xmath174 , @xmath175 and @xmath111 . there exist positive constants @xmath63 and @xmath64 such that every solution @xmath47 to eq.(3 - 2 ) satisfies @xmath444 $ ] for @xmath445 , and the function @xmath446 satisfies @xmath447\times [ \mathcal{l},\mathcal{u } ] ) \subset [ \mathcal{l},\mathcal{u}]\ ] ] set @xmath448 since @xmath449 then @xmath450 for @xmath451 , , i.e. , @xmath452 from the definition of @xmath64 we have @xmath453 write @xmath454 as @xmath455 , @xmath456 for @xmath457 . then for @xmath458 , @xmath459 that is , @xmath460 inspection of the proof of proposition [ prop : invariant and attracting ] reveals that , given that we have lemma [ lemma : 3 - 2 is bounded ] , the conclusion of the proposition is true concerning eq.(3 - 2 ) . the statement is given next . [ prop : invariant and attracting 3 - 2 ] at least one of the following statements is true : * every solution to eq.(3 - 2 ) converges to the equilibrium . * there exist @xmath57 , @xmath58 with @xmath461 s.t . * * @xmath60 $ ] is an invariant interval for eq.(3 - 2 ) , i.e. , @xmath61\times [ m^{*},m^ { * } ] ) \subset [ m^{*},m^{*}]$ ] . * * every solution to eq.(3 - 2 ) eventually enters @xmath60 $ ] . * * @xmath33 is coordinate - wise strictly monotonic on @xmath60^{2}$ ] . the proof of theorem [ th : 3 - 2-l ] may be reproduced here in its entirety with the only change being the elimination of the case @xmath113 , which presently does not apply . everything else in the proof applies to eq.(3 - 2 ) . the proof of theorem [ th : 3 - 2 ] is complete . | for nonnegative real numbers @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 such that @xmath6 and @xmath7 , the difference equation @xmath8 has a unique positive equilibrium . a proof is given here for the following statements : theorem 1 .
_ for every choice of positive parameters @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 , all solutions to the difference equation @xmath9 converge to the positive equilibrium or to a prime period - two solution .
_ theorem 2 .
_ for every choice of positive parameters @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 , all solutions to the difference equation @xmath10 converge to the positive equilibrium or to a prime period - two solution .
_ difference equation , rational , global behavior , global attractivity , period - two solution . |
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the continuous search for the physical mechanism which sets the cosmic acceleration of the universe and the corresponding conditions for stability has stimulated interesting and sometimes fascinating discussions on cosmological models @xcite . the dynamical mass attributed to neutrinos or dark matter @xcite can , for instance , regulate the time evolution of the dynamical dark energy providing the setup of the cosmic acceleration followed by the cosmological stability . in this context , the coupling of mass varying dark matter with neutrinos yields interesting relations between the present mass of neutrinos and the dark energy equation of state . in a previous issue @xcite , it was demonstrated that an effective generalized chaplygin gas ( gcg ) scenario @xcite can be reproduced in terms of a dynamical dark energy component @xmath0 with equation of state given by @xmath1 and a cold dark matter ( cdm ) component with a dynamical mass driven by the scalar field @xmath0 . dark matter is , most often , not considered in the mass varying neutrino ( mavan ) models . the treatment of dark energy and dark matter in the gcg unified scheme naturally offers this possibility . identifying sterile neutrinos as dark matter coupled with dark energy provides the conditions to implement such unified picture in the mavan formulation since active and sterile neutrino states are connected through the _ seesaw _ mechanism for mass generation . the constraints imposed by the _ seesaw _ mechanism allows one to establish an analytical connection to the gcg in terms of a real scalar field . the dynamics of the coupled fluid composed by neutrinos , dark matter and dark energy is driven by one single degree of freedom , the scalar field , @xmath2 . the simplest realization of mavan mechanisms @xcite consists in writing down an effective potential which , in addition to a scalar field dependent term , contains a term related to the neutrino energy density . it results in the so - called adiabatic condition , which sometimes leads to the stationary regime for the scalar field in respect with an effective potential @xcite . one indeed expect a tiny contribution from cosmological neutrinos to the energy density of the universe . mavan scenarios essentially predict massless neutrinos until recent times . when their mass eventually grows close to its present value , they form a non - relativistic ( nr ) fluid and the interaction with the scalar field stops its evolution . the relic particle mass is generated from the vacuum expectation value of the scalar field and becomes linked to its dynamics : @xmath3 . it is presumed that the neutrino mass has its origin on the vacuum expectation value ( vev ) of the scalar field and its behavior is governed by the dependence of the scalar field on the scale factor . in fact , it is well - known that the active neutrino masses are tiny as compared to the masses of the charged fermions . this can be understood through the symmetry of the standard model ( sm ) of electroweak interactions . it involves only left - handed neutrinos such that no renormalizable mass term for the neutrinos is compatible with the sm gauge symmetry @xmath4 . once one has assumed that baryon number and lepton number is conserved for renormalizable interactions , neutrino masses can only arise from an effective dimension five operator . it involves two powers of the vacuum expectation value of the higgs doublet . they are suppressed by the inverse power of a large mass scale @xmath5 of a sterile right - handed majorana neutrino , since it has hypercharge null in the sm . this super - massive majorana neutrino should be characteristic for lepton number violating effects within possible extensions beyond the sm . at our approach , the mass scale @xmath5 has its dynamical behavior driven by @xmath0 . in some other words , the sterile neutrino mass is characteristic of the aforementioned mass varying dark matter which indirectly results in mavan s , i. e. active neutrinos with mass computed through _ seesaw _ mechanism . in section ii , we report about the main properties of a unified treatment of dark matter and dark energy prescribed by the mass varying mechanism . one sees that model dependent choices of dynamical masses of dark matter allows for reproducing the conditions for the present cosmic acceleration in an effective gcg scenario . the stability conditions resulted from a positive squared speed of sound , @xmath6 , is recovered . in section iii , we discuss the neutrino mass generation mechanism in the context of the gcg model . following the simplest formulation of the _ seesaw _ mechanism , dirac neutrino masses with analytical dependencies on @xmath0 , @xmath7 and @xmath8 are considered . in section iv , we discuss the conditions for stability and the perturbative modifications on the accelerated expansion of the universe in the framework here proposed . one can state the conditions for reproducing the gcg scenario . we draw our conclusions in section v. to understand how the mass varying mechanism takes place for different particle species , it is convenient to describe the relevant physical variables as functionals of a statistical distribution @xmath9 . this counts the number of particles in a given region around a point of the phase space defined by the conjugate coordinates : momentum , @xmath10 , and position , @xmath11 . the statistical distribution @xmath9 can be defined in terms of the comoving momentum , @xmath12 , and for the case where @xmath9 is a fermi - dirac distribution , it can be written as @xmath13 } + 1\right\}^{\mi\1},\ ] ] where @xmath14 is the relic particle background temperature at present . in the flat frw scenario , the corresponding particle density , energy density and pressure can thus be depicted from the einstein s energy - momentum tensor @xcite as @xmath15 where @xmath16 is the scale factor ( cosmological radius ) for the flat frw universe , for which the metrics is given by @xmath17 . from the dependence of @xmath18 on @xmath16 one can obtain the energy - momentum conservation equation , @xmath19 that translates the dependence of @xmath5 on @xmath0 into a dynamical behavior , and where @xmath20 is the expansion rate of the universe and the _ overdot _ denotes differentiation with respect to time ( @xmath21 ) . simple mathematical manipulations allow one to easily demonstrate that @xmath22 from which , one can see that the strength of the coupling between relic particles and the scalar field is suppressed by the relativistic increase of pressure ( @xmath23 ) . as long as particles become ultra - relativistic ( @xmath24 ) the matter fluid and the scalar field fluid tend to decouple and evolve adiabatically separated . adding eq . ( [ gcg03 ] ) to the equation of motion for a cosmon field @xmath0 like @xmath25 rewritten as @xmath26 results in the equation of energy conservation for a unified fluid with a dark energy component and a mass varying dark matter component , @xmath27 for which we identify @xmath28 and @xmath29 . from this point , it is suggested that such a unified fluid corresponds to an effective description of a gcg universe . the gcg model is characterized by an exotic equation of state @xcite given by @xmath30 which can be obtained from a generalized born - infeld action @xcite . inserting the above equation of state into the unperturbed energy conservation eq . ( [ gcg06 ] ) and following a straightforward integration @xcite , one obtains @xmath31^{\1/(\1 \pl \al ) } , \label{gcg21}\ ] ] and @xmath32^{-\al/(\1 \pl \al)}. \label{gcg22}\ ] ] one of the most striking features of the gcg fluid is that its energy density interpolates between a dust dominated phase , @xmath33 , in the past , and a de - sitter phase , @xmath34 , at late times . this property makes the gcg model an interesting candidate for the unification of dark matter and dark energy . notice that for @xmath35 , gcg behaves always as matter whereas for @xmath36 , it behaves always as a cosmological constant . hence to use it as a unified candidate for dark matter and dark energy one has to exclude these two possibilities so that @xmath37 must lie in the range @xmath38 . furthermore , this evolution is controlled by the model parameter @xmath39 . assuming the canonical parametrization of @xmath18 and @xmath40 in terms of a scalar field @xmath0 , @xmath41 and following ref . @xcite , one can obtain through eq . ( [ gcg21]-[gcg22 ] ) the effective dependence of @xmath0 on @xmath16 implicitly given by @xmath42^{-\al/(\al \pl \1 ) } , \label{pap02}\ ] ] and explicit expressions for @xmath18 , @xmath40 and @xmath43 in terms of @xmath0 . assuming a flat evolving universe described by the friedmann equation @xmath44 ( with @xmath45 in units of @xmath46 and @xmath18 in units of @xmath47 , one obtains @xmath48 } , \label{pap03}\ ] ] where it is assumed that @xmath49}. \label{pap04}\ ] ] one then readily finds the scalar field potential , @xmath50^{\frac{\2}{\al \pl \1 } } + \left[\cosh{\left(3\bb{\alpha + 1 } \phi/2\right)}\right]^{-\frac{\2\al}{\al \pl \1 } } \right\}. \label{pap05}\ ] ] if one supposes that the energy density , @xmath18 , may be decomposed into a mass varying cdm component , @xmath51 , and a dark energy component , @xmath52 , connected by the scalar field equations ( [ gcg04])-([gcg05 ] ) , the equation of state ( [ gcg20 ] ) is just assumed as an effective description of the cosmological background fluid of the universe . since the cdm pressure , @xmath53 , is null , the dark energy component of pressure , @xmath54 , results in the gcg pressure , @xmath55 . assuming that dark energy behaves like a cosmological constant , that is , its equation of state is given by @xmath56 , the dark energy density can be parameterized by a generic quintessence potential , @xmath57 , since its kinetic component has to be null for a canonical formulation . it results in @xmath58 , where @xmath40 is the gcg pressure given by eq . ( [ gcg22 ] ) . by substituting the result of eq.([pap03 ] ) into the eq.([gcg22 ] ) , and observing that @xmath44 , with @xmath18 given by eq.([gcg21 ] ) , it is possible to rewrite the gcg pressure , @xmath40 , in terms of @xmath0 . it results in the following analytical expression for @xmath59 , @xmath60^{-\frac{\2 \al}{1 \pl \al } } , \label{pap08}\ ] ] which is consistent with the result for @xmath61 from eq . ( [ pap05 ] ) . since @xmath62 , the eq . ( [ gcg05 ] ) is thus reduced to @xmath63 and the problem is then reduced to finding a relation between the scalar potential @xmath59 and the variable mass @xmath64 . from the above equation , the effective potential governing the evolution of the scalar field is naturally decomposed into a sum of two terms , one arising from the original quintessence potential @xmath59 , and other from the dynamical mass @xmath64 . for appropriate choices of potentials and coupling functions satisfying eq . ( [ pap09 ] ) , the competition between these terms leads to a minimum of the effective potential . in the adiabatic regime , the matter and the scalar field are tightly coupled together and evolve as one effective fluid . at our approach , once one assumes a @xmath65-_like _ equation of state @xmath66 , without any additional constraint on cosmon field equations , eq . ( [ pap09 ] ) is naturally obtained . in the gcg cosmological scenario , the effective fluid description is valid for the background cosmology and for linear perturbations . the equation of state of perturbations is the same as that of the background cosmology where all the effective results of the gcg paradigm are maintained . ( [ pap08 ] ) leads to @xmath67 which , in the cdm limit , gives @xmath68 since the dependence of @xmath5 on @xmath16 is exclusively intermediated by @xmath2 , i. e. @xmath69 , from eqs . ( [ gcg21 ] ) , ( [ gcg22 ] ) and ( [ pap03 ] ) , after simple mathematical manipulations , one obtains @xmath70^{\frac { \2 \al}{1 \pl \al } } \label{pap11}\ ] ] which is consistent with eq . ( [ pap09 ] ) . from the above result , one can infers that the adequacy to the adiabatic regime is left to the mass varying mechanism which drives the cosmological evolution of the dark matter component . for mass varying cdm coupled with @xmath65-_like _ dark energy , with @xmath66 , the gcc leads to similar predictions for the equation of state , @xmath71 , independently of the scale parameter @xmath16 . the same is not true for hot dark matter ( hdm ) @xcite . unlike photons and baryons , cosmological neutrinos have not been observed , so arguments about their contribution to the total energy density of the universe are necessarily theoretical . otherwise , neutrinos coupled to dark energy can lead to a number of significant phenomenological consequences . the neutrino mass should be a dynamical quantity and neutrinos remain essentially massless until recent times . when their mass eventually increases close to its present value , their interaction with the background scalar field almost ceases @xcite . the energy of the scalar field becomes the dominant contribution to the energy density of the universe and cosmic acceleration ensues . without loss of generality , such a behavior can be easily implemented to a degenerate fermion gas ( dfg ) of neutrinos . in the limit where @xmath14 tends to @xmath72 in eq . ( [ gcg01 ] ) , the fermi distribution @xmath9 becomes a step function that yields an elementary integral for the above equations , with the upper limit equal to the fermi momentum here written as @xmath73 . it results in the equations for a dfg @xcite , which can be useful for parameterizing the transition between ultra - relativistic ( ur ) and non - relativistic ( nr ) thermodynamic regimes . the equation of state can be expressed in terms of elementary functions of @xmath74 and @xmath75 , @xmath76\\ p_m\bb{a } & = & \frac{1}{8 \pi^{\2 } } \left[\beta ( \frac{2}{3 } \beta^{\2 } - m^{\2})\sqrt{\beta^{\2 } + m^{\2 } } + \mbox{arc}\sinh{\left(\beta / m\right)}\right]\nonumber \label{gcg01d}\end{aligned}\ ] ] the relation given by eq . ( [ gcg02 ] ) can be verified for the above definition . besides the mass dependence , it is necessary to determine for which values of the scale factor the neutrino - scalar field coupling becomes important . the dfg approach take care of this . let us then turn back to the explanation for the smallness of the neutrino masses . according to the _ seesaw _ mechanism , the tiny masses , @xmath77 , of the usual left - handed neutrinos are obtained through a very massive , @xmath5 , sterile right - handed neutrino . the lagrangian density that describes the simplest version of the _ seesaw _ mechanism via yukawa coupling between a light scalar field and a single neutrino flavor is given by @xmath78 where it is shown that at scales well below the right - handed neutrino mass , one has the effective lagrangian density @xcite @xmath79 phenomenological consistency with the sm implies that logarithm corrections to the above terms are small , while it is well - know from the results of solar , atmospheric , reactor and accelerator neutrino oscillation experiments that neutrino masses given by @xmath80 lie in the sub-@xmath81 range . it is also clear that promoting the scalar field @xmath0 into a dynamical quantity leads to a mechanism in the context of which neutrino masses are time - dependent . associating the scalar field to the dark energy field allows linking nr neutrino energy densities to late cosmological times @xcite . this scenario can be implemented through the perturbative approach via eq . ( [ gcg06 ] ) @xcite . it is evident that this approach is fairly general , as well as independent of the choice of the equation of state and of the dependence of the neutrino mass on the scalar field . the form of @xmath64 and of the equation of state can indeed lead to quite different scenarios . in particular , we consider three cases of neutrino mass dependence on @xmath0 . in case 01 we set @xmath82 , in case 02 we have set @xmath83 , and in case 03 we have set @xmath84 . energy densities , @xmath18 , as function of the scale factor for the gcg fluid and for the decomposed components of the unified effective gcg background fluid , namely , the mass varying dark matter , @xmath85 , the cosmon-_like _ dark energy , @xmath52 , and the perturbative dfg of neutrinos , @xmath86 , are obtained in the fig . [ gcg-0 ] . the neutrino densities are computed for neutrinos masses correlated to the dark matter mass , @xmath64 , in correspondence with the gcg scenario with @xmath87 , that represents a realistic fraction of dark energy , and with @xmath88 and @xmath89 . in the fig . [ gcg - b ] we have taken into account the energy densities for photons ( @xmath90 ) and baryons ( @xmath91 ) in complement to our effective gcg scenario . then we have computed the relative modifications on the spectrum of the cosmic evolution of the dark sector components . conveniently in agreement with the phenomenological predictions , we have set the present values of the energy densities as : @xmath92 , @xmath93 , @xmath94 , @xmath95 , and @xmath96 . assuming that neutrinos are nr at present , and observing the behavior of eqs . ( [ gcg01d ] ) , we have set @xmath97 . this condition allows one to establish the correspondence between the values of @xmath16 for which the coupling transition takes place , i. e. the transition between relativistic and nr regimes , so that the neutrino masses achieve the present - day values . for our model such a transition is governed by the dfg behavior of eq . ( [ gcg01d ] ) . these scenarios are by no means the only possibilities . in particular , we choose them as they correspond to the simplest feasible possibility for active neutrino mass generation , given that the scalar field has mass dimension one @xcite . the essential information of the mass dependence on the scale factor is that , for case 03 , the neutrino mass increases with decreasing @xmath0 , in opposition to what happens for cases 01 and 02 . through the stability analysis performed in the next section we shall elucidate some important consequences due to such a distinct behavior . in the fig . [ gcg - c ] we verify how the energy density @xmath18 and the corresponding equations of state @xmath71 for the composed fluid deviate from the effective gcg scenario . for mass varying cdm coupled with dark energy with @xmath66 , the effective gcc leads to similar predictions for @xmath98 , independently of the scale factor @xmath16 . the same is not true for hdm which , in the dfg approach , while weakly coupling with dark energy , leads to the same behavior of the gcg just at late times ( @xmath99 ) . the possibility of adiabatic instabilities in cosmological scenarios was previously pointed out @xcite in a context of a mass varying neutrino model of dark energy . in opposition , in the usual treatment where dark matter are just coupled to dark energy , cosmic expansion together with the gravitational drag due to cold dark matter have a major impact on the stability of the cosmological background fluid . usually , for a general fluid for which we know the equation of state , the dominant effect on the sound speed squared @xmath6 arises from the dark sector component and not from the neutrino component . for the models where the adiabatic regime ( cf . ( [ pap09 ] ) ) implies an equation of state reproducing cosmological constant , @xmath65 , effects , @xmath100 , one obtains @xmath101 from the very start of the analysis . the effective gcg is free from this inconsistency . a detailed quantitative analysis of the stability conditions for the gcg in terms of the squared speed of sound is discussed in ref . positive @xmath6 implies that @xmath102 in the gcg equation of state . the coupling of the dark energy component with dynamical dark matter is responsible for removing such inconsistency by setting @xmath103 . in the fig . [ gcg - d ] we show the results for @xmath104 for a cosmological background scenario corresponding to the sum of the energy components of an effective gcg fluid : mass varying dark matter ( dm ) , cosmon-_like _ dark energy ( de ) and perturbative dfg of neutrinos ( @xmath105 ) . we have considered the gcg phenomenological given by @xmath87 and @xmath106 ( solid line ) , @xmath107 ( dot line ) , @xmath108 ( dash dot line ) , and @xmath89 ( dash line ) . the perturbative influence of neutrinos on the positiveness of @xmath104 has its magnitude measured by comparing the results with those for the gcg scenario with and and without neutrino perturbation for each one of the three cases in correspondence with fig . [ gcg-0 ] . in this scenario , the most important result illustrated by fig . [ gcg - d ] is that growing neutrino masses parameterized by @xmath109 ( case 03 ) results in instabilities at present times . this instability is characterized by a negative squared speed of sound for an effective coupled neutrino / dark energy fluid , and results in the exponential growth of small scale modes @xcite . a natural interpretation for this is that the universe becomes inhomogeneous with the neutrinos forming denser structures or lumps @xcite . effectively , the scalar field could mediate an attractive force between neutrinos leading to the formation of neutrino nuggets . anyway , these results are consistent with the accelerated expansion of the universe ruled by the dynamical masses of cdm and neutrinos since positive values for @xmath110 are observed , as we can notice in the fig . [ gcg - e ] . growing neutrinos coupled to mass varying dark matter plus cosmon-_like _ dark energy were studied assuming that the cosmological background unified fluid presents an effective behavior similar to that of the gcg . the essential ingredient of this class of models is a neutrino mass that , through the _ seesaw _ mechanism , depends on the cosmon field and grows in the course of the cosmological evolution . our setup is an effective gcg decomposed into two interacting components @xcite . the first one behaves like mass varying cdm since it is pressure - less . the second one corresponds to a cosmon-_like _ dark energy component @xmath0 with equation of state given by @xmath1 . apparently the model does not look different from the interacting quintessence models where one has two different interacting fluids . the present fraction of dark energy is set by a dynamical mechanism . as soon as dark matter become nr , their coupling to the cosmon triggers an effective stop ( or substantial slowing ) of the evolution of the cosmon . before this event , the quintessence field follows the tracking of the minimum of an effective potential , in a kind of stationary condition which drives the adiabatic regime , and for which the coupling strength is strong compared to gravitational strength . in the scope of finding a natural explanation for the cosmic acceleration and the corresponding adequation to stability conditions , we have proposed a systematic procedure to treat variations of fundamental parameters of neutrino cosmology ( @xmath0 , @xmath77 and @xmath5 ) independent of any particular theoretical model which enforces restrictive relations among these parameters . we have quantified the perturbative influence of such variations on the calculation of the squared speed of sound and on the cosmic acceleration of the universe , noticing that the former quantity is much more sensible to variations with respect to the exact gcg background cosmology . an interesting feature is however that the gcg has its cosmological evolution reproduced by a cdm mass varying mechanism consistent with the mass generation mechanism for neutrinos . most of dark sector models predict an entangled mixture of interacting dark matter and dark energy . we have consistently introduced the possibility of such a mixed coupling with active and sterile neutrinos , for which we have tested three hypothesis for _ seesaw _ masses . at our approach , the mass scale @xmath5 and scalar field @xmath0 are tightly coupled together and evolve as a unique effective fluid . the effective potential driving the evolution of the scalar field is decomposed into a sum of two terms , one arising from the original quintessence potential @xmath59 , and the other from the dynamical mass of the dark matter . the sterile neutrino mass produces the mass varying dark matter effects which indirectly result in mavan s , and the active neutrino has its mass perturbatively computed via _ seesaw _ relations . since the results for the cosmic acceleration followed by cosmological stability for gcg scenarios are well - defined , one can assert that the increase of neutrino mass acts as a cosmological clock or trigger for the crossover to the effective scenario here studied . in fact , a scalar field associated to dark energy in connection with the sm neutrinos and the electroweak interactions may bring important insights on the physics beyond the sm @xcite . the case of cosmological neutrinos , in particular , is a fascinating example where salient questions concerning sm particle phenomenology can be addressed and hopefully better understood . 99 i. zlatev , l. m. wang and p. j. steinhardt , phys . lett . * 82 * , 896 ( 1999 ) . l. m. wang , r. r. caldwell , j. p. ostriker and p. j. steinhardt , astrophys . j. * 530 * , 17 ( 2000 ) . p. j. steinhardt , l. m. wang and i. zlatev , phys . rev . * d59 * , 123504 ( 1999 ) . t. barreiro , e. j. copeland and n. j. nunes , phys . rev . * d61 * , 127301 ( 2000 ) . o. bertolami and p. j. martins , phys . rev . * d61 * , 064007 ( 2000 ) . r. bean , e. e. flanagan and m. trodden , arxiv:0709.1128 [ astro - ph ] . kari enqvist , steen hannestad , and martin s. sloth , phys . lett . * 99 * , 031301 ( 2007 ) ; 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r. n. mohapatra , _ unification and supersymmetry _ ( springer - verlag , berlin , 1986 ) . n. arkani - hamed , s. dimopoulos , g. r. dvali , and j. march - russell , phys . rev . * d65 * 024032 ( 2002 ) . n. afshordi , m. zaldarriaga and k. kohri , phys . * d72 * , 065024 ( 2005 ) . n. brouzakis and n. tetradis , jcap * 0601 * , 004 ( 2006 ) . a. e. bernardini and o. bertolami , phys . lett . * b684 * , 96 ( 2010 ) . b. patt and f. wilczek , arxiv : hep - ph/0605188 ; o. bertolami and r. rosenfeld , int . a23 * , 4817 ( 2008 ) . m. c. bento , a. e. bernardini and o. bertolami , j. phys . : conf . ser . * 174 * , 012060 ( 2009 ) . a. e. bernardini , phys . lett . * b684 * , 162 ( 2010 ) ; a. e. bernardini and o. bertolami , phys . * d81 * , 123013 ( 2010 ) . | neutrinos coupled to an underlying scalar field in the scenario for unification of mass varying dark matter and cosmon-_like _ dark energy is examined . in the presence of a tiny component of mass varying neutrinos ,
the conditions for the present cosmic acceleration and for the stability issue are reproduced .
it is assumed that _
sterile _ neutrinos behave like mass varying dark matter coupled to mass varying _ active _ neutrinos through the _ seesaw _ mechanism , in a kind of _ mixed _ dark matter sector .
the crucial point is that the dark matter mass may also exhibit a dynamical behavior driven by the scalar field .
the scalar field mediates the nontrivial coupling between the mixed dark matter and the dark energy responsible for the accelerated expansion of the universe .
the equation of state of perturbations reproduce the generalized chaplygin gas ( gcg ) cosmology so that all the effective results from the gcg paradigm are maintained , being perturbatively modified by neutrinos . |
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the role played by disorder on the nature of collective excitations in condensed matter physics has been the subject of intensive studies due to its relevance in defining general transport characteristics @xcite . usually , disorder induces localization of collective excitations thus degrading transport properties , an effect that is largely pronounced in low dimensions . in particular , the one - electron eigen - states in the one - dimensional anderson model with site - diagonal uncorrelated disorder are exponentially localized for any degree of disorder @xcite . however , several one - dimensional models with correlated disorder have been proposed which exhibit delocalized states @xcite . recently , it has been shown that the one - dimensional anderson model with long - range correlated disorder presents a phase of extended electronic states @xcite . these results have been confirmed by microwave transmission spectra of single - mode waveguides with inserted correlated scatters @xcite . the above results have motivated the study of further model systems that can be mapped onto the anderson model and , therefore , expected to present a similar transition between localized and extended collective excitations . recently , a study concerning the one - dimensional quantum heisenberg ferromagnet with exchange couplings exhibiting long - range correlated disorder reported some finite - size scaling evidences of the emergence of a phase of extended low - energy excitations @xcite . by using a renormalization group calculation the existence of such phase of extended spin - waves was confirmed and the scaling of the mobility edge with the degree of correlation was obtained @xcite . it was also shown that , associated with the emergence of extended spin - waves in the low - energy region , the wave - packet mean - square displacement exhibits a long - time ballistic behavior . the collective vibrational motion of one - dimensional disordered harmonic chains of @xmath5 random masses can also be mapped onto an one - electron tight - binding model @xcite . in such a case , most of the normal vibrational modes are localized . however , there are a few low - frequency modes not localized , whose number is of the order of @xmath6 , in which case the disordered chains behaves like the disorder - free system @xcite . futher , it was shown that correlations in the mass distribution produce a new set of non - scattered modes in this system @xcite . non - scattered modes have also been found in disordered harmonic chain with dimeric correlations in the spring constants @xcite . by using analytical arguments , it was also demonstrated that the transport of energy in mass - disordered ( uncorrelated and correlated ) harmonic chains is strongly dependent on non - scattered vibrational modes as well as on the initial excitation @xcite . for impulse initial excitations , uncorrelated random chains have a superdiffusive behavior for the second moment of the energy distribution [ @xmath7 , while for initial displacement excitations a subdiffusive spread takes place [ @xmath8 . the dependence of the second moment spread on the initial excitation was also obtained in ref . moreover , correlations induced by thermal annealing have been shown to enhance the localization length of vibrational modes , although they still present an exponential decay for distances larger than the thermal correlation length @xcite . recently the thermal conductivity on harmonic and anharmonic chains of uncorrelated random masses @xcite , as well as of the chain of hard - point particles of alternate masses @xcite , has been numerically investigated in detail . the main issue here is whether the systems display finite thermal conductivity in the thermodynamic limit , a question that remains controversial @xcite . in this paper we extend the study of collective modes in the presence of long - range correlated disorder for the case of vibrational modes . we will consider harmonic chains with long - range correlated random masses assumed to have spectral power density @xmath9 . by using a transfer matrix calculation , we obtain accurate estimates for the lyapunov exponent , defined as the inverse of the degree of localization @xmath10 of the vibrational modes . we show that , for @xmath11 , this model also presents a phase of extended modes in the low frequency region . this result is confirmed by participation ratio measurements from an exact diagonalization procedure and finite size scaling arguments . the spatial evolution of an initially localized excitation is also studied by computing the spread of the second moment of the energy distribution , @xmath3 . we find that , associated with the emergence of a phase of delocalized modes , a ballistic energy spread takes place . we consider a disordered harmonic chain of @xmath5 masses , for which the equation of motion for the displacement @xmath12 of the _ n_-th mass with vibrational frequency @xmath13 is @xcite @xmath14 here @xmath15 is the mass at the _ n_-th site and @xmath16 is the spring constant that couples the masses @xmath15 and @xmath17 . we use units in which @xmath18 . in the present harmonic chain model , we take the masses @xmath15 following a random sequence describing the trace of a fractional brownian motion @xcite : @xmath19^{1/2 } \cos{\left ( \frac{2\pi nk}{n } + \phi_k\right)},\ ] ] where @xmath1 is the wave - vector of the modulations on the random mass landscape and @xmath20 are @xmath21 random phases uniformly distributed in the interval @xmath22 $ ] . the exponent @xmath23 is directly related to the hurst exponent @xmath24 ( @xmath25 ) of the rescaled range analysis . in order to avoid vanishing masses we shift and normalize all masses generated by eq . ( 2 ) such to have average value @xmath26 and variance independent of the chain size ( @xmath27 ) . using the matrix formalism , eq . ( 1 ) can be rewritten as obtained using dean s method . the chain length is @xmath28 for all cases . the dos becomes less rough as @xmath23 is increased . for @xmath29 it displays a non - fluctuating part near the bottom of the band.,scaledwidth=45.0% ] @xmath30 for a specific frequency @xmath13 , a @xmath31 transfer matrix @xmath32 connects the displacements at the sites @xmath33 and @xmath34 to those at the site @xmath35 : @xmath36 once the initial values for @xmath37 and @xmath38 are known , the value of @xmath12 can be obtained by repeated iterations along the chain , as described by the product of transfer matrices @xmath39 as a function of @xmath40 for @xmath41 ( uncorrelated random chain ) and @xmath42 sites . the lyapunov coefficient is finite for non - zero frequencies ( localized states ) and vanishes as @xmath43 , @xmath44 ( inset).,scaledwidth=45.0% ] the localization length of each vibrational mode is taken as the inverse of the lyapunov exponent @xmath45 defined by @xcite @xmath46 where @xmath47 is a generic initial condition . typically , @xmath48 matrix products were used to calculate the lyapunov exponents . the nature of the vibrational modes can also be investigated by computing the participation ratio @xmath49 , since it displays a dependence on the chain size for extended states and is finite for exponentially localized ones . @xmath49 is defined by @xcite @xmath50 where the displacements @xmath12 are those associated with an eigenmodes @xmath13 of a chain of @xmath5 masses and are obtained by direct diagonalization of the @xmath51 secular matrix @xmath52 defined by @xmath53 , @xmath54 , and all other @xmath55 @xcite . the participation ratio calculations were averaged over @xmath56 samples . we compute the lyapunov exponents and the participation ratio for several values of the correlation exponent @xmath23 , and obtain the density of states ( dos ) using the numerical dean s method @xcite . strong fluctuations in the dos are related to the presence of localized states , whereas smooth a dos is usually connected with the emergence of delocalized states @xcite . in fig . 1 we show the normalized dos for chains with @xmath28 sites , and notice that it becomes less rough as @xmath23 is increased . in fig . 2 we display the plot of @xmath45 versus @xmath57 for @xmath58 ( uncorrelated random masses ) . the lyapunov coefficient is finite for all frequencies and vanishes at @xmath59 as @xmath60 , in agreement with ref . @xcite ( see inset of fig . 2 ) . the scaled participation ratio @xmath61 as a function of @xmath40 is shown in fig . 3 : @xmath62 remains finite in the thermodynamic limit , whereas for any finite frequency the vibrational modes are localized with @xmath63 as @xmath64 , in agreement with the above results obtained from the lyapunov coefficient calculations . to investigate the effect of weak long - range correlated disorder , we present in fig . 4(a ) the lyapunov coefficient as a function of @xmath40 for @xmath65 and @xmath42 . in spite of @xmath45 being very small in the bottom of the band , the scaled participation ratio for @xmath66 vanishes in the thermodynamic limit [ see fig . therefore , all modes with @xmath66 are still localized , a feature that holds for any @xmath67 . however , the nature of the low - frequency modes changes qualitatively for @xmath2 . in fig . 5(a ) we show @xmath45 versus @xmath57 for @xmath68 and @xmath42 sites . the lyapunov coefficient vanishes within a finite range of frequency values , thus revealing the presence of extended vibrational modes . the scaled participation ratio @xmath61 [ see fig . 5(b ) ] displays a well defined data collapse , confirming that the phase of extended low - frequency vibrational modes is stable in the thermodynamic limit . as a function of @xmath40 for @xmath41 ( uncorrelated disorder ) . from top to bottom , @xmath69 . for vibrational modes with @xmath70 , @xmath63 as @xmath5 diverges , thus confirming their localized nature.,scaledwidth=45.0% ] versus @xmath40 for @xmath71 and @xmath72 sites . ( b ) scaled participation ratio @xmath61 as a function of @xmath40 for @xmath71 . from top to bottom , @xmath73 . in spite of @xmath45 being very small in the bottom of the band , all modes with @xmath66 are localized.,scaledwidth=45.0% ] versus @xmath40 for @xmath74 and @xmath72 sites . the lyapunov coefficient vanishes within a finite range of frequency values , thus revealing the presence of extended vibrational modes . ( b ) scaled participation ratio @xmath61 as a function of @xmath40 for @xmath74 . from top to bottom , @xmath73 . the phase of extended vibrational modes is confirmed by the size independent plateau in the low - frequency region.,scaledwidth=45.0% ] in order to study the time evolution of a localized energy pulse , we calculate the second moment of the energy distribution @xcite . this quantity is related to the thermal conductivity by kubo s formula @xcite . the classical hamiltonian @xmath24 for an harmonic chain can be written as @xmath75 where the energy @xmath76 at the site @xmath34 is given by @xmath77~.\ ] ] here @xmath78 and @xmath79 define the momentum and displacement of the mass at the _ n_-th site . the hamilton s equations are @xmath80\ ] ] and @xmath81 the fraction of the total energy @xmath24 at the site @xmath34 is given by @xmath82 and the second moment of the energy distribution , @xmath3 , is defined by @xcite @xmath83,\ ] ] versus time @xmath84 for @xmath85 ( dotted line ) and @xmath86 ( dashed line ) with initial impulse excitation . for @xmath67 only superdiffusive behavior is found for long times.,scaledwidth=45.0% ] where an initial excitation is introduced at the site @xmath87 at @xmath88 . using the fourth - order runge - kutta method , we solve the differential equations for @xmath89 and @xmath90 and calculate @xmath3 . the second moment of the energy distribution @xmath3 has the same status of the mean - square displacement of the wavepacket of an electron in a crystal @xcite . in harmonic chains with an initial impulse excitation , the energy spread is faster than that in chains with an initial displacement excitation @xcite . we calculate @xmath3 for several @xmath23 values and two kinds of initial excitation : impulse excitation and displacement excitation . in fig . 6 we present the scaled second moment @xmath91 versus time @xmath84 for @xmath85 ( dotted line ) , which corresponds to the uncorrelated random chain , and @xmath86 ( dashed line ) . these results have been obtained after an initial impulse excitation , @xmath92 . in our calculations for @xmath85 , the self - expanded chain method with initial chain size @xmath93 was used to minimize end effects . throghout the numerical integration process we kept the fraction of the total energy @xmath24 at the ends of the chain [ @xmath94 and @xmath95 smaller than @xmath96 for all times . as shown in fig . 6 , we find a long - time superdiffusive behavior for @xmath41 , in agreement with previous analytical and numerical results for energy transport in harmonic chains with uncorrelated random masses under an impulse initial excitation @xcite . in contrast , for @xmath97 we can not use the self - expanded chain method due to the long - range character of the mass correlations . therefore , chains with @xmath98 masses were considered , and the runs stopped whenever the fraction of the total energy at the chain ends achieved @xmath96 . for @xmath71 the time - dependence of the scaled energy second moment @xmath91 typically represents a weak long - range correlated case . in such a case we also find superdiffusive behavior for long times . on the other hand , in the strong correlated regime , @xmath2 , a breakdown in the superdiffusive behavior 7 shows the time - dependence of the scaled energy second moment , @xmath99 , for @xmath29 ( dotted line ) and @xmath100 ( dashed line ) . associated with the emergence of extended vibrational modes in the low - energy region , the second moment @xmath3 displays a long - time ballistic behavior . versus time @xmath84 for @xmath29 ( dotted line ) and @xmath100 ( dashed line ) with initial impulse excitation . results were obtained by numerical integration in chains with @xmath98 sites . a ballistic behavior is found after an initial transient.,scaledwidth=45.0% ] the long time behavior of the second moment @xmath3 in uncorrelated random chains with initial displacement excitation is significantly different from the corresponding behavior with impulse initial excitation @xcite . analytical calculations predict that @xmath101 , a result that has been corroborated by numerical techniques @xcite . for @xmath85 we indeed reproduce this behavior , as shown in fig . 8 for the scaled second moment @xmath102 versus time @xmath84 with initial displacement excitation @xmath103 . again , we find that this asymptotic subdiffusive behavior remains true for @xmath104 ( dashed line in fig . for strong correlations ( @xmath2 ) , which induce the emergence of new extended vibrational modes in the low - energy region , the energy transport is faster than in the subdiffusive regime and again assumes a ballistic nature , as shown in fig . 9 . versus time @xmath84 for @xmath85(dotted line ) and @xmath86 ( dashed line ) with initially given displacement excitation . the subdiffusive behavior is found for long times.,scaledwidth=45.0% ] in this paper we have studied the nature of collective vibrational modes in harmonic chains with long - range correlated random masses @xmath15 , with spectral power density @xmath9 . by using a transfer matrix method and exact diagonalization , we have computed the localization length and the participation ratio of all normal modes . our results indicate that in the strong correlations regime , @xmath4 , there is a phase of extended low - energy vibrational modes . in this sense , long - range correlations in the mass distribution induce the emergence of a delocalization transition in harmonic chains similar to the one observed to occur with one - magnon excitations in ferromagnetic chains with random couplings @xcite and with one - electron eigen - states in the random hopping anderson model @xcite . we have also studied the energy transport in this harmonic chain model . the spread of the energy second moment @xmath3 is shown to be strongly dependent on the existence of non - scattered vibrational modes and initial excitation . we have also found that , associated with the emergence of a phase of low - energy extended collective excitations , @xmath3 displays a crossover from an anomalous sub- or super - diffusive regime ( depending on the initial impulse or displacement excitation , respectively ) to an asymptotic ballistic behavior . the above findings indicate that the thermal conductivity can be strongly influenced by the presence of long - range correlations in the random distribution of masses and we hope that the present work will stimulate further studies along this direction . this work was partially supported by cnpq , capes and finep ( brazilian agencies ) . mll also acknowledges the partial support of fapeal ( alagoas state agency ) . versus time @xmath84 for @xmath29 ( dotted line ) and @xmath100 ( dashed line ) with initially given displacement excitation . results obtained by numerical integration of diferential equation for chain with @xmath98 sites . a ballistic behavior is found for all times.,scaledwidth=45.0% ] 40 t. a. l. ziman , phys . lett . * 49 * , 337 ( 1982 ) . for a review see , e.g. , b. kramer and a. mackinnon , rep . . phys . * 56 * 1469 , ( 1993 ) . e. abrahams , p. w. anderson , d. c. licciardello , and t. v. ramakrishnan , phys . * 42 * , 673 ( 1979 ) . for a review see , e.g. , i. m. lifshitz , s. a. gredeskul and l. a. pastur , _ introduction to the theory of disordered systems _ ( wiley , new york , 1988 ) . j. c. flores , j. phys . : condens . matter * 1 * , 8471 ( 1989 ) . d. h. dunlap , h. l. wu , and p. w. phillips , phys . 65 * , 88 ( 1990 ) ; h .- l . wu and p. phillips , phys . lett . * 66 * , 1366 ( 1991 ) ; p. w. phillips and wu , science * 252 * , 1805 ( 1991 ) ; s. n. evangelou and d. e. katsanos , phys . a * 164 * , 456 ( 1992 ) . see also , s. n. evangelou , a. z. wang , and s. j. xiong , j. phys . : matter * 6 * , 4937 ( 1994 ) . f. a. b. f. de moura and m. l. lyra , phys . lett . * 81 * , 3735 ( 1998 ) . f. a. b. f. de moura and m. l. lyra , physica a * 266 * , 465 ( 1999 ) . f. m. izrailev and a. a. krokhin , phys . lett . * 82 * , 4062 ( 1999 ) ; f. m. izrailev , a. a. krokhin , and s. e. ulloa , phys . rev . b * 63 * , 41102 ( 2001 ) . u. kuhl , f. m. izrailev , a. krokhin , and h. j. stckmann , appl . 77 * , 633 ( 2000 ) . r. p. a. lima , m. l. lyra , e. m. nascimento , and a. d. de jesus , phys . b * 65 * , 104416 ( 2002 ) . f. a. b. f. de moura , m. d. coutinho - 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1 ( 2002 ) ; p. l. garrido and p. i. hurtado , phys . rev . lett . * 88 * , 249402 - 1 ( 2002 ) . j. feder , _ fractals _ ( plenum press , new york , 1988 ) ; a. tsonis , _ chaos : from theory to applications _ ( plenum press , new york , 1992 ) . a. r. osborne and a. provenzale , physica d * 35 * , 357 ( 1989 ) . n. p. greis and h. s. greenside , phys . a * 44 * , 2324 ( 1991 ) . s. n. evangelou and e. n. economou , j. phys . * 26 * , 2803 ( 1993 ) . p. dean , rev . * 44 * , 127 ( 1972 ) . p. b. allen and j. l. feldman , phys . b * 48 * , 12581 ( 1993 ) . | we study the nature of collective excitations in harmonic chains with masses exhibiting long - range correlated disorder with power spectrum proportional to @xmath0 , where @xmath1 is the wave - vector of the modulations on the random masses landscape . using a transfer matrix method and exact diagonalization ,
we compute the localization length and participation ratio of eigenmodes within the band of allowed energies .
we find extended vibrational modes in the low - energy region for @xmath2 . in order to study the time evolution of an initially localized energy input ,
we calculate the second moment @xmath3 of the energy spatial distribution .
we show that @xmath3 , besides being dependent of the specific initial excitation and exhibiting an anomalous diffusion for weakly correlated disorder , assumes a ballistic spread in the regime @xmath4 due to the presence of extended vibrational modes . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
we are interested in answering two very basic questions about continuous - time , discrete - symbol stochastic processes : * what are their minimal maximally predictive models their ? * what are information - theoretic characterizations of their randomness , predictability , and complexity ? for shorthand , we refer to the former as _ causal architecture _ and the latter as _ informational architecture_. minimal maximally predictive models of discrete - time , discrete - state , discrete - output processes are relatively well understood ; e.g. , see refs . some progress has been made on understanding minimal maximally predictive models of discrete - time , continuous - output processes ; e.g. , see refs . @xcite . relatively less is understood about minimal maximally predictive models of continuous - time , discrete - output processes , beyond those with exponentially decaying state - dwell times @xcite . the following is a first attempt at a remedy that complements the spectral methods developed in ref . @xcite , as we address the less tractable case of uncountably infinite causal states . we start by analyzing continuous - time renewal processes , as addressing the challenges there carries over to other continuous - time processes . ( elsewhere , we outline the wide interest and applicability of renewal processes in physics and the quantitative sciences generally @xcite . ) the difficulties are both technical and conceptual . first , the causal states are now continuous or hybrid discrete - continuous random variables , unless the renewal process is poisson . second , transitions between causal states are now described by partial differential equations . finally , and perhaps most challenging , most informational architecture quantities must be redefined . with these challenges addressed , we turn our attention to a very general class of continuous - time , discrete - alphabet processes stateful renewal processes generated by unifilar hidden semi - markov models . we identify their and find new expressions for entropy rate and other informational architecture quantities , extending results in ref . @xcite . our main thesis is rather simple : minimal maximally predictive models of continuous - time , discrete - symbol processes require a wholly new calculus . to develop it , sec . [ sec : background ] describes the required new notation and definitions that enable extending the framework which is otherwise well understood for discrete - time processes @xcite . sections [ sec : causalstates]-[sec : infoarch ] determine the causal and informational architecture of continuous - time renewal processes . section [ sec : uhsmms ] characterizes the and calculates the entropy rate and excess entropy of unifilar hidden semi - markov models . we conclude by describing potential applications to bayesian inference algorithms using new enumerations of topologies and to information measure estimation using the formulae of sec . [ sec : uhsmms ] . a continuous - time , discrete - symbol time series @xmath0 is described by a list of symbols @xmath1 in a finite alphabet @xmath2 and dwell times @xmath3 for those symbols . in this representation , we demand that @xmath4 to enforce a unique presentation of the time series . sections [ sec : causalstates]-[sec : infoarch ] focus on point processes for which @xmath5 . and so , in this case , we label the time series only with dwell times : @xmath6 . we view the time series @xmath7 as a realization of random variables @xmath8 . when the observed time series is strictly stationary and the process ergodic , in principle , we can calculate the probability distribution @xmath9 from a single realization @xmath7 . demarcating the present splits @xmath10 into two parts : the time @xmath11 since first emitting the previous symbol and the time @xmath12 to next symbol . thus , we define @xmath13 as the _ past _ and @xmath14 as the _ future_. ( to reduce notation , we drop the @xmath15 indices . ) the _ present _ @xmath16 itself extends over an infinitesimally small length of time . continuous - time renewal processes have a relatively simple generative model . _ interevent intervals _ @xmath17 are drawn from a probability density function @xmath18 . the _ survival function _ @xmath19 is the probability that an interevent interval is greater than or equal to @xmath20 and , in a nod to neuroscience , we define the _ mean firing rate _ @xmath21 as : @xmath22 the minimal generative model for a continuous - time renewal process is therefore a single causal - state machine with a continuous - value observable @xmath23 ; as shown in fig . [ fig : genmodel ] . of periods of silence ( corresponding to output symbol @xmath24 ) are drawn independently , identically distributed ( iid ) from probability density @xmath18 . ] a process _ forward - time causal states _ are defined , as usual , by the _ predictive _ equivalence relation @xcite , written here for the case of point processes : @xmath25 it is straightforward to write the predictive equivalence relation for continuous - time , discrete - alphabet point processes using the notation . this partitions the set of allowed pasts . each equivalence class of pasts is a forward - time causal state @xmath26 , in which @xmath27 is the function that maps a past to its causal state . the set of forward - time causal states @xmath28 inherits a probability distribution @xmath29 from the probability distribution over pasts @xmath30 . _ forward - time prescient statistics _ are any refinement of the forward - time causal - state partition . by construction , they are a sufficient statistic for prediction , but not necessarily _ minimal _ sufficient statistics @xcite . _ reverse - time causal states _ are essentially forward - time causal states of the time - reversed process . in short , reverse - time causal states @xmath31 are the classes defined by the retrodictive equivalence relation , written here for the case of point processes : @xmath32 it is , again , straightforward to write the predictive equivalence relation for continuous - time , discrete - alphabet point processes using the notation given above . and , similarly , reverse - time causal states @xmath33 inherit a probability measure @xmath34 from the probability distribution @xmath35 over futures . reverse - time prescient statistics are any refinement of the reverse - time causal - state partition . they are sufficient statistics for retrodiction , but not necessarily minimal . the main import of these definitions derives from the _ causal shielding _ relations : @xmath36 the consequence of these is illustrated in fig . [ fig : intuition2 ] . that is , arbitrary functions of the past and future do not shield the two aggregate past and future random variables from one another . so , these causal shielding relations are special to prescient statistics , causal states , and their defining functions @xmath27 and @xmath37 . forward and reverse - time generative models do not , in general , have state spaces that satisfy eqs . ( [ eq : cs1 ] ) and ( [ eq : cs2 ] ) . ( left oval , red ) , the future @xmath38 ( right oval , green ) , the forward - time causal states @xmath39 ( left circle , purple ) , and the reverse - time causal states @xmath40 ( right circle , blue ) . @xcite . ) the forward - time and reverse - time statistical complexities are the entropies of @xmath39 and @xmath40 , i.e. , the memories required to losslessly predict or retrodict , respectively . the excess entropy @xmath41 $ ] is a measure of process predictability ( central pointed ellipse , dark blue ) and theorem @xmath42 of ref . @xcite shows that @xmath43 $ ] by applying the causal shielding relations in eqs . ( [ eq : cs1 ] ) and ( [ eq : cs2 ] ) . ] the _ forward - time _ is that with state space @xmath39 and transition dynamic between forward - time causal states . the _ reverse - time _ is that with state space @xmath40 and transition dynamic between reverse - time causal states . defining these transition dynamics for continuous - time processes requires a surprising amount of care , as discussed in secs . [ sec : causalstates]-[sec : infoarch ] . we are broadly interested in information - theoretic characterizations of a process predictability , compressibility , and randomness . a list of current quantities of interest , though by no means exhaustive , is given in figs . [ fig : intuition2 ] and [ fig : intuition3 ] . curiously , many lose meaning when naively applied to continuous - time processes ; e.g. , see refs . @xcite . this section , as a necessity , will redefine many of these in relatively simple , but new ways to avoid trivial divergences and zeros . the _ forward - time statistical complexity _ $ ] is the cost of coding the forward - time causal states and the _ reverse - time statistical complexity _ $ ] is the cost of coding reverse - time causal states . when @xmath39 or @xmath40 are mixed or continuous random variables , one employs differential entropies for @xmath46 $ ] . the result , though , is that the statistical complexities are potentially negative or infinite or both ( * ? ? ? 8.3 ) , perhaps undesirable characteristics for a definition of process complexity . this definition , however , allows for consistency with complexity definitions for discretized continuous - time processes . @xcite for possible alternatives for @xmath46 $ ] . together , a process _ causal irreversibility _ @xcite is defined as the difference between the forward and reverse - time statistical complexities : @xmath47 if the forward- and reverse - time process are isomorphic i.e . , if the process is temporally reversible then @xmath48 . renewal processes are temporally symmetric : @xmath48 @xcite . as such , we will refer to forward - time causal states and the forward - time as simply causal states or the , with the understanding that reverse - time causal states and reverse - time will take the exact same form with slight labeling differences . we start by describing prescient statistics for continuous - time processes . the lemma which does this exactly parallels that of lemma @xmath42 of ref . the only difference is that the prescient statistic is the _ time _ since last event , rather than the number of @xmath24s ( count ) since last event . the time @xmath11 since last event is a prescient statistic of renewal processes . [ lem : ctrp_prescient ] from bayes rule : @xmath49 interevent intervals @xmath17 are independent of one another , so @xmath50 . the random variables @xmath11 and @xmath12 are functions of @xmath10 and the location of the present . both @xmath11 and @xmath12 are independent of other interevent intervals . and so , @xmath51 . this implies : @xmath52 the predictive equivalence relation groups two pasts @xmath53 and @xmath54 together when @xmath55 . we see that @xmath56 is a sufficient condition for this from eq . ( [ eq : lemproof ] ) . the lemma follows . some renewal processes are quite predictable , while others are purely random . a poisson process is the latter : interevent intervals are drawn independently from an exponential distribution and so knowing the time since last event provides no predictive benefit . a fractal renewal process can be the former . there , the interevent interval is so structured that the resultant process can have power - law correlations @xcite . then , knowing the time since last event can provide quite a bit of predictive power @xcite . intermediate between these two extremes is a broad class of renewal processes whose interevent intervals are structured up to a point and then fall off exponentially only after some time @xmath57 . these intermediate cases can be classified as either of the following types of renewal process , in analogy with ref . @xcite s classification . note that an eventually @xmath58-poisson process , but not an eventually poisson process , will generally have a discontinuous @xmath18 . an _ eventually poisson process _ has : @xmath59 for some @xmath60 and @xmath61 almost everywhere . we associate the eventually poisson process with the minimal such @xmath62 . [ def : ep ] an _ eventually @xmath58-poisson process _ with @xmath63 has an interevent interval distribution satisfying : @xmath64 for the smallest possible @xmath57 for which @xmath65 exists . [ def : edp ] a familiar example of an eventually poisson process is found in the spike trains generated by poisson neurons with refractory periods @xcite . there , the neuron is effectively prevented from firing two spikes within a time @xmath62 of each other the period during with its ion channels re - energize the membrane voltage to their nonequilibrium steady state . after that , the time to next spike is drawn from an exponential distribution . to exactly predict the spike train s future , we must know the time since last spike , as long as it is less than @xmath62 . we gain a great deal of predictive power from that piece of information . however , we do not care much about the time since last spike exactly if it is greater than @xmath62 , since at that point the neuron acts as a memoryless poisson neuron . these intuitions are captured by the following classification theorem . a renewal process has three different types of causal state : 1 . when the renewal process is not eventually @xmath58-poisson , the causal states are the time since last event ; 2 . when the renewal process is eventually poisson , the causal states are the time since last event up until time @xmath57 ; or 3 when the renewal process is eventually @xmath58-poisson , the causal states are the time since last event up until time @xmath57 and are the times since @xmath57 mod @xmath58 thereafter . [ the : renewcausalstates ] lemma [ lem : ctrp_prescient ] implies that two pasts are causally equivalent if they have the same time since last event , if @xmath66 . from lemma [ lem : ctrp_prescient ] s proof , we further see that two times since last event are causally equivalent when @xmath67 . in terms of @xmath18 , we find that : @xmath68 using manipulations very similar to those in the proof of thm . @xmath42 of ref . so , to find causal states , we look for @xmath69 such that : @xmath70 for all @xmath71 . to unravel the consequences of this , we suppose that @xmath72 without loss of generality . define @xmath73 and @xmath74 , for convenience . the predictive equivalence relation can then be rewritten as : @xmath75 for any @xmath71 , where @xmath76 . iterating this relationship , we find that : @xmath77 this immediately implies the theorem s first case . if a renewal process is _ not _ eventually @xmath58-poisson , then @xmath78 for all @xmath71 implies @xmath66 , so that the prescient statistics of lemma [ lem : ctrp_prescient ] are also minimal . to understand the theorem s last two cases , we consider more carefully the set of all pairs @xmath79 for which @xmath80 for all @xmath71 holds . define the set : @xmath81 and define the parameters @xmath57 and @xmath65 by : @xmath82 and : @xmath83 note that @xmath57 and @xmath65 defined in this way are unique and exist , as we assumed that @xmath84 is nonempty . when @xmath85 , then the process is eventually @xmath58-poisson . if @xmath86 , then the process must be an eventually poisson process with parameter @xmath57 . to see this , we return to the equation : @xmath87 and rearrange terms to find : @xmath88 as @xmath89 , we can take the limit that @xmath90 and we find that : @xmath91 the righthand side is a parameter independent of @xmath92 . so , this is a standard ordinary differential equation for @xmath18 . it is solved by @xmath93 for @xmath94 . theorem [ the : renewcausalstates ] implies that there is a qualitative change in @xmath39 depending on whether or not the renewal process is poisson , eventually poisson , eventually @xmath58-poisson , or not eventually poisson . in the first case , @xmath39 is a discrete random variable ; in the second case , @xmath39 is a mixed discrete - continuous random variable ; and in the third and fourth cases , @xmath39 is a continuous random variable . identifying causal states in continuous - time follows an almost entirely similar path to that used for discrete - time renewal processes in ref . the seemingly slight differences between the causal states of eventually poisson , eventually @xmath58-poisson , and not eventually @xmath58-poisson renewal processes , however , have surprisingly important consequences for continuous - time . as described by thm . [ the : renewcausalstates ] , there are often an uncountable infinity of continuous - time causal states . as one might anticipate from refs . @xcite , however , there is an ordering to this infinity of causal states that makes calculations tractable . there is one major difference between discrete - time and continuous - time : transition dynamics often amount to specifying the evolution of a probability density function over causal - state space . as such , a continuous - time constitutes an unusual presentation of a hidden markov model : they appear as a system of conveyor belts or , under special conditions , like conveyor belts with a trash bin or a second mini - conveyor belt . beyond the picaresque metaphor , in fact they operate like conveyor belts in that they transport the time since the last event , resetting it here and there in a stateful way . unsurprisingly , the exception to this general rule is given by the poisson process itself . the of a poisson process is exactly the minimal generative model shown in fig . [ fig : genmodel ] . at each iteration , an interevent interval is drawn from a probability density function @xmath95 , with @xmath96 . knowing the time since last event does not aid in predicting the time to next event , above and beyond knowing @xmath97 . and so , the poisson has only a single state . in the general setting , though , the dynamic describes the evolution of the probability density function over its causal states . how to represent this ? we might search for labeled transition operators @xmath98 such that @xmath99 , giving partial differential equations that govern the labeled - transition dynamics . , tracking the time since last event and depicted as the semi - infinite horizontal line , are isomorphic with the positive real line . if no event is seen , probability flows towards increasing time since last event , as described in eq . ( [ eq : mathcalo0 ] ) . otherwise , arrows denote allowed transitions back to the reset state or `` @xmath24 node '' ( solid black circle at left ) , denoting that an event occurred . ] the of a renewal process that is not eventually poisson takes the state - transition form shown in fig . [ fig : nedp ] . let @xmath100 be the probability density function over the causal states @xmath101 at time @xmath20 . our approach to deriving labeled transition dynamics parallels well - known approaches to determining fokker - planck equations using a kramers - moyal expansion @xcite . here , this means that any probability at causal state @xmath101 at time @xmath102 could only have come from causal state @xmath103 at time @xmath20 , if @xmath104 . this implies : @xmath105 however , @xmath106 is simply the probability that the interevent interval is greater than @xmath101 , given that the interevent interval is at least @xmath107 , or : @xmath108 together , eqs . ( [ eq : probflow1 ] ) and ( [ eq : probflow2 ] ) imply that : @xmath109 from this , we obtain : @xmath110 hence , the labeled transition operator @xmath111 given no event takes the form : @xmath112 the probability density function @xmath100 changes discontinuously after an event occurs , though . all probability mass shifts from @xmath113 resetting back to @xmath114 : @xmath115 in other words , an event `` collapses the wavefunction '' . the stationary distribution @xmath116 over causal states is given by setting @xmath117 to @xmath24 and solving . ( at the risk of notational confusion , we adopt the convention that @xmath116 denotes the stationary distribution and that @xmath100 does not . ) straightforward algebra shows that : @xmath118 from this , the continuous - time statistical complexity directly follows : @xmath119 this was the nondivergent component of the infinitesimal time - discretized renewal process statistical complexity found in ref . @xcite . are isomorphic with the real line only to @xmath120 $ ] , as they again denote time since last event . a leaky absorbing node at @xmath57 ( solid white circle at right ) corresponds to any time since last event after @xmath57 . if no event is seen , probability flows towards increasing time since last event or the leaky absorbing node , as described in eqs . ( [ eq : mathcalo0 ] ) and ( [ eq : mathcalo0b ] ) . when an event occurs the process transitions ( curved arrows ) back to the reset state node @xmath24 ( solid black circle at left ) . ] as thm . [ the : renewcausalstates ] anticipates , there is a qualitatively different topology to the of an eventually poisson renewal process , largely due to the continuous - time causal states being mixed discrete - continuous random variables . for @xmath121 , there is `` wave '' propagation completely analogous to that described in eq . ( [ eq : mathcalo0 ] ) of sec . [ sec : nedp ] . however , there is a new kind of continuous - time causal state at @xmath122 , which does not have a one - to - one correspondence to the dwell time . instead , it denotes that the dwell time is _ at least _ some value ; viz . , @xmath57 . new notation follows accordingly : @xmath100 , defined for @xmath123 , denotes a probability density function for @xmath121 and @xmath124 denotes the probability of existing in causal state @xmath125 . normalization , then , requires that : @xmath126 the transition dynamics for @xmath124 are obtained similarly to that for @xmath100 , in that we consider all ways in which probability flows to @xmath127 in a short time window @xmath128 . probability can flow from any causal state with @xmath129 or from @xmath122 itself . that is , if no event is observed , we have : @xmath130 the term @xmath131 corresponds to probability flow from @xmath122 and the integrand corresponds to probability influx from states @xmath132 with @xmath133 . assuming differentiability of @xmath124 with respect to @xmath20 , we find that : @xmath134 where @xmath135 is shorthand for @xmath136 . this implies that the labeled transition operator @xmath111 takes a piecewise form which acts as in eq . ( [ eq : mathcalo0 ] ) for @xmath121 and as in eq . ( [ eq : mathcalo0b ] ) for @xmath122 . as earlier , observing an event causes the `` wavefunction collapse '' to a delta distribution at @xmath114 . the causal - state stationary distribution is determined again by setting @xmath137 and @xmath138 to @xmath24 . equivalently , one can use the prescription suggested by thm . [ the : renewcausalstates ] to calculate @xmath139 via integration of the stationary distribution over the prescient machine given in sec . [ sec : nedp ] : @xmath140 if we recall that @xmath141 , we find that : @xmath142 the process continuous - time statistical complexity precisely , entropy of this mixed random variable is given by : @xmath143 this is the sum of the nondivergent @xmath144 component and the rate of divergence of @xmath144 of the infinitesimal time - discretized renewal process @xcite . -poisson renewal process : graphical elements as in the previous figure . the circular causal - state space at @xmath57 ( circle on right ) has total duration @xmath65 , corresponding to any time since last event after @xmath57 mod @xmath65 . if no event is seen , probability flows as indicated around the circle , as described in eq . ( [ eq : mathcalo0 ] ) . ] probability wave propagation equations , like those in eq . ( [ eq : mathcalo0 ] ) , hold for @xmath121 and for @xmath145 . at @xmath122 , if no event is observed , probability flows in from both @xmath146 and from @xmath147 , giving rise to the equation : @xmath148 unfortunately , there is a discontinuous jump in @xmath100 at @xmath122 coming from @xmath147 and @xmath149 . and so , we can not taylor expand either @xmath150 or @xmath151 about @xmath152 . again , we can use the prescription suggested by thm . [ the : renewcausalstates ] to calculate the probability density function over these causal states and , from that , calculate the continuous - time statistical complexity . below @xmath121 , the probability density function over causal states is exactly that described in sec . [ sec : nedp ] : @xmath153 . for @xmath154 , the probability density function becomes : @xmath155 recalling def . [ def : edp ] , we see that @xmath156 and so find that for @xmath157 : @xmath158 altogether , this gives the statistical complexity : @xmath159 [ cols= " < , < " , ] we define continuous - time information anatomy @xcite quantities as _ rates_. as mentioned earlier , the present extends over an infinitesimal time . to define information anatomy rates , we let @xmath160 be the symbols observed over an arbitrarily small length of time @xmath161 , starting at the present @xmath162 . it could be that @xmath160 encompasses some portion of @xmath163 ; the notation leaves this ambiguous . the entropy rate is now : @xmath164}{d\delta } ~. \label{eq : hmupercs}\end{aligned}\ ] ] this is equivalent to the more typical random - variable `` block '' definition of entropy rate @xcite : @xmath165 / \delta$ ] . similarly , we define the _ single - measurement entropy rate _ as : @xmath166}{d\delta } ~ , \label{eq : h0percs}\end{aligned}\ ] ] the _ bound information rate _ as : @xmath167}{d\delta } ~ , \label{eq : bmupercs}\end{aligned}\ ] ] the _ ephemeral information rate _ as : @xmath168}{d\delta } ~,\end{aligned}\ ] ] and the _ co - information rate _ as : @xmath169}{d\delta } ~.\end{aligned}\ ] ] in direct analogy to discrete - time process information anatomy , we have the relationships : @xmath170 so , the entropy rate @xmath171 , the instantaneous rate of information creation , again decomposes into a component @xmath172 that represents active information storage and a component @xmath173 that represents `` wasted '' information . ): information diagram for the past @xmath174 , infinitesimal present @xmath160 , and future @xmath175 . the measurement entropy rate @xmath176 is the rate of change of the single - measurement entropy @xmath177 $ ] at @xmath178 . the ephemeral information rate @xmath179 $ ] is the rate of change of useless information generation at @xmath178 . the bound information rate @xmath180 $ ] is the rate of change of active information storage . and , the co - information rate @xmath181 $ ] is the rate of change of shared information between past , present , and future . these definitions closely parallel those in ref . @xcite . ] prescient states ( not necessarily _ minimal _ ) are adequate for deriving all information measures aside from @xmath182 . as such , we focus on the transition dynamics of noneventually @xmath58-poisson and , implicitly , their bidirectional machines . to find the joint probability density function of the time to next event @xmath183 and time since last event @xmath184 , we note that @xmath185 is an interevent interval ; hence : @xmath186 the normalization factor of this distribution is : @xmath187 so , the joint probability distribution is : @xmath188 equivalently , we could have calculated the conditional probability density function of time - to - next - event given that it has been at least @xmath184 since the last event . this , by similar arguments , is @xmath189 . this would have given the same expression for @xmath190 . to find the excess entropy , we merely need calculate @xcite : @xmath191 \\ & = \int_0^{\infty } \int_0^{\infty } \mu\phi ( { { \sigma } } ^+ , { { \sigma } } ^- ) \log \frac{\mu \phi ( { { \sigma } } ^+ , { { \sigma } } ^-)}{\phi ( { { \sigma } } ^+)\phi ( { { \sigma } } ^- ) } d { { \sigma } } ^+ d { { \sigma } } ^- ~.\end{aligned}\ ] ] algebra not shown here gives : @xmath192 unsurprisingly @xcite , this agrees with the formula given in ref . @xcite , which was derived by considering the limit of infinitesimal time discretization . now , we turn to the more technically challenging task of calculating differential information anatomy rates . suppose that @xmath193 is a random variable for paths of length @xmath161 . each path is uniquely specified by a list of times of events . let @xmath194 be a random variable defined by : @xmath195 we first illustrate how to find @xmath176 , since the same technique allows calculating @xmath171 . we can rewrite the path entropy as : @xmath196 = { \operatorname{h}}[x_{\delta } ] + { \operatorname{h}}[\gamma_{\delta}|x_{\delta } ] ~.\end{aligned}\ ] ] for renewal processes , when @xmath21 can be defined , we see that : @xmath197 straightforward algebra shows that : @xmath198 & = \mu\delta - \mu\delta\log ( \mu\delta ) + o(\delta^2\log\delta ) ~.\end{aligned}\ ] ] we would like to find a similar asymptotic expansion for @xmath199 $ ] , which can be rewritten as : @xmath200 & = { \pr}(x_{\delta}=0 ) { \operatorname{h}}[\gamma_{\delta}|x_{\delta}=0 ] \nonumber \\ & \qquad + { \pr}(x_{\delta}=1 ) { \operatorname{h}}[\gamma_{\delta}|x_{\delta}=1 ] \nonumber \\ & \qquad + { \pr}(x_{\delta}=2 ) { \operatorname{h}}[\gamma_{\delta}|x_{\delta}=2 ] ~.\end{aligned}\ ] ] first , we notice that @xmath193 is deterministic given that @xmath201the path of all silence . so , @xmath202 = 0 $ ] . second , we can similarly ignore the term @xmath203 $ ] since @xmath204 is @xmath205 and , we claim , @xmath206 $ ] is @xmath207 : by standard maximum entropy arguments , @xmath206 $ ] is at most @xmath208 , and by noting that trajectories with only one event are a strict subset of trajectories with more than one event but with multiple events arbitrarily close to one another , @xmath206\geq { \operatorname{h}}[\gamma_{\delta}|x_{\delta}=1]$ ] which , by arguments below , is @xmath207 . thus , the term @xmath209 $ ] is @xmath210 at most . finally , to calculate @xmath211 $ ] , we note that when @xmath212 , paths can be uniquely specified by an event time , whose probability is @xmath213 . a taylor expansion about @xmath214 shows that @xmath215 for some @xmath216 in which @xmath217 for all @xmath218 . so , overall , we find that : @xmath219 where @xmath220 for any @xmath193 with at least one event in the path . the largest corrections to @xmath221 come from ignoring the paths with two or more events , rather than from approximating all paths with only one event as equally likely . in sum , we see that : @xmath200 & = \mu\delta\log\delta + o(\delta^2\log\delta ) ~.\end{aligned}\ ] ] together , these manipulations give : @xmath196 & = \mu\delta - \mu\delta\log\mu + o(\delta^2\log\delta ) ~.\end{aligned}\ ] ] this then implies : @xmath222}{d\delta } \\ & = \mu-\mu\log\mu ~.\end{aligned}\ ] ] a similar series of arguments helps to calculate @xmath223 defined in eq . ( [ eq : hmupercs ] ) , where now , @xmath21 is replaced by @xmath224 : @xmath225 which gives : @xmath226 algebra ( namely , integration by parts ) not shown here yields the expression : @xmath227 as expected , this is the nondivergent component of the expression given in eq . ( @xmath228 ) of ref . @xcite for the @xmath161-entropy rate of renewal processes . and , it agrees with expressions derived in alternative ways @xcite . we need slightly different techniques to calculate @xmath172 , as we no longer need to decompose a path entropy . from eq . ( [ eq : bmupercs ] ) , we have : @xmath229}{d\delta } ~.\end{aligned}\ ] ] let s develop a short - time @xmath161 asymptotic expansion for @xmath230 . first , we notice that @xmath231 , so that : @xmath232 we already can identify : @xmath233 to understand @xmath234 , we expand : @xmath235 recall that @xmath236 is @xmath205 , that : @xmath237 and that : @xmath238 then , straightforward algebra not shown gives : @xmath239 this can be used to derive : @xmath240 in nats . when @xmath241 , for instance , @xmath242 for all @xmath184 , confirming in a much more complicated calculation that poisson processes really are memoryless . this allows us to calculate the total @xmath172 as : @xmath243 in nats . and , from this , we find @xmath173 using : @xmath244 continuing , we calculate @xmath245 from : @xmath246 and , we calculate @xmath247 via : @xmath248 all these quantities are gathered in table [ tab:1 ] , which gives them in bits rather than nats . the of discrete - time , discrete - symbol processes are well understood and , as we now appreciate from secs . [ sec : causalstates]-[sec : infoarch ] , the predictive equivalence relation defining them readily applies to continuous - time renewal processes . this gives the latter s analogous maximally predictive models : continuous or hybrid discrete - continuous , when minimal . here , we introduce a new class of process generators that are unifilar versions of ref . @xcite s hidden semi - markov models , but whose dwell time distributions can take any form . ( note that general semi - markov models are a strict subset . ) roughly speaking , they are stateful renewal processes , but this needs to be clarified . many of their calculations reduce to those in secs . [ sec : causalstates]-[sec : infoarch ] . when appropriate , we skip these steps . we start by introducing the minimal generative models in fig . [ fig : uhsmm ] . let @xmath249 be the set of states in this generative model . each state @xmath250 emits a symbol @xmath251 and a dwell time @xmath252 for that symbol , and , based on the state @xmath253 and emitted symbol @xmath1 , transitions to a new state @xmath254 . we assume that the underlying generative model is _ unifilar _ : that the new state @xmath253 is uniquely specified by the prior state @xmath253 and emitted symbol @xmath1 . we introduce a perhaps unfamiliar restriction on the labeled transition matrices @xmath255 . define @xmath256 and @xmath257 . then , we focus only on generative models for which @xmath258 . this simply ensures that there is no uncertainty in when one dwell time finishes and another begins . for example , consider the generator in fig . [ fig : uhsmm](bottom ) : if states @xmath259 and @xmath260 were both to emit a @xmath24 in succession , it would be impossible to tease apart when the process switched from state @xmath259 to state @xmath260 . the restriction introduces no loss of generality for our purposes . , @xmath261 , @xmath262 , and @xmath263 are isomorphic to @xmath264 . during an event interval , symbol @xmath265 is emitted . a transition occurs to a new event when the state dwell time is exhausted at @xmath266 , which is distributed according to @xmath267 . ( bottom ) generative model with three hidden states ( @xmath268 , @xmath259 , and @xmath260 ) emits symbols @xmath269 for dwell times @xmath270 drawn from probability density functions @xmath271 , @xmath272,@xmath273 , and @xmath274 , respectively . ( transition labels as in fig . [ fig : genmodel ] . ) ] a prescient model of this combined process is shown in fig . [ fig : uhsmm](top ) . each state @xmath250 comes equipped with one or more renewal process - like tails ( semi - infinite spaces that act as continuous counters ) that generically take the form of fig . [ fig : nedp ] . the leakiness of these ( dissipative ) counters is given by @xmath275 , the probability density function from which the dwell time is drawn . this new form of state - transition diagram depicts the of these hidden semi - markov processes . moreover , if one or more of the dwell - time distributions gives an eventually poisson or an eventually-@xmath58 poisson structure , the presentation in fig . [ fig : uhsmm](top ) is a prescient machine , but not the . more generally , any such unifilar minimal generative model has a prescient machine with a `` node '' for each underlying hidden state @xmath253 and as many counters as needed one for every almost - everywhere unique @xmath276 . each counter leaks probability to the next underlying hidden state @xmath254 , which is completely determined by @xmath253 and @xmath1 . the presentation in fig . [ fig : uhsmm](top ) is a prescient machine for the process generated by the unifilar hidden semi - markov model of fig . [ fig : uhsmm](bottom ) . [ the:2 ] to show that this is a prescient machine , we need to show that the present model state consisting of hidden state @xmath253 , current emitted symbol @xmath277 , and dwell time @xmath270 is uniquely specified by the observed past almost surely . the observed symbol @xmath1 is given by the current symbol in the observed past . the restriction on successive emitted symbols ( that @xmath278 ) implies that the observed dwell time @xmath270 is exactly the observed length of @xmath1 . finally , the underlying hidden state @xmath253 is determined uniquely by a _ function _ of the past almost surely , in which all dwell - time information is removed , by assumption : the restriction mentioned earlier implies there is no uncertainty in when one dwell time finishes and another begins . and , the unifilarity of the dynamic on hidden states @xmath253 implies that the sequence of symbols in the observed past are sufficient to specify the hidden state @xmath253 almost surely . hence , @xmath253 is determined uniquely from the observed past almost surely . the theorem follows . theorem [ the:2 ] can be straightforwardly generalized to specify conditions under which the presentation is an , a minimal prescient machine , by incorporating the conditions of thm . [ the : renewcausalstates ] . the stationary distribution for @xmath279 directly follows the treatment for the continuous - time renewal processes in sec . [ sec : nedp ] , and so : @xmath280 where @xmath281 . then we note that : @xmath282 and so : @xmath283 to find @xmath284 , we again calculate the probability mass dumped at @xmath285 in terms of @xmath284 : @xmath286 after a straightforward substitution of eq . ( [ eq : foo ] ) and noting that @xmath287 , we find : @xmath288 so : @xmath289 let @xmath290 be the stationary distribution for the underlying discrete - state : @xmath291 where the eigenvector is normalized such that the sum of its entries is @xmath42 . then : @xmath292 or , rewriting and normalizing , we have : @xmath293 altogether , we find that the steady - state distribution is given by : @xmath294 using the formulae for entropies of mixed random variables @xcite , we find a statistical complexity of : @xmath295 \\ & = \left\langle { \operatorname{h}}[\rho(\tau|g , { { x } } ) ] \right\rangle_{p(g , { { x } } ) } + { \operatorname{h}}[p(g , { { x } } ) ] \\ & = \left\langle \int_0^{\infty } \mu_{g , { { x } } } \phi_{g , { { x } } } ( \tau ) \log \frac{1}{\mu_{g , { { x } } } \phi_{g , { { x } } } ( \tau ) } d\tau \right\rangle_{p(g , { { x } } ) } \\ & \qquad + { \operatorname{h}}\left [ \frac{\pi(g ) t_g^ { ( { { x } } ) } /\mu_{g , { { x } } } } { \sum_{g ' , { { x } } ' } \pi(g ' ) t_{g'}^ { ( { { x } } ' ) } /\mu_{g ' , { { x } } ' } } \right ] ~.\end{aligned}\ ] ] note that @xmath296 $ ] is the statistical complexity of the underlying discrete - time and that @xmath297 $ ] is the statistical complexity of a noneventually @xmath58-poisson renewal process with interevent distribution @xmath298 , averaged over @xmath253 and @xmath1 . hence , the statistical complexity of these unifilar hidden semi - markov processes differs from the statistical complexity of its `` components '' by : @xmath299 - { \operatorname{h}}[\pi(g ) ] ~.\end{aligned}\ ] ] whether this difference is positive or negative depends on both matrices @xmath300 . in general , we expect the difference to be positive . since there are multiple observed symbols @xmath301 generated by these machines , @xmath176 ( and so @xmath247 ) and @xmath172 as defined in sec . [ sec : infoarch ] diverge . however , the entropy rate @xmath171 and excess entropy @xmath302 as defined in sec . [ sec : infoarch ] do not diverge for processes generated by this restricted class of unifilar hidden semi - markov models . from the steady - state distribution given in eq . ( [ eq : steadystate ] ) and from the entropy rate expressions in eqs . ( [ eq : hmust])-([eq : hmufinal ] ) of sec . [ sec : infoarch ] , we immediately have the entropy rate for these unifilar hidden semi - markov models : @xmath303}{d\delta } \nonumber \\ & = \sum_{g , { { x } } } \rho(g , { { x } } ) \left(-\mu_{g , { { x } } } \int_0^{\infty } \phi_{g , { { x } } } ( \tau)\log \phi_{g , { { x } } } ( \tau ) d\tau\right ) \nonumber \\ & = -\frac{\sum_{g , { { x } } } \pi(g ) t_g^ { ( { { x } } ) } \int_0^{\infty } \phi_{g , { { x } } } ( \tau)\log \phi_{g , { { x } } } ( \tau ) d\tau } { \sum_{g ' , { { x } } ' } \pi(g ' ) t_{g'}^ { ( { { x } } ' ) } /\mu_{g ' , { { x } } ' } } ~. \label{eq : hmu_hsmm}\end{aligned}\ ] ] to ground intuition , recall that each state in the underlying for semi - markov processes corresponds to a unique observation symbol . hence , setting @xmath304 to @xmath305 and noting that each @xmath253 is uniquely associated to some @xmath1 in eq . ( [ eq : hmu_hsmm ] ) recovers the results of ref . @xcite for the entropy rate of semi - markov processes , though the notation differs somewhat . ) to a semi - markov process in reverse - time so that the underlying model is unifilar rather than co - unifilar ; but entropy rate is invariant to time reversal @xcite . ] the process excess entropy @xmath43 $ ] can be calculated if we can find the joint probability distribution @xmath306 of forward- and reverse - time causal states . to this end , we add an additional restriction on the generative model : we focus only on generative models for which @xmath307 . with this restriction on labeled transition matrices , the time - reversed of the process has the same form as the of the forward - time process , but with a different @xmath249 . the latter is related to the forward - time @xmath249 via manipulations described in ref . @xcite . as such , we can write down @xmath308 : @xmath309 where we obtain @xmath310 from standard methods @xcite applied to ( only ) the dynamic on @xmath249 . note that @xmath311 reduces to @xmath312 as @xmath254 and @xmath313 uniquely specify the distribution from which @xmath314 is drawn and since @xmath315 . we leave the the final steps to @xmath302 as an exercise . though the definition of continuous - time causal states parallels that for discrete - time causal states , continuous - time and information measures are markedly different from their discrete - time counterparts . similar technical difficulties arise more generally when describing minimal maximally predictive models of other continuous - time , discrete - symbol processes that are not the continuous - time markov processes analyzed in ref . the resulting do not appear like conventional hmms recall figs . [ fig : nedp]-[fig : edp ] and , especially , fig . [ fig : uhsmm](top)and most of the information measures excepting the excess entropy are reinterpreted as differential information rates . moreover , the continuous - time machinery gave us a new way to calculate these information measures . traditionally , expressions for such information measures come from calculating the time - normalized path entropy of arbitrarily long trajectories ; e.g. , as in ref . instead , we calculated the path entropy of arbitrarily short trajectories , conditioned on the past . this allowed us to extend the results of ref . @xcite for the entropy rate of continuous - time discrete - output processes to a previously untouched class of processes unifilar hidden semi - markov processes . there are two immediate practical benefits to an in - depth look at the of continuous - time hidden semi - markov processes . first , statistical model selection when searching through unifilar hidden markov models is significantly easier than when searching through nonunifilar hidden markov models @xcite , and these benefits should carry over to the case of continuous - time . second , the formulae in table [ tab:1 ] and those in sec . [ sec : uhsmms ] provide new approaches to binless plug - in information measure estimation ; e.g. , following ref . @xcite . the machinery required to use continuous - time is significantly different than that accompanying the study of discrete - time . our results here pave the way toward understanding the difficulties that lie ahead when studying the structure and information in continuous - time processes . the authors thank the santa fe institute for its hospitality during visits . jpc is an sfi external faculty member . this material is based upon work supported by , or in part by , the u. s. army research laboratory and the u. s. army research office under contract number w911nf-13 - 1 - 0390 . sm was funded by a national science foundation graduate student research fellowship , a u.c . berkeley chancellor s fellowship , and the mit physics of living systems fellowship . v. girardin . on the different extensions of the ergodic theorem of information theory . in r. baeza - yates , j. glaz , h. gzyl , j. husler , and j. l. palacios , editors , _ recent advances in applied probability theory _ , pages 163179 . springer us , 2005 . actually , we would apply eq . ( [ eq : hmu_hsmm ] ) to a semi - markov process in reverse - time so that the underlying model is unifilar rather than co - unifilar ; but entropy rate is invariant to time reversal @xcite . | we introduce the minimal maximally predictive models ( ) of processes generated by certain hidden semi - markov models .
their causal states are either hybrid discrete - continuous or continuous random variables and causal - state transitions are described by partial differential equations .
closed - form expressions are given for statistical complexities , excess entropies , and differential information anatomy rates .
we present a complete analysis of the of continuous - time renewal processes and , then , extend this to processes generated by unifilar hidden semi - markov models and semi - markov models .
our information - theoretic analysis leads to new expressions for the entropy rate and the rates of related information measures for these very general continuous - time process classes . |
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the proton magnetic form factor at large momentum transfer has been extensively analyzed within perturbative quantum chromodynamics ( pqcd ) over the last decade @xcite . the theoretical basis of these calculations is the hard scattering formula @xcite in which the proton form factor is generically expressed as a convolution of a hard - scattering amplitude @xmath5 and proton distribution amplitudes ( da ) @xmath6 which represent valence quark fock state wave functions integrated over quark transverse momenta ( defined with respect to the momentum of their parent proton ) : @xmath7 \int_{0}^{1}[dx^{\prime } ] |f_{n}(\mu _ { f})|^{2}\ , \phi ^{\star}(x^{\prime},\mu _ { f } ) t_{h}(x , x^{\prime},q,\mu ) \phi ( x,\mu _ { f } ) , \label{eq : gm}\ ] ] where @xmath1 is the invariant momentum transfer squared and @xmath8=dx_{1}dx_{2}dx_{3}\delta ( 1-\sum_{}^{}x_{i})$ ] , @xmath9 being the momentum fractions carried by the valence quarks . the renormalization scale is denoted by @xmath10 and the factorization scale by @xmath11 . the latter scale defines the interface between soft physics absorbed in the wave function and hard physics , treated explicitly within pqcd . the dimensionful constant @xmath12 represents the value of the proton wave function at the origin of the configuration space and has to be determined nonperturbatively @xcite . the residual ( mainly perturbative ) scale dependence of @xmath12 and that of the proton da is controlled by the evolution equation @xcite . to lowest order the hard scattering amplitude is calculated as the sum of all feynman diagrams for which the three quark lines are connected pairwise by two gluon propagators . this allows the quarks in the initial and final proton to be viewed as moving collinearly up to transverse momenta of order @xmath11 . it is then easy to show that @xmath13 , wherein @xmath0 is the running strong coupling constant in the one - loop approximation . the pauli form factor @xmath14 and hence the electric form factor @xmath15 can not be calculated within the hard scattering picture ( hsp ) , since they require helicity - flip transitions which are not possible for ( almost ) massless quarks . these form factors are dominated by sizeable higher twist contributions as we know from experiment @xcite . ( [ eq : gm ] ) is obtained by taking the + component of the electromagnetic vertex and represents the helicity - conserving part of the form factor . the choice of the renormalization scale in the calculation of the proton form factor is a crucial point . most authors @xcite use a constant @xmath0 outside the integrals over fractional momenta , with an argument rescaled by the characteristic virtualities for each particular model da . choosing @xmath10 that way and using das calculated by means of qcd sum rules distributions whose essential characteristic is a strong asymmetry in phase space results for @xmath16 have been obtained @xcite that compare fairly well with the data @xcite . on the other hand , the so - called `` asymptotic '' da @xcite @xmath17 into which any da should evolve with @xmath18yields a vanishing result for @xmath19 . however , for a renormalization scale independent of x , large contributions from higher orders are expected in the endpoint region , @xmath20 . indeed , for the pion form factor this has been shown explicitly , at least for the next - to - leading order @xcite . such large higher - order contributions would render the leading - order calculation useless . a more appropriate choice of the renormalization scale would be , e.g. , @xmath21 , since such a scale would eliminate the large logarithms arising from the higher - order contributions . unfortunately , this is achieved at the expense that @xmath0 becomes singular in the endpoint regions . it has been conjectured @xcite that gluonic radiative corrections ( sudakov factors ) will suppress that @xmath0-singularity and , therefore , in practical applications of the hsp one may handle this difficulty by cutting off @xmath0 at a certain value , typically chosen in the range 0.5 to 0.7 . another , semi - phenomenological recipe to avoid the singularity of @xmath0 is to introduce an effective gluon mass @xcite which cut - offs the interaction at low @xmath1 values . besides the extreme sensitivity of the form factors on the utilized da and besides the problem with higher - order contributions and/or the singularity of @xmath22 , there is still another perhaps more fundamental difficulty with such calculations . indeed , the applicability of ( [ eq : gm ] ) at experimentally accessible momentum transfer , typically a few gev , is not _ a priori _ justified . it was argued by isgur and llewellyn - smith @xcite and also by radyushkin @xcite that the hsp receives its main contributions from the soft endpoint regions , rendering the perturbative calculation inconsistent . recently , this criticism has been challenged by sterman and collaborators @xcite . based on previous works by collins , soper , and sterman @xcite , they have calculated sudakov corrections to the hard - scattering process taking into account the conventionally neglected transverse momentum , @xmath23 , of the quarks . the sudakov corrections damp those contributions from the endpoint regions in which transverse momenta of the quarks are not large enough to keep the exchanged gluons hard . moreover , as presumed , the sudakov corrections cancel the @xmath22-singularity without introducing additional _ ad hoc _ cut - off parameters as for instance a gluon mass . thus the modified hsp provides a well - defined expression for the form factor which takes into account the perturbative contributions in a self - consistent way , even for momentum transfers as low as a few gev however , an important element has not been considered in the analyses of refs . this concerns the inclusion of the intrinsic transverse momentum of the hadronic wave function . as it was recently shown by two of us @xcite for the case of the pion form factor , the inclusion of the transverse size of the pion extends considerably the self - consistency region of the perturbative contribution down to values of momentum transfer unreachable by the sudakov corrections alone . on the other hand , the incorporation of the @xmath23-dependence leads to a substantial decrease of the magnitude of the ( leading - order ) pion form factor . unfortunately , a clear - cut comparison with the available data is not possible because of their low quality and the uncertainty in the determination of the pion - nucleon coupling constant @xcite . nevertheless , it seems reasonable to expect that the pion form factor receives considerable soft contributions in the presently accessible gev region . the aim of the present paper is to perform an analysis for the proton form factor within the modified hsp . one of our objectives is to critically examine li s approach @xcite and to enlarge the theoretical framework by including the intrinsic @xmath23-dependence of the proton wave function . at the same time we want to clarify several technical points , which are absent in the pion case and are first encountered in the more complicated calculation of the proton form factor . the purpose of our analysis is to investigate how reliably the perturbative contribution to the proton form factor can be calculated and to answer the question whether there is a proton wave function modeled on the basis of qcd sum rules @xcite which is capable of providing , in a theoretically self - consistent way , a good agreement with the data within the modified hsp . it is clear that being able to identify the leading - order perturbative contribution reliably allows us to estimate the size of soft contributions to the proton form factor , contributions which are not accounted for in the modified hsp formalism . [ note that the @xmath23-dependent effects taken into account in the modified hsp represent also soft contributions of higher - twist type . ] sudakov suppression ( which can be viewed as the perturbative part of the transverse wave function ) and intrinsic @xmath23-dependence of the wave function may also have a lot of interesting consequences in other exclusive reactions . thus , for instance , sotiropoulos and sterman @xcite have applied these elements to near - forward proton - proton elastic scattering claiming that their interplay drives the transition of the fixed @xmath24 differential cross section from the @xmath25 behavior at moderate @xmath26 to the @xmath27 behavior at larger @xmath26 , as predicted by dimensional counting rules @xcite . the outline of the paper is as follows . in sec . ii we discuss the proton wave function . the modified hsp is treated in sec . the discussion of the infrared ( ir ) cut - off prescription in the sudakov factor and its effect on the @xmath0-singularities is given in sec . the numerical results are presented in sec . v and our conclusions are contained in sec . similarly to sotiropoulos and sterman @xcite , we write the valence quark component of the proton state with positive helicity in the form @xmath28 \int_{}^ { } [ d^{2}k_{\perp } ] \bigl\ { & \phantom { } & \!\!\!\!\!\ ! \psi _ { 123}\,{\cal m}_{+-+}^{a_{1}a_{2}a_{3 } } + \psi _ { 213}\,{\cal m}_{-++}^{a_{1}a_{2}a_{3 } } \nonumber \\ & - & \bigl(\psi _ { 132}\ , + \ , \psi _ { 231}\bigr){\cal m}_{++-}^{a_{1}a_{2}a_{3 } } \bigr\ } \epsilon _ { a_{1}a_{2}a_{3 } } , \label{eq:|p,+>}\end{aligned}\ ] ] where we have assumed the proton to be moving rapidly in the @xmath29-direction . hence , the ratio of transverse to longitudinal momenta of the quarks is small . the measure over the transverse momentum integration is defined by @xmath30 = \frac{1}{\left(16\pi ^{3}\right)^{2 } } \ , \delta ^{(2 ) } \left ( \sum_{i=1}^{3}\vec{k}_{\perp i } \right ) d^{2}k_{\perp 1 } d^{2}k_{\perp 2 } d^{2}k_{\perp 3}.\ ] ] in the zero binding energy limit , which is characteristic for the parton picture , one has @xmath31 the three quark state with helicities @xmath32 and colors @xmath33 is given by @xmath34 since the orbital angular momentum is assumed to be zero , the proton helicity is the sum of the quark helicities . the quark states are normalized as follows : @xmath35 from the permutation symmetry between the two u quarks and from the requirement that the three quarks have to be coupled to give an isospin @xmath36 state it follows that eq . ( [ eq:|p,+ > ] ) can be expressed in terms of only one independent scalar function @xcite . in the sequel , @xmath37 denotes the momentum space wave function . the subscripts on @xmath37 refer to the order of momentum arguments , for example @xmath38 . note that , in general , the wave function depends on the factorization scale @xmath11 . we make the following convenient ansatz for the wave function : @xmath39 the distribution amplitude @xmath40 ( in the notation of @xcite ) is defined in such a way that @xmath41\ , \phi _ { 123}(x,\mu _ { f } ) = 1 , \label{eq : danorm}\ ] ] where an obvious abbreviated notation has been introduced . the da can be expressed in terms of the eigenfunctions of the evolution equation @xcite , @xmath42 , which are linear combinations of appell polynomials . then the proton da can be cast into the form @xmath43 where the notations of @xcite are adopted . @xmath44 is the asymptotic da mentioned in the introduction . the exponents @xmath45 , driving the evolution behavior of the da , are related to the anomalous dimensions of trilinear quark operators with isospin @xmath36 ( see @xcite ) and resemble the @xmath46 in the brodsky - lepage notation @xcite . because they are positive fractional numbers increasing with n , higher - order terms in ( [ eq : phi ] ) are gradually suppressed . the constants @xmath45 are given in table 1 ; @xmath47 for three flavors . constraints on the da are obtained implicitly by restricting their few first moments within intervals determined from qcd sum rules @xcite , which are evaluated at some self - consistently determined normalization point @xmath11 of order 1 gev ( see , e.g. , @xcite ) : @xmath48x_{1}^{n_{1}}x_{2}^{n_{2}}x_{3}^{n_{3 } } \phi _ { 123}(x,\mu _ { 0})\ ] ] [ eq : moments ] in most model calculations , mentioned above , the moment constraints provided by qcd sum rules are used to determine the first five expansion coefficients @xmath49 , where @xmath50 due to normalization ( 2.7 ) . however , since the moments are burdened by errors , these expansion coefficients although mathematically uniquely determined by the moments of corresponding order @xcite in practice their numerical values can not be fixed precisely giving rise to different options for the proton da . in our calculation of form factors we employ amplitudes complying with the chernyak - ogloblin - zhitnitsky ( coz ) sum - rule moment constraints . it was shown in @xcite that such amplitudes constitute a finite orbit in the @xmath51 plane ranging from coz - like amplitudes @xcite with @xmath52 to the recently proposed @xcite heterotic one with @xmath53 . for the convenience of the reader , the qcd sum - rules constraints and the expansion coefficients @xmath49 of selected model amplitudes are compiled in table 1 . the @xmath23-dependence of the wave function is contained in the function @xmath54 which is normalized according to @xmath55 \omega _ { 123}(x,\vec{k}_{\perp } ) = 1 . \label{eq : omeganorm}\ ] ] due to ( [ eq : danorm ] ) and ( [ eq : omeganorm ] ) , @xmath12 is the value of the da at the origin of the configuration space . its evolution behavior is given by @xmath56 and its value has been determined to be @xmath57 @xcite . in eq . ( [ eq : ansatz ] ) @xmath58 represents the soft part of the proton wave function , which results by removing the perturbative part and absorbing it into the hard - scattering amplitude @xmath5 . the perturbative tail of the full wave function behaves as @xmath59 for large @xmath23 , whereas the soft part vanishes as @xmath60 or faster . the nonperturbative or intrinsic @xmath23-dependence of the soft wave function , being related to confinement , is parametrized as a simple gaussian according to @xmath61 . \label{eq : blhm - omega}\ ] ] this parametrization of the intrinsic @xmath23-dependence of the wave function , which is due to brodsky , huang , lepage , and mackenzie @xcite , seems to be more favorable than the standard form of factorizing @xmath62- and @xmath23-dependencies . at least for the case of the pion wave function , this has recently been effected by zhitnitsky @xcite on the basis of qcd sum rules . he finds that a factorizing wave function is in conflict with some general theoretical constraints which any reasonable wave function should comply . zhitnitsky s qcd sum - rule analysis of the pion wave function seems to indicate that the @xmath23-distribution may also show a double - hump structure , which means that small and large values of @xmath23 are favored relative to intermediate values . it is likely that the proton wave function may exhibit a similar behavior , though this kind of analysis has yet to be done . for the purposes of the present work we ignore this possibility in ( [ eq : blhm - omega ] ) the parameter @xmath63 controls the root mean square transverse momentum ( r.m.s . ) , @xmath64 , and the r.m.s . transverse radius of the proton valence fock state . from the known charge radius of the proton , we expect the r.m.s . transverse momentum to be larger than about @xmath65 mev . the actual value of @xmath64 may be much larger than @xmath65 mev , e.g. , 600 mev or so . indeed , sotiropoulos and sterman @xcite show that application of the modified hsp to proton - proton elastic scattering leads to an approximate @xmath25-behavior of the differential cross section at moderate @xmath66 . the behavior @xmath67 , predicted by dimensional counting , appears only at very large @xmath66 . at precisely which value of @xmath66 the transition from the @xmath25 to the @xmath27 behavior occurs , depends on the transverse size of the valence fock state of the proton . since the isr @xcite and the fnal @xcite data are rather compatible with a @xmath25-behavior of the differential cross section , sotiropoulos and sterman conclude that the transverse size of the proton is small , perhaps @xmath68 fm . correspondingly , the r.m.s . transverse momentum is larger than 600 mev . it is worth noting that such a value is supported by the findings of the emc group @xcite in a study of the transverse momentum distribution in semi - inclusive deep inelastic @xmath69 scattering . a phenomenologically successful approach to the hsp , in which baryons are viewed as bound states of a quark and an effective diquark , also uses a value of this size for @xmath64 @xcite . there is a second constraint on the wave function , _ viz . _ the probability for finding three valence quarks in the proton : @xmath70 \frac{2\left(\phi _ { 123}(x)\right)^{2 } + \phi _ { 132}(x ) \phi _ { 231}(x)}{x_{1}x_{2}x_{3 } } \leq 1.\ ] ] in our numerical analysis to be presented in sec . 5 , we make use of two different values of the r.m.s . transverse momentum , namely , one which is obtained by the requirement @xmath71 for a given wave function . [ this corresponds to the minimum value of the r.m.s . transverse momentum . ] the other option for the r.m.s . transverse momentum we consider is the rather large value of 600 mev . in the latter case , the probability for the valence quark fock state depends on the wave function . following li @xcite , we write the proton form factor in the form @xmath72[dx ' ] \int_{}^{}[d{}^{2}k_{\perp}][d{}^{2}k_{\perp}^{\prime } ] \sum_{j=1}^{2 } t_{h_{j}}(x , x',\vec{k}_{\perp},\vec{k}'_{\perp},q,\mu ) y_{j}(x , x',\vec{k}_{\perp},\vec{k}'_{\perp},\mu _ { f } ) . \label{eq : g_m}\ ] ] note , however , that our notation is slightly different compared to that of li . making use of the symmetry properties of the proton wave function under permutation , the contributions from the 42 diagrams involved in the calculation of the proton form factor in lowest order can be arranged into two reduced hard scattering amplitudes of the form @xmath73 \left [ x_{2}x_{2}'q^{2 } + ( \vec{k}_{\perp 2 } - \vec{k}_{\perp 2}')^{2 } \right ] } , \label{eq : t_h_1}\ ] ] @xmath74 \left [ x_{2}x_{2}'q^{2 } + ( \vec{k}_{\perp 2 } - \vec{k}_{\perp 2 } ' ) \right ] } , \label{eq : t_h_2}\ ] ] where @xmath75 is the casimir operator of the fundamental representation of @xmath76 . in the hard scattering amplitudes only the @xmath23-dependence of the gluon propagators is included , whereas that of the quark propagators has been neglected . it is expected that this technical simplification introduces only a minor error of about @xmath77 in the final result . for the case of the pion form factor this has been explicitly demonstrated by li @xcite . the functions @xmath78 in ( [ eq : g_m ] ) are short - hand notations for linear combinations of products of the initial and final state wave functions @xmath79 , weighted by @xmath9-dependent factors arising from the fermion propagators , namely : @xmath80 @xmath81 ignoring the transverse momenta in the hard scattering amplitudes ( [ eq : t_h_1 ] ) and ( [ eq : t_h_2 ] ) , and inserting ( [ eq : ansatz ] ) and ( [ eq : omeganorm ] ) , one arrives at the standard hsp result for the magnetic form factor . although this expression is correct in the asymptotic momentum domain , the transverse degrees of freedom are an essential ingredient of the formalism and neglecting them leads to inconsistencies in the endpoint regions , where one of the fractional momenta @xmath9 or @xmath82 tends to zero . after all , it is precisely this approximation that is responsible for the inconsistencies mentioned in the introduction . the power of combining the transverse momentum dependence of the hard scattering amplitude and radiative corrections in the form of sudakov form factors was realized by sterman and collaborators @xcite . ultimately , it leads to a suppression of contributions from the dangerous soft regions , where both the longitudinal and transverse momenta of the quarks are small . in order to include the sudakov corrections , it is advantageous to reexpress eq . ( [ eq : g_m ] ) in terms of the variables @xmath83 , which are canonically conjugate to @xmath84 and span the transverse configuration space . then @xmath72[dx ' ] \int_{}^{}\frac{d{}^{2}b_{1}}{(4\pi ) ^{2 } } \frac{d{}^{2}b_{2}}{(4\pi ) ^{2 } } \sum_{j}^{}\ , \hat{t}_{j}(x , x',\vec{b},q,\mu ) \hat{y}_{j}(x , x',\vec{b},\mu _ { f } ) \ , { \rm e}^{-s_{j } } , \label{eq : g_m(b)}\ ] ] where the fourier transform of a function @xmath85 is defined by @xmath86 since the hard scattering amplitudes depend only on the differences of initial and final state transverse momenta , there are only two independent fourier - conjugate vectors @xmath87 and @xmath88 . they are , respectively , the transverse separation vectors between quarks 1 and 3 and between quarks 2 and 3 . accordingly , the transverse separation of quark 1 and quark 2 is given by @xmath89 [ note that sotiropoulos and sterman @xcite define the transverse separations in a cyclic way which results in the interchange @xmath90 , as compared to our definition . ] the fact that there are only two independent transverse separation vectors is a consequence of the approximation made in the treatment of the hard scattering amplitudes ( [ eq : t_h_1 ] ) and ( [ eq : t_h_2 ] ) which disregards the @xmath23-dependence of the quark propagators . this approximation is justified by the enormous technical simplification it entails , given that the thereby introduced errors are very small . then by virtue of rotational invariance of the system with respect to the longitudinal axis , the form factor ( [ eq : g_m(b ) ] ) can be expressed in terms of a seven - dimensional integral instead of an eleven - dimensional one . physically , the relations @xmath91 , @xmath92 mean that the physical probe ( i.e. , the photon ) mediates only such transitions from the initial to the final proton state , which have the same transverse configurations of the quarks . the fourier - transformed hard scattering amplitudes appearing in eq . ( [ eq : g_m(b ) ] ) read @xmath93 @xmath94 where @xmath95 is the modified bessel function of order 0 and @xmath96 denotes the length of the corresponding vector . we have now chosen the renormalization scale in such a way that each hard gluon carries its own individual momentum scale @xmath97 as the argument of the corresponding @xmath0 . the @xmath97 are defined as the maximum scale of either the longitudinal momentum or the inverse transverse separation , associated with each of the gluons : @xmath98 , \nonumber \\ & t_{21 } & = { \rm max } \left [ \sqrt{x_{1}x_{1}^{\prime}q } , 1/b_{1 } \right ] , \nonumber \\ & t_{12 } & = t_{22 } = { \rm max } \left [ \sqrt{x_{2}x_{2}^{\prime}q } , 1/b_{2 } \right ] , \label{eq : t_ij}\end{aligned}\ ] ] one may think of other choices . however , they are not expected to lead to very different predictions for the form factor @xcite . the quantities @xmath99 contain the same combinations of initial and final state wave functions as those in ( [ eq : y_1 ] ) and ( [ eq : y_2 ] ) , the only difference being that now the products @xmath100 are replaced by corresponding products of fourier - transformed wave functions : @xmath101 . using ( [ eq : ansatz ] ) and ( [ eq : blhm - omega ] ) , the fourier transform of the wave function reads @xmath102 where the fourier - transform of the @xmath23-dependent part is given by @xmath103 \right\}. \label{eq : fourieromega}\ ] ] the exponentials @xmath104 in ( [ eq : g_m(b ) ] ) are the sudakov factors , which incorporate the effects of gluonic radiative corrections . because of this , ( [ eq : g_m(b ) ] ) is not simply the fourier transform of ( [ eq : g_m ] ) but an expression comprising an additional physical input . thus ( [ eq : g_m(b ) ] ) may be termed the `` modified hard - scattering formula '' . on the ground of previous works by collins and soper @xcite , botts and sterman @xcite have calculated a sudakov factor using resummation techniques and having recourse to the renormalization group . they find sudakov exponents of the form @xmath105 \nonumber \\ & + & \sum_{l=1}^{3 } \left [ s(x_{l}^{\prime},\tilde{b}_{l},q ) + \int_{1/\tilde{b}_{l}}^{t_{j2 } } \frac{d\bar{\mu}}{\bar{\mu } } \gamma _ { q}(g(\bar{\mu } ^{2 } ) ) \right ] , \label{eq : s}\end{aligned}\ ] ] wherein the sudakov functions @xmath106 are given by @xmath107 \nonumber \\ & - & \left [ \frac{a^{(2)}}{4\beta _ { 1}^{2 } } - \frac{a^{(1)}}{4\beta _ { 1 } } { \rm ln}\bigl({\rm e}^{2\gamma -1}/2\bigr ) \right ] { \rm ln}\left ( \frac{\hat{q}_{l}}{\hat{b}_{l } } \right ) \nonumber \\ & - & \frac{a^{(1)}\beta _ { 2}}{32\beta _ { 1}^{3 } } \left [ { \rm ln}^{2}(2\hat{q}_{l } ) - { \rm ln}^{2}(2\hat{b}_{l } ) \right ] . \label{eq : s}\end{aligned}\ ] ] here @xmath108 or @xmath109 ( @xmath110 ) and the variables @xmath111 and @xmath112 are defined as follows : @xmath113\ ] ] @xmath114.\ ] ] the coefficients @xmath115 and @xmath116 are @xmath117 @xmath118 where @xmath119 is the number of quark flavors and @xmath120 is the euler - mascheroni constant . in the sequel @xmath121 @xmath122 is the anomalous quark dimension in the axial gauge @xcite . the sudakov function , @xmath106 , in ( [ eq : s ] ) takes into account leading and next - to - leading gluonic radiative corrections of the form shown in fig . [ fig : gluonlines ] . the quantities @xmath123 ( @xmath110 ) are infrared cut - off parameters , naturally related to , but not uniquely determined by the mutual separations of the three quarks @xcite . a physical perspective on the choice of the ir cut - off is provided by the following analogy to ordinary qed . one expects that because of the color neutrality of a hadron , its quark distribution can not be resolved by gluons with a wave length much larger than a characteristic quark separation scale ; meaning that long wave length gluons probe the color singlet proton and hence radiation is damped . radiative corrections with wave lengths between the ir cut - off and an upper limit ( related to the physical momentum q ) yield to suppression ; it is understood that still softer gluonic corrections are already taken care of in the hadron wave function , whereas harder gluons are considered as part of @xmath5 . different choices of the ir cut - off have been used in the literature : thus , li @xcite chooses @xmath124 ( this choice hereafter is termed the `` l '' prescription ) , whereas hyer @xcite in his analysis of the proton - antiproton annihilation into two photons and of the time - like proton form factor as well as sotiropoulos and sterman @xcite take @xmath125 , @xmath126 , @xmath127 ( this choice is denoted the `` h - ss '' prescription ) . still another possibility , and the one proposed in the present work for reasons that will be explained below is to use as ir cut - off the maximum of the three interquark separations , i.e. , to set @xmath128 this choice , designated by `` max '' , is analogous to that in the meson case , wherein the quark - antiquark distance naturally provides a secure ir cut - off . the specific features of each particular cut - off choice will be discussed in detail in sec . [ sec : singularities ] . the integrals in ( [ eq : s ] ) arise from the application of the renormalization group equation ( rge ) . the evolution from one scale value to another is governed by the anomalous dimensions of the involved operators . the integrals combine the effects of the application of the rge on the wave functions and on the hard scattering amplitude . the range of validity of ( [ eq : s ] ) for the sudakov functions is limited to not too small @xmath129 values . whenever @xmath130 is large relative to the hard ( gluon ) scale @xmath131 , the gluonic corrections are to be considered as higher - order corrections to @xmath5 and hence are not contained in the sudakov factor but are absorbed in @xmath5 . for that reason , li @xcite sets any sudakov function @xmath106 equal to zero whenever @xmath132 . moreover , li holds the sudakov factor @xmath104 equal to unity whenever it exceeds this value , which is the case in the small @xmath123-region . actually , the full expression ( [ eq : s ] ) shows in this region a small enhancement resulting from the interplay of the next - to - leading logarithmic contributions to the sudakov exponents and the integrals over the anomalous dimensions . we follow the same lines of argument in our analysis . the ir cut - offs @xmath130 in the sudakov exponents mark the interface between the nonperturbatively soft momenta , which are implicitly accounted for in the proton wave function , and the contributions from soft gluons , incorporated in a perturbative way in the sudakov factors . obviously , the ir cut - off serves at the same time as the gliding factorization scale @xmath11 to be used in the evolution of the wave function . for that reason , li @xcite as well as sotiropoulos and sterman @xcite take @xmath133 . the `` max '' prescription ( [ eq : max ] ) , adopted in the present work , naturally complies with the choice of the evolution scale proposed in @xcite . it is well known that the inclusion of an @xmath62-dependent renormalization scale in the argument of @xmath0 within the standard hsp of brodsky - lepage @xcite presents the difficulty that the value of @xmath0 becomes singular in the endpoint regions . to render the form factors ( eq . ( [ eq : gm ] ) ) finite , additional external parameters , like an effective gluon mass @xcite or a cut - off prescription have to be introduced . technically , such parameters play the rle of ir regulators serving to regularize one of the gluon propagators , which may become soft along the boundaries of phase space ( see , e.g. , @xcite ) . one of the crucial advantages of the modified hsp , proposed by sterman and collaborators @xcite , is that there is no need for external regulators because the sudakov factor may suppress the singularities of the `` bare '' ( one - loop ) @xmath135 inherently . indeed in the pion case , it was shown @xcite that the transverse quark - antiquark separation is tantamount to an ir regulator which suffices to cancel all singularities from the soft region . concerning the proton form factor , the situation is much more complicated because more scales are involved and hence the choice of the appropriate ir cut - off parameters @xmath123 is not obvious , as discussed in sec . iii . as we shall effect in the following , the cancellation of the @xmath0-singularities by the sudakov factor depends sensitively on that particular choice . in fig . [ fig : sudakov ] we display the exponential of the sudakov function @xmath136 $ ] for @xmath137 by imposing li s requirement @xcite : @xmath138 whenever @xmath139 . ultimately , the cancellation of the @xmath0-singularities relies on the fact that whenever one of the @xmath0 tends to infinity ( owing to the limit @xmath140 ) , the sudakov factor @xmath104 rapidly decreases to zero . as it can be observed from fig . [ fig : sudakov ] this is not the case in the region determined by @xmath141 and simultaneously @xmath142 , where @xmath136 $ ] is fixed to unity . in the pion case this does not matter , since the other @xmath143\to 0 $ ] faster than any power of @xmath144 $ ] and , consequently , the sudakov factor drops to zero . in contrast , the treatment of the proton form factor is more subtle . in that case , @xmath104 does not necessarily vanish fast enough to guarantee the cancellation of the @xmath0-singularities . this can be illustrated by the following configuration : if , say , @xmath145 and @xmath146 then @xmath147 and @xmath148 can have any value between 0 and @xmath149 . since @xmath150 is unrestricted within the limits @xmath151 and @xmath152 , the corresponding exponentials of the sudakov functions @xmath153 $ ] and @xmath154 $ ] do not automatically fall off to zero in order to yield sufficient suppression of the @xmath0-singularities , unless all three @xmath123 are coerced to be equal . if the three @xmath123 are allowed to be different , then the sudakov factor provides suppression only through the contributions of the anomalous dimensions . according to the `` l '' and `` h - ss '' prescriptions , which , in general , allow for different @xmath123 in the sudakov functions , the integrand in ( [ eq : g_m(b ) ] ) has singularities behaving as @xmath155 for @xmath156 and @xmath157 hold fixed . the maximum degree of divergence is given by @xmath158 where the first term @xmath159 comes from the evolution of @xmath12 , ( [ eq : f_n ] ) and the constant @xmath160 is related to the anomalous dimension driving the evolution behavior of the proton da , see ( [ eq : phi ] ) and table 1 : @xmath160 is the maximum value of the @xmath161 within a given polynomial order of the expansion of the da . we reiterate that the @xmath45 are positive fractional numbers increasing with @xmath162 . thus the singular behavior of the integrand becomes worse as the expansion in terms of appell polynomials extends to higher and higher orders . the term @xmath163 in ( [ eq : kappa ] ) stems from the integrations over the anomalous dimensions in the sudakov factor @xmath104 ( see ( [ eq : s ] ) ) . finally , the term @xmath152 originates from that @xmath164 which becomes singular in ( [ eq : g_m(b ) ] ) , c.f . , ( [ eq : t_ij ] ) . which one of the @xmath0 couplings becomes actually singular , depends on the prescription imposed on the ir cut - off parameters @xmath123 . the integral ( [ eq : g_m(b ) ] ) does not exist if @xmath165 . as table 1 reveals , this happens already for proton das which include appell polynomials of order 1 , i.e. , for all das except for the asymptotic one : @xmath17 . thus application of the `` l '' and `` h - ss '' prescriptions on the choice of the ir cut - off parameters @xmath123 to the proton form factor entails the modified hsp to be invalid . in view of these results , li s analysis of the proton form factor @xcite seems to be seriously flawed . a simple recipe to bypass the singular behavior of the integrand is to ignore completely the evolution of the da or to `` freeze '' it at any ( arbitrary ) value larger than @xmath166 . hyer @xcite suggested to take for the factorization scale @xmath167 . in this case , the @xmath160 appears in ( [ eq : kappa ] ) only if all three @xmath123 tend to @xmath168 at once . but then at least one of the @xmath136 $ ] drops to @xmath151 faster than any power of @xmath169 . apparently , hyer s choice of the factorization scale avoids singularities of the form ( [ eq : sing ] ) , but seems to us physically implausible . since he only presents numerical results for the proton form factor in the time - like region , we can not compare with his results directly . another option , and actually the one proposed in this work , is to use a common ir cut - off not only for the evolution of the wave function but also in the sudakov exponent . for a common cut - off @xmath170 , the sudakov factors always cancel the @xmath0-singularities ; if , for a given @xmath171 , we are in the dangerous region , @xmath172 , @xmath173 , at least one of the other two sudakov functions lies in the region @xmath174 , @xmath173 ( @xmath175 ) and therefore provides sufficient suppression , as outlined above . in particular , we favor @xmath176 as the optimum choice ( `` max '' prescription ) , since it does not only lead to a regular integral but also to a non - singular integrand . the sudakov factor @xmath177 subject to the `` l '' and `` max '' prescriptions is plotted for a specific quark configuration in fig . [ fig : bcplots ] . this figure makes it apparent that the sudakov factor in connection with the `` max '' prescription is unencumbered by singularities in the dangerous soft regions . as a consequence of the regularizing power of the `` max '' prescription , the perturbative contribution to the proton form factor ( [ eq : g_m(b ) ] ) saturates in the sense that the results become insensitive to the inclusion of the soft regions . a saturation as strong as possible is a prerequisite for the self - consistency of the modified hsp , as will be discussed in sec . v. to demonstrate the amount of saturation , we calculate the proton form factor through ( [ eq : g_m(b ) ] ) , employing a cut - off procedure to the @xmath96-integrations at a maximum value @xmath178 . in fig . [ fig : g_m(b_c ) ] the dependence of @xmath16 on @xmath178 for the three choices , labeled : `` l '' , `` h - ss '' , and `` max '' is shown using , for reasons of comparison with previous works , the coz da and ignoring evolution . [ evolution has been dispensed with to avoid the concomitant singularity in @xmath179 as @xmath180 when imposing the `` l '' and `` h - ss '' prescriptions . ] as one sees from the figure , the `` max '' prescription leads indeed to saturation ; the soft region @xmath181 does not contribute to the form factor substantially . in fact , already @xmath182 of the result are obtained from the regions with @xmath183 . note that @xmath0 increases to a value of @xmath184 at @xmath185 . this indicates that a sizeable fraction of the contributions to the form factor is accumulated in the perturbative region . unfortunately , this saturation is achieved at the expense of a rather strong damping of the perturbative contribution to the proton form factor . using the two other prescriptions ( `` l '' and `` h - ss '' ) and ignoring evolution , we have found larger results for @xmath16 , but no indication for saturation : the additional contributions to the form factor gained this way are accumulated exclusively in the soft regions , i.e. , for values of @xmath186 near @xmath152 . these findings are in evident contradiction to li s results ( figure 5 in @xcite ) for which an acceptable saturation has been claimed . on the other hand , we can qualitatively confirm the saturation behavior of the proton form factor calculated by hyer @xcite in the time - like region . since we regard a saturation behavior as a stringent test for the self - consistent applicability of pqcd , calculations which accumulate large contributions from soft regions ( large @xmath178 ) can not be considered as theoretically legitimate . the rle of the evolution effect subject to the `` max '' prescription is also exhibited in fig . [ fig : g_m(b_c ) ] . it shows that the effect of evolution is large , although finite , owing to the strong suppression provided by the sudakov factor . note that according to our discussion in sec . iii , the factorization scale is @xmath187 . the significant feature of the evolution effect is that it tends to neutralize the influence of the ir cut - off . thus one obtains larger values of the proton form factor at the expense of a slightly worse saturation . in this section we give numerical results for the proton form factor . in these calculations we throughout employ the `` max '' prescription with evolution included , using @xmath188180 mev and @xmath189 gev . before proceeding with the presentation of our final results , let us investigate the effect of including the intrinsic transverse momentum in our calculations . the @xmath23-dependence of the proton wave function effectively introduces a confinement scale in the formalism , the importance of which may be appreciated by looking at fig . [ fig : g_m(q^2 ) ] . this figure shows results , obtained for the coz da without @xmath23-dependence and for two different values of @xmath190 . to describe the intrinsic @xmath23-dependence , one can use ( [ eq : blhm - omega ] ) or , after transforming to the transverse configuration space , ( [ eq : fourieromega ] ) . notice that in li s approach the gaussian in ( [ eq : fourieromega ] ) has been replaced by unity . the oscillator parameter @xmath63 is determined in such a way that either the normalization of the wave function @xmath191 is unity ( resulting into @xmath192 mev for the coz da ) , or by inputing the value of the r.m.s . transverse momentum . in the second case , we use a value of @xmath193 mev ( see the discussion in sec . ii ) , which implies @xmath194 . as can be seen from this figure , the predictions for the form factor are quite different for the three cases . the intrinsic @xmath23-dependence of the wave function leads to further suppression of the perturbative contribution , which becomes substantial if the r.m.s . transverse momentum is large . on the other hand , this suppression is accompanied by an increasing amount of saturation , since also the gaussian ( [ eq : fourieromega ] ) suppresses predominantly contributions from the soft regions , viz . , the large @xmath195-regions . in contrast to the sudakov factor , however , this suppression is @xmath196-independent . the interplay of the two effects , sudakov suppression and intrinsic transverse momentum , leads to a different @xmath196-behavior of the form factor depending on the value of the r.m.s transverse momentum , as can be seen from fig . [ fig : g_m(q^2 ) ] . the @xmath196-dependence beyond @xmath4 gev@xmath3 is rather weak , being approximately compatible with dimensional counting ( modulo logarithmic corrections ) . for very large values of @xmath196 beyond @xmath197 gev@xmath3 the three curves have approached each other within @xmath77 accuracy . this happens when the sudakov factor dominates the gaussian ( [ eq : fourieromega ] ) and selects those configurations with small interquark separations . in this region , which one may consider as the pure perturbative region , the results for the form factor are independent of the confinement scale introduced by the r.m.s . transverse momentum . the penalty of the additional suppression of the perturbative contribution caused by the gaussian ( [ eq : fourieromega ] ) is mitigated by the advantage that the perturbative contribution becomes more self - consistent than by the sudakov factor alone . this is indicated in the enhanced amount of saturation with increasing r.m.s . transverse momentum . adapting the criterion of self - consistency , originally suggested by li and sterman @xcite for the pion case , namely that @xmath198 of the results are accumulated at moderate values of the coupling constant , say , @xmath199 , we find self - consistency for @xmath200 gev@xmath3 ( for the coz da ) . finally , in fig . [ fig : strip ] , we demonstrate the effect of different proton das on the form factor . to this end , we investigate a set of @xmath201 das @xcite , which all respect the qcd sum - rule constraints @xcite . the results for the various das or more precisely wave functions , since we include their intrinsic transverse momentum dependence obtained under the `` max '' prescription with evolution included , form the shaded area shown in the figure . all wave functions are normalized to unity and the corresponding r.m.s . transverse momenta vary between @xmath202 mev and @xmath203 mev ( see table 1 ) . the theoretical form - factor predictions span a `` band '' congruent to the `` orbit '' of solutions found in @xcite . the upper bound of the `` band '' corresponds to the da coz@xmath204 , which yields the maximum value of the form - factor ratio @xmath205 in the standard hsp . the lower limit of the `` band '' is obtained using the da `` low '' ( sample 8 in @xcite ) with @xmath206 . explicitly shown are the results for the coz da @xcite , its optimized version ( with respect to the sum - rule constraints ) and the `` heterotic '' da , recently proposed by two of us in @xcite . we note that the differences among these curves practically disappear already at about @xmath207 gev@xmath3 , despite the fact that these amplitudes have distinct geometrical characteristics @xcite . since the true valence fock state probability is likely much smaller , or invariably the r.m.s . transverse momentum larger than of order of @xmath208 mev , the `` band '' describes rather _ maximal _ expectations for the ( leading - order ) perturbative contributions to the form factor ; at least for proton wave functions of the type we utilize . comparison with the experimental data reveals that the theoretical predictions amount , at best , to approximately @xmath198 of the measured values . this is the benchmark against which we have to discern novelties and aberrations . closing this discussion we note that , since we are calculating only the helicity - conserving part of the current matrix element it is not obvious whether we should compare the theoretical predictions with the data for the sachs form factor @xmath16 or the dirac form factor @xmath209 . therefore we have exhibited in fig . [ fig : strip ] both sets of data @xcite for comparison . since the two sets of data differ by only @xmath77 , our conclusions concerning the smallness of the theoretical results remain unaffected . the various model wave functions led to self - consistency of the perturbative contribution , i.e. , @xmath198 of the results are accumulated in regions where @xmath199 , in the range of @xmath1 between @xmath2 and @xmath4 gev@xmath3 . the objective of the present work has been to derive the proton magnetic form factor within the modified version ( sec . iii ) of the standard brodsky - lepage hsp @xcite , a scheme which takes into account gluonic radiative corrections @xcite in terms of transverse separations . this is done by incorporating in the formalism the sudakov factor , calculated by botts and sterman @xcite . there are already some interesting applications of the modified hsp @xcite . the significant element of this type of analyses is that the @xmath0-singularities , arising from hard - gluon exchange and evolution , can be cancelled without introducing free external parameters . we emphasize that in contrast to pure phenomenological recipies ( e.g. , the introduction of a gluon mass ) , the modified hsp provides an explicit scheme how the ir protection of the `` bare '' @xmath0 proceeds through gluonic radiation accumulated in the sudakov factor . thus , in the modified hsp , one may conceive of the ( finite ) ir - protected @xmath0 as being the effective coupling . by this procedure the potentially dangerous soft regions of momenta are suppressed entailing also a reduction of the perturbative contribution to the form factor . while in the pion case @xcite , it is fortunate that the cancellation of the @xmath0-singularities comes out naturally , li s approach to the proton form factor @xcite leads to a lack of complete cancellation of the @xmath0-singularities ( see sec . iv ) . without evolution of the proton wave function the emerging singularities in ( [ eq : g_m(b ) ] ) are still integrable , but logarithmic corrections due to evolution yield ultimately to uncompensated singularities . on the grounds of our discussion , we are reasonably confident that li s treatment can be cured within the modified hsp . we suggest to use a a common ir cut - off in the sudakov exponents ( [ eq : s ] ) and sudakov functions ( [ eq : s ] ) : viz . , the maximum transverse separation . this `` max '' prescription provides sufficient ir protection , since even with evolution , the integrand in ( [ eq : g_m(b ) ] ) remains finite . a significant feature of this treatment is that the proton form factor saturates , i.e. , it becomes insensitive to the contributions from large transverse separations . the other choices of the ir cut - off ( `` l '' , `` h - ss '' ) , we have discussed , do not lead to saturation . however , this reliable saturation and ir protection of the form factor is achieved at the expense of a strong reduction of the perturbative contribution to the form factor . the damping of the proton form factor becomes even stronger if one takes into account the intrinsic transverse momentum dependence of the proton wave function ( see fig . [ fig : g_m(b_c ) ] and fig . [ fig : g_m(q^2 ) ] ) . this has been done by assuming a non - factorizing @xmath62 and @xmath23-dependence of the wave function of the brodsky - lepage - huang - mackenzie @xcite type and fixing the value of @xmath190 either via the valence quark probability @xmath191 or by inputing the value @xmath193 mev by hand @xcite . a remarkable finding is that the form factor calculated within the modified hsp , appropriately extended to include the intrinsic transverse momentum of the proton wave function , shows only a mild dependence on the particular model da . the perturbative contribution to the form factor becomes self - consistent in all cases for momentum transfers larger than @xmath2 to @xmath4 gev@xmath3 . the actual value of the onset of self - consistency depends on the particular wave function and the r.m.s . transverse momentum chosen . self - consistency is defined such that @xmath198 of the result are accumulated in regions where @xmath210 is smaller than @xmath211 ( sec . v ) . comparing our theoretical results with the data , it turns out that they fall short by at least @xmath198 . this is true not only for the coz da , ( which we have exemplarily used to facilitate comparison with previous works ) but actually for the whole spectrum of amplitudes determined in @xcite and found to comply with the coz sum - rule requirements . depending on the actual value of the r.m.s . transverse momentum , the reduction of the perturbative contribution may be even stronger than @xmath198 . the fact that in all considered cases the self - consistently calculated perturbative contribution to the proton form factor fails to reproduce the existing data , is perhaps a signal that soft contributions ( higher twists ) not accounted for so far by the modified hsp should be included . such contributions comprise , e.g. , improved and/or more complicated wave functions , orbital angular momentum , higher fock components , quark - quark correlations ( diquarks ) , radiative corrections to the quark and gluon condensates , quark masses , etc . also remainders of genuine soft contributions , like vector - meson - dominance terms or the overlap of the soft parts of the wave functions ( feynman contributions ) , may still be large at accessible momentum transfers . the rather large value of the pauli form factor @xmath14 around @xmath4 gev@xmath3 , as found experimentally @xcite , indicates that sizeable higher - twist contributions still exist in that region of momentum transfer @xcite . one may suspect similar or even larger higher - twist contributions to the helicity non - flip current matrix element controlling @xmath209 and @xmath16 . large ( perturbative ) higher - order corrections to the hard - scattering amplitude can not be excluded as well , since their size has not yet been estimated . in analogy to the drell - yan process , these corrections may be condensed in a k - factor multiplying the leading - order perturbative result . however , with our choice of the renormalization scale , the k - factor is expected to be close to unity . at least for the case of the pion form factor , calculations of the k - factor to one - loop order exist @xcite , which indicate that choosing the renormalization scale analogously to ours , the value of the k - factor is indeed close to unity . in conclusion we note that it was not our primary aim to use the modified hsp to obtain best agreement with the data , although from our point of view this scheme represents a decisive step towards a deeper understanding of the electromagnetic form factors . in the present work the focus has been placed on theoretical problems , overlooked previously . lepage , s.j . brodsky : phys . d22 ( 1980 ) 2157 v.l . chernyak , i.r . zhitnitsky : nucl . b246 ( 1984 ) 52 m. gari , n.g . stefanis : phys . b175 ( 1986 ) 462 ; 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we parallel the analysis given previously by li and make apparent a number of serious objections .
the appropriate cut - off needed to render the form - factor calculation finite is both detailed and analyzed by considering different cut - off prescriptions .
the use of the maximum interquark separation as a common infrared cut - off in the sudakov suppression factor is proposed , since it avoids difficulties with the @xmath0-singularities and yields a proton form factor insensitive to the inclusion of the soft region which therefore can be confidently attributed to perturbative qcd .
results are presented for a variety of proton wave functions including also their intrinsic transverse momentum .
it turns out that the perturbative contribution , although theoretically self - consistent for @xmath1 larger than about @xmath2 gev@xmath3 to @xmath4 gev@xmath3 , is too small compared to the data .
= 0 |
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there are various models of collective opinion formation in which agents modify their opinions according to interaction with other agents @xcite . opinion formation is a dynamic process : for example , interaction between agents makes their opinions approach each other . an important problem in opinion dynamics is to examine when an agreement ( i.e. , consensus ) among all the agents occurs . complete agreement is rarely observed in the real world @xcite . however , it is an established fact that opinion dynamics under the voter model , a classical opinion model in statistical physics and probability theory , inevitably reaches agreement in finite populations @xcite . the majority rule model has a similar feature @xcite . partly motivated by this discrepancy , various extensions of voter and majority rule models and different models of collective opinion formation have been proposed to account for the disagreement in finite populations . examples include the deffuant model @xcite , language competition models @xcite , voter - like models on adaptive networks @xcite , voter model under partisan bias ( the assumption that agents naturally prefer one opinion ) @xcite , and variations of axelrod s cultural dynamics ( see @xcite for references ) . theoretical models have also been proposed in social sciences to explain disagreement in the context of polarization . for example , prior beliefs or initially received signals can cause disagreement between agents , even if they receive the same public signals from then on @xcite . although there is a plethora of studies addressing the problem of agreement and disagreement in opinion dynamics , we propose a model incorporating two factors that are relevant to human behavior : bayesian belief updating and confirmation bias . bayesian belief updating is commonly used in studies of the decision making of agents receiving uncertain information @xcite . the confirmation bias is a psychological bias inherent in humans , in which an agent inclined towards an opinion tends to misperceive incoming signals as supporting the agent s belief @xcite . a non - bayesian model with the confirmation bias was previously proposed for explaining the influences of media and interactions between agents @xcite . we are not the first to study opinion formation under the bayesian updating and confirmation bias . in the framework of single agent opinion formation , rabin and schrag showed that the confirmation bias triggers overconfidence and can cause the individual to hold incorrect beliefs , even if it receives a series of external signals suggesting the true state of the world @xcite . orlan studied the bayesian dynamics of agents subjected to the confirmation bias , interacting through the mean field @xcite . the model yields agreement or disagreement depending on the parameter values . in this study , motivated by the rabin - schrag model @xcite , we propose a model of collective opinion formation with a confirmation bias . we model direct peer - to - peer interactions between agents ( not through the mean field ) and their effects on the bayesian updating of each agent . to study the pure effects of interactions among agents , we do not assume that agents receive signals from the environment as in previous studies @xcite . we numerically simulate the model to reveal the conditions under which the populations of agents agree and disagree , depending on the values of parameters such as the strength of the confirmation bias , fidelity of the signal , and the system size . our model modifies the bayesian decision - making model proposed by rabin and schrag @xcite in two main ways . first , we consider a well - mixed population of bayesian agents that interact with each other ; rabin and schrag focused on the case of the single agent . second , agents do not receive external signals from the environment in our model . in the rabin - schrag model , such an external signal , which represents the correct " answer in the binary choice situation ( i.e. , the true state of nature ) , is assumed . by making the two changes , we concentrate on collective opinion formation by bayesian agents , whereby there are two possible alternative opinions of equal attractiveness . we label the @xmath0 agents @xmath1 and denote the opinion of agent @xmath2 ( @xmath3 ) by @xmath4 , where a and b are the alternative opinions . we assume that agents are not perfectly confident in their opinions . to model this factor , we adopt the bayesian formalism used by rabin and schrag @xcite . we denote by @xmath5 the strength of the belief ( hereafter , simply the belief ) with which agent @xmath2 believes in opinion a. a parallel definition is applied to @xmath6 . it should be noted that @xmath7 , @xmath8 , and @xmath9 . if @xmath10 , agent @xmath2 is indifferent to either opinion . we update the agent s belief as follows . the time @xmath11 starts from @xmath12 . upon every updating of an agent s belief , we add @xmath13 to @xmath11 such that the belief of each agent is updated once per time unit on average . in an updating event , we select an agent @xmath2 to be updated with equal probability @xmath13 . agent @xmath2 refers to agent @xmath14 s opinion for updating @xmath2 s belief @xmath5 , where @xmath14 ( @xmath15 ) is selected with equal probability @xmath16 from the population . agent @xmath14 imparts a signal @xmath17 , where @xmath18 and @xmath19 correspond to @xmath14 s opinions a and b , respectively . we assume that the probabilities that agent @xmath14 imparts @xmath20 and @xmath21 are given by @xmath22 and @xmath23 respectively , where @xmath24 represents the reliability of the signal , and @xmath25 . if @xmath14 is confident in its own opinion and the transformation from @xmath14 s belief [ i.e. , @xmath26 to @xmath14 s output signal ( i.e. , @xmath18 or @xmath19 ) is reliable , signals @xmath18 and @xmath19 are likely to indicate opinions a and b , respectively . in the limit @xmath27 , @xmath28 and @xmath29 . if @xmath30 , @xmath31 such that @xmath32 does not convey any information about @xmath14 s belief . we implicitly assume that all the agents share the same value of @xmath33 and that they know this fact when performing the bayesian update , as described below . when agent @xmath14 imparts signal @xmath34 , agent @xmath2 is assumed to perceive a subject signal @xmath35 , where @xmath36 and @xmath37 correspond to @xmath38 and @xmath39 , respectively . the flow of the signal conversion is depicted in fig . [ fig : signals ] . if agent @xmath2 is not subject to the confirmation bias , @xmath36 and @xmath37 are equal to @xmath18 and @xmath19 , respectively . otherwise , agent @xmath2 may misinterpret the signal imparted by agent @xmath14 , depending on the prior exposure of agent @xmath2 to other signals . following rabin and schrag @xcite , we define @xmath40 = \pr[\sigma=\beta | s = b,\pr(x_i={\rm a } ) \le 1/2 ] = 1 \label{eq : s_to_sigma_bias_1}\end{aligned}\ ] ] and @xmath41 = \pr[\sigma=\beta | s = a,\pr(x_i={\rm a } ) < 1/2 ] = q , \label{eq : s_to_sigma_bias_2}\end{aligned}\ ] ] where @xmath42 ( @xmath43 ) parameterizes the strength of the confirmation bias . equation states that an agent preferring opinion a misinterprets an arriving @xmath19 signal as @xmath38 ( i.e. , @xmath44 ) with probability @xmath42 . if @xmath45 , the confirmation bias is absent , and @xmath20 and @xmath21 are always converted to @xmath44 and @xmath46 , respectively . if @xmath47 , the agent perceives the signal that is consistent with its current preference [ i.e. , @xmath36 if @xmath48 and @xmath37 if @xmath49 , irrespective of the signal imparted by agent @xmath14 ( i.e. , @xmath18 or @xmath19 ) . the other conditional probabilities can be readily derived from eqs . ( [ eq : s_to_sigma_bias_1 ] ) and ( [ eq : s_to_sigma_bias_2 ] ) . for example , eq . ( [ eq : s_to_sigma_bias_1 ] ) implies @xmath50 = 1 - \pr[\sigma=\alpha|s = a , \pr(x_i={\rm a})>1/2 ] = 0 , \label{eq : s_to_sigma_bias_3}\end{aligned}\ ] ] and eq . ( [ eq : s_to_sigma_bias_2 ] ) implies @xmath51 = 1- \pr[\sigma=\beta|s = a , \pr(x_i={\rm a})<1/2 ] = 1-q . \label{eq : s_to_sigma_bias_4}\end{aligned}\ ] ] then , by using the bayes theorem , we update agent @xmath2 s belief @xmath52 on the basis of the old belief @xmath5 [ @xmath53 and the perceived signal ( i.e. , @xmath36 or @xmath37 ) . the perceived signal may be different from the received signal ( i.e. , @xmath18 or @xmath19 ) because of their confirmation bias [ eqs . and ] . we assume that agents are not aware that they may be subject to the confirmation bias . agents use the subjective conditional probabilities given by @xmath54 and @xmath55 to perform the bayesian update . the posterior belief @xmath52 is given by @xmath56 it should be noted that @xmath57 . then , we increment the time by @xmath13 such that each agent is updated once per unit time on average . iterative application of eq . ( [ eq : pia_update ] ) leads to @xmath58 and @xmath59 where @xmath60 ( @xmath61 ) is the accumulated number of signals @xmath44 ( @xmath46 ) that agent @xmath2 has perceived . the state of each agent @xmath2 is uniquely determined by @xmath62 , which is consistent with basic bayesian theory @xcite . unless otherwise stated , we set @xmath63 and assume a neutral initial condition @xmath64 ( @xmath65 ) , or , equivalently , @xmath66 ( @xmath67 ) . the agents exchange signals and update their beliefs , possibly under a confirmation bias . after a transient , the agents believe in either opinion with a strong confidence , i.e. , @xmath68 or @xmath69 . we halt a run when @xmath70 is satisfied for all @xmath2 for the first time , where @xmath71 is the threshold . in other words , a run continues if at least one agent @xmath2 has the @xmath72 value smaller than @xmath71 . we first consider the case without a confirmation bias ( i.e. , @xmath45 ) . we investigate the dynamics of the mean belief @xmath73 at time @xmath11 by drawing a return map , i.e. , @xmath74 as a function of @xmath75 @xcite . the return map for @xmath76 , @xmath77 , and @xmath78 based on @xmath79 runs is shown in fig . [ fig : meanp_ba ] . because @xmath80 when @xmath81 and @xmath82 when @xmath83 , the dynamics is in accordance with majority rule behavior . all @xmath79 runs finished with an agreement of opinion a [ i.e. , @xmath84 for all @xmath2 ] or opinion b [ i.e. , @xmath86 for all @xmath2 ] . each case occurred approximately half the time . we turn on the confirmation bias to examine the possibility that it induces disagreement among agents . at least for large @xmath42 ( i.e. , @xmath87 ) , disagreement is expected to be reached because the first perceived signal would determine the final belief of each agent and is equally likely to be @xmath36 and @xmath37 for many agents . in the following numerical simulations , we measured the degree of disagreement , which we defined as follows . we determined that agreement was reached in a run if the final signs of @xmath62 were the same for @xmath3 . otherwise , we said that disagreement was reached . we denoted the fraction of runs that finished with disagreement by @xmath88 . we set @xmath78 and the number of runs to @xmath79 . in figs . [ fig : r_bm](a ) and [ fig : r_bm](b ) , @xmath88 is shown as a function of @xmath42 and @xmath33 for @xmath89 and @xmath90 , respectively . first , @xmath88 monotonically increases with @xmath42 and decreases with @xmath33 for both @xmath89 and @xmath76 . it should be noted that disagreement occurred in at least one run in the regions right to the solid fractured lines in fig . [ fig : r_bm ] . second , @xmath88 for @xmath89 [ fig . [ fig : r_bm](a ) ] is smaller than @xmath88 for @xmath76 [ fig . [ fig : r_bm](b ) ] for all the @xmath42 and @xmath33 values . therefore , disagreement seems to be a likely outcome of the model for large @xmath0 , particularly for large @xmath42 and small @xmath33 . when @xmath76 , perfect agreement , i.e. , @xmath91 , is realized only for @xmath42 close to zero . in other words , even a small degree of confirmation bias elicits disagreement among the agents . to obtain analytical insights into the model , we performed an annealed approximation for @xmath89 by averaging out fluctuations of the dynamics for different times and runs . the configuration of the population is specified by @xmath92 . the stochastic dynamics of the model can be mapped to a random walk on the two - dimensional lattice ; a walker is initially located at @xmath93 and randomly hops to one of the four neighboring lattice points in each time step . we defined @xmath94 , @xmath95 , @xmath96 , and @xmath97 as the probabilities that the walker located at @xmath98 moves to @xmath99 , @xmath100 , @xmath101 , and @xmath102 , respectively . the four probabilities are given by @xmath103 \dfrac{\pr(s = a | m_2)}{2 } & ( m_1 = 0 ) , \\[6pt ] \dfrac{(1-q ) \pr(s = a | m_2)}{2 } & ( m_1 \le -1 ) , \end{cases } \label{eq : fr}\end{aligned}\ ] ] @xmath104 @xmath105 \dfrac{\pr(s = a | m_1)}{2 } & ( m_2 = 0 ) , \\[6pt ] \dfrac{(1-q ) \pr(s = a | m_1)}{2 } & ( m_2 \le -1 ) , \end{cases } \label{eq : fu}\end{aligned}\ ] ] and @xmath106 where @xmath107 is the probability that agent @xmath14 with @xmath108 imparts signal @xmath20 . @xmath94 and @xmath96 increase with @xmath109 and @xmath110 , and @xmath95 and @xmath97 decrease with @xmath109 and @xmath110 . in the following , we study the mean dynamics of the random walk driven by the drift terms . because the transition probability of the random walk is symmetric with respect to the lines @xmath111 and @xmath112 , we focus on the region given by @xmath113 . we define @xmath114 and @xmath115 , which are not integers in general , as the values satisfying @xmath116 and @xmath117 , respectively . they are given by @xmath118^{-1}}{\ln \dfrac{\theta}{1-\theta}}. \label{eq : r2_fr = fl}\end{aligned}\ ] ] note that @xmath114 and @xmath115 exist if and only if @xmath119 , i.e. , @xmath120 first , we consider the case @xmath121 . we partition the upper quadrant of the lattice ( given by @xmath113 ) into five regions : region @xmath69 ( @xmath122 ) , region @xmath123 ( @xmath124 ) , region @xmath125 ( @xmath126 ) , region @xmath127 ( @xmath128 ) , and region @xmath129 ( @xmath130 ) , as shown in fig . [ fig : schematicview_fr - fl_fu - fd](a ) . we obtain from the condition @xmath131 @xmath132 in region @xmath69 , @xmath133 in region @xmath123 , @xmath134 in region @xmath125 , @xmath135 in region @xmath127 , and @xmath136 in region @xmath129 . the probability flow of the walker after the annealed approximation , i.e. , ( @xmath137 ) inferred from eqs . - is shown schematically in fig . [ fig : schematicview_fr - fl_fu - fd](a ) . if the walker is in the second quadrant ( i.e. , regions @xmath125 , @xmath127 , and @xmath129 ) where the two agents disagree with each other , the random walker is likely to eventually escape and enter the first quadrant ( i.e. , region @xmath69 ) where the two agents agree with each other . in fact , fig . [ fig : vectorplot](a ) , which shows the actual probability flow , indicates that the agreement necessarily occurs . therefore , agreement is the expected outcome when @xmath138 . second , if @xmath139 , regions @xmath127 and @xmath129 are absent because @xmath140 and @xmath141 diverge . regions @xmath69 and @xmath123 , in which inequalities and are satisfied , respectively , are the same as those in the case @xmath138 . region @xmath125 , in which inequality is satisfied , is modified to @xmath142 . the probability flows are schematically shown in fig . [ fig : schematicview_fr - fl_fu - fd](b ) . @xmath143 and @xmath144 are satisfied in region @xmath125 . therefore , once the walker is deep in the second quadrant , it is likely to move toward @xmath145 and @xmath146 , which implies that two agents finally disagree . the actual probability flow shown in fig . [ fig : vectorplot](b ) is consistent with this prediction . the transition line @xmath147 is shown by the dashed line in fig . [ fig : r_bm](a ) . it accurately predicts the parameter region in which disagreement can occur , i.e. , the region right to the solid line . the same transition line is also derived for the rabin - schrag model , which is concerned with a single agent subjected to a confirmation bias @xcite . in their model , the agent forms a belief by repetitively receiving a stochastic signal @xmath34 from nature , according to @xmath148 . rabin and schrag calculated the probability that the agent eventually misunderstands the state of the nature ( i.e. , a or b ) , starting from neutral belief . this probability is equal to zero when @xmath149 and positive when @xmath150 ( see proposition @xmath127 in @xcite ) . our results obtained in this section are consistent with theirs because disagreement in our model roughly corresponds to misunderstanding in the rabin - schrag model . in general , there are @xmath151 disagreement configurations , as distinguished by the number of agents that finally believe in opinion a , which ranges from @xmath69 to @xmath151 . to distinguish different disagreement configurations , we examined the fraction of agents that believed in the minority opinion at the end of a run . we averaged this fraction over the runs ending with disagreement . we called this quantity the average size of the minority . figures [ fig : r_minority_minoritymin](a ) and [ fig : r_minority_minoritymin](b ) show the average size of the minority for @xmath152 and @xmath76 , respectively . the black regions indicate the parameter values for which the average size of the minority is undefined because all @xmath79 runs end with agreement . when @xmath42 is small , the average size of the minority monotonically decreases with @xmath42 and monotonically increases with @xmath33 for both @xmath152 and @xmath90 . therefore , small @xmath42 and large @xmath33 values allow only balanced disagreement configurations , in which the numbers of the agents believing in the opposite opinions are close to @xmath153 . however , the average size of the minority increases when @xmath42 is large . this is particularly the case for @xmath76 [ fig . [ fig : r_minority_minoritymin](b ) ] . this increase occurs for the following reason . with a strong confirmation bias , agents end up with an opinion consistent with a small number of signals perceived in the early stages , and both signals are equally likely to be observed in the early stages under neutral initial conditions . in the extreme case in which @xmath47 , agents reinforce the opinion that is consistent with their first perceived signal . therefore , unbalanced disagreement configurations are rarely realized when @xmath42 is large . figures [ fig : r_bm ] and [ fig : r_minority_minoritymin ] suggest that the agreement is unlikely to be reached in a large population . to examine the effect of the population size , we defined @xmath154 as the value of @xmath42 such that a threshold number of runs among @xmath155 runs end with agreement . for a given @xmath33 value , we determined @xmath154 by the bisection method . the number of agreement runs may not monotonically change in @xmath42 because the number of runs is finite . therefore , the bisection method does not perfectly work in general . however , we corroborated that the following results were negligibly affected by the lack of monotonicity . the dependence of @xmath154 on @xmath0 is shown in fig . [ fig : qcr_vs_n](a ) for three threshold values . for example , the results for the threshold value @xmath90 ( shown by circles ) indicate that at least @xmath90 runs among the @xmath155 runs end up with disagreement when @xmath156 . we set @xmath77 and @xmath78 . to explore the possibility of disagreement in large populations , we set @xmath33 close to @xmath69 . it should be noted that fig . [ fig : r_bm ] indicates that the probability of disagreement is small for a large @xmath33 value . in fig . [ fig : qcr_vs_n](a ) , @xmath154 quickly decreases for @xmath157 and gradually decreases for @xmath158 . disagreement often occurs for large @xmath0 unless @xmath42 is small . nevertheless , fig . [ fig : qcr_vs_n](a ) suggests that the range of @xmath42 for which agreement always occurs survives for diverging @xmath0 . in generating fig . [ fig : qcr_vs_n](a ) , we used an initial condition in which all the agents had a neutral belief [ i.e. , @xmath159 . to check the effect of the initial condition , we investigated the dependence of @xmath154 on @xmath0 under two other initial conditions . in the bimodal initial condition , we initially set @xmath160 ( @xmath161 ) and @xmath162 ( @xmath163 ) . we assumed that @xmath0 was even for this initial condition . in the so - called most unbalanced initial condition , we set @xmath164 and @xmath162 ( @xmath165 ) . the numerical results for the two initial conditions are shown in figs . [ fig : qcr_vs_n](b ) and [ fig : qcr_vs_n](c ) . the parameter values @xmath77 , @xmath78 are the same as those used in fig . [ fig : qcr_vs_n](a ) . the transition point @xmath154 decreases with @xmath0 more rapidly with the bimodal initial condition [ fig . [ fig : qcr_vs_n](b ) ] than with the neutral initial condition [ fig . [ fig : qcr_vs_n](a ) ] . this result is intuitive : the bimodal initial condition paves the way to disagreement . in contrast , @xmath154 under the most unbalanced initial condition is almost constant near @xmath166 irrespective of @xmath0 . therefore , disagreement is highly unlikely unless the confirmation bias is strong ( i.e. , @xmath42 is greater than @xmath166 ) . the results shown in fig . [ fig : qcr_vs_n ] suggest that the eventual behavior of the model strongly depends on the initial condition even after the results are averaged over runs . our numerical results are summarized as follows . when the confirmation bias is absent ( i.e. , @xmath45 ) , the opinion dynamics under the bayesian update rule leads to the complete agreement among agents . the behavior of the model is similar to majority rule dynamics ( fig . [ fig : meanp_ba ] ) . when the confirmation bias is present , disagreement is a likely outcome , particularly for a strong confirmation bias ( i.e. , large @xmath42 ) . disagreement is also more likely for a lower fidelity of the signal ( i.e. , @xmath167 ) and a larger system size . the transition line separating the parameter region in which both agreement and disagreement can occur and that in which only agreement occurs is approximately given by @xmath168 when @xmath89 . this line is identical to the one determined by rabin and schrag for their model for a single agent s decision making @xcite . finally , the behavior of the model strongly depends on the initial condition . our model and results are different from orlan s @xcite , although orlan s model employs multiple agents that perform the bayesian updates under a confirmation bias . first , the belief of each agent is binary in orlan s model , whereas our model introduces an infinite range of discrete beliefs , as in @xcite . second , interaction between agents is introduced differently in the two models . in orlan s model , each agent refers to the global fraction of agents believing in one of the two opinions . in our model , agents refer to other opinions by peer - to - peer interaction , i.e. , by receiving a binary signal that is correlated with the belief of the sender . third , the stochastic dynamics of orlan s model is ergotic when the collective opinion does not reach agreement . the collective opinion obeys a stationary distribution , irrespective of the initial condition . in contrast , in all our simulations , the stochastic dynamics of our model was nonergotic , such that the final configuration depended on the initial condition in a wide parameter region . in social science studies of polarization , several authors analyzed bayesian models in which different agents receiving a series of common signals end up in disagreement . the proposed mechanisms governing disagreement include different initial beliefs or factors that affect perception of later incoming signals @xcite , different update rules @xcite , and ambiguity aversion @xcite . these models and ours are different in three major ways . first , a ground truth opinion corresponding to the state of nature is assumed in these models but not in ours . second , public signals commonly received by different agents are assumed in these models but not in ours . third , the agents do not have direct peer - to - peer interaction in these models , but they do in ours . models with interacting bayesian agents , which show disagreement ( reviewed in ref . @xcite ) , are also different from our model in the first respect . it should be noted that zimper and ludwig discussed confirmation bias with their bayesian model @xcite . however , they derived a confirmation bias from their model , rather than assuming one , such that their results pertaining to confirmation bias were also distinct from ours . extending our model to the case of networks is straightforward . for example , we can select a recipient of the signal with probability @xmath13 and then select the sender with equal probability among the neighbors of the recipient on the network . another possible update rule is to select the sender first and then the recipient among the sender s neighbors . yet another possibility is to select a link with equal probability and designate one of the two agents as sender and the other as recipient . on heterogeneous networks , the results may depend on the update rule because it is the case in the voter model @xcite . extension of the model to the case of confirmation bias heterogeneity may also be interesting . neurological evidence shows that different individuals have different confirmation bias strengths @xcite . the strength of the confirmation bias and the position of the node in a social network may be correlated and affect the dynamics . it is also straightforward to extend the model to the case of multiple opinion cases . these and other extensions , along with the study of analytically tractable models that capture the essence of the present study , warrant future work . we thank mitsuhiro nakamura , taro takaguchi , and shoma tanabe for critical reading of the manuscript . this work is supported by grants - in - aid for scientific research [ grant 23681033 and innovative areas systems molecular ethology " ( grant no . 20115009 ) ] from mext , japan . , @xmath45 , and @xmath77 . we recorded the values of @xmath169 ) for @xmath170 for @xmath79 runs and divided the recorded pairs into @xmath171 classes . the @xmath172th class ( @xmath173 ) was composed of the pairs satisfying @xmath174 . we obtained the mean value @xmath175 for the @xmath172th class by averaging @xmath74 over all the pairs contained in the @xmath172th class . finally , we plotted @xmath175 against @xmath176 for @xmath173 . the diagonal @xmath177 is also shown as a guide . ] . ( a ) @xmath89 . ( b ) @xmath76 . solid lines represent the boundary between @xmath91 and @xmath178 . the dashed line in ( a ) represents @xmath168 . the dashed line is not drawn in ( b ) because this theoretical estimate is valid only for @xmath89 . in ( a ) , the two lines almost overlap each other . the initial belief of each agent was assumed to be neutral [ i.e. , @xmath179 , @xmath67 ] . ] . vector @xmath183 is shown by an arrow of proportional size at each position of the random walker @xmath184 . ( a ) @xmath185 and @xmath186 , which satisfies @xmath138 . ( b ) @xmath187 and @xmath186 , which satisfies @xmath188 . the size of the vectors is manually normalized for clarity , independently for the two panels . ] and ( b ) @xmath76 . the initial belief of each agent is assumed to be neutral [ i.e. , @xmath179 , @xmath67 ] . the black region represents the case where all the @xmath79 runs end with agreement such that the average size of the minority is undefined . ] 99ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop ( , , ) @noop ( , , ) @noop ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.100.108702 [ * * , ( ) ] link:\doibase 10.1103/physreve.77.016102 [ * * , ( ) ] link:\doibase 10.1103/physreve.74.056108 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.158701 [ * * , ( ) ] @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) @noop ( , , ) @noop ( ) @noop ( , , ) @noop * * , ( ) @noop * * ( 5 ) , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) | we propose a collective opinion formation model with a so - called confirmation bias .
the confirmation bias is a psychological effect with which , in the context of opinion formation , an individual in favor of an opinion is prone to misperceive new incoming information as supporting the current belief of the individual .
our model modifies a bayesian decision - making model for single individuals [ m. rabin and j. l. schrag , q. j. econ . *
114 * , 37 ( 1999 ) ] for the case of a well - mixed population of interacting individuals in the absence of the external input .
we numerically simulate the model to show that all the agents eventually agree on one of the two opinions only when the confirmation bias is weak .
otherwise , the stochastic population dynamics ends up creating a disagreement configuration ( also called polarization ) , particularly for large system sizes .
a strong confirmation bias allows various final disagreement configurations with different fractions of the individuals in favor of the opposite opinions .
pacs numbers : : 87.23.ge , 02.50.ey , 02.50.le 0 |
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let us consider the inclusive process of the pion creation in the fragmentation region of polarized proton at high energy proton - proton collisions @xmath2 where @xmath3 is the transversal to beam ( implied by cms ) axes spin of initial proton @xmath4 where @xmath5 is the energy fraction of pion , @xmath6 are the invariant masses of the jets , which we will assume to be of the order of nucleon mass @xmath7 . we study the two jet kinematics with jets @xmath8 , @xmath9 moving along the initial hadron directions . the jet created by the transversely polarized proton is supposed to contain the detected pion . moreover , we consider the case when its production is not related with the creation of nucleon resonances . in terms of pion transverse components it corresponds to the condition @xmath10 > m_{res}^2-m^2,\\ { \nonumber}{\mathbf{p}}+{\mathbf{k_1}}+{\mathbf{p_1'}}=0,\end{gathered}\ ] ] where @xmath11 is the transfer momentum between protons . through this paper we use sudakov parameterization of 4momenta of the problem @xmath12 production @xmath13 . ] the azimuthal one - spin asymmetry arises from the interference of the amplitudes with one and two gluon exchanges between nucleons ( see fig . 1 ) @xmath14 where @xmath15 is the strong coupling constant , @xmath16 is the azimuthal angle between the 2vectors @xmath17 and @xmath18 , transverse to the beam axis . the functions @xmath19 , @xmath20 as well as the color factors @xmath21 will be specified below . for convenience we put here the alternative form for the phase volume of pion @xmath22 the one gluon exchange matrix element has a form @xmath23 the currents @xmath24 , @xmath25 are associated with the jets created by particles 1 , 2 and index @xmath3 describes the color state of the jet . using gribov representation of the metric tensor @xmath26 and the gauge condition for the currents @xmath27 we express @xmath28 in the form @xmath29 @xmath30 the quantities @xmath31 are the invariant masses of the jets . we imply that the jet @xmath8 does not contain the detected pion . at this point we need some model describing the jets . we use the heavy fermion model , i.e. , we consider the jet as a result of heavy fermion decay . we assume the coupling constant of the interaction of the pion with a nucleon and with heavy fermion to be the same . we do not specify it as well as it is cancelled in the asymmetry ( [ eq:5 ] ) . so we have @xmath32 @xmath33 are the generators of color group @xmath34 and @xmath35,\quad d_2=-\frac{m^2}{\beta}[\rho^2+\beta^2 ] . { \nonumber}\end{aligned}\ ] ] we use here the spin density matrices of the jets ( i.e. , heavy fermions ) @xmath36 it is important to note that we impose the gauge invariant form of matrix element and after that we use the heavy fermion model . this operations do not commute as well as the heavy fermion currents do not satisfy the current conservation condition . it is a specific of the considered model . for the lowest order differential cross section we obtain @xmath37 with the explicit expressions for the impact factors @xmath38 , @xmath39 given in appendix a and @xmath40 . the lowest order spin dependent contribution to the cross section arises from the interference of imaginary part of 1loop radiative correction ( rc ) of feynman diagram ( fd ) fig 1b , c with the born amplitude fig 1a . we do not consider the rc from fd fig . 1 , believing that such kind fd contribute to the nucleon resonances formation . besides it do not contribute in the leading logarithmical approximation ( lla ) @xmath41 we obtain in the lowest order @xmath42 with @xmath43 and the color factor @xmath44=\frac{(n^2 - 1)(n^2 - 2)}{8n}.\ ] ] the impact factors @xmath45 are given in appendix b. impact factors @xmath45 contain the new mass parameters @xmath46 which are intermediate jet state masses . we had shown that the lowest order unpolarized and polarized cross sections can be expressed in terms of impact factors of projectiles moving in opposite directions which where introduced first in the papers of h. cheng and t. t. wu @xcite . the calculation of rc to them can be done following the method developed by j. balitski and l. n. lipatov @xcite . it was shown by these authors that in the lla , the cross section has as well the form of conversion of impact factors of colliding particles with some universal kernel . physically it corresponds to the replacement of exchanged gluons by the reggeized gluons . the reggeization states are taking into account in two factors . first the regge factor @xmath47 must be introduced , where @xmath48 is the regge trajectory of gluon with the momentum squared @xmath49 . the second factor takes into account the contribution of inelastic processes of emission of real gluons . these both contributions suffer from the infrared divergences , however the total sum is free of them . , @xmath50 dependence of a partial contribution to the pomeron intercept . ] , @xmath50 dependence of a partial contribution to the odderon intercept . ] for the rc to the unpolarized cross section we have @xmath51 with color factor @xmath52 and @xmath53,\end{aligned}\ ] ] with @xmath54 and @xmath55 given in appendix c. this formula can be inferred from the result for the non forward high energy scattering amplitude obtained in the paper @xcite @xmath56 with definition @xmath57\frac{\phi^{bb'}({\mathbf{k'}},{\mathbf{q } } ) } { { \mathbf{k'}{\lower-.2em\hbox{}^{2}}}({\mathbf{q}}-{\mathbf{k'}})^2}\\ { \nonumber}&\quad - \frac{\phi^{bb'}({\mathbf{k}},{\mathbf{q}})}{({\mathbf{k}}-{\mathbf{k'}})^2 } \bigg[\frac{{\mathbf{k}}^2}{{\mathbf{k'}{\lower-.2em\hbox{}^{2}}}+({\mathbf{k}}-{\mathbf{k'}})^2}+ \frac{({\mathbf{q}}-{\mathbf{k}})^2}{({\mathbf{q}}-{\mathbf{k'}})^2+({\mathbf{k}}- { \mathbf{k'}})^2}\bigg]\bigg\}.\end{aligned}\ ] ] the formula ( [ eq:17 ] ) can be obtained from the last general one by putting @xmath58 . note that in the formula obtained in @xcite the used color group was @xmath59 . let now consider the lla rc to the polarized part of the differential cross section . there are presented three types of contributions corresponding to three different choices of two gluons which are involved in the reggeization procedure in the lowest order rc @xmath60,\ ] ] with the color factor @xmath61 symmetry reasons lead to the conclusion that @xmath62 and @xmath63 contributions are equal to @xmath64 one . for @xmath64 we have @xmath65\right . { \nonumber}\\ & \quad -\left . \frac{\phi_2^{(12)}({\mathbf{k}},{\mathbf{k_1}})}{({\mathbf{k}}-{\mathbf{k'}})^2 } \left[\frac{{\mathbf{k}}^2}{{\mathbf{k'}{\lower-.2em\hbox{}^{2}}}+({\mathbf{k}}-{\mathbf{k'}})^2}+ \frac{({\mathbf{k_1}}-{\mathbf{k}})^2}{({\mathbf{k_1}}-{\mathbf{k'}})^2 + ( { \mathbf{k}}-{\mathbf{k'}})^2}\right]\right\}.\end{aligned}\ ] ] the details of impact factor calculations are given in the appendices a d . in the appendix e we give some details used for performing the integration over the transversal component of the loop momenta . the quantities @xmath66 may be interpreted as a partial contributions to the odderon and pomeron intercepts . their dependence on @xmath67 , @xmath50 is illustrated in fig . [ pom ] and fig . [ ode ] . the jet s masses was supposed to be larger than 1gev . the size of contributions to the polarized and unpolarized differential cross sections depends on the used jet model as well as on the choice of the vertices which describe the transition of the nucleon to the jet and on the choice of the vertex function which describes the conversion of one sort of jet to the jet of another sort . because of the here used choice @xmath68 , we are forced to put on the gauge conditions . another possible choice , @xmath69/m$ ] , leads to the zero contribution to the asymmetry ( in the limit of infinite large @xmath70 ) . we see that one - spin effect is rather large . another mechanisms of one - spin asymmetry associated with the nucleon resonances in intermediate state and final state interaction in @xmath71 process was considered in papers @xcite where the effect was of the same order . the work was supported by intas-00366 . e. z. is gratefull to rfbr for the grant 0001 - 00617 . the explicit expressions for the lowest order impact factors in the unpolarized case are @xmath72 to simplify the calculation of the traces we write down here the useful relations @xmath73 which follow from the on mass shell conditions . explicit expressions for @xmath38 and @xmath39 are @xmath74\big[\beta^2m^2+({\mathbf{p}}-{\mathbf{k_1 } } \beta)^2\big ] } , { \nonumber}\\\phi_{02}{\mathbf{k_1}})&=\frac{2{\mathbf{k_1}}^2\big [ { \mathbf{k_1}{\lower-.2em\hbox{}^{2}}}+(m_2-m)^2\big]}{\big({\mathbf{q}}^2+m_2 ^ 2-m^2\big)^2}\end{aligned}\ ] ] the lowest order contribution to the impact factor corresponding to the polarized proton with 4-momentum @xmath75 is @xmath76 the quantity @xmath77 has a form @xmath78 the sudakov decomposition of the 4-vectors @xmath79 is given in ( a.2 ) . the exchanged gluon expansions are @xmath80 the quantity @xmath81 is equal @xmath82 with @xmath83 given in ( [ eq:11 ] ) and @xmath84.\ ] ] the quantity @xmath85 has a form @xmath86 with @xmath87\ ] ] and with the same expression for @xmath79 . the relevant representation for the exchanged gluon 4momenta is following @xmath88 and the same expression for @xmath79 . impact factor for the unpolarized proton @xmath55 has a form @xmath89 for then calculation of the trace for @xmath55 we used sudakov representation @xmath90 the impact factors for the case of unpolarized protons are @xmath91 where @xmath92 let us now give the expressions for the impact factors in the case of rc to the polarized cross sections . for the @xmath93 in ( [ eq:21 ] ) we have @xmath94 with @xmath95 and @xmath96 with the substitutions similar to ones for @xmath77 from appendix b for the momenta including @xmath97 and the substitutions similar to ones for @xmath85 for @xmath98 . besides @xmath99 the impact factor of unpolarized proton in this case have a form @xmath100 with @xmath101 and @xmath102 here we give some details used in performing the loop momenta integration . when calculating the relevant trace we use the shouthen identity @xmath103 this identity permits to express all conversions with levi - chivita tensor in the standard form . for instance @xmath104 the second term in the right side of this equation can be expressed through the @xmath105 using the relation @xmath106 | single - spin asymmetry appears due to the interference of single and double gluon exchange between protons .
a heavy fermion model is used to describe the jet production in the interaction of gluon with the proton implying the further averaging over its mass . as usually in one - spin correlations ,
the imaginary part of the double gluon exchange amplitude play the relevant role .
the asymmetry in the inclusive set - up with the pion tagged in the fragmentation region of the polarized proton does not depend on the center of mass energy in the limits of its large values .
the lowest order radiative corrections to the polarized and unpolarized contributions to the differential cross sections are calculated in the leading logarithmical approximation . in general , a coefficient at logarithm of the ratio of cms energy to the pion mass depends on transversal momentum of the pion .
this ratio of the lowest order contribution to the asymmetry may be interpreted as the partial contribution to the odderon intercept .
the ratio of the relevant contributions in the unpolarized case can be associated with the partial contribution to the pomeron intercept .
the numerical results given for the model describe the jet as a heavy fermion decay fragments .
@xmath0 _ join institute for nuclear research , 141980 dubna , russia , + @xmath1 _ leningrad institute for nuclear physics , gatchina , russia _ _ |
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spontaneous parametric down - conversion ( spdc ) is the basic source of non - classical light in experimental quantum optics @xcite , testing foundations of the quantum theory @xcite , and implementing protocols for quantum information information processing and communication @xcite . the essential feature of spdc is the guarantee that the photons are always produced in pairs , and suitable arrangements allow one to generate various types of classical and quantum correlations within those pairs . the physics of spdc depends strongly on optical properties of nonlinear media in which the process is realized . this leads to an interplay between different characteristics of the source and usually imposes trade - offs on its performance . for example , many experiments require photon pairs to be prepared in well - defined single spatio - temporal modes . in contrast , photons generated in typical media diverge into large solid angles and are often correlated in space and time , as shown schematically in fig . [ fig : source ] . specific modes can be selected afterwards by coupling the output light into single - mode fibers and inserting narrowband spectral filters . however , it is usually not guaranteed that both the photons in a pair will always have the matching modal characteristics , and in many cases only one of the twin photons will get coupled in @xcite . this effect , which can be modelled as a loss mechanism for the produced light , destroys perfect correlations in the numbers of twin photons . these losses come in addition to imperfect detection , and can be described jointly using overall efficiency parameters . is pumped with a laser beam @xmath0 . generated photons are highly correlated and useful modes @xmath1 and @xmath2 are typically selected by narrow spatial and frequency filters @xmath3 . ] the effects of losses become more critical when the spdc source is pumped with powers so high that it is no longer possible to neglect the contribution of events when multiple pairs have been simultaneously produced @xcite . such a regime is necessary to carry out multiphoton interference experiments , it can be also approached when increasing the production rate of photon pairs . one is then usually interested in postselecting through photocounting the down - conversion term with a fixed number of photon pairs and observing its particular quantum statistical features @xcite . in the presence of losses the same number of photocounts can be generated by higher - order terms when some of the photons escape detection . however , the statistical properties of such events can be completely different , thus masking the features of interest . although some quantum properties may persist even in this regime , with a notable example of polarization entanglement @xcite , their extraction and utilization becomes correspondingly more difficult . the present paper is an experimental study of multiphoton events in spontaneous parametric down - conversion with particular attention paid to the effects of filtering and losses . the multiple - pair regime is achieved by pumping the nonlinear crystal by the frequency - doubled output of a 300 khz titanium - sapphire regenerative amplifier system . the kilohertz repetition rate has allowed us to count the number of the photons at the output with the help of the loop detector @xcite . using a simplified theoretical description of the spdc source we introduce effective parameters that characterize its performance in multiphoton experiments . the obtained results illustrate trade - offs involved in experiments with multiple photon pairs and enable one to select the optimal operation regime for specific applications . this paper is organized as follows . first we describe a theoretical model for spdc statistics in sec . [ sec : spdcstat ] . [ sec : parameters ] introduces effective parameters to characterize spdc sources . the experimental setup and measurement results are presented in sec . [ sec : exp ] . finally , sec . [ sec : conclusions ] concludes the paper . we will start with a simple illustration of the effects of higher - order terms in spdc . suppose for simplicity that the source produces a two - mode squeezed state which can be written in the perturbative expansion as @xmath4 , where @xmath5 measures squeezing and is assumed to be real . for two - photon experiments , the relevant term is @xmath6 and the contribution of the higher photon number terms can be neglected as long as @xmath7 . this enables postselecting the two - photon term and observing associated quantum effects , such as hong - ou - mandel interference . suppose now that each of the modes is subject to losses characterized by @xmath8 , where @xmath9 is the overall efficiency . losses may transform the term @xmath10 into @xmath11 or @xmath12 , whose presence will lower the visibility of the hong - ou - mandel interference . the two - photon term now occurs with the probability @xmath13 , while the four - photon term effectively produces one of the states @xmath11 or @xmath12 with the total probability equal to @xmath14 . this constitutes a fraction of @xmath15 of the events that come from single pairs produced by the source . this fraction can easily become comparable with one , especially when the losses are large . let us now develop a general model of photon statistics produced by an spdc source . in the limit of a classical undepleted pump the output field is described by a pure multimode squeezed state . by a suitable choice of spatio - temporal modes , called characteristic modes , such a state can be brought to the normal form @xcite in which modes are squeezed pairwise . denoting the annihilation operators of the characteristic modes by @xmath16 and @xmath17 , the non - vanishing second - order moments can be written as : @xmath18 where @xmath19 is the squeezing parameter for the @xmath20th pair of modes . because the state of light produced in spdc is gaussian , these equations , combined with the fact that first - order moments vanish @xmath21 define fully quantum statistical properties of the output field . let us first consider the case when the spatial and spectral filters placed after the source select effectively single field modes . the annihilation operators @xmath22 and @xmath23 are given by linear combinations of the characteristic modes : @xmath24 where @xmath25 and @xmath26 describe amplitude transmissivities of the filters for the characteristic modes and @xmath27 . because the complete multimode state is gaussian , the reduced state of the modes @xmath22 and @xmath23 is also gaussian , and it is fully characterized by the average numbers of photons @xmath28 and @xmath29 , and the moment @xmath30 . these quantities can be written in terms of the multimode moments given in eq . ( [ eq:<aa>,<ab > ] ) , but we will not need here explicit expressions . it will be convenient to describe the quantum state of the modes @xmath22 and @xmath23 with the help of the wigner function : @xmath31 where @xmath32 and @xmath33 is the correlation matrix composed of symmetrically ordered second order moments : @xmath34 given the wigner function of the reduced state for the modes @xmath22 and @xmath23 in the gaussian form , the calculation of the joint count statistics @xmath35 is straightforward . it will be useful to introduce an operator representing the generating function of the joint count statistics @xmath36 whose expectation value over the quantum state expanded into the power series yields the joint count statistics @xmath37 : @xmath38 because the operator @xmath39 is formally equal , up to a normalization constant , to product of density matrices describing thermal states of modes @xmath22 and @xmath23 with average photon numbers @xmath40 and @xmath41 respectively , the corresponding wigner function has a gaussian form : @xmath42 using the above expression it is easy to evaluate the generating function for the joint count statistics by integrating the product of the respective wigner functions : @xmath43 where we introduced the following three parameters : @xmath44 these three parameters have a transparent physical interpretation in the regime when @xmath45 . then , it is easy to check that the generating function given in eq . ( [ eq : xi(x , y)=squeezedwlosses ] ) describes the count statistics of a plain two - mode squeezed state whose two modes have been sent through lossy channels with transmissivities @xmath9 and @xmath46 , and the average number of photons produced in each of the modes was equal to @xmath47 . note that the inequality @xmath45 implies nonclassical correlations between the modes @xmath22 and @xmath23 . if the opposite condition is satisfied , the state can be represented as a statistical mixture of coherent states in modes @xmath22 and @xmath23 with correlated amplitudes . in this case , it is not possible to carry out an absolute measurement of losses . in a realistic situation , the spectral filters employed in the setup are never sufficiently narrowband to ensure completely coherent filtering . therefore a sum of counts originating from multiple modes will be observed . we will model this effect by assuming that the detected light is composed of a certain number of @xmath48 modes with identical quantum statistical properties described in the preceding section . the generating function @xmath49 for count statistics is therefore given by the @xmath48-fold product of the expectation value @xmath50 calculated in eq . ( [ eq : xi(x , y)=squeezedwlosses ] ) : @xmath51 the parameter @xmath48 , which we will call the equivalent number of modes , can be read out from the variance of the count statistics in one of the arms characterized by the generating function @xmath49 . for a single mode source the variance is equal to that of a thermal state with @xmath52 . it is easily seen that for @xmath48 equally populated modes the variance becomes reduced to the value @xmath53 . solving this relation for @xmath48 leads us to a measurable parameter that will help us to characterize the effective number of detected modes : @xmath54 note that @xmath48 is closely related to the inverse of the mandel parameter @xcite . the second parameter we will use to characterize the spdc source measures the overall losses experienced by the produced photons . let us first note that in the perturbative regime , when all the squeezing parameters @xmath55 , each one of the quantities @xmath56 , @xmath57 , and @xmath58 appearing in eq . ( [ eq : r2,eta = onemode ] ) is proportional to the pump intensity . in this regime the efficiencies can be approximated by the ratios @xmath59 and @xmath60 that are independent of the pump intensity . on the other hand , the average photon numbers @xmath61 and @xmath62 in both the arms are linear in the pump power . following early works on squeezing @xcite , we will introduce here a parameter that quantifies the subpoissonian character of correlations between the counts @xmath63 and @xmath64 in the two arms of the setup . first we define a stochastic variable : @xmath65 this definition takes into account the possibility of different losses in the two arms through a suitable normalization of the count numbers . the subpoissonian character of the correlations can be tested by measuring the average @xmath66 . the semiclassical theory predicts @xmath67 , while for two beams with count statistics characterized by the generating function @xmath68 one obtains : @xmath69 in the case of equal efficiencies @xmath70 this expression reduces simply to @xmath71 . we will use the last relation as a method to measure the average overall efficiency of detecting the state produced by the spdc source . as discussed at the beginning of sec . [ sec : spdcstat ] , the initial quantum state of light used for many experiments should ideally be in a state @xmath6 or @xmath10 . in the perfect case of two - mode squeezing and negligible losses , such states can be isolated through postselection of the spdc output on the appropriate number of counts . in practice , the postselected events will also include other combinations of input photon numbers . because typical detectors have limited or no photon number resolution , one needs to take into account also the deleterious contribution of higher total photon numbers . this leads us to the following definition of the parameter measuring the contamination of photon pairs with other terms that can not be in general removed through postselection : @xmath72 an analogous definition can be given for quadruples of photons , which ideally should be prepared in a state @xmath10 : @xmath73 in fig . [ fig:2imp ] we depict contour plots of the contamination parameters @xmath74 and @xmath75 as a function of the production rates and the overall efficiency . it is clearly seen that the non - unit efficiency imposes severe bounds on the production rates that guarantee single or double photon pair events sufficiently free from spurious terms . the graphs also imply that strong pumping is not a sufficient condition to achieve high production rates , but it needs to be combined with a high efficiency of collecting and detecting photons . ( upper plot ) and double photon pairs @xmath74 ( lower plot ) as a function of the overall efficiency @xmath9 and the respective production rates @xmath76 and @xmath77 . the calculations have been carried out in the regime of single selected modes when @xmath78.,title="fig : " ] ( upper plot ) and double photon pairs @xmath74 ( lower plot ) as a function of the overall efficiency @xmath9 and the respective production rates @xmath76 and @xmath77 . the calculations have been carried out in the regime of single selected modes when @xmath78.,title="fig : " ] the experimental setup is depicted in the fig . [ fig : setup ] . the master laser ( rega 9000 from coherent ) produces a train of 165 fs fwhm long pulses at a 300 khz repetition rate centered at the wavelength 774 nm , with 300 mw average power . the pulses are doubled in the second harmonic generator xsh based on a 1 mm thick beta - barium borate ( bbo ) crystal cut for a type - i process . ultraviolet pulses produced this way have 1.3 nm bandwidth and 30 mw average power . they are filtered out of the fundamental using a pair of dichroic mirrors dm and a color glass filter bg ( schott bg39 ) , and imaged using a 20 cm focal length lens il on a downcoversion crystal x , where they form a spot measured to be 155 @xmath79 m in diameter . the power of the ultraviolet pulses , prepared in the polarization perpendicular to the plane of the setup , is adjusted using neutral density filters nd and a motorized half waveplate . the type - i down - conversion process takes place in a 1 mm thick bbo crystal x cut at @xmath80 to the optic axis , and oriented for the maximum source intensity . the down - converted light emerging at the angle of @xmath81 to the pump beam is coupled into a pair of single mode fibers placed at the opposite ends of the down - conversion cone . the fibers and the coupling optics define the spatial modes in which the down - conversion is observed @xcite . the coupled photons enter the loop detector @xcite in which light from either arm can propagate towards one of the detectors through eight distinct paths . the minimal delay difference between two paths is 100 ns , more than twice the dead time of the detectors . finally the photons exit the fiber circuit , go through interference filters if and are coupled into multimode fibers which route them directly to single photon counting modules spcm ( perkinelmer spcm - aqr-14-fc ) connected to fast coincidence counting electronics ( custom - programmed virtex4 protype board ml403 from xilinx ) detecting events in a proper temporal relation to the master laser pulses . the measurement series are carried out with pairs of interference filters of varying spectral widths . the measurement proceeds as follows : for each pair of interference filters the loop detector is calibrated using data collected at a very low pump light intensities , when the chance of more than one photon entering the fiber circuit is negligible compared to the rate single photons appear . this allows one to calculate the complete matrix of conditional probabilities @xmath82 of observing @xmath20 detector clicks with @xmath63 initial photons @xcite for one arm , and analogously @xmath83 for the second arm . the losses in the detectors and in the fiber circuit are assumed to contribute to the overall efficiencies @xmath9 and @xmath46 . thus @xmath82 and @xmath83 describe lossless loop detectors for which @xmath84 . after the calibration , the counts are collected for approximately @xmath85 master laser pulses for each chosen intensity of the pump , which yields the probabilities @xmath86 of observing @xmath20 clicks in one arm and @xmath87 in the other one . these probabilities are related to the joint count probability @xmath35 corrected for combinatorial inefficiencies of the loop detector through the formula : @xmath88 the probabilities @xmath35 can be retrieved from experimentally measured @xmath86 using the maximum likelihood estimation technique @xcite . an exemplary joint photon number distribution reconstructed from the experimental data is shown in fig . [ fig : exprho ] . the results of the reconstruction are subsequently used to calculate the parameters of the source discussed in the preceding section . = 0.15 and @xmath89=0.18 . ] in a single arm as a function of the average photon number @xmath90 , for measurements carried out without interference filters ( circles ) , with 10 nm fwhm filters ( squares ) and 5 nm fwhm filters ( crosses ) . ] in fig . [ fig : m ] we depict the reconstructed equivalent number of modes @xmath48 in single arm for different filtering and pump intensities . this quantity can be also understood as a measure of how many incoherent modes a photon from the source occupies . naturally , @xmath48 drops with application of narrowband spectral filtering since it erases the information on the exact time the photon pair was born in the nonlinear crystal . it also is seen that @xmath48 is practically independent on the pumping intensity , as predicted in sec . [ sec : parameters ] . as a function of the total average photon number @xmath90 , measured without interference filters ( circles ) , with 10 nm fwhm filters ( squares ) and 5 nm fwhm filters ( crosses ) . ] the average overall efficiency of the squeezed state calculated as @xmath91 is shown in fig . [ fig : eta ] . it is seen that the efficiency decreases with an application of narrowband filtering , which again is easily understood . in addition , @xmath9 exhibits a very weak dependence on the pump intensity which again agrees with theoretical predictions for the regime when the average number of photons is much less than one . a relatively large difference in @xmath9 between 10 nm and 5 nm interference filters can be explained by the fact that in the latter case the selected bandwidth becomes narrower than the characteristic scale of spectral correlations within a pair . consequently , the filter in one arm selects only a fraction of photons conjugate to those that have passed through the filter placed in the second arm . this observation is consistent with the determination of the parameter @xmath92 , which shows that for 5 nm filters effectively single spectral modes are selected . for single photon pairs as a function of the pair production rate , measured without interference filters ( circles ) , with 10 nm fwhm filters ( squares ) and 5 nm fwhm filters ( crosses ) . the parameters of the fitted theoretical curves are : @xmath93 and @xmath94 ( red solid line ) , @xmath95 and @xmath96 ( red dashed line ) and @xmath97 and @xmath98 ( red dash - dot line ) . ] for double photon pairs as a function of the production rate , measured without interference filters ( dots ) . the parameters of the fitted theoretical curve are : @xmath99 and @xmath100 ( red dashed line ) . measurements with interference filters did not provide statistically significant data . ] finally , in figs . [ fig:2impexp ] and [ fig:4impexp ] we plot the respective contamination parameters for single @xmath74 and double @xmath75 photon pairs , that describes non - postselectable contributions of other photon terms to the output . as expected , the contamination grows with decreasing overall efficiency @xmath9 that corresponds to narrowing the spectral bandwidth , as well as with the increasing pair production rates . it is noteworthy that the measurement results agree well with fitted theoretical curves whose parameters match those that can be read out from figs . [ fig : m ] and [ fig : eta ] . it is seen that in the regime the experiment was carried out the contamination is rather significant , and it would affect substantially effects such as two - photon interference . this limits in practice the energy of pump pulses and consequently the production rates that result in photon pairs sufficiently free from spurious terms . the effect is particularly dramatic in the case of @xmath75 , where genuine double pairs appear only in approximately half of all the events . despite the simplicity of the basic concept , the application of spdc sources in more complex quantum optics and quantum information processing experiments requires a careful choice of operating conditions . the actual output state is a result of a subtle interplay between the pump strength , spatial and spectral filtering of the output , and the losses experienced by the signal . in this paper we concentrated on the features of the photon number distribution and developed an effective theoretical description . starting from the fact that the spdc output processed by arbitrary passive linear optics is given by a gaussian state , we introduced effective parameters characterizing the joint photon number distribution . the first parameter is the effective number of orthogonal modes that impinge on each of the detectors . this parameter carries information about the modal purity of the photons produced in each arm . the second parameter characterizes average losses experienced by the spdc output , and can be calculated as a suitably normalized variance of the count difference . finally , we also proposed and determined experimentally a measure of the contamination of photon pairs by other spurious terms that in general can not be rejected by postselection and may contribute to an unwanted background . we showed that the presence of such a background puts stringent requirements on the detection efficiency if a bright source with high pair generation probability is desired . these constraints become very challenging when designing multiple - pair experiments . we acknowledge helpful discussions with c. silberhorn and i. a. walmsley . this work has been supported by the polish budget funds for scientific research projects in years 2005 - 2008 and the european commission under the integrated project qubit applications ( qap ) funded by the ist directorate as contract number 015848 . the experiment has been carried out in the national laboratory for atomic , molecular , and optical physics in toru , poland . p. g. kwiat , e. waks , a. g. white , i. appelbaum , and p. h. eberhard , phys . a * 60 * , r773 ( 1999 ) ; t. paterek , a. fedrizzi , s. grblacher , t. jennewein , m. ukowski , m. aspelmeyer , and a. zeilinger , phys . * 99 * , 210406 ( 2007 ) ; c. branciard , a. ling , n. gisin , c. kurtsiefer , a. lamas - linares , and v. scarani , _ ibid . _ * 99 * , 210407 ( 2007 ) . j. l. obrien , g. j. pryde , a. g. white , t. c. ralph , d. branning , nature * 426 * , 264 ( 2003 ) ; p. walther , k. j. resch , t. rudolph , e. schenck , h. weinfurter , v. vedral , m. aspelmeyer , and a. zeilinger , _ ibid . _ * 434 * , 169 ( 2005 ) ; m. halder , a. beveratos , n. gisin , v. scarani , c. simon , and h. zbinden , nature physics * 3 * , 692 ( 2007 ) ; g. vallone , e. pomarico , f. de martini , and p. mataloni , phys . lett . * 100 * , 160502 ( 2008 ) . arvind , b. dutta , n. mukunda , and r. simon , pramana - j . * 45 * , 471 ( 1995 ) ; r. s. bennink and r. w. boyd , phys . a * 66 * , 053815 ( 2002 ) ; a. botero and b. reznik , _ ibid . _ * 67 * , 052311 ( 2003 ) ; s. l. braunstein , _ ibid . _ * 71 * , 055801 ( 2005 ) ; w. wasilewski , a. i. lvovsky , k. banaszek , and c. radzewicz , _ ibid . _ * 73 * , 063819 ( 2006 ) . f. a. bovino , p. varisco , a. m. colla , g. castagnoli , g. di giuseppe , and a. v. sergienko , opt . comm . * 227 * , 343 ( 2003 ) ; s. castelletto , i. p. degiovanni , a. migdall , and m. ware new j. phys . * 6 * , 87 ( 2004 ) ; a. dragan , phys . a * 70 * , 053814 ( 2004 ) . | we present an experimental characterization of the statistics of multiple photon pairs produced by spontaneous parametric down - conversion realized in a nonlinear medium pumped by high - energy ultrashort pulses from a regenerative amplifier .
the photon number resolved measurement has been implemented with the help of a fiber loop detector .
we introduce an effective theoretical description of the observed statistics based on parameters that can be assigned direct physical interpretation .
these parameters , determined for our source from the collected experimental data , characterize the usefulness of down - conversion sources in multiphoton interference schemes that underlie protocols for quantum information processing and communication . |
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in nano - electronics , quantitative evaluation of impurity effects is crucial because device properties are strongly influenced by or even built on such effects . experimentally , the impurities exist and can be doped in nano - devices without knowing their exact locations , so theoretically it is important to predict the averaged transport quantities such as conductance over impurity configurations . the most direct way to obtain the averaged conductance is to generate many different configurations , then calculate the conductance for each configuration , and finally take the mean value . such a brute - force method is usually used in the mesoscopic systems from diffusive regime to localized regime because it is an exact calculation . but in order to get good statistics , huge number of configurations has to be generated making it very time - consuming especially for the calculation of conductance fluctuation . when the disorder strength is weak , it is not necessary to use the brute - force method since some analytic approximate method is superior in speed while maintaining the same accuracy . for this purpose considerable effort has been made to develop approximate techniques , within which the most widely used technique is the coherent potential approximation ( cpa ) , which is a useful tool to evaluate the configurational averaged one - electron green s function@xcite @xmath6 , and has also been extended to determine the so - called vertex corrections"@xcite for quantities involving two green s functions . cpa approach has been implemented in the korringa - kohn - rostoker@xcite and linear muffin - tin orbital@xcite for first principles calculations and has many successful applications@xcite . the central idea of cpa is to find a coherent potential " such that the one - electron green s function evaluated under such potential approximately equals the configurational averaged green s function . as an extension , cpa can also be used to determine the so - called vertex corrections"@xcite for the product of two green s functions . later , levin _ et al _ also proposed an elegant diagrammatic method to evaluate the hall coefficient which relates to the direct multiple of three green s functions@xcite . importantly , the cpa approach and its extensions can be combined with local - orbital based dft to calculate the physical properties , such as the band structure and the density of states , of realistic materials . one example is the development of the so called kkr - cpa " , used to study the band structure and density of states of cu - ni@xcite , ag - pd@xcite , and cu - pd@xcite alloys . the linear muffin - tin orbital ( lmto ) method has also been proposed@xcite and used to study the electronic structures of metal alloys@xcite . cpa combined with lmto works very well and has many successful applications . examples are the investigation of transport properties in disordered magnetic multilayers@xcite , structure of sn - ge alloys@xcite , the electronic structure of non - stoichiometric compounds@xcite , and doped semiconductors@xcite . the latest development of cpa extended its range of application to non - equilibrium quantum transport problems where impurity average has to be performed . one prominent work is the non - equilibrium vertex correction " ( nvc ) discussed in ref . . it has been shown by zhuravlev _ _ et al.__@xcite that this nvc formalism can be interpreted in terms of the bttiker voltage - probe model so that it is not merely a correction to the electronic structure.@xcite generally speaking , this site - oriented algorithm to evaluate the average conductance is well developed and adopted by different groups.@xcite in the presence of disorder cpa - nvc approach allows one to calculate non - equilibrium transport properties such as i - v curve and other quantities involving two green s functions . however , it can not be applied directly to investigate equilibrium transport properties involving four green s functions such as conductance fluctuation and shot noise . since the fluctuation of transport properties of nano - devices , known as `` variation '' of nano - devices , is a very important quantity in nano - electronics and it provides the information on how much the specific device configuration could deviate from the mean value . we notice that a quantified experiment has been reported to measure such kind of fluctuation@xcite recently . therefore , it is timely to develop a theoretical formalism that is capable of treating disorder average of four green s functions . to the best of our knowledge , so far this is still an outstanding problem yet to solve based on cpa approach . one possible reason is that , the nvc could be regarded as a perturbation expansion approach based on cpa to evaluate the conductance by including the ladder diagrams . for conductance fluctuation , however , such a partial summation is not good enough . in this paper , we develop a direct perturbation expansion with respect to the single - site - t - matrix " up to a given order which is a good approximation for weak disorder strength or small doping concentration . we carry out benchmark calculation of average conductance , shot noise , and conductance fluctuation using the direct expansion method on a graphene system and a two - dimensional lattice model with anderson impurities as well as random dopants . we have compared our results with the brute - force calculation . we find that a six - order expansion can give very good results for conductance fluctuation and shot noise when disorder strength @xmath0 is comparable to the hopping strength @xmath2 , @xmath7 . in the presence of doping , our results also show good agreement with that obtained from brute - force method at low doping concentration . we note that our method can be easily implemented in the first principles transport calculation in nanostructures . the rest of this paper is organized as the following . in section ii , we briefly revisit cpa formalism and introduce our direct expansion approach to calculate disorder average of four green s functions . an expansion view on nvc method is also provided . in section iii , we compare our results with that obtained from the brute - force method on a graphene system and square lattice of size @xmath8 for two types of disorder : anderson disorder and different doping concentration . the results for average conductance , shot noise@xcite , and the conductance fluctuation are also presented . finally we conclude our work in section [ conclusion ] . we consider a tight - binding mode on a square lattice model described by the following hamiltonian : @xmath9 where @xmath2 is the nearest neighbor hopping energy and @xmath10 and @xmath11 are electron annihilation and creation operators on atomic site @xmath12 respectively . we choose @xmath13 as the energy unit . the on - site energy chosen as @xmath14 is a convention that the energy bottom of the 2d band structure to be zero . we also assume that the structure of the left and right leads has a similar interaction . the effect of leads can be taken into account by self - energy@xcite @xmath15 for the left and @xmath16 for the right . the self - energy of leads can be calculated numerically@xcite . although the hamiltonian in eq.([h_cent ] ) is very simple , our direct expansion in principle can handle more complicated hamiltonians as long as it only contains single particle interactions . it is also straightforward to generalize our approach to the case of multi - orbital per site . here we consider diagonal disorder"@xcite with disorder strength @xmath17 on @xmath12th atomic site . different types of disorder can be described by introducing a probability function " for @xmath17 . we consider two different types of disorder . one is anderson disorder " with the probability function given by @xmath18 where @xmath19 is called the strength of anderson disorder . another is to dope the system with different type of atom : @xmath20 here @xmath21 is the doping concentration , and @xmath22 is the energy difference between the dopant and the original atom . in the theoretical formalism we can general types of diagonal disorder including these two types of disorder . in this subsection , we revisit the well - developed single - site cpa " , because this is the starting point of our direct expansion approach . cpa is an approximation to evaluate the averaged single - particle retarded or advanced green s function ( @xmath23 or @xmath24 ) , and it is known to be good in homogeneous ensembles@xcite . in realistic nano - devices with small concentration , it has been shown that the nvc which based on cpa also works very well.@xcite . in cpa approximation , the disorder effect renormalizes the on - site energy by adding a coherent potential " @xmath25 on each atomic site , such that @xmath26 where @xmath27 denotes equilibrium green s function in the absence of disorder and can be expressed as @xmath28^{-1},\end{aligned}\ ] ] in which @xmath29 is the total self - energy due to the leads , and @xmath30 is a infinitesimal positive number . for a given disorder configuration , the green s function @xmath31 is related to @xmath27 by a t - matrix " , @xmath32 in which the t - matrix " is used to describe one specific disorder configuration , and it can also be understood as the irreducible " self - energy induced by the disorder . taking configurational average on both sides , and compare with eq.([cpa_equation ] ) , we require @xmath33 however , to implement cpa , we need a further approximation , which is usually referred as weak overall scattering approximation " or single - site approximation " , and either of them can lead to the cpa condition : @xmath34 where @xmath35 is a matrix with only one non - vanishing element @xmath36 and @xmath37^{-1}-(g_e^r)_{ii}\}^{-1}$ ] . taking average on t - matrix , we have @xmath38^{-1}-g_{e , ii}^r}=0\ ] ] from which the self - consistent equation for @xmath39 can be obtained @xcite @xmath40}.\ ] ] this equation is easy to converge . with the definition of the linewidth function , @xmath41 , we can define the transmission matrix @xmath42 . the averaged conductance ( set @xmath43 ) is defined as @xmath44 , and the averaged dc shot noise is proportional to @xmath45 , while the conductance fluctuation reads @xmath46 ^ 2\rangle-\langle\mathrm{tr}(\mathcal{t})\rangle^2}$ ] . the averaged conductance is usually calculated within the nvc approximation . while the shot noise and conductance fluctuation involve four green s function and nvc approach can not apply here . our direct expansion approach is to expand them according to eq.([tmat ] ) , together with the t - matrix expansion with respect to single - site - t - matrix @xmath35 as the following : @xmath47 notice that the multiple summation in eq.([tmat_expansion ] ) requires that the successive index should not be the same . plugging this expansion into the expression of conductance fluctuation or shot noise and generate all the diagrams up to a certain order in @xmath48 and then store them once for all . here we think @xmath48 is a natural expansion parameter because it describes the on - site scattering and it is a small quantity under small disorder strength and low doping concentration . with all the diagram generated , the average value of shot noise and conductance fluctuation can be calculated for different systems numerically . considering the @xmath49 term in the dc shot noise that involves the average of four green s functions @xmath50 , we substitute eq.([tmat ] ) into this expression and it generates several terms up to the fourth order in t - matrix . the terms with only one t - matrix vanish due to eq.([general_cpa_condition ] ) . in the following we illustrate how to use direct expansion method to generate diagrams for the other terms involving multi - t - matrices . as an example , one typical term containing three t - matrices is @xmath51 . we focus on the average part @xmath52 , where @xmath53 and @xmath54 are independent of randomness . we expand this average using eq.([tmat_expansion ] ) and truncate the resulting series to a certain order in @xmath48 ( we have obtained 8th order ) . for this three t - matrices term the lowest order in @xmath48 is three because there is no zero - order term in eq.([tmat_expansion ] ) , and all higher order terms ( we will call them diagrams from now on ) in @xmath48 up to our target order can be generated . _ symbolically _ , we write @xmath55 = \sum_{n , m , l } c_{n , m , l } ( t_i^r)^n ( t_j^a)^m ( t_k^r)^l\ ] ] this equation is symbolic so there is no summation over site indices i , j , and k. here @xmath56 represents all the diagrams with the same order of @xmath57 in @xmath58 ( @xmath59 ) contributed from different site indices i , j , k . since @xmath48 is a matrix and does not commute with @xmath60 , we have to keep both indices @xmath61 . obviously , we need to find two things : ( 1 ) how many combinations of @xmath57 we have ; ( 2 ) how many diagrams are there for a particular @xmath57 due to different site indices @xmath62 . for instance , up to the 6th order ( @xmath63 ) , we have @xmath64 , @xmath65 along with all their permutations and @xmath66 , totally 10 different combinations . @xmath56 can be calculated by counting different combinations of @xmath62 and for @xmath67 it is obtained from the following expression , @xmath68\rangle,\end{aligned}\ ] ] which is a six - multiple summation and can be handled using single - site cpa . the evaluation of disorder average of eq.([expand_2 ] ) seems to be impossible . however , we note that each @xmath48 is a matrix with only one matrix element ( it becomes a diagonal block matrix in the multi - orbital case , e.g. , if spin - orbit interaction is considered ) , as in eq.([ti_component ] ) . this simplifies calculation drastically . in addition , the cpa condition eq.([cpa_condition ] ) , indicates that if the summation index appears only once , the average vanishes . so we have to find out all possible combinations of those six site indices , and there are many possibilities . for example , we can have @xmath69 , and this combination gives the following contribution to eq.([expand_2 ] ) @xmath70\rangle \nonumber \\ & & = \sum'_{ijk}(x_1)_{ii } ( g_e^a)_{ij } ( g_e^a)_{jk } ( x_2)_{kk } ( g_e^r)_{kj } ( x_3)_{ji } \nonumber \\ & & \times \langle\tau_i^r\tau_i^a\rangle\langle \tau_j^r\tau_j^a\rangle\langle\tau_k^r\tau_k^a\rangle,%|i\rangle\langle j|,\end{aligned}\ ] ] where the prime on top of @xmath71 means the indices in the summation are mutually different and @xmath72 is defined after eq.([ti_component ] ) . another possible combination is @xmath73 which gives @xmath74 alternatively , we can have a much simpler diagrammatic representation of our expansion on the averaged shot noise . this representation is very similar to that of levin @xcite . as an example , eq.([t3_eg1 ] ) can be diagrammatically expressed as fig.[diagram_shotnoise](a ) while the diagram corresponding to eq.([t3_eg2 ] ) is shown in fig.[diagram_shotnoise](b ) . the thick lines in diagrams of fig.[diagram_shotnoise ] represent the known matrix @xmath75 , and the black dots represent the single site t - matrix @xmath48 . diagrammatically , expansion up to sixth order means that we only take into account those diagrams with the number of such black dots less than six . the thin line between two black dots represents either @xmath27 or @xmath76 , depending on the configuration . the site indices such as @xmath12 , @xmath77 and @xmath78 should be different one from another , and we should also keep in mind that the indices of two ends of a thin line can not be identical , from eq.([tmat_expansion ] ) . furthermore , we have to connect the repeated site indices with the dashed lines , like fig.[diagram_shotnoise](d ) when we have four t matrices . by constructing such a diagrammatic rule , our expansion can be carried out by finding all the topologically distinct diagrams in which the number of black dots(single site t - matrix ) is not more than six . numerically , this procedure can be implemented by computer from which we can calculate the average conductance and shot noise . ) . ( b ) the diagram corresponding to eq.([t3_eg2 ] ) . ( c ) examples of other sixth order diagrams on @xmath79 terms . ( d ) examples of sixth order diagrams on @xmath80 terms . , width=288,height=172 ] comparing with the averaged shot noise discussed in the last subsection , the calculation of conductance fluctuation is different . this is because the shot noise contains one trace while the conductance fluctuation has two traces as can be seen below , @xmath81 ^ 2\rangle = \langle\mathrm{tr}[g^r\gamma_l g^a\gamma_r]\mathrm{tr}[g^r\gamma_l g^a\gamma_r]\rangle.\end{aligned}\ ] ] if we still use the same idea as that of shot noise , we will find the calculation becomes more complicated because we can only write the above equation as @xmath81 ^ 2\rangle = \sum_{ij}\langle(g^r\gamma_l g^a\gamma_r)_{ii}(g^r\gamma_l g^a\gamma_r)_{jj}\rangle\notag\\ = \sum_{ij}\langle\mathrm{tr}[g^r\gamma_l g^a\gamma_r p^{ij } g^r\gamma_l g^a\gamma_r p^{ji}]\rangle,\end{aligned}\ ] ] in which the matrix @xmath82 is the extremely sparse matrix with only one non - zero element , @xmath83 . it turns out that the mean value of @xmath84 will cost a factor of @xmath85 to the time scale as to evaluate shot noise . even if we take into account from physics the propagation modes@xcite , @xmath86 , we still have @xmath87 ^ 2\rangle \notag \\ = & & \sum_{mn}\langle\mathrm{tr}[g^r\gamma_l g^a s^{mn } g^r\gamma_l g^a ( s^{mn})^\dagger]\rangle,\end{aligned}\ ] ] where @xmath88 represents the @xmath89th non - evanescent mode of right lead and @xmath90 is defined as @xmath91 . in this case , the factor of the computational cost is the square of the number of the non - evanescent modes , still difficult . however , in our direct expansion approach , we can get rid of this difficulty by taking the advantage of the property of @xmath48 , eq.([ti_component ] ) , see below . as before , we substitute eq.([tmat_expansion ] ) into the above equation and expand it in terms of t - matrix . here we take the term involving four t - matrices as an example , which is @xmath92\mathrm{tr}[g_e^rt^rg_e^r\gamma_l g_e^at^ag_e^a\gamma_r]\rangle\\ & & = \langle\mathrm{tr}[t^r x_1 t^a x_2]\mathrm{tr}[t^r x_1 t^a x_2]\rangle.\end{aligned}\ ] ] up to the sixth - order in @xmath48 , there are many diagrams with different ways of contraction for site indices . now considering a particular diagram ( 12,21 ) where the first two indices are in the first trace and the second two are in the second trace and a specific index contraction @xmath93 as an example , fig.([fluct_diagram_1221 ] ) , whose contribution is @xmath94\mathrm{tr}[t_k^r g_e^r t_j^r x_1 t_k^a x_2]\rangle \nonumber \\ = & & \sum'_{ijk}(x_1)_{ii}(g_e^a)_{ij}(x_2)_{ji}(g_e^r)_{kj}(x_1)_{jk}(x_2)_{kk } \nonumber \\ \times & & \langle\tau_i^r\tau_i^a\rangle\langle\tau_j^r\tau_j^a\rangle\langle\tau_k^r\tau_k^a\rangle,\end{aligned}\ ] ] where we have used eq.([ti_component ] ) to deal with two traces . in order to calculate the conductance fluctuation , we need to evaluate both @xmath95 ^ 2\rangle$ ] and @xmath95\rangle$ ] . we notice that @xmath95\rangle$ ] can be calculated accurately using nvc while @xmath95 ^ 2\rangle$ ] can only be obtained in direct expansion . to make sure the accuracy of conductance fluctuation , we have to treat these two terms on the equal footing and use direct expansion on both terms . as an example , if we expand @xmath96 to sixth order but still use cpa+nvc to evaluate @xmath97 , the fluctuation obtained is not very accurate . in fig.[fig1](d ) , at @xmath98 , we get @xmath99 from sixth - order expansion and @xmath100 from cpa+nvc , the fluctuation evaluated from these results is then larger than @xmath101 , which deviates from the exact result @xmath102 quite a lot . however , our sixth - order cumulant expansion directly on fluctuation gives the result @xmath103 , better agreement compared with the exact one . actually , in order to get the conductance fluctuation , a better way is to do cumulant expansion , which discards all the disconnected diagrams " @xcite . the advantages of such cumulant expansion " include the following separate aspects : 1 . we can directly attack the fluctuation instead of expand both @xmath96 and @xmath97 , so the computational cost is reduced to nearly a half . 2 . in this way we can naturally evaluate @xmath96 and @xmath97 on the same footing without to evaluate either of them , and also avoid the error stated in the above paragraph . this cumulant expansion only include connected diagrams , making the physical meaning more clear because that the connected diagrams only contributes to @xmath104 , which is never needed when we concentrate on the conductance fluctuation . in our case , in one specific index combination , if the indices in the first trace do not connect to those in the second trace , then it is a disconnected terms . for example , in the decomposition ( 11,22 ) , one disconnected term is @xmath105 , while @xmath106 is a connected term . to a certain order , the sum of all the connected terms give the square of the conductance fluctuation . with anderson disorder and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=115 ] with anderson disorder and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=115 ] + with anderson disorder and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=115 ] with anderson disorder and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=115 ] + with @xmath4 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=111][fig2a ] with @xmath4 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=111][fig2b ] + with @xmath4 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=111][fig2c ] with @xmath4 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=111][fig2d ] + before we show the numerical results we wish to mention the computational cost of our approach . as we can see from the algorithm , first we need to generate all the topologically inequivalent diagrams of @xmath48 up to certain order . secondly we have to generate all the possible index contractions for a given diagram . as we go to higher order , both number of @xmath48 and the number of contractions for each @xmath48 grow exponentially . note that due to the cpa condition , eq.([cpa_condition ] ) , a diagram does not contribute if an index appears only once . hence each index has to appear at least twice in the summation . thus , up to the @xmath108th order , the largest number of different indices in the summation is @xmath109 which dominates the computational cost . in general an additional index will cost about @xmath110 times computational time , with @xmath110 being the number of atoms . for this reason , although we have generated all the diagrams up to the 8th order in @xmath48 , we can only apply our approach to a small sized system such as a 10-by-10 system in 2d in a reasonable amount of cpu time . in this paper , we apply our formalism to 40-by-40 and 60-by-60 systems in 2d up to 6th order in @xmath48 . below we show the results of conductance , shot noise , and conductance fluctuations where we consider anderson disorder with different disorder strength and doping with low ( 1% ) and high ( 10% ) doping concentrations . fig.[fig1 ] - fig.[fig11 ] depict our results . each figure has four panels . in panel ( a ) , we compare our result of average conductance expanded at different orders with that of the brute - force method ( blue circle ) . in the panel ( b ) , we compare our result up to the 6th order with results obtained from the brute - force method as well as the nvc method . the panel ( c ) and ( d ) show the averaged shot noise and the conductance fluctuation , respectively , where we compare our results with that of brute - force method . in the brute - force calculation , we have collected @xmath111 random configurations for each data point on the curve . with @xmath112 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=113 ] with @xmath112 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=113 ] + with @xmath112 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=113 ] with @xmath112 doping concentration and fixed energy @xmath107 . ( a ) conductance , direct expansion at different orders vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion at different orders vs brute force . ( d ) conductance fluctuation , direct expansion at different orders vs brute force.,title="fig:",height=113 ] + in fig.[fig1 ] we show the results on @xmath8 square lattice with anderson disorder . we have fixed the fermi energy to @xmath107 where we have twenty incoming channels . we see from fig.[fig1](a ) that up to the 4th or 5th order , our expansion result agrees with that of the brute - force method for disorder strength up to @xmath113 . for the 6th order , the good agreement is extended to @xmath114 . we note that up to @xmath115 the method of nvc and brute - force give the same result ( fig.[fig1](b ) ) . for the shot noise ( fig.[fig1](c ) ) , the 4th and 5th orders seems to give almost the same result and up to @xmath116 good agreement is reached . for the 6th order expansion the agreement is better for @xmath0 up to 0.5 . we see that the direct expansion method underestimate the conductance and overestimate the shot noise . for the conductance fluctuation , the situation is different . from fig.[fig1](d ) we see that the conductance fluctuation is of order @xmath117 which is a well known result in mesoscopic physics . it is interesting to see that the 4th order expansion is better than 5th and 6th orders . the range of @xmath0 to have good agreement is @xmath113 . one thing to note . although here we benchmark our result on the lattice model with anderson disorder , the previous knowledge such as anderson localization , the universal conductance fluctuation and the percolation theory can not be expected from our approach because those physics require that the strength of disorder large enough and the system enters diffusive and even localization region , but our method can not reach that region due to its perturbative nature . now we dope the system with a fixed impurity strength @xmath0 and two different doping concentrations . for @xmath4 doping ( fig.[fig2 ] ) , very good agreement can be obtained for conductance among three methods : nvc , brute - force , and direct expansion up to 6th order in the window of @xmath118 . for the shot noise and conductance fluctuation , 6th order expansion can give good agreement for @xmath0 up to 1 . when we increase the doping concentration , our results deviate from that of the brute - force . at @xmath5 doping concentration ( fig.[fig3 ] ) we find that for average conductance , the range of @xmath0 decreases to @xmath119 while for shot noise and conductance fluctuation the agreement is not good beyond @xmath120 . one word on the computational time . in our proposed expansion method , the time cost is dominated by solving the cpa self - consistent equation . as an example , for 2d 40 by 40 lattice model , 10% doping case , we need 11 steps to obtain the cpa solvent and each step 2.5 seconds . after that , we spend approximately 40 seconds to obtain the fluctuation . however , this time used together can only be used to calculate approximately 50 configurations , from which even the mean value can not be surely given . as the system goes larger , our time advantages becomes more obvious . we have also studies the average conductance , shot noise , and conductance fluctuation in a disordered graphene ribbon system of size @xmath121 with hard - wall boundary condition perpendicular to the transport direction . here we use the simplest non - spin tight - binding hamiltonian on the honeycomb lattice , which is @xmath122 in graphene , the nearest hopping energy is @xmath123 , and we set @xmath13 as the energy unit , then both the fermi energy and and the disorder strength are measured according to it . besides , in the above hamiltonian , @xmath124 denote the nearest neighbor hopping , with the nearest - neighbor unit vector @xmath125 , @xmath126 , @xmath127 , and the lattice constant @xmath128 nm . in the following calculation , we fix the fermi energy @xmath129 where there are 15 incoming channels . for anderson disorder , we see from fig.[fig9 ] that for average conductance good agreement is obtained for disorder strength up to @xmath116 . for shot noise , however , the deviation can be seen when @xmath120 . to our surprise , the conductance fluctuation from direct expansion method is good for @xmath0 as large as 0.4 . for low doping concentration at @xmath4 , fig.[fig10 ] shows that good agreement between our method and brute - force method can be reached for average conductance and shot noise with disorder strength up to @xmath115 while for conductance fluctuation reasonable agreement is obtained for @xmath0 up to @xmath130 . for larger doping concentration , the agreement is good for smaller disorder strength . for instance at @xmath5 doping ( fig.[fig11 ] ) , the average conductance is good up to @xmath131 while for shot noise and conductance fluctuation @xmath0 is about 0.4 for a reasonable agreement compared with brute - force method . . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] + . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] + . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] + . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] + . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] + . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] . ( a ) conductance , direct expansion up to different order vs brute force . ( b ) conductance , direct expansion up to 6th order vs brute force vs nvc . ( c ) averaged shot noise , direct expansion up to different order vs brute force . ( d ) conductance fluctuation , direct expansion up to different order vs brute force.,title="fig:",height=114 ] in this paper , we have developed a direct expansion approach to deal with the average shot noise and the conductance fluctuation for disordered systems . two kinds of disorder were considered : anderson disorder and the random dopant . we have bench marked our results on a graphene system and a two dimensional square lattice model . our results can be summarized as follows . we find that our expansion method up to the 6th order is comparable , although not as good as nvc method for the calculation of averaged conductance . up to the sixth order , our results of shot noise and conductance fluctuation agree well with the brute - force method for anderson impurities with disorder strength up to @xmath132 for the square lattice and @xmath133 for the graphene system . in the presence of dopant at small doping concentration ( 1% ) our results are good when @xmath0 is around 0.9 . in general , up to the same order of expansion , average conductance gives better result than the shot noise and conductance fluctuation while the shot noise is the least accurate quantity . one can improve the accuracy by going to higher order expansion at the expenses of more cpu time . since our method is an expansion approach , it can not deal with large disorder strength and high doping concentration . our formalism can be combined with lmto type of first principles calculation , which can give quantitative prediction to the conductance fluctuation for nano - devices . in the realistic device calculations , such comparisons with brute force method are also in principle available . for example , in the realistic doping devices , one can generate a large number of random configurations at the given concentration , and the averaged shot noise as well as conductance fluctuation can be exactly evaluated . thus our method is controllable and should be successful as long as cpa itself is valid . the authors would like to thank l. zhang , g. b. liu , y. wang , y. zhu , and h. guo for their helpful discussions . we gratefully acknowledge the support from research grant council ( hku 705611p ) and university grant council ( contract no . aoe / p-04/08 ) of the government of hksar . this research is conducted using the hku computer centre research computing facilities that are supported in part by the hong kong ugc special equipment grant ( seg hku09 ) . p. soven , phys . rev . * 156 * , 809 ( 1967 ) . b. velick , s. kirkpatrick , and h. ehrenreich , phys . rev . * 175 * , 747(1968 ) . w. m. temmerman , b. l. gyorffy and g. m. stocks , j. phys . f * 8 * , 2461 ( 1978 ) . g. m. stocks and h. winter , z. phys , b * 46 * , 95 ( 1982 ) . n. stefanou , r. zeller , and p. h. dederichs , solid state commun . * * ( * * 62 ) , 735 ( 1987 ) . i. turek _ et al . , electronic structure of the disordered alloys , surfaces and interfaces _ ( kluwer , boston , 1997 ) . j. kudrnovsk and v. drchal , phys . b. * 41 * , 7515 ( 1990 ) . j. kudrnovsk , v. drchal , and j. masek , phys . b * 35 * , 2487 ( 1987 ) . m. a. korotin , n. a. skorikov , v. m. zainullina , e. z. kurmaev , a. v. lukoyanov , and v. i. anisimov , jetp lett . , * 94 * , 806 ( 2012 ) . j. j. pulikkotil , a. chroneos , and u , schwingenschloegl , journal of applied physics , * 110 * , 036105 ( 2011 ) . k. carva , i. turek , j. kudronovsk , and o. bengone , phys . b * 73 * , 144421 ( 2006 ) . k. levin , b. velick , and h. ehrenreich , phys . b. * 2 * , 1771(1970 ) . y. wang , f. zahid , y. zhu , l. liu , j. wang , and h. guo , applied physics letters * 102 * , 132109 ( 2013 ) . m. csar , y. ke , w. ji , h. guo , and z. mi , appl . phys . 98 , 202107 ( 2011 ) . y. ke , k. xia , and h. guo , phys . lett . * 100 * , 166805 ( 2008 ) and its supplementary material . zhuravlev , a. v. vedyayev , k. d. belashchenko , and e. y. tsymbal , phys . b * 85 * , 115134 ( 2012 ) . a. v. kalitsov , m. g. chshiev , and julian p. velev , phys . rev . b * 85 * , 235111(2012 ) . l. tang , mod . lett b * 26 * , 1250205 ( 2012 ) . z. bai , y. cai , l. shen , m. yang , v. ko , g. han , and y. feng , appl . lett * 100 * , 022408 ( 2012 ) . yang , j - r . hwang , h - m . chen , j - j . shen , s - m . yu , y. li , and d. d. tang , in vlsi technol . symp . dig . 208 , june ( 2007 ) . m. blanter , and m. bttiker , phys . 336 , 1 ( 2000 ) . s. datta , electronic transport in mesoscopic systems ( cambridge university press , cambridge ) . d. h. lee and j. d. joannopoulos , phys . rev . b 23 , 4997 ( 1981 ) . m. p. lopez sancho et al . , j. phys . f 14 , 1205 ; 15 , 851 ( 1985 ) . p. sheng , _ introduction to wave scattering , localization and mesoscopic phenomena _ ( springer , new york , 2006 ) . y. zhu , l. liu , and h. guo , unpublished ( 2012 ) . r. kubo , j. phys . jpn , * 17 * , 1100 ( 1962 ) . | we report the investigation of conductance fluctuation and shot noise in disordered graphene systems with two kinds of disorder , anderson type impurities and random dopants . to avoid the brute - force calculation which is time consuming and impractical at low doping concentration
, we develop an expansion method based on the coherent potential approximation ( cpa ) to calculate the average of four green s functions and the results are obtained by truncating the expansion up to 6th order in terms of single - site - t - matrix " .
since our expansion is with respect to single - site - t - matrix " instead of disorder strength @xmath0 , good result can be obtained at 6th order for finite @xmath0 .
we benchmark our results against brute - force method on disordered graphene systems as well as the two dimensional square lattice model systems for both anderson disorder and the random doping .
the results show that in the regime where the disorder strength @xmath0 is small or the doping concentration is low , our results agree well with the results obtained from the brute - force method .
specifically , for the graphene system with anderson impurities , our results for conductance fluctuation show good agreement for @xmath0 up to @xmath1 , where @xmath2 is the hopping energy . while for average shot noise
, the results are good for @xmath0 up to @xmath3 .
when the graphene system is doped with low concentration @xmath4 , the conductance fluctuation and shot noise agrees with brute - force results for large @xmath0 which is comparable to the hopping energy @xmath2 . at large doping concentration @xmath5
, good agreement can be reached for conductance fluctuation and shot noise for @xmath0 up to @xmath1 .
we have also tested our formalism on square lattice with similar results .
our formalism can be easily combined with linear muffin - tin orbital first - principles transport calculations for light doping nano - scaled systems , making prediction on variability of nano - devices . |
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one of the major discoveries in physics during the last two decades was the accelerated expansion of the universe . general relativity and the standard model of particle physics fail to explain this phenomenon . this situation calls for new alternative ideas able to give a satisfactory explanation of the cosmological observations . one of the possibilities is to go beyond general relativity and to consider more general theories of gravity . among the most natural generalizations of the original einstein theory are the scalar - tensor theories @xcite . these theories are viable gravitational theories and can pass all known experimental and observational constraints . in addition , they can explain the accelerated expansion of the universe . the scalar - tensor generalizations of the original einstein theory naturally arise in the context of the modern unifying theories as string theory and kaluza - klein theories . in scalar - tensor theories the gravitational interaction is mediated not only by the spacetime metric but also by an additional scalar field . from a physical point of view this scalar field plays the role of a variable gravitational constant . general relativity ( gr ) is well - tested in the weak - field regime , whereas the strong - field regime remains largely unexplored and unconstrained . in the strong - field regime one expects the differences between gr and alternative theories of gravity to be more pronounced . the natural laboratories for testing the strong - field regime of gravitational theories are compact stars and black holes . there exist scalar - tensor theories which are indistinguishable from gr in the weak - field regime but which can differ significantly from gr in the strong - field regime . an example of such a phenomenon is the so - called spontaneous scalarization , observed in a certain class of scalar - tensor theories . when spontaneous scalarization takes place , in addition to the general relativistic solutions with a trivial scalar field , there exist further solutions with a nontrivial scalar field . in fact , these scalarized solutions are energetically more favorable than their gr counterparts . spontaneous scalarization was first observed for neutron stars @xcite , where _ spectacular changes _ were seen in static equilibrium configurations for a given nuclear equation of state . more recently , spontaneous scalarization was also observed in rapidly rotating neutron stars @xcite , where the deviations of the rapidly rotating scalar - tensor neutron stars from the general - relativistic solutions were even significantly larger than in the static case . spontaneous scalarization was also observed for static uncharged and charged boson stars @xcite . the first purpose of the present paper is to study rapidly rotating boson stars in scalar - tensor theories , and to establish the phenomenon of spontaneous scalarizarion for these stationary compact objects . the second purpose of this paper is to address the existence of scalarized hairy black holes . in general relativity ( gr ) rotating vacuum black holes are described in terms of the kerr solution . this solution specifies the full spacetime in terms of only two parameters , its mass and its angular momentum . hairy black holes appear , when suitable matter fields are included . examples are chiral fields , yang - mills and higgs fields , yielding hairy static black holes @xcite as well as rapidly rotating hairy black holes @xcite . recently it was noted , that also a single complex scalar field allows for hairy black holes , provided the black holes are rotating @xcite . in fact , these solutions maybe viewed as a generalization of rotating boson stars , that are endowed with a horizon . the regular boson stars form part of the boundary of the domain of existence of this new type of hairy black holes . the other parts of the boundary exist of extremal hairy black holes and scalar clouds . here we show , that besides these rapidly rotating hairy black holes , already present in gr , scalar - tensor theory again allows for the phenomenon of scalarization . in particular , we study the physical properties of these scalarized hairy black holes , and map their domain of existence . denoting the gravitational scalar by @xmath0 , the gravitational action of scalar - tensor theories in the physical jordan frame is given by @xmath1 , \end{aligned}\ ] ] where @xmath2 is the bare gravitational constant , @xmath3 is the spacetime metric , @xmath4 is the ricci scalar curvature , and @xmath5 $ ] denotes the action of the matter fields . the functions @xmath6 , @xmath7 and @xmath8 are subject to physical restrictions : we require @xmath9 , since gravitons should carry positive energy , and @xmath10 ^ 2 \ge 0 $ ] , since the kinetic energy of the saclar field should not be negative . the matter action @xmath11 depends on the matter field @xmath12 and on the space - time metric @xmath3 . the matter action does not involve the gravitational scalar field @xmath0 in order to satisfy the weak equivalence principle . variation of the action with respect to the spacetime metric and the gravitational scalar as well as the matter field leads to the field equations in the jordan frame . however , these field equations are rather involved . it is therefore easier to consider a mathematically equivalent formulation of scalar - tensor theories in the conformally related einstein frame with metric @xmath13 @xmath14 in the einstein frame the action then becomes ( up to a boundary term ) @xmath15 , \end{aligned}\ ] ] where @xmath16 is the ricci scalar curvature with respect to the einstein metric @xmath13 , @xmath17 represents the new scalar field defined via @xmath18 with the new functions @xmath19 by varying this action with respect to the metric in the einstein frame @xmath13 , the scalar field @xmath17 , and the matter field @xmath20 , we find the set of field equations in the einstein frame . in particular , we here consider a scalar tensor theory with potential @xmath21 and function @xmath22 @xcite @xmath23 we present a systematic study for the parameter value @xmath24 , but we have also done calculations for larger values of @xmath25 . for the matter action we choose a complex boson field @xmath26 @xmath27 = - \int d^4x \sqrt{-g } \left [ \frac{1}{2 } { \cal a}^2(\varphi ) g^{\mu\nu } \left ( \psi _ { , \ , \mu}^ * \psi _ { , \ , \nu } + \psi _ { , \ , \nu}^ * \psi _ { , \ , \mu } \right ) + { \cal a}^4(\varphi ) u ( \left| \psi \right| ) \right]\ ] ] with self - interaction potential @xmath28 to obtain stationary hairy black hole and boson star solutions we employ the line element @xmath29 + \bar{r}^2 \sin^2 \theta \left [ d\varphi - f_3 dt \right]^2 \right ) , \label{metric}\ ] ] with the metric functions @xmath30 , @xmath31 , and @xmath32 where @xmath33 denotes the horizon parameter . for the boson stars we seet @xmath34 , i.e. @xmath35 . likewise , we parametrize the gravitational scalar field @xmath0 by @xmath36 . for the boson field @xmath26 we adopt the stationary ansatz @xmath37 where @xmath38 is a real function , @xmath39 denotes the boson frequency , and as required by the single - valuedness of the scalar field @xmath40 is an integer representing a rotational quantum number . in the lagrangian for the boson field @xmath26 we employ a quartic self - interaction potential ( [ smatu ] ) with coupling constant @xmath41 . while we present our results for the value @xmath42 , we have also performed calculations for other values of @xmath41 . as in @xcite we introduce a new radial coordinate @xmath43 such that @xmath44 , and the event horizon is located at @xmath45 . the functions satisfy the following set of boundary conditions , obtained from the requirements of asymptotic flatness as well as of regularity at the origin and the event horizon in the case of the boson star , resp . black hole solutions @xmath46 and @xmath47 , resp . @xmath48 for boson star and black hole solutions . the mass @xmath49 and the angular momentum @xmath50 of stationary asymptotically flat space - times can be obtained in scalar - tensor theory - analogously to gr - from the asymptotic behavior of the metric functions @xmath51 and @xmath52 , respectively , @xmath53 and @xmath54 since the action is invariant under the global phase transformation @xmath55 , a conserved current arises @xmath56 it is associated with the noether charge @xmath57 representing the particle number , @xmath58 at the event horizon of the hairy black holes the killing vector @xmath59 @xmath60 is null , and @xmath61 represents the horizon angular velocity . hairy black holes satisfy @xcite @xmath62 we denote the horizon area by @xmath63 in the einstein frame , and define the areal horizon radius by @xmath64 the horizon temperature by @xmath65 is obtained from the surface gravity @xmath66 , @xmath67 and @xmath68 . we note , that the limit @xmath69 comprises two different types of configurations : * extremal black holes are obtained , when @xmath70 , * globally regular solutions are obtained , when @xmath71 . the latter correspond to rotating boson stars . for boson stars @xmath72 . we have obtained rapidly rotating scalarized boson stars and hairy black hole solutions for a sequence of rotational quantum numbers @xmath73 . to this end we have solved the set of coupled non - linear partial differential equations subject to the appropriate boundary conditions with a numerical algorithm based on the newton - raphson method @xcite . compactifying space by introducing the radial coordinate @xmath74 we have discretized the equations on a non - equidistant grid in @xmath75 and @xmath76 , with typical grid sizes on the order of @xmath77 , covering the integration region @xmath78 and @xmath79 . in the following we will used scaled quantities @xmath80 , @xmath81 , @xmath82 @xmath83 , and @xmath84 , with @xmath85 , @xmath86 , @xmath87 , @xmath88 , where @xmath89 denotes the plack mass . the globally regular ordinary boson star solutions form a large part of the boundary of the domain of existence of the hairy black hole solutions @xcite . let us therefore first consider the boson star solutions . in fig . [ fig1](a ) the scaled mass @xmath80 of the boson star solutions is exhibited versus the scaled boson frequency @xmath82 for rotational quantum numbers @xmath90 , 2 , 4 , 6 and 8 . the families of ordinary boson star solutions emerge from the vacuum at @xmath91 . they form a first ( and at least in part classically stable ) branch , until the mass reaches its maximal value . we note , that this maximal value of the mass increases rapidly with @xmath40 . beyond the maximal mass the families of ordinary boson star solutions continue in a spiral - like manner for the lowest values of @xmath40 . they end in a merger solution , where a branch of extremal hairy black hole solutions is encountered @xcite . for the higher @xmath40 , however , they feature only a single further branch , their second branch , before they merge with a branch of extremal hairy black hole solutions . interestingly , each of these higher @xmath40 second branches of boson star solutions ends close to an extremal kerr black hole solution , possessing almost the same mass and the respective horizon angular velocity @xmath92 . when considering the scaled angular momentum @xmath93 of these ordinary boson star solutions versus the mass @xmath80 as exhibited in fig . [ fig1](b ) together with the line @xmath94 of extremal kerr black holes , the branches form cusps at extremal values of the mass . the lower @xmath40 solutions therefore feature several cusps , whereas the higher @xmath40 solutions have a single cusp . clearly , for the higher @xmath40 families of boson star solutions the second branches approach the extremal kerr value closely , when they end in a merger solution , where a branch of extremal hairy black hole solutions is encountered . in addition , fig . [ fig1 ] exhibits the scalarized boson star solutions associated with the ordinary boson star solutions . for a given value of @xmath40 the scalarization arises at a critical value of the boson frequency , @xmath95 , where a branch of scalarized boson star solutions emerges from the first branch of ordinary boson star solutions . interestingly , @xmath95 is rather independent of the rotational quantum number @xmath40 . the families of scalarized boson stars then extend up to a second critical value @xmath96 , where they merge again into the respective second branch of ordinary boson stars . since @xmath96 decreases with @xmath40 , the domain of existence of rapidly rotating scalarized boson stars increases with @xmath40 . for @xmath90 , the critical point @xmath95 is close to but slightly below the maximal value of the mass of the ordinary boson stars . since the mass of the scalarized boson stars decreases monotonically until the critical point @xmath96 is reached , and since the same holds for the particle number @xmath57 , the ordinary boson stars are stable with respect to scalarization along their first branch . indeed , for a given value of @xmath57 along their first branch the mass of the ordinary boson stars is always lower than the mass of the scalarized boson stars , when these exist . for @xmath97 the situation is analogous . for @xmath98 , however , the scalarized boson stars assume their maximal mass no longer at @xmath95 , but at a smaller value of @xmath39 . thus they form a ( potentially ) stable branch , starting at the first minimum of the mass and extending until their global maximum . along this branch , the scalarized boson stars represent the energetically favored solutions . thus ordinary boson stars will be unstable with respect to scalarization in this range of frequencies . we note , that the maximal mass of the scalarized boson stars increases with @xmath40 . however , unlike the case of rapidly rotating neutron stars , where the maximal mass reached for scalarized neutron stars significantly exceeds the maximal mass of ordinary neutron stars , the maximal mass of scalarized boson stars does not deviate too strongly from the one of ordinary boson stars . considering the end point of the families of scalarized boson star solutions , we note that for the larger values of @xmath40 , with increasing @xmath40 the end point gets closer to the end point of the respective family of ordinary boson star solutions . in particular , the second branches of the ordinary boson star solutions shorten with increasing @xmath40 . for @xmath99 , the end points of both ordinary and scalarized boson star solutions are rather close to each other . let us now turn to the hairy black holes . starting from a generic ordinary boson star solution , a sequence of hairy black holes emerges , when the presence of a small horizon is imposed , and the horizon is then increased in size . the domain of existence of hairy black holes is then mapped by varying the horizon size and the horizon angular velocity . for ordinary hairy black holes , the domain of existence has been studied before for @xmath90 and 2 @xcite , employing only a mass term for the boson field . there it was shown , that the boundary of the domain of existence of these solutions consists of * the family of boson stars , and the associated families of * extremal hairy black holes and * scalar clouds @xcite . we exhibit in fig . [ fig2 ] the domain of existence of ordinary hairy black holes for the case of the @xmath100 potential , and rotational quantum numbers @xmath90 , 4 and 8 . in particular , we here show the scaled mass @xmath80 ( left column ) and the scaled particle number @xmath81 ( right column ) of the hairy black holes versus the scaled boson frequency @xmath82 . we note , that for black holes @xmath101 . here we have included the case @xmath90 , to allow for direct comparison with the case without self - interaction @xcite . studies of the rotational quantum numbers @xmath102 were not reported before , neither with nor without self - interaction . in all these figures , the beige regions labelled gr represent the domain of existence of the ordinary hairy black holes . the families of extremal hairy black hole solutions possess two endpoints . at one endpoint they join the respective branch of globally regular boson stars solutions in a merger solution . at the other endpoint they join precisely the respective branch of scalar cloud solutions . these latter endpoints are marked in the figures by an asterisk . in these figures the sets of extremal hairy black hole solutions have been obtained by extrapolation , where the horizon parameter @xmath33 was decreased towards zero . case . ] the hair of these extremal black hole solutions becomes evident , for instance , when inspecting the scaled particle number @xmath81 of these extremal solutions : @xmath57 is always finite ( see fig . [ fig2 ] ) . as seen in fig . [ fig2 ] for @xmath90 , the extremal hairy black hole solutions still form part of a spiral , whereas for the higher values of @xmath40 this spiralling pattern is lost . for @xmath103 and in particular for @xmath99 , the mass of the extremal hairy black hole solutions is close to the mass of the extremal kerr black holes . the particle number of these extremal hairy black holes is , however , clearly finite and reaches zero only when the scalar cloud solutions are reached . the study of @xmath57 therefore helps to clarify the domain of existence and its boundaries . let us now consider the domain of existence of the scalarized hairy black holes . it is also exhibited in fig . [ fig2 ] and marked by the green region labelled stt . the upper boundary of this domain of existence is always given by the regular scalarized boson stars . the lower boundary is reached somewhere within the domain of existence of ordinary hairy black holes , at the moment that the scalarization disappears . thus we do not observe extremal scalarized hairy black holes . there is always a part of the domain of existence of scalarized hairy black holes , where there are no ordinary hairy black holes . this part increases with increasing @xmath40 . for the lower @xmath40 , scalarized black holes exist for smaller boson frequencies , for the higher @xmath40 scalarized hairy black holes reach higher values of the mass and the particle number than ordinary hairy black holes . kerr black holes satisfy the bound @xmath104 . but this bound may be exceeded by hairy black holes @xcite . we demonstrate this in fig . [ fig3 ] for several families of hairy black holes . in particular , we exhibit the scaled angular momentum @xmath105 versus the areal horizon radius @xmath84 ( [ r_h ] ) for hairy black holes with @xmath106 , @xmath103 and @xmath99 in gr and with scalarization ( stt ) . we note , that scalarized hairy black holes can also exceed the kerr bound . we exhibit in fig . [ fig4 ] contour plots of the component @xmath107 of the energy momentum tensor ( left column ) and of the particle number density @xmath108 ( right column ) of a hairy black hole in gr ( top ) and a scalarized hairy black hole ( bottom ) . for comparison , we have chosen the same rotational quantum number @xmath99 , boson frequency @xmath109 and mass @xmath110 for these black holes . the central black hole in gr has a horizon area of @xmath111 and is thus much smaller than the scalarized black hole with @xmath112 . both @xmath113 and @xmath114 are concentrated in tori around the central black hole . the maximal value of @xmath113 is considerably larger for the gr black hole than for the scalarized one , while the maximal value of @xmath114 is only slightly larger . in the figures showing @xmath114 also the ergosurfaces are indicated . ordinary hairy black holes can feature ergosurfaces consisting of an ergosphere and an ergoring , forming together an ergo - saturn @xcite . this is also the case for the gr black hole shown . here we note , that the same phenomenon may hold as well for scalarized hairy black holes . their ergosurface may also represent an ergo - saturn , as depicted in the figure . scalar - tensor theories of gravity offer several observable consequences ( see e.g. , @xcite and references therein ) . here we have concentrated on the effect scalarization . first , we have shown that scalarization occurs for rapidly rotating boson stars ( with a fourth order self - interaction ) . rotating boson stars possess a rotational quantum number , an integer @xmath40 . constructing families of boson stars for @xmath115 , we have shown , that with increasing @xmath40 , the scalarization becomes more pronounced . we expect this trend to continue . subsequently , we have constructed hairy black holes . after mapping out the domain of existence of hairy black holes in gr , which is bounded by boson stars , extremal hairy black holes and scalar clouds , we have surveyed the domain of existence of scalarized black holes . one boundary of their domain of existence is formed by scalarized boson stars . the other boundary , however , is formed by ordinary hairy black holes . here the scalar field simply vanishes , thus reducing the solutions to general relativistic solutions with a trivial scalar field . we have shown that the physical properties of the scalarized hairy black holes resemble in many respects those of hairy black holes in gr . for instance , they may substantially exceed the kerr bound @xmath104 , and they can exhibit ergosurfaces , that consist of two parts , forming an ergo - saturn . the scalarization of rapidly rotating boson stars and hairy black holes allows their mass and particle number to exceed the maximal values allowed for their general relativistic counterparts . this effect seen here for the larger values of the rotational quantum number may be viewed as a downscaled version of what has been observed for neutron stars . it should be interesting to increase the rotational quantum number further , to see how strong the effect of scalarization may become for rotating hairy black holes . we gratefully acknowledge support by the dfg within the research training group 1620 models of gravity and by fp7 , marie curie actions , people , international research staff exchange scheme ( irses-606096 ) . we gratefully acknowledge discussions with e.radu . j.k . gratefully acknowledges discussions with c.lmmerzahl . 99 p. jordan , nature * 164 * ( 1949 ) 637 . m. fierz , helv . acta * 29 * ( 1956 ) 128 . p. jordan , z. phys . * 157 * ( 1959 ) 112 . c. brans and r. h. dicke , phys . * 124 * ( 1961 ) 925 . r. h. dicke , phys . * 125 * ( 1962 ) 2163 . t. damour and g. esposito - farese , phys . lett . * 70 * , 2220 ( 1993 ) . d. d. doneva , s. s. yazadjiev , n. stergioulas and k. d. kokkotas , phys . d * 88 * , no . 8 , 084060 ( 2013 ) . d. d. doneva , s. s. yazadjiev , n. stergioulas , k. d. kokkotas and t. m. athanasiadis , phys . d * 90 * , no . 4 , 044004 ( 2014 ) . a. w. whinnett , phys . d * 61 * , 124014 ( 2000 ) . m. alcubierre , j. c. degollado , d. nunez , m. ruiz and m. salgado , phys . d * 81 * , 124018 ( 2010 ) . m. ruiz , j. c. degollado , m. alcubierre , d. nunez and m. salgado , phys . d * 86 * , 104044 ( 2012 ) . c. a. r. herdeiro and e. radu , phys . lett . * 112 * , 221101 ( 2014 ) . s. hod , phys . d * 86 * , 104026 ( 2012 ) [ erratum - ibid . d * 86 * , 129902 ( 2012 ) ] . c. herdeiro , e. radu and h. runarsson , phys . b * 739 * ( 2014 ) 302 . c. herdeiro and e. radu , arxiv:1501.04319 [ gr - qc ] . w. schnauer and r. wei , j. comput . 27 , 279 ( 1989 ) 279 ; + m. schauder , r. wei and w. schnauer , the cadsol program package , universitt karlsruhe , interner bericht nr . 46/92 ( 1992 ) | in the presence of a complex scalar field scalar - tensor theory allows for scalarized rotating hairy black holes .
we exhibit the domain of existence for these scalarized black holes , which is bounded by scalarized rotating boson stars and ordinary hairy black holes .
we discuss the global properties of these solutions . like their counterparts in general relativity , their angular momentum may exceed the kerr bound , and their ergosurfaces may consist of a sphere and a ring , i.e. , form an ergo - saturn . |
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the past 15 years have seen large advances in the capabilities of ground - based gamma ray detection , from the pioneering observation of the crab nebula by the whipple observatory in 1989@xcite to the new generation of air cherenkov telescope arrays such as hess@xcite , veritas@xcite , and cangaroo@xcite and large area air cherenkov telescopes such as stacee @xcite , celeste@xcite , and magic@xcite . there are now at least 10 known sources of very - high - energy ( vhe ) gamma rays@xcite . the physics of these objects is astounding : from spinning neutron stars to super - massive black holes , these objects manage to accelerate particles to energies well in excess of 10 tev . how this acceleration occurs is not well understood and there is not universal agreement on what particles are being accelerated in some of these sources . at lower energies egret has detected over 270 sources of high - energy gamma rays@xcite and glast is expected to detect several thousand sources . in addition there are transient sources such as gamma - ray bursts that have to date eluded conclusive detection in the vhe regime ( despite some tantalizing hints@xcite ) . the paucity of vhe sources can be traced to the nature of the existing instruments : they are either narrow field instruments that can only view a small region of the sky at any one time and can only operate on clear moonless nights ( whipple , hegra , etc . ) or large field instruments with limited sensitivity ( milagro , tibet array ) . the milagro observatory has pioneered the use of a large area water cherenkov detector for the detection of extensive air showers . since an extensive air shower ( eas ) array directly detects the particles that survive to ground level it can operate continuously and simultaneously view the entire overhead sky . with the observation of the crab nebula and the active galaxies mrk 421 and mrk 501 , milagro has proven the efficacy of the technique and its ability to reject the cosmic - ray background at a level sufficient to detect sources@xcite . at the same time the tibet group@xcite has demonstrated the importance of a high - altitude site and what can be accomplished with a classical scintillator array at extreme altitudes . a detector with the all - sky and high - duty factor capabilities of milagro , but with a substantially lower energy threshold and a greatly increased sensitivity , would dramatically improve our knowledge of the vhe universe . reasonable design goals for such an instrument are : * ability to detect gamma - ray bursts to a redshift of 1.0 * ability to detect agn to a redshift beyond 0.3 * ability to resolve agn flares at the intensities and durations observed by the current generation of acts * ability to detect the crab nebula in a single transit this paper describes a design for a next generation all - sky vhe gamma - ray telescope , the hawc ( high altitude water cherenkov ) array , that satisfies these requirements . to quantify the definition of observing `` short '' flares from agn , previous measurements of flare intensities and durations by air cherenkov telescopes can be used . to date the shortest observed flares have had @xmath015 minute durations with an intensity of 3 - 4 times that of the crab@xcite . the low energy threshold needed to accomplish these goals requires that the detector be placed at extreme altitudes ( hawc would be situated at an altitude of @xmath04500 meters ) and the required sensitivity demands a large area detector - of order 40,000 m@xmath2 . section [ sec : particle_detection ] discusses the limiting performance of an eas array based on the properties of the eas , section [ sec : detector_description ] gives a physical description of the hawc and section [ sec : detector_performance ] details the expected performance of hawc . the ultimate performance of an eas array will be determined by the number , type , position , arrival time , and energy of the particles that reach the ground . here these properties of air showers are investigated to arrive at the limiting performance of eas arrays . to attain this level of performance an eas array would have to measure each of the above parameters with good precision . the most well - studied aspect of eas is the dependence of the number of particles to reach ground level on the observation altitude . for electromagnetic cascades , approximation b is a good estimator of the average number of particles in an eas as a function of atmospheric depth . however , at the threshold of an eas array , it is the fluctuations in the shower development that determines the response of the detector . to incorporate the effect of shower fluctuations the event generator corsika ( version 6.003@xcite ) is used to generate eas from gamma rays . the gamma rays were generated with an @xmath3 spectrum beginning at 10 gev , and uniformly over over the sky with zenith angles from 0 to 45 degrees . four different observation altitudes were studied : 2500 m , 3500 m , 4500 m , and 5200 m . figure [ fig : f1-altitude_effect ] shows the fraction of primary gamma rays that generated an air shower where more than 100 particles with energy above 10 mev survived to the observation level . the requirement that a particle have at least 10 mev in energy is imposed as a reasonable detection criterion and the requirement that 100 such particles survive enables one to examine the effects of altitude on the fluctuation spectrum of the air showers . this figure is a reasonable indication of the relative effective area of a fixed detector design as a function of the altitude of the detector . at high energies each km in altitude results in a factor of 2 - 3 increase in effective area . at low energies ( of relevance for extragalactic sources such as grbs ) the increase with altitude is larger . note that for primary energies between 100 gev and 500 gev a detector placed at 5200 m has @xmath4 times more effective area than the identical detector placed at 2500 m altitude ( close to the altitude of milagro ) . from this figure alone one can estimate that a detector roughly 10 times the size of milagro placed at 5200 m altitude would be of order 22 ( @xmath5 ) times more sensitive than milagro ( assuming the background scaled in a similar fashion ) . this level of sensitivity satisfies the requirements listed above . -0.3 in , title="fig : " ] it is well known that gamma rays outnumber electrons and positrons in an air shower by a large factor . figure [ fig : gamma - electron_ratio ] shows the ratio of gamma rays to electrons ( @xmath110 mev ) for three observation altitudes : 3500 m , 4500 m , and 5200 m . over this altitude range the ratio of gamma rays to electrons is relatively independent of the observation level . at these low energies gamma rays outnumber electrons and positrons by an order of magnitude . therefore to attain the lowest possible energy threshold the detector must be sensitive to the gamma rays in the air shower . an important feature of any gamma - ray detector is its ability to reject a significant fraction of the cosmic - ray background . the milagro detector has demonstrated that detecting and characterizing the penetrating component of an air shower is a reliable method of rejecting the cosmic - ray background@xcite . in figure [ fig : hadron_muon ] we show the average number of hadrons and muons in an air shower as a function of primary proton energy . an observation altitude of 5200 m is assumed in this figure . it should be noted that the fluctuations about these mean values are * not * poisson . but it is clear from this figure that the ability to detect showering hadrons in the air shower is important for the discrimination of the cosmic - ray background . this is especially true at low energies where the hadronic component is larger than the muon component . ( hadrons are defined to be protons , anti - protons , neutrons , and pions . ) -0.3 in , title="fig : " ] -0.3 in , title="fig : " ] after an event is recorded its direction in the sky must be reconstructed . to establish a limiting angular resolution for an eas array gamma - ray induced eas are generated using corsika as above ( on an @xmath3 spectrum beginning at 10 gev ) , however only vertical showers were generated . this simplifies the reconstruction of the air shower . in what follows it is assumed that the position , time , and energy of each particle are measured exactly . since the direction of the individual particle is not utilized in what follows it may be possible to develop a detector with improved angular resolution , though at a rather large expense . it is beyond the scope of this paper to give a detailed description of the fitting procedure . the leading edge of the eas is curved : particles farther from the core of the air shower arrive late relative to the particles at the core . studies of the corsika showers show that this curvature is dependent upon the individual energies of the shower particles . lower energy particles are delayed longer than higher energy particles . before using the arrival times of the particles to fit the direction of the primary gamma ray the particle arrival times must be corrected for this effect . to demonstrate the size of this effect figure [ fig : particle - point - spread ] shows the point spread function before and after this correction is applied to the particle arrival times ( for an observation altitude of 5200 m ) . the median angle error before the corrections are applied is 5.1 degrees and after the corrections are applied the median angle error is reduced to 0.8 degrees ( 71% of the events survive the complete fitting procedure ) . figure [ fig : particle - angle_comparealtitude ] shows the energy dependence of the mean of the point - spread function as a function of primary energy . for a fixed primary energy the angular reconstruction is improved by moving the detector to higher altitude . a detector at 4500 m would have the same angular resolution as a detector at 5200 m for gamma - ray primaries with 50% more energy . a detector at 3500 m altitude would require over a 100% increase in primary energy to obtain a similar angular resolution . -0.3 in , title="fig : " ] -0.3 in , title="fig : " ] because of the fluctuations in the shower development ( mainly the depth of the first interaction ) eas arrays have relatively poor energy resolution . to investigate the limiting energy resolution of an eas array primary gamma rays were generated on an @xmath3 spectrum from the zenith and the total number of particles with more than 10 mev that reach the ground within 100 m of the shower core were counted . figure [ fig : eres - particles_compare ] shows the mean ( and the error on the mean ) number of particles on the ground as a function of primary energy for observation altitudes of 3500 m , 4500 m , and 5200 m . at an altitude of 5200 m there is a good correlation between the mean number of particles and primary energy at primary energies as low as 50 gev . at an altitude of 4500 m this correlation begins at about 100 gev in primary energy and at 3500 m altitude there is no correlation until the primary energy is about 250 gev . to estimate the energy resolution , the curves in figure [ fig : eres - particles_compare ] are fit to a power law in the number of particles that reach the ground ( @xmath6 , where @xmath7 is the number of particles on the ground ) . figure [ fig : particle_eres_gt_50gev ] shows the distribution of @xmath8 for primary energies above 50 gev ( for an observation altitude of 5200 m ) . the resulting distribution is non - gaussian , but is fit reasonably well by a landau distribution ( shown on the figure ) . the asymmetric tail is an over - estimation of the particle energy and is inherent in the nature of the fluctuations in the depth of the first interaction . with a full width at half maximum of 0.5 the equivalent size of the 1-sigma error bar would be @xmath025% ( if the distribution were gaussian ) . as one goes to higher energy primaries the distribution becomes narrower and more gaussian . at energies below 50 gev the energy resolution is significantly worse , with an rms greater than 1 . since the fit is to the mean energy and there is a large positive tail , the peak in the distribution is negative . -0.3 in , title="fig : " ] -0.3 in , title="fig : " ] this section concludes with a brief summary of the best performance that can be expected from an extensive air shower array operating at high altitude . * an energy threshold below 50 gev * an angular resolution of 0.25 degrees at median energy * an energy resolution @xmath025% above 50 gev ( with a non - gaussian tail ) based on the above study and experience with the milagro detector a complete monte carlo simulation of a detector that is similar to milagro has been performed . the need to detect the gamma rays in the eas and the penetrating component of muons and hadronic cascades leads to a thick active detector . water is a natural and inexpensive choice of detecting medium . the cherenkov photons are relatively plentiful , prompt , and have large mean free paths ( relative to the water depth ) in clean water . since the cherenkov angle in water is 41@xmath9 , a detector spacing of roughly twice the detector depth ensures that there are no geometrical blind areas . ( the depth and spacing were chosen to match that of milagro , though more work is needed to optimize these parameters . ) the choice of a two layer design was again driven by experience with milagro . the deep layer makes a good calorimeter , whereas a shallow layer does not . in the shallow layer the detected light level depends critically on the exact geometry of the incoming particles , their distance from the photomultiplier tube ( pmt ) and incident angle . these fluctuations caused by the geometry of the incoming particles dilutes the energy measuring capability of a shallow layer of pmts . the ability to measure the energy of the shower particles is needed to determine the penetrating component of the air shower . in milagro this is the most useful method of rejecting the abundant cosmic - ray background . a shallow layer may be required to obtain the best possible angular resolution ( also important for rejecting the cosmic - ray background ) . since the speed of light in water is substantially different from the speed at which the particles propagate ( the speed of light in vacuum ) , the light and the particles diverge . this leads to an increase in the dispersion of the arrival time of the measured shower front as the depth of the measuring device is increased and the angular resolution is proportional to this spread . this effect is somewhat compensated for by the fact that a deeper layer typically makes more measurements of the shower front ( because each pmt can see a larger area of the water surface ) . the top layer may also be useful in the rejection of the cosmic ray background ( described below ) . work in better understanding these effects is still underway , but it may be possible to eliminate the shallow layer and maintain the performance of the detector . in the remainder of this paper , the following detector characteristics are used . * 40,000m@xmath2 physical area ( 200m@xmath10200 m ) * 2.7 m spacing of pmts ( 74@xmath1074 pmts per layer ) * hamamatsu 20 cm pmts ( r5912sel ) * two layers of pmts . a shallow layer under 1.5 m of water and a deep layer under 6.5 m of water * an attenuation length of 20 meters , composed solely of absorption . * trigger requires 50 pmts in the top layer to be struck by at least 1 photo - electron ( pe ) * a detector latitude of 36 degree ( that of milagro ) corsika 6.003@xcite was used to generate the air showers and geant 3.21 was used to track the particles through the water , generate the cherenkov light in the water , and to model the detector response ( pmts , reflectivity of the material , scattering and absorption of light in water , etc . ) . for the remainder of this paper this design will be referred to as the high altitude water cherenkov ( hawc ) array . to obtain realistic estimates of the sensitivity of hawc , the individual events must be reconstructed - their core position , direction , type ( hadronic or gamma ray ) , and energy need to be determined . preliminary algorithms have been developed to determine all of these parameters with the exception of the primary energy . work is ongoing in improving the existing algorithms and developing one to determine the energy of the primary gamma ray . in what follows , the events were generated beginning at 10 gev on an e@xmath11 spectrum ( similar to that observed for the crab nebula@xcite ) uniformly over incident direction . as discussed above , the shower front of an eas is not a true plane . therefore , the first step in performing an angular reconstruction is the determination of the core of the air shower . a simple center - of - mass algorithm using the pulse heights recorded in the pmts in the bottom layer of the detector is used to determine the shower core . figure [ fig : hawc - core_resolution4572 ] shows the distribution of core reconstruction errors in meters . for all events that trigger the detector and are successfully reconstructed ( see below ) the mean core error is 36 meters . despite the simplicity of the algorithm the core resolution is good enough that it does not degrade the angular resolution . the events with large core errors are those whose cores fall outside of the detector . the dashed line in the figure shows the core error for events whose core falls within the detector , the mean of this distribution is 20 meters . -0.3 in , title="fig : " ] the angular reconstruction is performed in an analogous manner to that described above , however the timing correction that was applied based on the individual particle energies is replaced by a correction based upon the number of pes measured in each pmt . after applying this correction and a correction for the shower front curvature ( proportional to the distance from the pmt to the core of the shower ) the resultant times and positions to a plane . in addition the weight associated with each pmt is determined by the measured pulse height . though the weight also depends upon the distance to the shower core , this effect has been ignored in the following analysis . a 20 - 30% improvement in the angular resolution may be expected with an improved algorithm . this improvement should arise when the correlations between the two corrections ( core distance and pulse height ) are utilized and the weights assigned to each pmt incorporate the core distance . the fitting procedure is iterative , where the pmts that made large contributions to the chi - square are removed in subsequent iterations . events with more than 20 pmts surviving in the final iteration of the fitting procedure are considered to be successfully fit . the space angle difference between the true direction and the fit direction is shown in figure [ fig : hawc - angular_resolution4572 ] . using this point - spread function the signal to noise ratio is maximal for an analysis bin size of 1.2 degrees . if the point - spread function were gaussian this bin size would correspond to an angular resolution of 0.75 degrees@xcite . -0.3 in , title="fig : " ] the ability to discriminate between gamma - ray induced air showers and hadronic air showers is critical to the success of any gamma - ray detector . unlike air cherenkov telescopes the background rejection discussed here is essentially independent of angular resolution . based on angular resolution alone with a 2sr field - of - view hawc ( and milagro ) remove roughly 99.93% of the background in their field - of - view . the background rejection discussed below is based purely on differences in the air showers regardless of their incoming direction . figure [ fig : events ] shows three gamma - ray induced events and three proton induced events in hawc . the figure shows the pulse height distribution of the pmts in the bottom layer of the detector . the area of each square is proportional to the number of detected photoelectrons ( pes ) in the pmt . the small black dots represent the pmts and the square is the position of the shower core . the figure caption gives the primary energy for each event . there are clear differences in morphology between proton and gamma - ray events . though there are some proton induced events that have a similar morphology to gamma - rays induced events , the events shown here are typical in the sense that most events have similar properties . note that while some of these events have very low primary energies , they are easily detected in hawc , striking many pmts . -0.3 in , title="fig : " ] milagro utilizes a parameter known as compactness that is defined as the ratio of the number of pmts in the bottom layer with more than 2 pes detected to the maximum number of pes measured in a single pmt in the bottom layer@xcite . hadronic showers tend to have small values of compactness and gamma ray showers a large value of compactness . the studies performed here indicate that as one moves to higher elevations the ability of this parameter to discriminate between hadronic and gamma - ray initiated cascades degrades . this is consistent with the observations of the milagro collaboration , who found that the background rejection capabilities of compactness improved as the zenith angle of the primary particle increased@xcite . however , a modified form for the compactness parameter , where the number of pmts with more than 2 pes is replaced with the number of pmts that are struck in the bottom layer , has a similar background rejection capability to that obtained by milagro . figure [ fig : hawc - compactness4572 ] shows this modified compactness parameter ( which will be referred to simply as compactness in what follows ) for proton ( dashed line ) and gamma ray ( solid line ) showers . if events are required to have a compactness greater than 8.3 one retains 66% of the gamma ray events and only 17% of the proton events , yielding a quality factor of 1.6 . ( the quality factor is defined as the fraction of gamma - ray events that survive the cut divided by the square root of the number of proton events that survive the cut , and is the relative improvement in sensitivity of the detector . ) another parameter that appears promising in discriminating gamma rays from background is the ratio of the number of pmts struck in the top layer ( @xmath12 ) to the number of pes in the brightest pmt in the bottom layer , where the pmts within 20 m of the fit shower core are excluded from the search for the maximum ( @xmath13 ) . this distribution is shown in figure [ fig : hawc - nxtop ] for gamma ray and proton induced events . the requirement @xmath14 retains 97.7% of the gamma ray events while rejecting 52.5% of the proton induced events . the important feature of this parameter is that essentially all of the gamma ray events survive , so even though the quality factor is only 1.4 one has retained all of the signal events ( regardless of their energy ) . the requirement @xmath15 yields a quality factor of 1.6 ( retaining 85% of the gamma ray induced events and 29% of the proton induced events ) , similar to the compactness criterion , but retaining a larger fraction of the gamma ray events . unlike the compactness parameter defined above ( and that used by milagro ) there is very little energy dependence to this criterion . in milagro compactness rejects low - energy gamma rays@xcite . -0.3 in , title="fig : " ] -0.3 in for proton induced events and gamma - ray induced events . see the text for the definitions of the parameters . [ fig : hawc - nxtop],title="fig : " ] the performance of any detector is determined by its effective area as a function of energy , its angular resolution , the level of the background and the ability to eliminate this background . these considerations can be parameterized in the following function : @xmath16 where @xmath17 is the `` sensitivity '' of the detector , @xmath18 is the angular resolution of the detector as a function of energy and zenith angle , @xmath19 is the differential spectral index of the source , @xmath20 is the effective area of the detector to gamma rays as a function of energy and zenith angle , @xmath21 is the zenith angle of the source as a function of time , @xmath22 is the effective area of the detector to the cosmic - ray background as a function of energy and zenith angle , and @xmath23 is the observation time . in determining the functions @xmath24 the background rejection should be included . the time dependence of the zenith angle is required since the energy response of the detector is dependent upon the atmospheric overburden . to obtain the sensitivity of hawc , equation [ eqn : sensitivity ] is evaluated with the aid of the monte carlo . events are generated on an e@xmath11 spectrum for gamma ray primaries and @xmath25 for proton primaries . after following the particles and the cherenkov light through the detector a list of pmts that are struck , the number of pes in each of these pmts , and the arrival time of the first pe in each pmt is written to a file . the events are generated over a distribution of zenith angles to mimic an isotropic flux . for a given source declination the time spent at each zenith angle is found and the effective area as a function of energy for that zenith angle is integrated to give a number of detected events as a function zenith angle for a given source declination . figure [ fig : hawc - effective_area ] shows the effective area as a function of energy for gamma - ray primaries for hawc at two different detector altitudes , 4572 m asl and 5200 m asl . the zenith angle averaging in this figure is as if the source was spread uniformly over the sky between zenith angles of 0 and 45 degrees . this figure is used for illustration only . to calculate the sensitivity to a source the integral in equation [ eqn : sensitivity ] is evaluated over the source transit ( given the declination of the source ) . in the remainder of the paper ( with the exception of the calculation of the sensitivity to gamma - ray bursts ) it is assumed that the source is at the declination of the crab ( 22 degrees ) and the detector is at a latitude of 36 degrees north . only events that are reconstructed within 1.2 degrees of their true direction are included in the calculation of the effective area . ( this accounts for the first term @xmath18 in equation [ eqn : sensitivity ] . ) at high energies ( @xmath01 tev ) hawc at 5200 m would have about 170 times the effective area of milagro to gamma rays and at 100 gev about 1000 times the effective area of milagro . as the energy decreases below this , the ratio of effective areas continues to increase , though there are insufficient monte carlo events at low energies for milagro to determine the ratio below 100 gev . -0.3 in 50 pmts struck in the top layer , @xmath26 pmts used in the angular reconstruction , and the reconstructed angle within 1.2 degrees of the true direction.[fig : hawc - effective_area],title="fig : " ] not withstanding the above discussion , the `` sensitivity '' of a detector is a poorly defined concept . since different astrophysical sources have different properties ( energy spectrum , duration , prior knowledge of their existence ) a single number called sensitivity is not useful . in what follows the sensitivity of hawc to three distinct phenomena is estimated : galactic sources with an energy spectrum of @xmath27 , active galaxies at various redshifts ( to investigate the effect of the absorption of vhe photons by the extragalactic background light ) , and gamma ray bursts . the latter is a major motivation for this type of detector , since acts by their nature have no opportunity to the prompt phase of grb emission . the crab nebula is the prototypical galactic tev gamma ray source . in reality the spectrum of the crab nebula is not well represented by a power law over the energy range 10 gev to 10 tev . however , for simplicity and to enable a comparison to other instruments the spectrum of the crab nebula@xcite is assumed to be @xmath28 m@xmath29s@xmath30 over the energy range of hawc . using the procedure outlined above the number of detected gamma rays per source transit in hawc is calculated . to estimate the background level in a bin of the same size with the same transit , the identical procedure is carried out with proton primaries ( the criterion that the events must be reconstructed with 1.2 degrees of their true direction is dropped ) . the identical procedure is than performed for the milagro monte carlo . the ratio of the simulated events observed in hawc to the simulated events observed in milagro is used to scale the actual number of events detected in milagro and predict the background rate in hawc . for an observation altitude of 5200 m the background rate will be @xmath085 times that of milagro and for an observation altitude of 4572 m the background rate will be @xmath060 times larger than milagro s . these numbers apply to the raw data , before the application of any background rejection . table [ tab : crab_rates ] gives the expected number of source and background events for a single transit of the crab . the predicted excess for milagro is somewhat larger than is actually observed by milagro . with two years of data milagro observes roughly 4@xmath31 on the crab nebula . the discrepancy is accounted for by three detector related effects ( the dead - time in milagro is about 12% , the on time is about 90% , and at any given time about 6% of the pmts are not working ) and one astrophysical effect , the spectrum of the crab nebula turns over at the higher energies@xcite . when these effects are properly accounted for the measurement of the crab flux by milagro is in good agreement with the measurement by hegra@xcite . table [ tab : crab_rates ] shows that a detector at 5200 m elevation is about 40% more sensitive than the same detector at 4572 m elevation . this is in agreement with the results given above on the effect of altitude on the particles at the ground . but both elevations result in a detector that can observe the crab nebula at high significance in a single transit . this is an important feature which allows one to quickly verify all aspects of the detector response . current acts such as the whipple telescope obtain roughly 5@xmath32 on the crab nebula in one hour of observation . though the calculated transit here consists of 6 hours , the bulk of the signal arrives in a 4-hour span , so the sensitivity for the same observation period of hawc will be roughly 1/2 that of the whipple telescope . with one year of observation a 50 mcrab source would yield a 5@xmath32 detection . in contrast the veritas 7-telescope array will detect a 7 mcrab source at 5@xmath32 with a 50 hour observation@xcite . with this sensitivity level veritas would require 3 - 4 years of dedicated time to survey 2sr of the sky at the level of 50 mcrab . ( this assumes no change in sensitivity for veritas for sources 1 degree off - axis . ) it is worth noting that these two surveys are fundamentally different , in that hawc would obtain a measurement ( or upper limit ) which is an average over a year , while the veritas measurement ( or upper limit ) would result from a 7-minute snapshot over a 3 - 4 year period . given the transient nature of many vhe sources , such a snapshot may be a limited value . the gamma - ray rate ( in hawc ) from the crab will be roughly 0.4 hz or 24/minute for a detector at 5200 m and 0.23hz ( or 14/minute ) for a detector at 4572 m asl . this number compares favorably with veritas and is significantly higher than that currently observed by the whipple telescope . .expected signals from the crab nebula in hawc at 5200 m and 4572 m elevation . milagro is at an elevation of 2700 m and is @xmath33 the size of hawc . [ cols="^ , < , < , < , < , < , < " , ] [ tab : crab_rates ] for more distant sources one must account for the absorption of high - energy photons caused by interactions with the extragalactic background light ( ebl)@xcite . while a direct measurement of the ebl has proved elusive , several models exist . in what follows the `` fast evolution '' model of stecker and de jager@xcite is used to estimate the effect as a function the redshift of the source on the sensitivity of hawc . for nearby objects ( redshift less than 0.05 ) there is a negligible difference between this model and the `` baseline '' model discussed in the reference . at a redshift of 0.2 the baseline model for the ebl results in @xmath015% more signal events detected by hawc than the model used here . figure [ fig : hawc - ebl - energy ] shows the number of signal events detected as a function of energy for sources at various redshifts . only events that satisfied the background rejection criterion , @xmath34 , had more the 20 pmts survive the angular reconstruction , and were reconstructed within 1.2 degrees of their true direction where used to calculate the curves in the figure . a source differential spectral index of -2.49 and an observation altitude of 5200 m have been assumed in this figure . an integration of these curves gives the total number of detected events for each source . -0.3 in are used to calculate the curves in the figure.[fig : hawc - ebl - energy],title="fig : " ] using the modified spectra given in figure [ fig : hawc - ebl - energy ] the sensitivity of hawc to distant sources can be calculated . figure [ fig : hawc - agn - time5 ] shows the number of days required to detect a source at the 5@xmath32 level as a function of source intensity for several different source redshifts . the source intensity is given in units of the crab flux and is the flux that would be present at the top of the atmosphere if there was no absorption of the high - energy gamma rays . hawc could detect a source at a redshift of 0.3 with a flux at the top of the atmosphere ( before accounting for absorption ) one tenth that of the crab nebula with less than one year of observation . -0.3 in as a function of the source intensity for sources at various redshifts . [ fig : hawc - agn - time5],title="fig : " ] the study of the high - energy spectrum of gamma - ray bursts ( grbs ) is perhaps the strongest motivation for an all - sky vhe telescope . while glast will have good sensitivity to grbs up to several 10 s of gev , a ground - based instrument , with its much larger effective area , could have good sensitivity up to the highest energy gamma rays that reach the earth . the ability to study the prompt phase of vhe emission could lead to exciting new physics . given its low energy threshold , hawc would have sensitivity to grbs at a redshift 1 . to estimate the sensitivity of hawc to grbs requires knowledge of any absorption at the source and the inevitable absorption of the vhe photons as they traverse the cosmos . since one does not know the inherent spectrum of grbs , in what follows the grb energy spectrum is modeled as a power law with a sharp cutoff . this cutoff could arise from absorption at the source or from the ebl . by investigating the sensitivity of hawc to different cutoff energies it is possible to estimate the sensitivity to grbs at different redshifts . figure [ fig : hawc_grb ] shows the sensitivity of hawc ( at an observation altitude of 4572 m asl ) to grbs as a function of burst duration for several energy cutoffs . the spectra are assumed to be @xmath3 up to the indicated cutoff energy , where the photon flux falls to zero . these curves give the flux level needed to observe a grb at the 5@xmath32 level . no background rejection has been applied to these data . the black circles are data from the batse detector , giving the observed distribution of grb fluence as a function of burst duration . the lower solid line is the sensitivity of glast and the upper solid line the sensitivity of egret . to detect a 1-second duration burst hawc would require 50 events from a grb over a background level of 85 events . the signal to noise level in a detected burst would be quite high . -0.3 in detection in hawc for various cutoff energies ( as given in the legend ) . for comparison the sensitivities of egret and glast are shown.[fig : hawc_grb],title="fig : " ] figure [ fig : hawc_grb ] can be used in conjunction with the known redshift distribution of grbs and the known grb rate to estimate how many grbs / year hawc would detect . assuming that the intrinsic grb spectrum extends to at least 50 gev , one sees that roughly 20% of all grb s are bright enough to be detected by hawc . if half of all bursts lie within a redshift of 1 and assuming a grb rate of 1 per day , hawc ( which views 1/6 of the sky ) would detect at least 6 grbs per year . if the intrinsic grb spectra extend beyond 50 gev hawc would detect more bursts . a design for the next generation all - sky vhe gamma - ray telescope has been presented . this instrument , hawc , would be over 20 times more sensitive than the milagro detector . with the ability to continuously view the entire overhead sky hawc will be an excellent complement to both glast and the coming generation of air cherenkov telescopes ( hess , magic , veritas , and cangaroo iii ) . with comparable sensitivity to glast it will be the only instrument capable of monitoring the many thousands of sources that glast is expected to detect at higher energies . in addition to searching the sky for galactic sources ( the vhe complement to the 150 egret unidentified objects ) , and active galaxies , the low - energy sensitivity of a detector placed at high altitude ensures that such an instrument will detect any vhe emission from gamma - ray bursts . perhaps most importantly an open aperture instrument with this level of sensitivity could discover completely new and unexpected phenomena that have so far eluded detection . 0 aharonian , f.a . , et al . , 2000 , _ apj _ , * 539 * , 317 . alexandreas , d.e . , et al . , 1993 , _ nim _ , * a328 * , 570 . amenomori , m , et al . , 1999 , _ apj _ , * 525 * , l93 . atkins , r.a . , et al . , 2000 , _ apj _ , * 533 * , l119 . atkins , et al . , 2003 , _ apj _ , * 595 * , 803 . gaidos , j. , et al . , 1996 , _ nature _ , * 383 * , 319 . hanna , d.s . , et al . , 2002 , _ nim _ , * a491 * , 126 . hartman , r.c . , et al . , 1999 , _ apjs _ , * 123 * , 79 . hillas , a.m. , et al . , _ apj _ , * 503 * , 744 . hofmann , w. , et al . , 2003 , _ proceedings of the 28@xmath35 icrc _ , * 5 * , 2811 . horan , d. and weekes , t.c . , 2003 , astro - ph/030391v1 . knapp , j. and heck , a. , 1993 , extensive air shower simulation with corsika : a user s manual ( kfk 5196 b ; karlsruhe : kernforschungszentrum karlsruhe ) . martnez , m. , et al . , 2003 , _ proceedings of the 28@xmath35 icrc _ , * 5 * , 2815 . par , e. , et al . , 2002 , _ nim _ , * a490 * , 71 . primack , j. r. , somerville , r. s. , bullock , j. s. , and devriendt , j. e. g. , 2000 , aip conference proceedings , 558 , 463 , aip : f. a. aharonian and h. j. v " olk stecker , f. and de jager , o. c. , 2002 , _ apj _ , * 566 * , 738 . weekes , t.c . , et al . , 1989 , _ apj _ , * 342 * , 379 . weekes , t.c . , et al . , 2002 , _ astroparticle physics _ , * 17 * , 221 . yoshikoski , t , et al . , 1999 , _ astroparticle physics _ , * 11 * , 267 . | the study of the universe at energies above 100 gev is a relatively new and exciting field .
the current generation of pointed instruments have detected tev gamma rays from at least 10 sources and the next generation of detectors promises a large increase in sensitivity .
we have also seen the development of a new type of all - sky monitor in this energy regime based on water cherenkov technology ( milagro ) . to fully understand the universe at these extreme energies requires a highly sensitive detector capable of continuously monitoring the entire overhead sky .
such an instrument could observe prompt emission from gamma - ray bursts and probe the limits of lorentz invariance at high energies .
with sufficient sensitivity it could detect short transients ( @xmath015 minutes ) from active galaxies and study the time structure of flares at energies unattainable to space - based instruments . unlike pointed instruments a wide - field instrument can make an unbiased study of all active galaxies and enable many multi - wavelength campaigns to study these objects .
this paper describes the design and performance of a next generation water cherenkov detector . to attain a low energy threshold and have high sensitivity
the detector should be located at high altitude ( @xmath1 4 km ) and have a large area ( @xmath040,000 m@xmath2 ) .
such an instrument could detect gamma ray bursts out to a redshift of 1 , observe flares from active galaxies as short as 15 minutes in duration , and survey the overhead sky at a level of 50 mcrab in one year . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this note we consider three reactions @xmath0 + h + [ lrg ] + p \,\ , ; \label{r1}\\ p + p & \longrightarrow & x_1 + [ lrg ] + h + [ lrg ] + x_2 \,\ , ; \label{r2}\\ p + p & \longrightarrow & b + \bar b + x + [ lrg ] + p \,\,;\label{r3}\end{aligned}\ ] ] where lrg denotes the large rapidity gap between produced particles and @xmath1 corresponds to a system of hadrons with masses much smaller than the total energy . the first two reactions are so called double pomeron production of higgs meson while the third is the single diffraction production of bottom - antibottom pair . the goals of this note are the following : 1 . to give the highest from reasonable estimates for the cross sections of reactions and ; 2 . to summarize all uncertainties which we see in doing these estimates ; 3 . to show that there is a new mechanism of diffractive heavy quark production ( ) which is suppressed in dis and dominates in hadron - hadron collision at the tevatron ; 4 . to estimate the value of the cross section of reaction due to this new mechanism and to show that all attempts to compare the diffraction dissociation in hadron - hadron collisions and dis@xcite look unreliable without a detail experimental study of this process at fermilab . inclusive higgs production has been studied in many details @xcite for the tevatron energies . the main source for higgs is gluon - gluon fusion which gives @xmath2 for higgs with mass @xmath3 @xcite . the reference point for our estimates is the cross section of higgs production due to w and z fusion which is equal to @xmath4 @xcite . in this process we also expect the two lrg @xcite and in some sense this is a competing process for reactions of and . let us estimate the simplest digram for the dp higgs production , namely , fig.1 without any of s - channel gluons . this diagram leads tpo the amplitude [ m1 ] m ( q q q h q ) = @xmath5 for reaction of , @xmath6 and therefore , [ m2 ] m ( q + q q + h + q ) . has an infrared divergency that is regularized by the size of the colliding hadrons . in other words , one can see that the simplest diagrams shows that dp higgs production is a typical soft " process . in fig.1 one can see that we have two sets of gluon which play a different role . the first one is the gluons that connect @xmath7-channel lines . their contribution increases the value of cross section @xcite [ g1 ] |_y = 0 = @xmath8 can be rewritten in the form [ g2 ] |_y = 0 = @xmath9 which is convenient for numeric estimates . however , first we need to find the value of @xmath10 . in inclusive production the value of @xmath11 has been calculated @xcite [ hx ] g^2_h= g_f ^2_s(m^2_h ) n^2/9 ^2 . however , i think that the scale of @xmath12 for our process is not the mass of higgs but the soft " scale ( @xmath13 with @xmath14 ) . indeed , using blm procedure @xcite we can include the bubbles with large number of light quarks only in @xmath7-channel gluon line which carry the `` soft '' transverse momenta . this gives a sizable effect in numbers , since @xmath15 for @xmath16 is equal to 1.16 pb ( @xmath17 ) and to 20 pb ( @xmath18 ) . taking the last value we have [ n1 ] to gluon emission suppressed the value of the cross section . actually , we have to multiply the cross section of by two factors to obtain the estimate for the experimental cross section [ sp1 ] d ( pp pp h)d y|_y = 0 @xmath19 the first factor is the probability that there is no inelastic interaction of the spectators in our process . i the situation with calculation of this factor has been reported in this workshop @xcite and the conclusion is that this factor @xmath20 at the tevatron energies . the discussion for double pomeron processes you can find in ref . @xcite the second factor in describe the probability that there is no parasite emission in fig.1 which leads to a process with hadrons in central rapidity region which do not come from the higgs decay . the generic formula for @xmath21 is [ sp2 ] s^2_spect = e^- < n_g(y = ln(m^2_h / s_0 ) > where @xmath22 is the mean number of gluon in interval @xmath23 . in pqcd this number is large @xcite @xmath24 which leads to very small cross section for higgs production . for `` soft '' double pomeron production we can estimate the value of @xmath25 assuming that the hadron production is two stage process : ( i ) production of mini jet with @xmath26 and ( ii ) minijet decay in hadrons which can be taken from @xmath27 process . finally , [ sp3 ] < n_g(y ) > = 2 3 , which gives @xmath28 . finally , we have [ god1 ] |_y = 0 = 0.02 pb we can increase the cross section , measuring reaction of . its cross section is equal to [ god2 ] ( pp \rightarrow pp h)}{d y}|_{y = 0 } \left ( \frac{\sigma^{sd}\cdot b_{el}(\sqrt{s}/m_h)}{4\,\sigma_{el}\cdot b_{dd}(\sqrt{s}/m_h ) } \right)^2 = \ ] ] @xmath29 and are our results . i firmly believe that they give the maximum values of the cross sections which we could obtain from reasonable estimates . however , i would like to summarize the most sensitive points in our estimates : 1 . the scale for running coupling qcd constant in cross section of higgs production . we took the `` soft '' scale for our estimates . however , it is a point which needs more discussion and even more it looks in contradiction with our feeling , as i have realised during our last meeting . my argument is the blm procedure but more discussions are needed ; 2 . we took @xmath30 for double pomeron processes the same as for hard " lrg process . the justification for this is eikonal type model @xcite , but it could be different opinions as well as direct experimental data ; 3 . the estimates for @xmath21 is very approximate and we need to work out better theory for this suppression . we would like also to mention that the new ideas on high energy interaction such as the saturation of the gluon density at high energy@xcite , will give a more optomistic estimates for the process of interest . the main observation is that there are two contributions for heavy quark diffractive production ( see ): ( i ) the first is so called ingelman - schlein mechanism @xcite which described by fig.2-a and ( ii ) the second one is closely related to coherent diffraction suggested in ref . @xcite and which corresponds to fig . the estimates of both of them have been discussed in ref . the main conclusion is that the main contribution for the tevatron energies stems from cd ( see also @xcite while the is mechanism leads to the value of the cross section in one order @xcite less than cd one . on the other hand in dis the cd contribution belongs to the high twisdt and because of that it is rather small @xcite . our conclusion is very simple . at the tevatrom we has a good chance to measure a new contribution to hard " diffraction which is small in dis . the typical values of the cross section is @xmath31 [ hq1 ] 10 ^ -4 10 ^ -10 forp_t , min = 550gev one can find all details in ref . i am very grateful to a. gotsman and u. maor for encouraging optimism and their permanent discussions on the subject . my special thanks goes to larry mclerran and his mob at the bnl for very creative atmosphere and fruitful discussions . this research was supported in part by the israel science foundation , founded by the israeli academy of science and humanities , and bsf @xmath32 9800276 . this manuscript has been authorized under contract no . de - ac02 - 98ch10886 with the u.s . department of energy . 99 l. alvero , j.c . collins and j.j . whitmore , psu - th-200,hep - ph/9806340 ; + l. alvero , j.c . collins , j. terron and j.j . whitmore , phys . d59 ( 1999 ) 074022 . z. kunszt and w.j . stirling , aachen ecfa ws , p.428 , 1990 . dokshitzer , v.a . khoze and t. sjostrand , phys . b274 ( 1992 ) 116 . marciano and s. willenbrock , phys . d49 ( 1994 ) 1354 . a. bialas and p.v . landshoff , phys . b256 ( 1991 ) 540 . b. muller and alec j. schramm , nucl a523 ( 1991 ) 677 . j - r cudell and oscar f. hernandez , nucl . b471 ( 1996 ) 471 . v. barger , r.j.n.phillips and d.zeppenfeld , phys . b346 ( 1995 ) 106 . s. dawson , nucl . b359 ( 1991 ) 283 ; a. djouadi , m. spira and p. zerwas , phys . lett . b264 ( 1991 ) 440 . brodsky , p. lepage and p. b. mackenzie , phys.rev . d28 ( 1983 ) 228 . e. levin , talk at this ws . e. gotsman , e. levin and u. maor , phys.lett . b353 ( 1995 ) 526 . martin , m.g . ryskin and v.a . khoze , phys.rev . d56 ( 1997 ) 5867 ; phys.lett . b401 ( 1997 ) 330 . l. v. gribov , e. m. levin , and m. g. ryskin , phys . rep . * 100 * , 1 ( 1983 ) ; + mclerran and r. venugopalan , phys . rev . * d 49 * ( 1994 ) 2233 ; * 49 * ( 1994 ) 3352 ; * 50 * ( 1994 ) 2225 ; + yu . v. kovchegov , phys * d 54 * 5463 ( 1996 ) ; * d 55 * 5445 ( 1997 ) ; + kovner , l. mclerran , and h. weigert , phys . rev . * d52 * , 3809 ( 1995 ) ; 6231 ( 1995 ) ; yu . v. kovchegov and dirk h. rischke , phys * c56 * , 1084 ( 1997 ) . g. alves , e. levin and a. santoro , phys . dd55 ( 1997 ) 2683 . g. ingelman and p. schlein , phys . b152 ( 1985 ) 256 . j. collins , l. frankfurt and m. strikman , phys . b307 ( 1993 ) 161 . d. e. soper , talk at dis97 , hep - ph/9707384 . m. wusthoff and a.d . martin , hep - ph/9909362 . f. yuan and k .- chao , hep - ph/9810340 ; hep - ph/9811285 . | in this note we give the highest of reasonable estimates for the value of cross section of the double pomeron higgs meson production and suggest a new mechanism for heavy quark diffractive production which will dominate at the tevatron energies .
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( [ # 1 ] ) bnl - nt-99/9 + taup-2615 - 99 + |
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a recent idea on possible application of topological superconductors to quantum information processing has attracted both theoretical and experimental interest @xcite . according to this idea , a quantum information unit , qubit , can be formed and propagated by means of majorana mode ( see , for e.g. , ref . ) , localized at the end of a one - dimensional ( 1d ) chain hosting a topological superconductor @xcite . recent investigations suggest several detection mechanisms of such a majorana mode @xcite such as an existence of a central peak in the tunneling current through a topological superconductor ( s)- normal metal ( n)junction and fractional period of the josephson current in s - n - s junctions . most recently experimental systems involving 1d semiconductor wire has been shown to host such modes ; the mechanism of the appearance of such modes arise from the combination of strong soi , proximity - induced superconducting gap , chemical potential and applied zeeman field in these wires @xcite . two majorana modes in the josephson junction , formed between two topological insulator edges or one - dimensional superconducting nanowires separated by barrier , hybridize resulting in splitting of the zero energy modes . this splitting energy depends not only on the phase difference of the two superconductors but also on the relative direction of the spin polarization at the two side of the junction . the oscillations of a josephson current between two such superconductors separated by insulator or metal as a function of their phase difference , with @xmath9 periodicity instead of a conventional @xmath10 periodicity due to hybridization of majorana states was predicted by kitaev @xcite for a idealized model of an 1d spinless p - wave superconductor . following this , kwon _ et al_. @xcite proposed that the similar effect can be observed between quasi-1d or 2d unconventional superconducting tunnel barrier junctions where the superconductors are separated by an insulating region , usually modeled by a delta function potential barrier . these systems did not have soi or zeeman field ; majorana - like modes appeared in such systems from the unconventional nature of the pairing potential . further it was realized in ref . that a signature of the fractional josephson effect constitutes in having a halved josephson frequency , @xmath11 , in the presence of a dc voltage @xmath12 applied across the junction . these effects have been interpreted in terms of the josephson current being carried by electrons rather than cooper pairs @xcite . further , it was shown that a fractional josephson effect may be realized at topological insulator edge @xcite . this prediction has later been extended to different systems @xcite . recent activities have established that a topological insulator with proximity - induced coupling to a s - wave superconductor exhibits a superconductivity - magnetism duality @xcite , revealing the fractional periodicity not only with superconducting phase difference but also with the orientation of zeeman magnetic field . in this case , the magnetic field on one side of the junction rotates in the plane normal to the direction of an effective magnetic field of the soi ; consequently , the majorana - mediated josephson current reverses sign after @xmath13 rotation of the magnetic field orientation and reveals an unconventional @xmath9 periodic magneto - josephson oscillation in response to variation of the magnetic field orientation in a topological insulator edge @xcite . furthermore , a dissipationless fractional josephson effect mediated by with @xmath14 periodicity has been also predicted @xcite at the edge of a quantum spin hall insulator . the josephson effect in consisting of topological superconducting ( s ) and normal ( n ) regions , has been reported in @xcite . these works also reveal a signature of majorana bound states located at s - n edges , producing a fractional josephson current with @xmath9 periodicity @xcite . these previous works in the field have pointed out the importance of the fractional josephson and the magneto - josephson effect in 1d superconducting junctions or atop the surface of a topological insulator . however , the role of spin - orbit coupling and the external magnetic field behind these effects has not been systematically investigated in these earlier works . such a systematic study is the main aim of the present work . to this end , we study the josephson effect between two 1d nanowires oriented along @xmath0 with proximity induced @xmath1-wave superconducting pairing and separated by a narrow dielectric with a rashba spin - orbit interaction ( soi ) of strength @xmath2 and zeeman fields ( @xmath3 along @xmath4 and @xmath5 in the @xmath6 plane ) . a schematic representation of the proposed setup is shown in fig . [ josephson ] . the main results of our study are as follows . first , we develop a general method for computing the andreev bound states energy in these junctions . such a method constitutes a generalization of the method of ref . to junctions with zeeman magnetic fields and spin - orbit coupling . second , using this method , we obtain analytical expressions for the energy of the andreev bound states in several asymptotic cases and discuss their implication on the josephson current . for example , we find that in the absence of the magnetic fields the energy gap between these bound states decreases with increasing rashba soi constant leading eventually to level touching while in the absence of rashba soi , they display oscillatory behavior with orientational angle of @xmath15 . third , we present analytic expressions for the dc josephson current charting out their dependence on both @xmath5 and @xmath3 and the soi interaction strength . fourth , we demonstrate the existence of finite spin - josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on @xmath2 , @xmath7 and @xmath3 . finally , we study the ac josephson effect in the presence of the soi ( for @xmath8 ) and an external radiation and show that the width of the resulting shapiro steps in such a system can be tuned by varying @xmath2 . we discuss experiments which can test our theoretical results . the plan of the rest of the paper is as follows . in sec . [ sec2 ] , we describe the model and present explicit form of hamiltonian is presented . the hybridization energy of edge states is calculated in sec . [ sec3 ] , where several asymptotic expressions for the josephson coupling energy are obtained . this is followed by a discussion of the dc josephson effect in sec . [ sec4 ] . the ac josephson effect in these system and the dependence of the shapiro step on soi strength is studied in sec . finally we conclude in sec . some details of our calculations are specified in the appendices . we consider a junction of two 1d nanowires with proximity induced @xmath1-wave pairing symmetry in the presence of rashba spin - orbital interaction and external magnetic fields . the schematic representation of such a junction is shown in fig . [ josephson ] where the proximate superconductors are not shown for clarity . -like dielectric potential under magnetic fields @xmath16 and @xmath5 co - planar and perpendicular to spin - orbit interaction respectively . the bulk s - wave superconductors which induces superconductivity in the wires are not shown for clarity.,height=245 ] in what follows we assume the pairing is induced by two proximate @xmath1-wave superconductors which leads to effective pairing potentials @xmath17 and @xmath18 in the two wires . the hamiltonian for such a system reads @xmath19 where @xmath20 is hamiltonian of the nanowire in the presence of external magnetic fields and @xmath21 represents rashba soi . the former term is given by @xmath22 \sigma_0 + h \sigma_z + b \{[\sigma_x \cos \phi_1 + \sigma_y \sin \phi_1 ] \theta(-x ) + [ \sigma_x \cos \phi_2 + \sigma_y \sin \phi_2 ] \theta(x ) \ } \big ) \psi_{\sigma'}(x ) \nonumber\\ & & + ( \delta_1 \theta(-x ) + \delta_2 \theta(x ) ) \psi_{\uparrow}^{\dag}(x ) \psi_{\downarrow}^{\dag}(x)+ { \rm h.c . } \big\ } , \label{h - sc}\end{aligned}\ ] ] where @xmath23 denotes the electron kinetic energy as measured from the fermi energy @xmath24 , @xmath25 is the electron annihilation operator , @xmath3 and @xmath5 are external zeeman magnetic fields in @xmath26 direction and in the @xmath6 plane respectively , @xmath27 is the heaviside step function , and @xmath28 and @xmath29 denote pauli and identity matrices respectively in spin space . note that the magnetic field @xmath5 forms an angle @xmath30 with wire which can be tuned externally . in what follows , we choose @xmath5 in the left side of the junction to be aligned along the wire ( @xmath31 ) while in the right side it is chosen to make an angle @xmath30 with it ( @xmath32 ) . in eq . ( [ h - sc ] ) , the pairing potential @xmath18 in the right of the junction is chosen to have a phase difference @xmath33 compared to its left counterpart : @xmath34 and @xmath35 . the potential barrier @xmath36 represents the barrier potential between two superconductors located at @xmath37 . the hamiltonian of rashba soi can be written as @xmath38\psi_{\sigma'}(x ) , \label{rashba - o}\end{aligned}\ ] ] where @xmath2 is the strength of rashba soi which is chosen to be the same for both wires . in what follows , we shall look for the localized subgap andreev bound states with @xmath39 for the josephson junction of two nanowires described by eq.([h ] ) . in this section , we first obtain solution for the andreev bound states for junction described by eq . ( [ h ] ) . to do this , it is advantageous to use a four component field operator given by @xmath40 here the third subscript of the annihilation operator ( which we shall designate henceforth as @xmath41 ) labels the right- ( @xmath42 ) and the left - moving @xmath43 ) quasiparticles respectively while the index @xmath44 denotes either right ( @xmath45 ) or left ( @xmath46 ) superconductor . in terms of the field operator given by eq . ( [ op1 ] ) , the hamiltonian ( eq . ( [ h ] ) ) can be written as @xmath47 using the pauli matrices @xmath48 in spin- and @xmath49 in particle - hole spaces . from eqs . ( [ h ] ) and ( [ h - sc ] ) , we find @xmath50 and @xmath51 . in eq.([h2 ] ) , the energy spectrum of the electrons are linearized around the positive and negative fermi momenta leading to @xmath52 , where @xmath53 is the fermi energy . note that the hamiltonians @xmath54 acquires a magnetism - superconductivity duality @xcite in the absence of the kinetic term , implying that it becomes invariant under the transformation @xmath55 . the existence of a magneto - josephson effect in a topological insulator is known to be a result of this duality @xcite . we shall see that for the system we study , the magneto - josephson effect takes place even in the presence of the additional quadratic kinetic energy term of the electrons . the energy spectrum of quasi - particles in a bulk superconductor in the presence of soi and external magnetic fields and its expression for different asymptotic is calculated in appendix [ appa ] . note that in our case , all energies are measured from the fermi energy ; thus the condition for realization of a topological superconducting phase with effective @xmath56-wave pairing is @xmath57 , @xcite . however , the existence of such a topological phase requires strong @xmath58 or @xmath3 and so interaction so that only the electron band of a single spin species remains below the fermi surface . in what follows we shall focus on the other regime where the bands of both spin species are below the fermi surface and the superconductivity is still s - wave . the bogolyubov - de gennes ( bdg ) equations for the superconductors in the right- and left parts of the barrier are written as @xmath59 where @xmath60 denotes the bdg wave function . for a barrier modeled by the delta function potential @xmath61 , they satisfy the boundary condition @xmath62 where @xmath63 and the transmission coefficient @xmath64 is expressed through @xmath65 as @xmath66 . for obtaining the energy of the andreev bound states for our system , we first note that analysis of the energy spectrum of a bulk superconductor ( see , eq . ( [ eo ] ) ) shows that the rashba soi splits the energy spectrum shifting it along the momentum axis leading to four fermi momenta at @xmath67 ( see , eqs.([e100 ] ) and ( [ k100 ] ) ) . the contribution to the andreev bound states comes from momenta around these fermi points . the external magnetic field splits spin - up and spin - down electrons ( see , eqs . ( [ e011 ] ) and ( [ k011 ] ) ) , and the amplitudes of the electron wavefunction are redistributed around four fermi points due to the presence of such a field . finally , the presence of a barrier between the two superconductors leads to superposition of the right and left moving quasiparticles . therefore , the bdg wavefunction @xmath60 can be written as a linear superposition of its right and left moving components around each fermi momentum and with two different spins . since we look for bound state solutions , the general solution of eq . ( [ sch ] ) with ( [ h2 ] ) can be written as @xmath68 % \end{displaymath } \label{wave}\end{aligned}\ ] ] where @xmath69 denotes the localization length of the bound states , and @xmath70 for @xmath71 . henceforth , we shall rename the coefficients as @xmath72 , @xmath73 , and @xmath74 , @xmath75 for clarity . substituting the wave functions ( [ wave ] ) into the boundary conditions ( [ bc ] ) one gets eight linear homogeneous equations for @xmath76 , @xmath77 , @xmath78 , and @xmath79 with @xmath80 which can be represented in terms of a @xmath81 matrix @xmath82 and a column vector @xmath83 as @xmath84 . the details of this procedure is charted out in appendix [ appa2 ] . the energy of the andreev bound states can then be obtained from @xmath85 . we note that since the momentum splitting @xmath86 vanishes in the absence of soi and magnetic field ; in this limit , either @xmath87 and @xmath88 or both @xmath78 and @xmath79 vanish . the elements of four columns of the @xmath81 determinant , depending on @xmath89 become equal to other four column elements as @xmath90 , and the determinant @xmath82 vanishes as @xmath91 and @xmath92 . _ andreev bound states at @xmath93 : _ in this limit , the andreev bound states are determined using @xmath94 determinant written for electron and hole pairs with opposite spins @xcite . the boundary conditions ( [ bc ] ) for the wave function ( [ wave ] ) , written in the absence of the soi induced momentum splitting yield again eight equations for four coefficients @xmath95 and @xmath96 ; these equations are bdg equations for a s - wave superconductor with spin - dependent eigenfunctions @xmath97 and @xmath98 , where the overline of an index ( e.g. , @xmath99 ) means an opposite direction or sign . one chooses four equations corresponding to an electron - hole pair with opposite spins . the determinant corresponding to the matrix ( defined as @xmath100 in appendix[appa2 ] ) in the front of the coefficients @xmath76 and @xmath77 is calculated to give @xmath101 where @xmath102 \left[\frac{\eta^{\ast}_{+ , \uparrow,+}}{\eta_{+,\downarrow,- } } -\frac{\eta^{\ast}_{- , \uparrow,+}}{\eta_{-,\downarrow,-}}\right]-\\ \nonumber ( 1-d ) \left[\frac{\eta^{\ast}_{+ , \uparrow,-}}{\eta_{+,\downarrow,+ } } -\frac{\eta^{\ast}_{- , \uparrow,+}}{\eta_{-,\downarrow,-}}\right ] \left[\frac{\eta^{\ast}_{- , \uparrow,-}}{\eta_{-,\downarrow,+ } } -\frac{\eta^{\ast}_{+ , \uparrow,+}}{\eta_{+,\downarrow,-}}\right ] . \label{energy0}\end{aligned}\ ] ] equating this determinant to zero one gets a condition to find the energy spectrum @xcite . note that the other four equations yields the same expression with only spin being interchanged leading to @xmath103 . it is easy to see that the condition to determine the andreev bound state energy in this limit , where @xmath82 constitutes two @xmath94 blocks , is given by equating @xmath104 to zero . in order to get the explicit expressions for the wave functions @xmath97 and @xmath98 we write eq . ( [ sch ] ) for finite @xmath58 , @xmath3 and @xmath2 as @xmath105 eqs . ( [ sch1]) .. ([sch4 ] ) allow us to calculate all possible ratios @xmath106 , @xmath107 , and @xmath108 , @xmath109 . furthermore , we note that only the ratio @xmath110 is non - zero for @xmath111 . we shall return to this case below . next , we note from eqs . ( [ sch1]) .. ([sch4 ] ) that the dependencies of these equations on @xmath30 and @xmath33 are completely removed by transforming the wave function as @xmath112 in the transformed basis one has @xmath113 the different ratios that appear in the left - side of eqs.[dagup - up] .. [up - up ] can be understood as follows . the ratio @xmath106 corresponds to the amplitude of conventional andreev reflection channel which constitutes reflection of an electron - like quasiparticle to a hole - like quasiparticle with opposite spin on a n - s interface . in contrast , the ratio @xmath114 which is finite only in the presence of soi and/or magnetic field , represents amplitude of andreev reflection channel where the electron - like quasiparticle incident on the interface is reflected to a hole - like quasiparticle state with the same spin orientation . finally , the ratio @xmath115 represents a usual reflection channel of an electron - like quasiparticle on the boundary without creation of a cooper pair in a superconducting part of the junction . since these ratios enter the expressions of @xmath116 , these also represents andreev and normal reflection processes involving electron - like and hole - like quasiparticles in the opposite ( @xmath117 ) and same ( @xmath118 ) spin sector . we note that the ratio of wavefunctions in eq . ( [ dagup - up ] ) depend on both @xmath30 and @xmath33 while those in eqs . ( [ dagup - down ] ) and ( [ up - up ] ) depend on either @xmath33 or @xmath30 . this suggests that the ratios ( [ dagup - up ] ) and ( [ dagup - down ] ) are responsible for the dependence of observable parameters on the order parameter phase difference @xmath33 , whereas the ratios ( [ dagup - up ] ) and ( [ up - up ] ) are responsible for the dependence on the magnetic field orientation angle @xmath30 . the ratios @xmath119 and @xmath120 are determined from eqs . ( [ sch1])-([sch4 ] ) as @xmath121 where the upper ( lower ) sign @xmath122 ( @xmath123 ) corresponds to spin @xmath124 ( @xmath125 ) . using eq . ( [ wave - up - down ] ) , one obtains , after a few lines of algebra , the expressions for @xmath126 and @xmath127 for general @xmath5 , @xmath3 and @xmath2 as @xmath128 ^ 2 \nonumber\\ & & -4d|\delta|^2 ( e^2 + \alpha^2 k^2)^2 \sin^2 \frac{\varphi}{2}\big\ } ( |\delta|^2(e^2+\alpha^2 k^2)^2)^{-1 } , \label{f - up - down } \nonumber\\\end{aligned}\ ] ] at @xmath129 a contribution to the bound state energy comes only from expression of @xmath130 , and all other ratios vanish . by equating @xmath131 ( eq . ( [ f - up - down ] ) ) to zero and using the expressions ( [ e000 ] ) and ( [ k000 ] ) for the energy and momentum in this limit , one gets an expression for the bound state energy in consistent with kwon _ et al_. result @xcite , @xmath132 thus our formalism reproduces the earlier known result in the literature in this limit . in the absence of the magnetic fields a contribution to the bound energy due to soi comes from the _ conventional _ andreev reflection connecting electron - like and hole - like quasiparticles with opposite spins . these can be expressed as @xmath133 where @xmath134 can be obtained using eqs . [ f - up - down ] and [ energy0 ] . in contrast , the main tunneling channel in the presence of the magnetic field constitutes an electron - like quasiparticle with a given spin polarization being andreev reflected to a hole - like quasiparticle state with the same spin . the contribution to the bound state energy from this channel is @xmath135 where @xmath136 is given by @xmath137 we note that @xmath138 ( or @xmath139 ) in eq . ( [ feqss ] ) is determined by eq.([energy0 ] ) after replacing the ratio @xmath140 in @xmath141 by @xmath142 . the expressions for @xmath114 can be obtained from eqs . ( [ sch1])-([sch4 ] ) @xmath143 where the upper(lower ) signs correspond to @xmath144 . these ratios can be used to obtain @xmath145 as @xmath146 finally , the contribution to the bound energy from the channel given by ( [ up - up ] ) can be expressed as @xmath147 where @xmath148 a procedure , similar to the one outlined above yields @xmath149 \left[\frac{\eta_{+ , \uparrow,-}}{\eta_{+,\downarrow,- } } - \frac{\eta_{- , \uparrow,-}}{\eta_{-,\downarrow,-}}\right ] \nonumber\\ & & -(1-d ) \left[\frac{\eta_{+ , \uparrow,+}}{\eta_{+,\downarrow,+ } } -\frac{\eta_{- , \uparrow,-}}{\eta_{-,\downarrow,-}}\right ] \left[\frac{\eta_{- , \uparrow,+}}{\eta_{-,\downarrow,+ } } -\frac{\eta_{+ , \uparrow,-}}{\eta_{+,\downarrow,-}}\right ] \nonumber\\ & = & \frac{16 b^2}{m^2_-(k)}\left[\alpha^2 k^2-d(e^2 + \alpha^2 k^2 ) \sin^2\frac{\phi}{2}\right ] . \label{fmag2}\end{aligned}\ ] ] the expression for @xmath150 differs from @xmath151 by replacing @xmath152 in ( [ fmag2 ] ) . by equating to zero the sum of the expressions ( [ ksy ] ) , ( [ cont1 ] ) , ( [ cont2 ] ) , and ( [ cont3 ] ) yields the andreev bound state energy in the presence of soi and magnetic fields . in what follows , we shall discuss two limiting case where a simple analytical expressions for these bound states can be obtained . _ absence of rashba soi _ : in this case , @xmath153 and @xmath154 , the main contribution , which depends on the magnetic field orientation , yields the expression ( [ cont2 ] ) with ( [ energym ] ) for @xmath155 and @xmath156 . although the contribution from ( [ cont1 ] ) does depend on the magnetic field , it does not depend on the field orientation @xmath30 . a few lines of algebra then leads to the equation for the energy of the andreev bound states , obtained by equating the sum of ( [ ksy ] ) , ( [ cont1 ] ) and ( [ cont2 ] ) to zero , using ( [ e011 ] ) and ( [ k011 ] ) for the energy spectrum and momentum in this limit , given by @xmath157\right\ } = 0 , \label{eq - mag - add}\end{aligned}\ ] ] where the second and third terms come from ( [ cont2 ] ) and ( [ cont1 ] ) corresponding to the reflection mechanisms ( [ dagup - up ] ) and ( [ dagup - down ] ) . if we neglect the third contribution , which can be done for @xmath158 , eq.([eq - mag - add]),can be written @xmath159 we find that eq . ( [ eq - mag ] ) leads to the following features of the andreev bound states . first , @xmath160 decreases with increasing the magnetic field . second , the kwon et al . result @xcite is recovered as @xmath161 . ( [ eq - mag ] ) can be solved approximately . we replace the energy under square root by its zero - approximation value ( [ e0 ] ) , which yields @xmath162 with @xmath163 , where @xmath164 the second term in the bracket of eq . ( [ eq - maga])depends on the magnetic field as @xmath165 for @xmath166 . we note here that @xmath167 oscillates both with the superconducting phase difference @xmath33 and the angle orientation @xmath30 of @xmath5 with a period @xmath10 as shown in fig . [ magnet ] . note that all parameters in the figures presented below are dimensionless ones in the scale of @xmath168 , i.e. @xmath169 , @xmath170 , @xmath171 . at @xmath172 , when kwon et al . @xcite result is recovered for @xmath1-wave superconducting junction and the andreev bound state energy oscillates with @xmath13 periodicity ( see , fig . [ magnet]a ) for barrier transparency @xmath173 . the electron - like and hole - like energy branches corresponding to @xmath174 , touch each other at maximal transmission when @xmath175 , creating a zero - energy state at the center of the brillouin zone . the variation of @xmath58 and @xmath3 changes a character of @xmath33- and @xmath176-dependencies * of @xmath177*. note that since the gap between them vanishes at @xmath178 , it might be possible to have a @xmath9 periodic component of the josephson current in case of landau - zener transitions with a finite transmission probability between two states . this case will be investigated somewhere else . [ h ! ] ) , on the order parameter phase difference at @xmath179 and ( a ) @xmath180 , @xmath181 ; ( b ) @xmath182 , @xmath183 , @xmath184.,title="fig:",height=188 ] ) , on the order parameter phase difference at @xmath179 and ( a ) @xmath180 , @xmath181 ; ( b ) @xmath182 , @xmath183 , @xmath184.,title="fig:",height=188 ] _ absence of in - plane zeeman field _ : next , we consider the andreev bound states for @xmath185 , but @xmath186 . we find that eqs . ( [ sch1])- ( [ sch4 ] ) in this case link only @xmath187 and @xmath188 and are hence greatly simplified . a few lines of algebra shows that the andreev bound states energy in this case can be expressed as @xmath189 ^ 2 + \nonumber \\ f_{\uparrow , \downarrow}^{\ast}(k_+ ) f_{\downarrow , \uparrow}^{\ast}(k_- ) - f_{\uparrow , \downarrow}^{\ast}(k_-)f_{\downarrow , \uparrow}^{\ast}(k_+)=0 . \label{energyb=0}\end{aligned}\ ] ] the expression for @xmath190 in this limit is calculated in appendix [ appb ] and is given by eq.([apf ] ) . the expression for @xmath191 at @xmath111 is obtained from eq.([apf ] ) by replacing @xmath192 and @xmath193 . below we will study two asymptotic solutions of eq . ( [ energyb=0 ] ) at @xmath194 , @xmath195 and @xmath196 , @xmath197 . in the former case , eq . ( [ energyb=0 ] ) with ( [ apf ] ) yields the following equation @xmath198 solution of this equation provides a simple expression for the josephson energy @xmath199 where the sign @xmath200 in the front of the expression signifies an electron and hole energies , whereas the sign @xmath201 characterizes rashba splitting of the electron and hole states . this expression shows that @xmath202 depends nonlinearly on the soi coupling constant @xmath2 , and kwon et al . result @xcite is recovered as @xmath91 . according to ( [ soi ] ) , @xmath203 oscillates still with @xmath13 period for @xmath204 and @xmath205 , which is presented in fig . [ solutions](a ) at @xmath206 and @xmath207 . possible solutions for the energy spectrum according to the expression ( [ soi ] ) as a function of the order parameter phase difference at @xmath208 and @xmath175 is presented in fig.[solutions](b ) . it shows touching of all four branches at @xmath209 . the electron- and hole energy branches approach each other faster for non - zero soi . the dependence of the @xmath210 energy branches on the order parameter phase difference @xmath33 at fixed transmission coefficient @xmath64 and different values of the soi strength @xmath2 , is presented in the left panel of fig . [ e - p - dep](a ) . in fig.[e - p - dep](b ) , we present the dependence of the andreev bound state energies on @xmath64 for fixed @xmath2 . we note that both the branches approach zero as @xmath2 or @xmath64 is varied . _ absence of @xmath58 and @xmath2 _ : next , we consider the case @xmath211 but @xmath212 . in this case , eq . ( [ energyb=0 ] ) reduces to [ h ! ] ) as a function of the order parameter phase difference at ( a ) @xmath206 and @xmath207 , and ( b ) @xmath208 and @xmath175.,title="fig:",height=188 ] ) as a function of the order parameter phase difference at ( a ) @xmath206 and @xmath207 , and ( b ) @xmath208 and @xmath175.,title="fig:",height=188 ] [ h ! ] and different values of the soi strength , and ( b ) @xmath206 and different values of the transmission coefficient @xmath64.,title="fig:",height=226 ] and different values of the soi strength , and ( b ) @xmath206 and different values of the transmission coefficient @xmath64.,title="fig:",height=226 ] [ h ! ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] @xmath218 whose solutions read @xmath219 where @xmath201 . we note that the particle - like and the hole - like branches touch at zero energy ; in order to investigate the possible existence of a zero energy mode , which may create a @xmath220 oscillatory component of the josephson current in the landau - zenner sense , we introduce a dimensionless magnetic field @xmath221 . it is easy to see from eq . ( [ h ] ) that the condition for the particle and the hole states to cross at a phase difference @xmath33 is given by @xmath222 which yields @xmath223\sin^2\frac{\varphi}{2}}.\ ] ] for @xmath224 , the value of the critical @xmath225 for spin - up ( @xmath226 ) and spin - down ( @xmath227 ) states are @xmath228 and @xmath229 , correspondingly . we note here that the bands touch each other at @xmath230 but do not cross ; thus the andreev states still have @xmath231 periodic dispersion . the variation of @xmath232 with @xmath213 , the dependence of @xmath213 on @xmath64 , the touching of the @xmath233 and @xmath234 energy branches at @xmath215 and @xmath216 , and that between @xmath235 and @xmath236 energy branches at d=0.5 and @xmath217 are plotted in fig . [ crossing ] . in fig.[crossing](a ) , where the dependence of the spin - up particle energy branch on @xmath33 at different @xmath213 is presented , we find that the amplitude of the energy oscillation increases with @xmath213 , and additionally , the character of dependence around @xmath237 is changed . in fig . [ crossing](b ) we show the mutual optimal values of @xmath213 and @xmath64 at which electron - like and hole - like energy branches touche each other . finally , the touching of the two branches @xmath238 and @xmath239 for @xmath215 and @xmath216 is presented in fig . [ crossing](c ) . as it was mentioned above , these feature might be responsible for a @xmath240 periodicity in case of landau - zener transitions with a finite transmission probability between two states . the contribution of the andreev bound state to the josephson current can be calculated using to the expression @xmath241 where @xmath242 signifies all states which give a contribution to the current , and @xmath243 is the fermi occupation number corresponding to the @xmath242-th states . we note that since only the andreev bound states depend explicitly on the phase difference @xmath33 , their expression can be used to determine the dc josephson current using eq . ( [ current ] ) . in the absence of soi a contribution to the total equilibrium current gives electron and hole states , each of which is split into two levels due to zeeman effect @xmath244 where the expression for @xmath177 is given by eq . ( [ eq - maga ] ) , and @xmath245 \label{des}\end{aligned}\ ] ] with @xmath246 the current - phase relation at magnetic field @xmath182 calculated by using expressions ( [ currentt ] ) , ( [ des ] ) and ( [ cph ] ) is presented in fig . [ 44 ] . we note , that changes in @xmath3 does not make an essential effect at @xmath247 . next , we consider the spin - josephson current which is generated as response to rotation of the magnetic field @xmath248 in @xmath249 plane@xcite . as shown in ref . , the spin current can be defined as a derivative of the tunneling energy with respect to the magnetic field orientation @xmath30 and is given by @xmath250 according to the formulas ( [ currentt])-([cph]),height=188 ] , @xmath182 , @xmath251 , @xmath252 and two values of magnetic filed @xmath183 ( curve 1 ) and @xmath253 ( curve 2 ) . calculations are done according to the formulas ( [ j - mag ] ) , ( [ j - mag2 ] ) and ( [ eq - maga]).,height=188 ] where @xmath254 with @xmath255 for @xmath256 and @xmath257 for @xmath258 . as it is seen from formulas ( [ des ] ) and ( [ j - mag2 ] ) , the product @xmath259 increases with @xmath58 at @xmath166 as @xmath260 . instead in the opposite limit when @xmath261 this product decreases with increasing @xmath3 as @xmath262 . on the other hand , in the high temperature limit , when @xmath263 , one can expand @xmath264 function for small argument @xmath265 as @xmath266 . therefore , the amplitude of the supercurrent @xmath267 , given by eq.([des ] ) , and of the spin current @xmath268 , given by eq.([j - mag2 ] ) , will depend on the magnetic field exactly in the same form as described above for two limiting cases . the change of @xmath269direction can rotate the direction of spin current . spin current as a function of magnetic field orientation at two values of magnetic filed @xmath183 and @xmath253 is shown in fig . [ 47 ] . calculations are done according to the formulas ( [ j - mag ] ) , ( [ j - mag2 ] ) and ( [ eq - maga ] ) . the josephson current in other limiting case when @xmath180 and @xmath195 is calculated by replacing @xmath177 with @xmath270 given by ( [ soi ] ) in the expression ( [ currentt ] ) @xmath271 \sqrt{\left(1-d \sin^2\frac{\varphi}{2}\right)\left[1-s\frac{4v_f\alpha}{(v_f+s\alpha)^2}d\sin^2\frac{\varphi}{2}\right ] } } \tanh\left(\frac{e_s^{soi}}{2k_bt}\right).\ ] ] the corresponding plots demonstrated a strong variation of current - phase relation with parameter of spin - orbital coupling @xmath2 are presented in fig . [ the figure demonstrates a crucial breaking of the sinusoidal current - phase relation with increase in spin - orbital coupling . it shows a singular behavior at small @xmath33 . at @xmath215 ( formula ( 49 ) ) . numbers show the values of parameter spin - orbital coupling.,height=226 ] in this section , we compute the ac josephson effect for the tunnel junctions mentioned above . if there is the voltage in josephson junction @xmath272 , then from josephson relation @xmath273 we get @xmath274 , \label{phaseeq1}\end{aligned}\ ] ] we shall now use this relation to obtain the shapiro step width for @xmath180 and demonstrate that the step - width depends on the strength of the spin - orbit coupling . to do this we first consider the case @xmath251 for which @xmath275 $ ] is given at @xmath276 by @xmath277 substituting eq . ( [ phaseeq1 ] ) into eq . ( [ iexp ] ) , one gets @xmath278 using the identity @xmath279 where @xmath280 means imaginary part , @xmath242 is an integer and @xmath281 denotes bessel function of the first kind , one gets @xmath282/2 } } \label{ieq3}\ ] ] here @xmath283 means the real part . the shapiro steps thus occur when @xmath284 for integer @xmath285 ; at these values of the applied radiation frequency , the ac component of the supercurrent vanishes leading to an extra contribution to the dc current in the circuit . the magnitude of the extra dc current from @xmath286 can be read off from eq . ( [ ieq3 ] ) as @xmath287/2 } } \label{idc1}\ ] ] from eq . ( [ idc1 ] ) , we find that both the shapiro step width and the position of maxima / minima of @xmath288 depends on @xmath64 . let us assume that the maxima and minima occur at @xmath289 . note that @xmath290 can be obtained from the solution of @xmath291 and equals @xmath292 for @xmath293 . in terms of @xmath294 , one obtains the step width as @xmath295/2 } } \label{sstep1}\ ] ] which clearly shows the @xmath64 dependence of the step - width . one can now carry out a similar analysis for the case where @xmath180 and @xmath296 ( eq . ( [ soi ] ) ) . starting from eq.([current ] ) , the ac josephson current at @xmath276 can be obtained as @xmath297^{3/2}\left[1-d ( 1-\cos \varphi(t))/2\right]^{1/2 } } \label{acjos1}\ ] ] where @xmath298 and @xmath299 . similar straightforward algebra , as carried out earlier in this section , leads to steps at @xmath300 with @xmath301^{1/2 } } \sum_{s=\pm } \frac{(1-\eta_s)}{\left[1-\eta_s d ( 1-j_{n_0}(\omega ) \cos \varphi_0)/2\right]^{3/2 } } \label{dcjos1}\ ] ] as before , the minimum and maximum of the dc component of the occurs at @xmath302 which can be obtained as the solution of @xmath303 . the step width can thus be expressed in terms of @xmath304 as @xmath305^{1/2 } } \\ \sum_{s=\pm } \frac{(1-\eta_s)}{\left[1-\eta_s d ( 1-j_{n_0}(\omega ) \cos \varphi_0^{n_0 \alpha})/2\right]^{3/2}}\label{swidthso}\ ] ] thus we find the step width depends on the magnitude of the spin - orbit coupling . indeed , [ plots - width](a ) demonstrates this effect of transparency and spin - orbital coupling on the @xmath33-dependence of the shapiro step width according to formula ( [ swidthso ] ) . we also note that for @xmath293 , the maxima and minima of the dc current occur for @xmath306 and eq . ( [ swidthso ] ) simplifies to yield @xmath307 for small @xmath308 , it is easy to see by expanding @xmath309 in power of @xmath310 , that @xmath311 which demonstrates the dependence of step width on the so coupling @xmath2 . comparison of these three plots according to eqs . ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2 ] ) is presented in fig.[plots - width](b ) . as we can see , the results of approximations ( [ swidthdll1 ] ) and ( [ swidthdll2 ] ) demonstrate more sharper increasing of shapiro step width with @xmath2 in compare with formula ( [ swidthso ] ) . it s clear that the difference disappears in the limit @xmath312 . the obtained dependence of the ss width on the spin - orbit coupling may be used for the experimental estimation of its value . -dependence of the shapiro step width according to the formula ( [ swidthso ] ) ; ( b ) demonstration of @xmath2-dependence of shapiro step width in different approximations according to the formulas ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2]).,title="fig:",height=188 ] -dependence of the shapiro step width according to the formula ( [ swidthso ] ) ; ( b ) demonstration of @xmath2-dependence of shapiro step width in different approximations according to the formulas ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2]).,title="fig:",height=188 ] , @xmath206 without radiation ( curve 1 ) and under external radiation ( curve 2),height=226 ] to investigate the effect of soi on the amplitude dependence of shapiro step width , we have calculated the i - v curves for the junction under external radiation using equation ( [ acjos1 ] ) . this result is presented in fig . [ iv_curve ] , where we show the i - v curve of the junction at @xmath215 , @xmath206 under external electromagnetic radiation with frequency @xmath313 and amplitude @xmath314 . in this figure we include for comparison the i - v characteristics without radiation also . the i - v curve demonstrates the main shapiro step at @xmath315 and its harmonics . [ amp_dep](a ) shows the amplitude dependence of shapiro step width in case @xmath316 ( line 1 ) and @xmath206 ( line 2 ) under external radiation with frequency @xmath313 . calculation is provided for value of transparency @xmath215 . we see that the value of the soi parameter has a noticeable effect on the shapiro step width and its dependence on amplitude of the external radiation . these results of i - v characteristics simulations coincide qualitatively with the conclusion followed from fig . [ plots - width ] . we see that in case with @xmath316 the width of shapiro step is larger than case @xmath206 . the similar effect can be seen in amplitude dependence of critical current @xmath317 , which is shown in fig.[amp_dep](b ) . dependence of @xmath317 for @xmath215 and @xmath64dependence of @xmath317 for @xmath318 at @xmath313 , @xmath314.,height=226 ] the transparency coefficient @xmath64 also effects the critical current value . to distinguish and clarify the effect of soi we have calculated the @xmath2 and @xmath64dependence of @xmath317 , which is demonstrated in figures [ alpha_d - dep ] ( a ) and ( b ) . these results might be used for the comparison with future experimental results . in this paper we study the josephson current between 1d superconducting nanowires separated by an insulating barrier in the presence of rashba soi and the magnetic fields @xmath5 and @xmath3 . the presence of the soi and zeeman magnetic fields leads to four distinct fermi points in each bulk superconductor . therefore , the study of josephson effect in these junctions requires construction of an incident quasiparticle wave function which is in a linear superposition state of plane waves around each fermi points . in our study , we have developed a theoretical method to study josephson effect in such systems ; our work thus constitutes a generalization of analysis of ref . to systems with soi and zeeman fields . we have provided analytical results for the andreev bound states in several asymptotic limits from our analysis , demonstrated the presence of spin - josephson current in these junctions , and studied the dependence of shapiro steps on soi interaction strength @xmath2 in the presence of external radiation . moreover , we have demonstrated the existence of magneto - josephson effect in these systems . we note that although the existence of the magneto - josephson effect in a topological superconductor has been predicted recently @xcite , the question of whether this effect is observable in superconducting junctions with quadratic electronic dispersion and the absence of soi was not addressed before . we show in the paper the magneto - josephson effect takes place even in the absence of soi . experimental verification of our work would require experiments conducted on josephson junctions in 1d nanowires analogous to ones studied in ref . . we predict that the variation of the angle @xmath30 of the in - plane magnetic field @xmath5 would lead to a spin - josephson current as shown in fig . furthermore , ac josephson effect measurement in these junction , analogous to those done in ref . , should reveal a quadratic dependence of the shapiro step - width as a function of @xmath2 for small @xmath319 as shown in fig . [ alpha_d - dep ] . our work allows for several possible future direction . first , a numerical solution of the condition @xmath320 yielding andreev bound state energies in the regime where @xmath321 may lead to a better understanding of the interplay between these parameters to shape the characteristics of the bound state energies . second , the formalism that we develop here may be extended to regime of strong @xmath2 where the presence of majorana bound states shapes the characteristics of the josephson current . third , our formalism may be applied to cases where the superconducting pair - potential is unconventional ( for example p - wave ) ; indeed , interplay of such unconventional pair - potentials and so coupling may lead to additional interesting characteristics in the josepshon current . we intend to explore these issues in future work . in conclusion , we have studied josephson effect in a unction between two 1d nanowires in the presence of soi and zeeman fields . we have analyzed the josephson current in these junctions and provided analytical expressions of the andreev bound states in several limiting cases . we have also demonstrated the presence of magneto - josephson effect in these junctions and studied the shapiro step width in ac josephson effect on the soi strength . our theoretical predictions are shown to be verifiable by straightforward experiments on these systems . the authors thank v. osipov for discussion of this paper and support . the reported study was funded partially by azerbaijan - jinr collaboration , the science development foundation under the president of the republic azerbaijan - grant no eif - ketpl-2 - 2015 - 1(25)-56/01/1 , the rfbr according to the research projects 165245011@xmath322india , 155161011@xmath322egypt , 152901217 and dst - rfbr grant . the expression @xmath323 for the energy spectrum is written @xmath324 where @xmath325 , @xmath326 , @xmath327 , and @xmath328 . calculation of this determinant yields the energy spectrum of a `` bulk '' @xmath329 superconductor @xmath330 this expression contains a linear in energy term , which is a result of an alignment of @xmath16 and the effective magnetic field of the soi @xmath331 . we consider different limiting cases below . * * the case of * @xmath332 . the energy spectrum looks @xmath333 the energy levels of bdg quasi - particles lie in the gap , symmetrical to the fermi level , with momentum @xmath334 * * the case of * @xmath172 , but @xmath197 and @xmath195 . the energy spectrum ( [ eo ] ) in this limiting case is factorized @xmath335\left[(e - h)^2+(v_f+\alpha)^2k^2-|\delta|^2\right]=0 . \label{e100}\ ] ] one gets for the quasi - particles energy @xmath336 where @xmath163 . the momenta is expressed as @xmath337 soi and/or magnetic field @xmath3 split both electron and hole levels due to rashba momentum - shifting and/or zeeman effect . the fermi points around @xmath338 and @xmath339 are split also due to these effects . * * the limit of * @xmath153 , and @xmath340 , @xmath197 . expression ( [ eo ] ) under these conditions reads @xmath341\left[\left(e-\sqrt{b^2+h^2}\right)^2+v_f^2k^2-|\delta|^2\right]=0,\ ] ] yielding the following expression for the energy spectrum @xmath342 the momenta around the fermi points @xmath338 and @xmath339 split also @xmath343 the expressions for the energy and momentum in the limits of @xmath153 , @xmath111 but @xmath197 or of @xmath153 , @xmath194 but @xmath344 are easily obtained from ( [ e011 ] ) and ( [ k011 ] ) . note that a topological superconducting gapped phase is realized when @xmath345 in consistent with ref.@xcite . in this section , we chart out the expression for @xmath82 . substituting the wave functions ( [ wave ] ) into the boundary conditions ( [ bc ] ) one gets eight linear homogeneous equations for @xmath76 , @xmath77 , @xmath78 , and @xmath79 with @xmath80 as explained in the main text . we can represent these equations in terms of a @xmath346 matrix @xmath82 and a column vector @xmath347 as @xmath84 . the energy of the andreev bound states can then be obtained from @xmath348 . the expression for the matrix @xmath82 , obtained from some straightforward algebra , is given by @xmath349 where @xmath350 and @xmath351 takes values @xmath352 . we note that it is difficult to obtain analytical expression of @xmath353 for general values of @xmath58 , @xmath2 and @xmath3 . however , the physical content of the several terms in this determinant can be understood as follows . we define the minors of the selected blocks of @xmath82 as @xmath354 , @xmath355 , @xmath356 , @xmath357 . furthermore we define the @xmath358 matrices @xmath359 the determinants of these matrices are denoted by @xmath360 and @xmath361 . similarly one can also construct expressions for @xmath362 and @xmath363 . note that all these blocks are interpreted to correspond to a definitive physical process as explained in the main text . all of these determinants enter the expressions of the andreev bound states as discussed in sec.[sec3 ] of the main text . in this section we look into the expression of andreev bound states for @xmath185 . equations ( [ sch1])-([sch4 ] ) are strongly simplified in this link providing only a link between @xmath187 and @xmath188 @xmath364 then , one gets for @xmath126 according to eq . ( [ energy0 ] ) @xmath365 \sin^2\frac{\varphi}{2}\right\}. \label{apf}\ ] ] the expression for @xmath366 differs from that for @xmath190 by replacing @xmath367 and @xmath193 in eq . ( [ apf ] ) . the tunneling energy in this case receives its contribution from the expression @xmath368 ^ 2 + f_{\uparrow , \downarrow}^{\ast}(k_+ ) f_{\downarrow , \uparrow}^{\ast}(k_- ) - f_{\uparrow , \downarrow}^{\ast}(k_- ) f_{\downarrow , \uparrow}^{\ast}(k_+)=0 \label{apfeq}\ ] ] with energy spectrum obtained from eq . ( [ sch1b0 ] ) @xmath369 and from eq . ( [ sch2b0 ] ) @xmath370 this expression has been used to analyze eq . 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we formulate a general method for computing the andreev bound states energy which allows us to obtain analytical expressions for the energy of these states in several asymptotic cases .
we find that in the absence of the magnetic fields the energy gap between the andreev bound states decreases with increasing rashba soi constant leading eventually to touching of the levels . in the absence of rashba soi , the andreev bound states depend on the magnetic fields and display oscillatory behavior with orientational angle of b leading to magneto - josephson effect .
we also present analytic expressions for the dc josephson current charting out their dependence on @xmath5 , @xmath3 , and @xmath2 .
we demonstrate the existence of finite spin - josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on @xmath2 , @xmath7 and @xmath3 .
finally , we study the ac josephson effect in the presence of the soi ( for @xmath8 ) and an external radiation and show that the width of the resulting shapiro steps in such a system can be tuned by varying @xmath2 .
we discuss experiments which can test our theoretical results . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
ce- and yb - based @xmath18-electron kondo - lattice ( kl ) systems have shown the most drastic forms of non - fermi - liquid ( nfl ) behaviour @xcite . in these heavy - fermion ( hf ) materials , the ground state sensitively depends on the balance between two competing interactions , which are both determined by the strength of the @xmath19-conduction electron hybridisation @xmath20 : whereas the kondo interaction leads to a screening of the local moments below a kondo temperature @xmath21 , resulting in a paramagnetic ( pm ) ground state with itinerant @xmath19-electrons , the indirect exchange coupling ( rkky interaction ) can mediate long - range magnetic ordering @xcite . + one of the explanations for such nfl phenomena is the presence of a quantum critical point ( qcp ) at a particular @xmath22 : if the transition temperature @xmath23 of the long - range magnetic order is continuously shifted to zero by an external parameter @xmath24 , e.g. pressure , magnetic field or chemical substitution , a @xmath25 order quantum phase transition ( qpt ) takes place at @xmath26 and @xmath27 , to which a qcp is associated . here , the typical length and time scales of order parameter fluctuations diverge when approaching the transition point . these fluctuations are believed to be responsible for the observed nfl corrections to the fl prediction for the heat capacity @xmath28 , magnetic susceptibility @xmath29 , electrical resistivity @xmath30 etc . qcps are not the only mechanism to provide nfl behaviour . if the magnetic order changes from long - range to short - range and disorder comes into play , spatial regions ( also called `` rare regions '' ) can show local magnetic order , although the bulk system is in a pm state @xcite . the order parameter fluctuations of these regions can become strong enough to destroy the qpt and can give rise to nfl effects @xcite . here , the length scale of order parameter fluctuations becomes finite , while the time scale still diverges . as a consequence in , e.g. , the quantum griffiths phase ( qgp ) scenario , power - law corrections , @xmath31 and @xmath5 with @xmath7 , are expected in a broad region across the qcp and not only at the qcp itself @xcite . the global phase transition is then smeared @xcite , as observed in doped ferromagnetic ( fm ) materials , as , e.g. , the itinerant zr@xmath2nb@xmath3zn@xmath32 @xcite , the @xmath33-based hf system urh@xmath2ru@xmath3ge @xcite and the ce - based cepd@xmath2rh@xmath3 @xcite . the ground state of such systems depends on several factors and can be very exotic , as in ceni@xmath2cu@xmath3 , where a percolative cluster scenario has been proposed @xcite . + recently , we reported on the formation of a `` kondo - cluster - glass '' state in cepd@xmath2rh@xmath3 for @xmath34 , caused by the freezing of clusters with predominantly fm coupling : they form below a temperature @xmath35 and freeze at a lower temperature @xmath36 @xcite . in cepd@xmath2rh@xmath3 , the chemical substitution of the ce - ligand pd by rh induces not just a negative volume effect , but , more importantly , it locally increases the hybridisation strength @xmath20 of the cerium @xmath19 electrons , leading to a strong enhancement of the local @xmath21 . simultaneously , disorder is introduced to the system , which induces a statistical distribution of @xmath21 . both effects create regions where @xmath18 moments are still unscreened and can form fm clusters because of the rkky interaction @xcite . in the temperature region @xmath37 we have observed power - law behaviour of several thermodynamic quantities , i.e. @xmath38 , @xmath12 , @xmath14 @xcite , and we concluded that the quantum griffiths phase scenario might be realised in cepd@xmath2rh@xmath3 for rh content @xmath39 . for @xmath40 the system is too close to the fm long - range order to be considered in the qgp . in this article we take a closer look at the magnetic susceptibility @xmath9 and magnetisation @xmath0 in single crystals with rh content @xmath41 within the frame of the qgp scenario . + before starting with the analysis of the results , we have to consider that the qgp is restricted to a small region of the temperature - magnetic field ( @xmath42 ) phase diagram . this region is limited to the ranges @xmath37 and @xmath43 , where @xmath44 and @xmath45 are , respectively , the magnetic fields necessary to destroy the glass state and to remove the effect of the cluster formation ( cf . inset of fig . [ fig1 ] ( b ) ) . both @xmath36 and @xmath44 have very low values of the order of mk and mt . to study the dynamic processes in the region close to @xmath46 , @xmath9 was measured down to 0.02 k ( fig . [ fig1 ] ) , at a frequency of 113 hz , with a modulation field of @xmath47 t . the dc magnetisation @xmath0 was measured with a squid ( quantum design ) at high temperatures and with a high - resolution faraday magnetometer , in magnetic fields as high as 11 t and at temperatures down to 0.05 k ( fig . [ fig2 ] and [ fig3])@xcite . the magnetic field has been applied along the @xmath48 axis in all the measurements presented here . it is worth mentioning that the magnetic anisotropy of the system is low for @xmath49 @xcite and poly- and single crystals exhibit similar behaviour @xcite . + in frame ( a ) of fig . [ fig1 ] , the real part of the ac susceptibility is plotted as a function of @xmath10 in a double - logarithmic scale for four crystals of cepd@xmath2rh@xmath3 with @xmath50 . the peak temperature is @xmath36 . in frame ( b ) , @xmath9 is shown only for the crystal with @xmath46 at different fields . k has been determined as the temperature where the field - cooled ( fc ) and zero - field - cooled ( zfc ) measurements of @xmath0 split , as illustrated in the inset of the same frame . as in polycrystals , @xmath9 follows a @xmath52 power - law function above @xmath36 where the exponent @xmath11 varies systematically with @xmath13 between -0.12 and 0.45 @xcite , as expected in the qgp scenario . the value of the exponents seems to be independent of the impurity phases observed at about 3 k ( see curve for @xmath53 ) . the negative @xmath11 value for the sample with @xmath46 , different from the one measured in a polycrystal with the same @xmath13 ( @xmath54 ) , suggests that this sample is still too close to the fm instability . moreover , the error in estimating the composition by energy dispersive x - ray spectrometry amounts to 1 at . taking into account that the sample with slightly higher rh content @xmath53 shows @xmath55 , it is not possible from this measurement to discern whether the sample with @xmath46 can be considered in the qgp or not . furthermore , the exponent is affected by the freezing which already takes place at @xmath56 k. focussing on the temperature range between @xmath36 and @xmath35 , the @xmath9 vs @xmath10 plot changes slope slightly at about 3.5 k , and again at about 1.5 k ( indicated by arrows ) . we have fitted this range to extract @xmath11 , considering that in the 10 mt curve the fit range can be expanded to even lower temperatures . + to verify the presence of a qgp in this sample , we have measured @xmath0 vs @xmath8 at different temperatures . in fig . [ fig2 ] the isotherms are shown : in frame ( a ) , the curve at 2 k is plotted together with those for two fm concentrations , @xmath57 and @xmath58 , to emphasise the strong decreasing of the magnetic moment with @xmath13 ; in frame ( b ) , the isotherms at 0.05 k and 2 k are compared . the magnetisation at 0.05 k shows a very small hysteresis ( inset of fig . [ fig2 ] ) , due to freezing at @xmath56 k , and it is far from reaching saturation at 10 t. we consider the value of the coercive field to be @xmath59 t. since @xmath51 k @xmath60 , we can assume @xmath45 to be close to 12 t , as the magnetic moment @xmath61 at 12 t is only @xmath62 . it is thus plausible to look for qgp in @xmath0 vs @xmath8 for fields between 0.1 and 12 t. @xmath0 vs @xmath63 is plotted in fig . [ fig3 ] at three temperatures ; below @xmath36 ( 0.05 k ) , just above it ( 0.5 k ) and close to @xmath35 ( 2 k ) . the curves follow a power - law behaviour , with an almost constant value of @xmath17 , close to those observed in @xmath12 for the other crystals , as expected in the qgp . since @xmath0 vs @xmath8 plots are less sensitive to paramagnetic impurities , and for @xmath64 the freezing does not affect the power law , these plots can be considered as signatures of a qgp in cepd@xmath15rh@xmath16 . + as discussed in the introduction , there is a fundamental difference between the nfl behaviour given by long - range and short - range order fluctuations . the presence of clusters and the power - law corrections to susceptibility and magnetisation in cepd@xmath2rh@xmath3 indicate that the expected qpt at @xmath41 is replaced by disordered phases , possibly like the griffiths one . + we wish to thank t. vojta for helpful discussions . supported by the dfg research unit 960 `` quantum phase transition '' . 9 von lhneysen h , rosch a , vojta m and wlfle p 2007 _ rev . phys . _ * 79 * 1015 and references therein doniach s 1977 _ physica b _ * 91 * 231 vojta t 2006 _ j. phys . a : math . gen . _ * 39 * r143-r205 miranda e , dobrosavljevi v and kotliar g 1997 _ phys . rev . lett . _ * 78 * 290 castro neto a h , castilla g and jones b a 1998 _ phys . lett . _ * 81 * 3531 castro neto a h and jones b a 2000 _ phys . b _ * 62 * 14975 vojta t and schmalian j 2005 _ phys . rev . b _ * 72 * 045438 hoyos j a and vojta t 2008 _ phys . lett . _ * 100 * 240601 ; hoyos j. a. , kotabage c. and vojta t 2007 _ phys . lett . _ * 99 * 230601 sokolov d a , aronson m c , gannon w and fisk z 2006 _ phys . lett . _ * 96 * 116404 huy n t , gasparini a , klaasse j c p , de visser a , sakarya s and van dijk n h 2007_phys . b _ * 75 * 212405 sereni j g , westerkamp t , kchler r , caroca - canales n , gegenwart p and geibel c 2007 _ phys . rev . b _ * 75 * 024432 marcano n , gmez sal j c , espeso j i , de teresa j m , algarabel p a , paulsen c and iglesias j r 2007 _ phys . lett . _ * 98 * 166406 westerkamp t , deppe m , kchler r , brando m , geibel c , gegenwart p , pikul a p and steglich f 2009 _ phys . lett . _ * 102 * 206404 dobrosavljevi v and miranda e 2005 _ phys . lett . _ * 94 * 187203 pikul a p , caroca - canales n , deppe m , gegenwart p , sereni j g , geibel c and steglich f _ j. phys . condens . matter _ * 18 * l535 sakakibara h , mitamura h , tayama t and amitsuka h 1994 _ jpn . phys . _ * 33 * 5067 deppe m , pedrazzini p , caroca - canales n , geibel c and sereni j g 2006 _ physica b _ * 378 - 380 * 96 | the magnetic field dependence of the magnetisation ( @xmath0 ) and the temperature dependence of the ac susceptibility ( @xmath1 ) of cepd@xmath2rh@xmath3 single crystals with @xmath4 are analysed within the frame of the quantum griffiths phase scenario , which predicts @xmath5 and @xmath6 with @xmath7 .
all @xmath0 vs @xmath8 and @xmath9 vs @xmath10 data follow the predicted power - law behaviour .
the parameter @xmath11 , extracted from @xmath12 , is very sensitive to the rh content @xmath13 and varies systematically with @xmath13 from -0.1 to 0.4 .
the value of @xmath11 , derived from @xmath14 measurements on a cepd@xmath15rh@xmath16 single crystal , seems to be rather constant , @xmath17 , in a broad range of temperatures between 0.05 and 2 k and fields up to about 10 t. all observed signatures and the @xmath11 values are thus compatible with the quantum griffiths scenario . |
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due to the exploding popularity of all things wireless , the demand for wireless data traffic increases dramatically . according to a cisco report , global mobile data traffic will increase 13-fold between 2012 and 2017 @xcite . this dramatic demand puts on pressure on mobile network operators ( mnos ) to purchase more spectrum . however , wireless spectrum is a scarce resource for mobile services . even if the continued innovations in technological progress relax this constraint as it provides more capacity and higher quality of service ( qos ) , the shortage of spectrum is still the bottleneck when the mobile telecommunications industry is moving toward wireless broadband services @xcite . to achieve a dominant position for future wireless services , thus , it is significant how new spectrum is allocated to mnos . since the spectrum is statically and infrequently allocated to an mno , there has been an ongoing fight over access to the spectrum . in south korea , for example , the korea communications commission ( kcc ) planed to auction off additional spectrum in both 1.8 ghz and 2.6 ghz bands . the main issue was whether korea telecom ( kt ) acquires the contiguous spectrum block or not . due to the kt s existing holding downlink 10 mhz in the 1.8 ghz band , it could immediately double the existing long term evolution ( lte ) network capacity in the 1.8 ghz band at little or no cost . this is due to the support of the downlink up to 20 mhz contiguous bandwidth by lte release 8/9 . to the user side , there is no need for upgrading their handsets . lte release 10 ( lte - a ) can support up to 100 mhz bandwidth but this requires the carrier aggregation ( ca ) technique , for which both infrastructure and handsets should be upgraded . if kt leases the spectrum block in the 1.8 ghz band , kt might achieve a dominant position in the market . on the other hand , other mnos expect to make heavy investments as well as some deployment time to double their existing lte network capacities compared to kt @xcite . thus , the other mnos requested the government to exclude kt from bidding on the contiguous spectrum block to ensure market competitiveness . although we consider the example of south korea , this interesting but challenging issue on spectrum allocation is not limited to south korea but to most countries when asymmetric - valued spectrum blocks are auctioned off to mnos . spectrum auctions are widely used by governments to allocate spectrum for wireless communications . most of the existing auction literatures assume that each bidder ( i.e. , an mno ) only cares about his own profit : what spectrum block he gets and how much he has to pay @xcite . given spectrum constraints , however , there is some evidence that a bidder considers not only to maximize his own profit in the event that he wins the auction but to minimize the weighted difference of his competitor s profit and his own profit in the event that he loses the auction @xcite . this strategic concern can be interpreted as a _ spite motive _ , which is the preference to make competitors worse off . since it might increase the mno s relative position in the market , such concern has been observed in spectrum auctions @xcite . in this paper , we study bidding and pricing competition between two competing / spiteful mnos with considering their existing spectrum holdings . given that asymmetric - valued spectrum blocks are auctioned off to them , we developed an analytical framework to investigate the interactions between two mnos and users as a three - stage dynamic game . in tage i , two spiteful mnos compete in a first - price sealed - bid auction . departing from the standard auction framework , we address the bidding behavior of the spiteful mno . in tage ii , two competing mnos optimally set their service prices to maximize their revenues with the newly allocated spectrum . in tage iii , users decide whether to stay in their current mno or to switch to the other mno for utility maximization . our results are summarized as follows : * _ asymmetric pricing structure _ : we show that two mnos announce different equilibrium prices to the users , even providing the same quality in services to the users . * _ different market share _ : we show that the market share leader , despite charging a higher price , still achieve more market share . * _ impact of competition _ : we show that the competition between two mnos leads to some loss of their revenues . * _ cross - over point between two mno s profits _ : we show that two mnos profits are switched . the rest of the paper is organized as follows : related works are discussed in ection ii . the system model and three - stage dynamic game are described in ection iii . using backward induction , we analyze user responses and pricing competition in ections vi and v , and bidding competition in ection vi . we conclude in section ii together with some future research directions . in wireless communications , the competition among mnos have been addressed by many researchers @xcite@xcite . yu and kim @xcite studied price dynamics among mnos . they also suggested a simple regulation that guarantees a pareto optimal equilibrium point to avoid instability and inefficiency . niyato and hossain @xcite proposed a pricing model among mnos providing different services to users . however , these works did not consider the spectrum allocation issue . more closely related to our paper are some recent works @xcite@xcite . the paper @xcite studied bandwidth and price competition ( i.e. , bertrand competition ) among mnos . by taking into account mnos heterogeneity in leasing costs and users heterogeneity in transmission power and channel conditions , duan _ et al_. presented a comprehensive analytical study of mnos spectrum leasing and pricing strategies in @xcite . in @xcite , a new allocation scheme is suggested by jointly considering mnos revenues and social welfare . x. feng _ et al . _ @xcite suggested a truthful double auction scheme for heterogeneous spectrum allocation . none of the prior results considered mnos existing spectrum holdings even if the value of spectrum could be varied depending on mnos existing spectrum holdings . we consider two mnos ( @xmath0 and @xmath1 ) compete in a first - price sealed - bid auction , where two spectrum blocks @xmath2 and @xmath3 are auctioned off to them as shown in ig . 1 . note that @xmath2 and @xmath3 are the same amount of spectrum ( i.e. , 10 mhz spectrum block ) . without loss of generality , we consider only the downlink throughput the paper . note that both mnos operate frequency division duplex lte ( fdd lte ) in the same area . due to the mnos existing spectrum holdings ( i.e. , each mno secures 10 mhz downlink spectrum in the 1.8 ghz band ) , the mnos put values on spectrum blocks @xmath2 and @xmath3 asymmetrically . if mno @xmath4 leases @xmath2 , twice ( 2x ) improvements in capacity over his existing lte network capacity are directly supported to users . in third generation partnership project ( 3gpp ) lte release 8/9 , lte carriers can support a maximum bandwidth of 20 mhz for both in uplink and downlink , thereby allowing for mno @xmath4 to provide double - speed lte service to users without making many changes to the physical layer structure of lte systems @xcite . on the other hand , mno @xmath5 who leases @xmath3 should make a huge investment to double the capacity after some deployment time @xmath6 . without loss of generality , we assume that mno @xmath4 leases @xmath2 . to illustrate user responses , we define the following terms as follows . * definition 1 . * ( asymmetric phase ) _ assume that mno @xmath5 launches double - speed lte service at time @xmath6 . when @xmath7 , we call this period asymmetric phase due to the different services provided by mnos @xmath4 and @xmath5 . _ * definition 2 . * ( symmetric phase ) _ assume that @xmath8 denotes the expiration time for the mnos new spectrum rights . when @xmath9 , we call this period symmetric phase because of the same services offered by mnos @xmath4 and @xmath5 . _ we investigate the interactions between two mnos and users as a three - stage dynamic game as shown in ig . 2 . in tage i , two spiteful mnos compete in a first - price sealed - bid auction where asymmetric - valued spectrum blocks @xmath2 and @xmath3 are auctioned off to them . the objective of each mno is maximizing his own profit when @xmath2 is assigned to him , as well as minimizing the weighted difference of his competitor s profit and his own profit when @xmath3 is allocated to him . in tage ii , two competing mnos optimally announce their service prices to maximize their revenues given the result of tage i. the analysis is divided into two phases : asymmetric phase and symmetric phase . in tage iii , users determine whether to stay in their current mno or to switch to the new mno for utility maximization . to predict the effect of spectrum allocation , we solve this three - stage dynamic game by applying the concept of backward induction , from tage iii to tage i. each user subscribes to one of the mnos based on his or her mno preference . let us assume that mnos @xmath4 and @xmath5 provide same quality in services to the users so they have the same reserve utility @xmath10 before spectrum auction . each mno initially has 50% market share and the total user population is normalized to 1 . in asymmetric phase , the users in mnos @xmath4 and @xmath5 obtain different utilities , i.e. , @xmath11 where @xmath12 is a user sensitivity parameter to the double - speed lte service than existing one . it means that users care more about the data rate as @xmath13 increases . the users in mno @xmath5 have more incentive to switch to mno @xmath4 as @xmath13 increases . when they decide to change mno @xmath4 , however , they face switching costs , the disutility that a user experiences from switching mnos . in the case of higher switching costs , the users in mno @xmath5 have less incentive to switch . the switching cost varies among users and discounts over time . to model such users time - dependent heterogeneity , we assume that the switching cost is heterogeneous across users and uniformly distributed in the interval @xmath14 $ ] at @xmath15 , where @xmath16 denotes the discount rate @xcite . this is due to the fact that the pays for the penalty of terminating contract with operators decrease as time passes . now let us focus on how users churn in asymmetric phase . a user @xmath17 in mno @xmath5 , with switching cost , @xmath18 , observes the prices charged by mnos @xmath4 and @xmath5 ( @xmath19 and @xmath20 ) . a user @xmath17 in mno @xmath5 will switch to mno @xmath4 if and only if @xmath21 thus the mass of switching users from mno @xmath5 to @xmath4 is = -2mu = -0.5mu @xmath22 where @xmath23 is a uniform @xmath24 random variable and @xmath25 denotes the initial market share . since the market size is normalized to one , each mno s market share in asymmetric phase is as follows : = 1mu = 1mu @xmath26 given users responses ( 4 ) , mnos @xmath4 and @xmath5 set their service prices @xmath27 and @xmath28 to maximize their revenues , respectively , i.e. , @xmath29 the nash equilibrium in this pricing game is described in the following proposition . * proposition 1 . * _ when @xmath30 and @xmath31 , there exists a unique nash equilibrium , i.e. , @xmath32 _ * proof . * in asymmetric phase , two competing mnos try to maximize their revenues @xmath33 and @xmath34 , respectively , given users responses , i.e. , @xmath35 a nash equilibrium exists by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath36 @xmath37 @xmath38 proposition 1 shows two mnos equilibrium prices in asymmetric phase . intuitively , @xmath27 increases as @xmath13 increases . with larger @xmath13 , users care more about the data rate . thus , mno @xmath4 increases his service price to obtain more revenue . on the other hand , @xmath39 decreases as @xmath13 increases . it means that mno @xmath5 tries to sustain the revenue margin by lowering the service price and holding onto market share . an interesting observation is that both mnos decrease their service prices as @xmath40 increases . due to the discount factor ( @xmath16 ) , the users in mno @xmath5 are not locked - in and tries to maximize their utilities by churning to mno @xmath4 as switching costs decrease over time . therefore , mno @xmath4 lowers his service price to maximize his revenue , which forces mno @xmath5 to decrease the service price . this phenomenon is consistent with the previous results @xcite , @xcite in that the reduction of switching costs intensifies the price - down competition between two mnos . if @xmath41 , then all users in mno @xmath5 churn to mno @xmath4 . however , it is an unrealistic feature of the mobile telecommunication industry so we add the constraint @xmath42 . next we will show how each mno s market share changes in asymmetric phase . inserting the equilibrium prices ( 6 ) into ( 4 ) , each mno s market share can be calculated as follows : @xmath43 intuitively , mno @xmath4 takes mno @xmath5 s market share more as @xmath40 increases or @xmath13 increases . to hold onto or take mno @xmath4 s market share , the time to launch double - speed lte service @xmath6 is of great importance to mno @xmath5 . when mno @xmath5 launches double - speed lte service at time @xmath6 , each mno s total revenue in asymmetric phase is given by = -0.5mu @xmath44 @xmath45 similar to the analysis of market share , equation ( 8) shows that mno @xmath5 should launch double - speed lte service as quickly as possible to narrow the revenue gap between mno @xmath4 and mno @xmath5 ( see the last term of the revenues ( 8) ) . since mno @xmath5 launches double - speed lte service in symmetric phase , we assume that the users in mnos @xmath4 and @xmath5 obtain same utility , i.e. , @xmath46 for better understanding of user responses in symmetric phase , we first discuss the effect of switching costs on market competition . given the same services offered by two mnos , an mno s current market share plays an important role in determining its price strategy . each mno faces a trade - off between a low price to increase market share , and a high price to harvest profits by exploiting users switching costs . the following emma examines this trade - off and characterizes each mno s price strategy , which is directly related to user responses in symmetric phase . * lemma 1 . * _ in a competitive market with switching costs , the market share leader ( i.e. , mno @xmath4 ) charges a high price to exploit its current locked - in users while the marker share followers ( i.e. , mno @xmath5 ) charge low prices to increase market share for revenue maximization , respectively , given the same services offered by them . _ * proof . * we prove emma 1 by contradiction . suppose that mno @xmath5 charges a higher price than mno @xmath4 ( i.e. , @xmath47 . the mass of switching users from mno @xmath5 to @xmath4 is = -1mu = 0mu = 0mu @xmath48 where @xmath49 is the market share of mno @xmath5 at the end of asymmetric phase . then , each mno s market share is given by @xmath50 @xmath51 where @xmath52 is the market share of mno @xmath4 at the end of asymmetric phase . following the same steps of the roposition 1 , we can find the nash equilibrium by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath53 @xmath54 = 0,\nonumber\end{aligned}\ ] ] which yields the solution given as follows @xmath55 thus , this contradicts to our assumption , completing the proof . @xmath38 with emma 1 , let us illustrate the process of user churn in symmetric phase . the mass of switching users from mno @xmath4 to @xmath5 is = -1mu @xmath56 where @xmath57 is the market share of mno @xmath4 at the end of asymmetric phase . then each mno s market share in symmetric phase is given by @xmath58 @xmath59 where @xmath60 is the market share of mno @xmath5 at the end of asymmetric phase . as noted in emma 1 , mno @xmath5 charges a lower price than mno @xmath4 in symmetric phase . following the same procedure ( 5 ) , the nash equilibrium is described in the following proposition . * proposition 2 . * _ when @xmath61 , there exists a unique nash equilibrium , i.e. , = -0.5mu = 0mu @xmath62 _ * proof . * following the same steps of the roposition 1 , a nash equilibrium exists by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath63 = 0 , \nonumber\end{aligned}\ ] ] @xmath64 @xmath38 , @xmath65 ) . other parameters are @xmath66 , @xmath67 , @xmath68 , and @xmath69.,width=326 ] , @xmath65 ) . other parameters are @xmath66 , @xmath67 , @xmath68 and @xmath69.,width=326 ] roposition 2 states the mnos equilibrium prices in symmetric phase . as described in emma 1 , mno @xmath4 , the market share leader announces a higher service price up to @xmath70 than mno @xmath5 . to further investigate the effect of competition under the same quality in services , let us calculate each mno s falling price level in the neighborhood of the point @xmath6 . from ( 6 ) and ( 15 ) , each mno s falling price level ( i.e. , @xmath71 and @xmath72 ) is = 0mu = 0mu = 0mu @xmath73 @xmath74 because @xmath75 , mnos @xmath4 and @xmath5 always decrease their prices up to @xmath71 and @xmath76 at the starting point of the symmetric phase , respectively . perhaps counter - intuitively , it shows that mno @xmath5 always lowers his price despite launching double - speed lte service at the starting point of the symmetric phase . it can be interpreted as follows . since mno @xmath5 loses his market share in asymmetric phase , mno @xmath5 attempts to maximize his revenue by lowering his service price and increasing his market share , which forces mno @xmath4 to drop the service price at the same time . this means that the mnos competition under the same quality in services lead to some loss of their revenues , which , known as a _ price war _ , is consistent with our previous work @xcite . 3 shows @xmath77 and @xmath78 as a function of @xmath40 under two different user sensitivities ( @xmath79 , @xmath65 ) . note that mno @xmath4 s falling price level is more sensitive to @xmath13 . next we show that how each mno s market share varies in symmetric phase . from ( 14 ) and ( 15 ) , each mno s market share is @xmath80 unlike the asymmetric phase , each mno s market share only depends on the deployment time of carrier aggregation @xmath6 in symmetric phase . an interesting observation is that the market share leader ( i.e. , mno @xmath4 ) , despite charging a higher price , still achieves more market share up to @xmath81 than mno @xmath5 . in terms of market share , mno @xmath4 always gains a competitive advantage over mno @xmath5 if mno @xmath5 was forced to lease less - valued spectrum block . this explains how critical new spectrum is allocated to the mnos , and how struggling they are over access to the spectrum for improving market competitiveness for future wireless services . 4 shows user responses as a function of @xmath40 under two different user sensitivities ( @xmath79 , @xmath65 ) . if the new spectrum rights expire at @xmath82 , each mno s total revenue in symmetric phase is = 2mu @xmath83 = -1mu=-1mu=-1mu @xmath84 @xmath85 @xmath86 using ( 8) and ( 18 ) , we examine the two mnos aggregate revenues when mno @xmath4 leases @xmath2 and mno @xmath5 leases @xmath3 . each mno s aggregate revenue at @xmath82 is given in ( 19 ) . when mno @xmath5 decides to launch double - speed lte service , the optimal deployment time of the carrier aggregation @xmath87 should be studied . the following lemma describes the mno @xmath5 s optimal deployment time . * lemma 2 . * _ the market share followers ( i.e. , mno @xmath5 ) should launch double - speed lte service as quickly as possible not only for maximizing their own revenues but also for minimizing the market leader s revenue . _ * proof . * by taking the derivative of the two mno s aggregate revenues @xmath88 and @xmath89 with respect to @xmath6 , respectively , it can be checked that @xmath90 and @xmath91 . we omit the details of the derivations here . @xmath38 emma 2 states that the revenue of mno @xmath5 is strictly decreasing over @xmath6 while the reverse is for mno @xmath4 . to gain more insight into the effect of the allocation of asymmetric - valued spectrum blocks , let us define the revenue gain as follows : @xmath92 ig . 5 shows the revenue gain as a function of @xmath13 under two different deployment times ( @xmath68 , @xmath93 ) . as expected , the revenue gain is strictly increasing over @xmath6 and @xmath13 . in terms of @xmath13 , it can be checked directly by following the same steps of the emma 2 . such result explains why each mno should spitefully bid in a first - price sealed - bid auction to achieve a dominant position or compensate the revenue gap , which we will discuss these points in the next section . in tage i , two spiteful mnos @xmath4 and @xmath5 compete in a first - price sealed - bid auction where asymmetric - valued spectrum blocks @xmath2 and @xmath3 are auctioned off to them . for fair competition , each mno is constrained to lease only one spectrum block ( i.e. , @xmath2 or @xmath3 ) . we assume that the governments set the reserve prices @xmath94 and @xmath95 to @xmath2 and @xmath3 , respectively . note that the reserve price is the minimum price to get the spectrum block . since @xmath2 is the high - valued spectrum block , we further assume that two spiteful mnos are only competing on @xmath2 to enjoy a dominant position in the market . mnos @xmath4 and @xmath5 bid @xmath2 independently as @xmath96 and @xmath97 , respectively . in this case , @xmath3 is assigned to the mno who loses in the auction as the reserve price @xmath95 . because the mno who leases @xmath3 should make huge investments to double the existing lte network capacity compared to the other mno , we also assume the only mno who leases @xmath3 incurs the investment cost @xmath98 . under two different times ( @xmath68 , @xmath93 ) . other parameters are @xmath66 , @xmath67 and @xmath69.,width=326 ] when asymmetric - valued spectrum blocks are allocated to the mnos , there is a trade - off between self - interest and spite . to illustrate this trade - off , we first restrict ourselves to the case where spite is not present . if mno @xmath4 is _ self - interested _ , his objective function is as follows . = 1mu=1mu=1mu @xmath99 \cdot i_{b_i \geqslant b_j } + \pi^{b}(t_1,t_2 ) \cdot i_{b_i < b_j } , \end{aligned}\ ] ] where @xmath100 is the indicator function and @xmath101 is the profit when leasing @xmath3 . this case is the standard auction framework in that mno @xmath4 maximizes his own profit without considering the other mno s profit . in the real world , however , there is some evidence that some mnos are _ completely malicious_. the german third generation ( 3 g ) spectrum license auction in 2000 is a good example @xcite . german telekom kept raising his bid to prevent his competitors from leasing spectrum . if mno @xmath4 is completely malicious , his objective function can be changed as follows . = -1mu=-1mu=-1mu @xmath99 \cdot i_{b_i \geqslant b_j } - \left [ { r^a(t_1,t_2 ) - b_j } \right ] \cdot i_{b_i < b_j } .\end{aligned}\ ] ] it means that mno @xmath4 gets disutility as much as the profit of mno @xmath5 when he loses the auction . the minus term in ( 22 ) implies this factor . to reflect this strategic concern , our model departs from the standard auction framework in that each spiteful mno concerns about maximizing his own profit when he leases @xmath2 , as well as minimizing the weighted difference of his competitor s profit and his own profit when he leases b. combining ( 21 ) and ( 22 ) , we define each mno s objective function as follows . * definition 3 . * _ assume that two spiteful mnos ( i.e. , @xmath102 and @xmath103 ) compete in a first - price sealed - bid auction . the objective function that each mno tries to maximize is given by : = 0.5mu=0.5mu=0mu @xmath104 \cdot i_{b_i \geqslant b_j } \nonumber\\ & + & \left [ { ( 1 - \alpha _ i ) \pi ^b ( t_1 , t_2 ) - \alpha _ i ( r^a ( t_1 , t_2 ) - b_j ) } \right ] \cdot i_{b_i < b_j } \nonumber\\\end{aligned}\ ] ] = 3mu=3mu=2mu where @xmath100 is the indicator function , @xmath105 is the mno s profit when leasing @xmath3 , and @xmath106 $ ] is a parameter called the spite ( or competition ) coefficient . _ as noted , mno @xmath4 is self - interested and only tries to maximize his own profit when @xmath107 . when @xmath108 , mno @xmath4 is completely malicious and only attempts to obtain more market share by forcing mno @xmath5 to lease the less - valued spectrum block . for given @xmath106 $ ] and @xmath109 $ ] , we can derive the optimal bidding strategies that maximize the objective function in efinition 3 as follows . * proposition 3 . * _ in a first - price sealed - bid auction , the optimal bidding strategy for a spiteful mno @xmath110 and @xmath111 is : = 0.5mu=0.5mu=0.5mu @xmath112 @xmath113 _ * proof . * without loss of generality , suppose that mno @xmath4 knows his bid @xmath96 . further , we assume that mno @xmath4 infer that the bidding strategy of mno @xmath5 on @xmath2 is drawn uniformly and independently from @xmath114 $ ] . the mno @xmath4 s optimization problem is to choose @xmath96 to maximize the expectation of = 0.5mu=0.5mu=0.5mu @xmath115 } { \rm { } } f(b_j ) db_j \nonumber \\ & & + \int\limits_{b_i } ^{r^a ( t_1 , t_2 ) } { \left [ { ( 1 - \alpha _ i ) ( \pi ^b ( t_1 , t_2 ) ) - \alpha _ i ( r^a ( t_1 , t_2 ) - b_j ) } \right ] } { \rm { } } f(b_j ) db_j . \nonumber \\\end{aligned}\ ] ] differentiating equation ( 25 ) with respect to @xmath96 , setting the result to zero and multiplying by @xmath116 give = 5mu=5mu=5mu @xmath117 since the same analysis can be applied to the mno @xmath5 , the proof is complete . @xmath38 roposition 3 states that the mnos equilibrium bidding strategies . intuitively , the more spiteful the mno is , the more aggressively the mno tends to bid . for consistency , we assume that @xmath118 . then we can now calculate mno @xmath4 s profit and mno @xmath5 s profit as follows = 2mu=2mu=2mu @xmath119 where @xmath120 is calculated by substraction of the bidding price @xmath121 of ( 24 ) from @xmath88 of ( 19 ) . under two different costs ( @xmath122 , @xmath123 ) . other parameters are @xmath66 , @xmath67 , @xmath65 , @xmath68 , @xmath69 , @xmath124 , and @xmath125.,width=326 ] under two different spite coefficients ( @xmath126 , @xmath127 ) . other parameters are @xmath66,@xmath67 , @xmath65 , @xmath69 , @xmath123 , @xmath124 , and @xmath125.,width=326 ] to get some insight into the properties of the mnos equilibrium profits , let us define @xmath128 is different from @xmath129 of ( 19 ) where @xmath130 is the revenue gain from @xmath2 relative to @xmath3 without considering any cost . ] , which can be interpreted as the profit gain from @xmath2 relative @xmath3 . when @xmath131 , the profit of mno @xmath4 is higher than that of mno @xmath5 . it implies that mno @xmath4 could gain a competitive advantage over mno @xmath5 in both market share and profit . when @xmath132 , the situation is reversed . mno @xmath5 could take the lead in the profit despite losing some market share to mno @xmath4 . if the role of the government is to ensure fairness in two mnos profits , the government may devise two different schemes : setting appropriate reserve prices and imposing limits on the timing of the double - speed lte services . according to the ofcom report , setting the reserve prices closer to market value might be appropriate @xcite . it indicates that the government set @xmath94 and @xmath95 by estimating the value asymmetries between spectrum blocks @xmath2 and @xmath3 ( i.e. , @xmath133 ) and the spite coefficient @xmath134 . ig . 6 shows the profit gain as a function of @xmath134 under two different reserve prices for @xmath2 ( i.e. , @xmath122 , @xmath123 ) . for example , if @xmath135 , the government should set the reserve prices @xmath123 , @xmath124 . on the other hand , the government should set the reserve prices @xmath122 , @xmath124 when @xmath136 . besides setting appropriate reserve prices , the government can impose limits on the timing of the double - speed lte service . in south korea , for instance , korea telecom ( kt ) who acquired the continuous spectrum spectrum is allowed to start its double - speed lte service on metropolitan areas immediately in september 2013 , other major cities staring next march , and nation - wide coverage starting next july @xcite . this scheme implies to reduces @xmath6 by limiting the timing of the double - speed lte service to the mno who acquires spectrum block @xmath2 . 7 shows the profit gain as a function of @xmath6 under two different spite coefficients ( i.e. , @xmath126 , @xmath127 ) . in this paper , we study bidding and pricing competition between two spiteful mnos with considering their existing spectrum holdings . we develop an analytical framework to investigate the interactions between two mnos and users as a three - stage dynamic game . using backward induction , we characterize the dynamic game s equilibria . from this , we show the asymmetric pricing structure and different market share between two mno . perhaps counter - intuitively , our results show that the mno who acquires the less - valued spectrum block always lowers his price despite providing double - speed lte service to users . we also show that the mno who acquires the high - valued spectrum block , despite charging a higher price , still achieves more market share than the other mno . we further show that the competition between two mnos leads to some loss of their revenues . with the example of south korea , we investigate the cross - over point at which two mnos profits are switched , which serves as the benchmark of practical auction designs . results of this paper can be extended in several directions . extending this work , it would be useful to propose some methodologies for setting reserve prices @xcite , @xcite . second , we could consider an oligopoly market where multiple mnos initially have different market share before spectrum allocation , where our current research is heading . m. shi , j. chiang , and b .- d . price competition with reduced consumer switching costs : the case of `` wirelss number portability '' in the cellular phone industry , " , vol . 1 , pp . 2738 , 2006 . dotecon and aetha , spectrum value of 800mhz , 1800mhz , and 2.6ghz , " a dotecon and aetha report , jul . 2012 . available : http://stakeholders.ofcom.org.uk/binaries/consultations/award800mhz/ statement / spectrum value.pdf . | we study bidding and pricing competition between two spiteful mobile network operators ( mnos ) with considering their existing spectrum holdings . given asymmetric - valued spectrum blocks
are auctioned off to them via a first - price sealed - bid auction , we investigate the interactions between two spiteful mnos and users as a three - stage dynamic game and characterize the dynamic game s equilibria .
we show an asymmetric pricing structure and different market share between two spiteful mnos . perhaps counter - intuitively ,
our results show that the mno who acquires the less - valued spectrum block always lowers his service price despite providing double - speed lte service to users .
we also show that the mno who acquires the high - valued spectrum block , despite charing a higher price , still achieves more market share than the other mno .
we further show that the competition between two mnos leads to some loss of their revenues . by investigating a cross - over point at which the mnos profits are switched
, it serves as the benchmark of practical auction designs . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the `` pitch - drop - experiment '' @xcite , which received the ig - nobel price in physics 2005 , has brought to the attention that a fast process like drop formation @xcite can be retarded considerably if instead of a standard liquid like water it has a viscosity of @xmath0 at @xmath1c a material like pitch , with a viscosity around @xmath2 is selected . the funnel was filled in 1930 @xcite , and today `` finally the ninth pitch drop has fallen from the world s longest running lab experiment '' @xcite and the 10th is awaited within the next 14 years . here the question arises whether those high viscosities may give access to so far not resolved phenomena . in a vessel with diameter of 120 mm . the picture is taken from @xcite . a movie showing the formation of rosensweig patterns can be accessed at @xcite.,width=323 ] in the following we are investigating this question for the case of the well known rosensweig or normal field instability @xcite . it is observed in a horizontal layer of magnetic fluid ( mf ) @xcite with a free surface , when a critical value @xmath3 of the vertically oriented magnetic induction is surpassed . figure [ fig : peaks ] presents a photo of the final hexagonal arrangement of static liquid peaks . beside the threshold , beyond which the instability occurs , two quantities characterizing the emerging pattern have been in the focus of various studies : the critical wave number of the peaks and the corresponding growth rate , where both are strongly influenced by the viscosity of the magnetic fluid . that essential role of the viscosity for the dynamics of the pattern formation is reflected in the course of the analyses devoted to the rosensweig instability . for an inviscid magnetic fluid ( the dynamic viscosity @xmath4 is zero ) and an infinitely deep container , @xcite provide a linear stability analysis already in the very first description of the normal field instability to find the critical threshold @xmath3 and the critical wave number @xmath5 . this approach has been extended later by @xcite to fluids with non - zero viscosity , where the growth rate depends on @xmath4 , and to a finite depth of the container by @xcite . first experimental investigations on the growth of the pattern are provided by @xcite , who also derive the growth rate for the case of a viscous magnetic fluid and an arbitrary layer thickness @xmath6 . this theoretical analysis has been later extended to the case of a nonlinear magnetization curve @xmath7 by @xcite . whereas so far the growth rate of the emerging rosensweig pattern has been measured utilizing ferrofluids with @xmath8 @xcite and @xmath9 @xcite we are tackling here the growth process in a ferrofluid which is a thousand times more viscous than the first one . such a ferrofluid is being created by cooling a commercially available viscous ferrofluid ( apg e32 from ferrotec co. ) down to @xmath10c . the ferrofluid has now a viscosity of @xmath11 . in such a cooled rosensweig ( sloppy _ frozensweig _ ) instability @xcite the growth of the pattern takes 60 seconds and can be measured with high temporal resolution in the extended system using a two - dimensional x - ray imaging technique @xcite . that technique provides the full surface topography , as opposed to the fast , but one dimensional hall - sensor array , which had to be utilized for the low viscosity ferrofluids @xcite . the potential of the retarded instability was demonstrated before @xcite , when the coefficients of nonlinear amplitude equations were determined in this way . in addition a sequence of localized patches of rosensweig pattern could be uncovered most recently @xcite with that technique . here we exploit a higher viscosity to investigate the linear growth rate in a regime , which was hitherto not accessible . this expectation is based on a scaling analysis presented in ref.@xcite . for supercritical inductions larger than @xmath12 ( the dimensionless kinematic viscosity @xmath13 is defined in eq . ( [ eq : scaling_visc ] ) below ) the behavior of the growth rate is characterized by a _ square - root _ dependence on those inductions , as confirmed in @xcite . contrary , for supercritical inductions smaller than @xmath12 the behavior of the growth rate is characterized by a _ linear _ dependence . in the present experiment we increase @xmath12 by six orders of magnitude due to the high viscosity of the ferrofluid apg e32 at @xmath10c . thus a new territory of linear scaling is open for exploration . the outline of the paper is as follows : the experimental setup and the measurements are sketched in the next sect . [ sec : experiment ] . the theoretical analysis is presented in sect . [ sec : theory ] and compared with the experimental findings in the subsequent sect . [ sec : results ] . in this section we describe the experimental setup ( sect.[subsec : setup ] ) , the ferrofluid ( sect.[subsec : fluid ] ) , the protocol utilized for the measurements ( sect.[subsec : measurements ] ) , and the way the linear growth rate is extracted from the recorded data ( sect.[subsec : extracting ] ) . the experimental setup for the measurements of the surface topography consists of an tailor made x - ray apparatus described in detail before @xcite . an x - ray point source emits radiation vertically from above through the container filled with the mf . underneath the container , an x - ray camera records the radiation passing through the layer of mf . the intensity at each pixel of the detector is directly related to the height of the fluid above that pixel , as sketched in fig . [ fig : setup](a ) . therefore , the full surface topography can be reconstructed after calibration @xcite . the container , which holds the mf sample , is depicted in fig . [ fig : setup](b ) . it is a regular octagon machined from aluminium with a side length of @xmath14 and two concentric inner bores with a diameter of @xmath15 . these circular holes are carved from above and below , leaving only a thin base in the middle of the vessel with a thickness of @xmath16 . on top of the octagon , a circular aluminium lid is placed , which closes the hole from above , as shown in fig . [ fig : setup](b ) . each side of the octagon is equipped with a thermoelectric element ` qc-127 - 1.4 - 8.5ms ` from quick - ohm , as shown in fig . [ fig : setup](c ) . the latter are powered by a @xmath17 kepco ` klp-20 - 120 ` power supply . the hot side of the peltier elements is connected to water cooled heat exchangers . the temperature is measured at the bottom of the aluminium container with a pt100 resistor . the temperature difference between the center and the edge of the bottom plate does not exceed @xmath18k at the temperature @xmath19c measured at the edge of the vessel . a closed loop control , realized using a computer and programmable interface devices , holds @xmath20 constant within @xmath21 . the container is surrounded by a helmholtz - pair - of - coils , thermally isolated from the vessel with a ring made from the flame resistant material ` fr-2 ` . the size of the coils is adapted to the size of the vessel in order to introduce a `` magnetic ramp '' at the edge of the vessel . this technique , as described more detailed in ref.@xcite , serves to minimize distortions by compensating partly the jump of the magnetization at the container edge . filling the container to a height of @xmath22 with ferrofluid enhances the magnetic induction in comparison with the empty coils for the same current @xmath23 . therefore @xmath24 is measured immediately beneath the bottom of the container , at the central position , and serves as the control parameter in the following . 0.5 cm + 0.5 cm ( a ) 5.5 cm ( b ) + + 0.5 cm ( c ) the vessel is filled with the commercial magnetic fluid apge32 from ferrotec co. up to a hight of 5 mm . the material parameters of this mf are listed in tab . [ tab : apge32_parameters ] . the density was measured using a ` dma 4100 ` density meter from anton paar . the surface tension was measured using a commercial ring tensiometer ( lauda ` te 1 ` ) and a pendant drop method ( dataphysics ` oca 20 ` ) . both methods result in a surface tension of @xmath25 , but when the liquid is allowed to rest for one day , @xmath26 drops down to @xmath27 . this effect , which is not observed in similar , but less viscous magnetic liquids like the one used in ref.@xcite , gives a hint that the surfactants change the surface tension at least on a longer time scale , when the surface is changed . since indeed the pattern formation experiments do change the surface during the measurements , the uncertainty of the surface tension is @xmath28 , as given in tab . [ tab : apge32_parameters ] . lcdcc quantity & & value & error & unit + density at @xmath29 & @xmath30 & 1168.0 & @xmath31 & @xmath32 + surface tension at @xmath29 & @xmath26 & 30.9 & @xmath33 & @xmath34 + viscosity at @xmath35 & @xmath4 & 4.48 & @xmath36 & @xmath37 + saturation magnetization & @xmath38 & 26.6 & @xmath39 & @xmath40 + initial susceptibility at @xmath35 & @xmath41 & 3.74 & @xmath42 & + fit of @xmath7 with the model by ref.@xcite & & & & + exponent of the @xmath43-distribution & @xmath44 & 3.8 & @xmath31 & + typical diameter of the bare particles & @xmath45 & 1.7 & @xmath46 & @xmath47 + volume fraction of the magnetic material & @xmath48 & 5.96 & @xmath46 & % + fit of @xmath49 with the model by ref.@xcite & & & & + mean diameter of the bare particle & @xmath50 & 15 & & @xmath47 + volume fraction of the magnetic material & @xmath48 & 21.4 & @xmath46 & % + critical induction for a semi - infinite layer @xcite & @xmath51 & 10.5 & @xmath36 & @xmath52 + [ tab : apge32_parameters ] . the black dashed line is a fit with the model by ref.@xcite . the blue solid line marks an extrapolation to @xmath53c according to this model . ] [ [ magnetization - curve ] ] magnetization curve + + + + + + + + + + + + + + + + + + + the magnetization has been determined using a fluxmetric magnetometer ( lakeshore model 480 ) constructed to deal with larger samples of high viscosity at a temperature of @xmath54 . figure[fig : magkurve ] shows the data , which have been fitted by the modified mean field model of second order @xcite , marked by the dashed black line . for a comparison with the pattern formation experiments performed at @xmath55 , this curve is extrapolated utilizing this model ( blue line ) . the deviation between both curves is tiny , which was corroborated with a vibrating sample magnetometer ( lakeshore vsm 7404 ) at @xmath54 and @xmath56 . note that the vsm offers the advantage that it can be tempered , but has a lower resolution in comparison to the fluxmetric device because of the smaller sample volume . to take into account the nonlinear @xmath7 , an effective susceptibility @xmath57 is defined by a geometric mean @xmath58 with the tangent susceptibility @xmath59 and the chord susceptibility @xmath60 @xcite . for any field @xmath61 the effective susceptibility @xmath57 can be evaluated , when the magnetization @xmath7 curve is known . [ [ viscosity ] ] viscosity + + + + + + + + + the viscosity @xmath4 deserves special attention for the experiments , as it influences the time scale of the pattern formation . it has been measured in a temperature range of @xmath62 , using a commercial rheometer ( mcr301 , anton paar ) with a shear cell featuring a cone - plate geometry . at room temperature , the magnetic fluid with a viscosity of @xmath63 is @xmath64 times more viscous than water . the value of @xmath4 can be increased by factor of @xmath65 when the liquid is cooled to @xmath66 . the temperature dependent viscosity data can be nicely fitted with the well - known vogel - fulcher law @xcite @xmath67 with @xmath68 and @xmath69 , as described in detail in ref.@xcite . for the present measurements , we chose a temperature of @xmath55 , where the viscosity amounts to @xmath70 according to eq . ( [ eq : vogelfulcher ] ) . [ [ magnetoviscosity ] ] magnetoviscosity + + + + + + + + + + + + + + + + for a shear rate of @xmath71 . the @xmath72 ( @xmath73 ) mark measurements for increasing ( decreasing ) @xmath74 and the solid line is a fit by eq . ( [ eq : shliomis1972 ] ) . the upper abscissa displays the applied magnetic induction @xmath75 measured in the air gap beneath the magnetorheological cell . ] the growth and decay of ferrofluidic spikes takes place in a magnetic field , which is known to alter the viscosity . furnishing the rheometer with the magnetorheological device mrd 170 - 1 t from anton paar we exemplary measure the magnetoviscous behaviour for a shear rate of @xmath76 . we use a plate - plate configuration with a gap of @xmath77 , where the upper plate has a diameter of 20 mm . figure[fig : mag.viscosity ] displays the measured data together with a fit by @xmath78 which describes the magnetoviscosity according to shliomis @xcite . here @xmath79 , denotes the ratio between the magnetic energy of the dipole in the field @xmath74 and the thermal energy @xmath80 , where @xmath81 is the domain magnetisation of saturated magnetite @xcite , and @xmath82 the magnetic active volume . moreover @xmath83 captures the viscosity without a magnetic field , @xmath84 the additional rotational viscosity due to the presence of the magnetic field @xmath85 in the ferrofluid , and @xmath86 is the hydrodynamic volume fraction of the magnetite particles . the brackets @xmath87 indicate a spatial average over the inclosed quantity . note that in case of fig.[fig : mag.viscosity ] the angle @xmath88 between @xmath85 and the vorticity of the flow is @xmath89 . for the fit the internal field was obtained via solving @xmath90 , assuming a demagnetization factor of @xmath91 . the fit yields a hydrodynamic volume fraction of @xmath92 and @xmath93 . from @xmath82 one estimates a mean diameter of @xmath94 for the magnetic particles . this is almost a factor of ten larger than @xmath95 obtained from the magnetisation curve ( cf.table [ tab : apge32_parameters ] ) . assuming a spherical layer of oleic acid molecules of thickness @xmath96 around the magnetic particles @xcite , the volume fraction of the magnetic active material is @xmath97 . this is more than three times larger than the value obtained via the magnetisation curve ( cf.table [ tab : apge32_parameters ] ) . the elevated values of @xmath98 and @xmath99 may be a consequence of magnetic agglomerates , which are not taken into account by eq.([eq : shliomis1972 ] ) . for @xmath100 ( red ) , @xmath101 ( black ) , and @xmath102 ( blue ) . the crosses mark the measured data ( for clarity only every 5th data point is shown ) , whereas the solid lines display fits by eq.([eq : sisko ] ) . ] to test the flow behaviour of the ferrofluid , the viscosity was measured versus the shear rate for three exemplary magnetic inductions , as presented in fig.[fig : shear.thinning ] . all curves exhibit a decay of the viscosity for increasing @xmath103 , i.e. shear thinning which is typical for dispersions @xcite . for a quantitative description of this effect the measured data are fitted by the sisko equation @xcite @xmath104 adapted to the limit @xmath105 , where @xmath106 . moreover @xmath107 denotes a factor and @xmath108 a scaling exponent . table [ tab : sisko ] displays the fitting parameters obtained for the three inductions . .the parameters obtained by fitting eq.([eq : sisko ] ) to the experimental data . [ cols="<,^,^,^",options="header " , ] [ tab : fit.para ] next we focus as well on the experimental data for the decay , which are plotted together with the growth data in fig . [ fig : growthrate_comparison ] . the decay rates ( @xmath109 ) are scattering more widely in comparison to the growth rates ( @xmath110 ) . this may be due to the fact , that the decay rates could not be resolved in the bistability range , and thus not in the immediate vicinity of @xmath111 , in contrast to the growth rates . the black dashed line marks the outcome of a fit of eq([eq : hat_omega2_hatm ] ) to _ all _ experimental values . also in this extended range the fit describes the measured growth and decay rates to some extent . in table [ tab : fit.para ] we present in line two the fit parameters for viscosity and surface tension . the fitted surface tension is well within the error bars of the measured value , whereas the fitted viscosity is about 20 % above the measured one . ) and decay ( @xmath112)rates @xmath113 , respectively , of the pattern amplitude as a function of the magnetization @xmath114 . the symbols represent the measured data . the black dashed line shows a fit of eq.([eq : hat_omega2_hatm ] ) to the experimental growth and decay rates , with the parameters given in the line two of table [ tab : fit.para ] . the orange dashed line marks as well a fit by eq.([eq : hat_omega2_hatm ] ) , but is taking into account a growth rate dependent surface tension , as described by eq.([eq : dyn.sigma ] ) . for the parameters see line three of table [ tab : fit.para ] . the solid blue line displays a fit taking into account a growth - rate dependent viscosity according to eq . ( [ eq : dyn.eta ] ) . for parameters see line four of table [ tab : fit.para].,title="fig:",width=298 ] 0.4 cm most importantly , inspecting the measured data more closely , one observes a different inclination for growth and decay rates with respect to @xmath114 . obviously this systematic deviation is not matched by eq.([eq : hat_omega2_hatm ] ) . as a possible origin for the different inclinations one may suspect that utilizing the static surface tension in eq.([eq : hat_omega2_hatm ] ) is not a sufficient approximation . indeed during the growth of the peaks new surface area is generated , and the diffusion of surfactants from the bulk of the ferrofluid towards the surface may lag behind . similarly during the decay of the peaks surface area is annihilated and the surface density of surfactants may there exceed the equilibrium concentration . therefore we adopt a growth - rate - dependent dynamic surface tension according to @xmath115 where @xmath116 denotes the static surface tension and @xmath117 a coefficient of dimension @xmath118 . in fig.[fig : growthrate_comparison ] the orange dashed line marks the outcome of the fit . it follows the black dashed line , and thus can not explain the different inclinations . in a next attempt to describe the different inclinations we postulate a growth rate dependent viscosity in the form of @xmath119 where @xmath120 is a coefficient of dimension @xmath121 . in fig.[fig : growthrate_comparison ] a fit by eq.([eq : dyn.eta ] ) is marked by the solid blue line . obviously this phenomenological ansatz meets the data remarkeably well . a possible explanation of this complex behaviour is based upon the formation of chains of magnetic particles , which is indicated by the enhanced shear thinning as recorded in fig.[fig : shear.thinning ] . the chain formation will be most prominent in the higher magnetic field in the spikes at the starting amplitude @xmath122 , marked in fig.[fig : frozenprotocol ] . these chains are then increasing the magnetoviscosity during the decay of the spikes , which retards the decay ( cf.path @xmath123__2 _ _ and @xmath123__3b _ _ in fig.[fig : frozenprotocol ] ) . during the decay they are partially destroyed . as a consequence , after switching again to an overcritical induction , the growth of the spikes ( path @xmath124__3a _ _ ) is comparatively faster . in contrast , our theory is based on newtonian fluids . an extension to shear thinning and structured liquids has still to be developed . we are next comparing the critical inductions in the last column of table [ tab : fit.para ] . the static fit of the growth process yields @xmath125 and deviates by only 1% from the mean value @xmath126 obtained by a fit of the full dynamics by means of amplitude equations in ref.@xcite . all other values for @xmath3 underestimate this value slightly more ( cf.line 2 - 4 ) . in the latter three cases the growth _ and _ decay was taken into account . this is a conformation , that mainly the decay is affected by chain formation in the spikes . eventually we will not hide _ four _ further effects which may have impact on our experiment : _ first _ , the experiments are performed in a finite container which comprises only 27 spikes on a hexagonal lattice , whereas the theory considers a laterally infinite layer . our finite circular size does indeed suppress the onset of a hexagonal pattern , due to the ramp described above . _ second _ , by seeding a regular hexagonal pattern at large amplitude the selected wavelength may differ from the wavelength of maximal growth . this can in principle shift the experimental threshold towards higher values . however , it was demonstrated by linear stability analysis that this effect can be neglected in the limit of high viscosities @xcite . _ third _ , magnetophoresis may take place in the crests of the pattern , in this way creating an inhomogeneous distribution of magnetite . even though the timescale for separation in a low viscous mf comprise days @xcite and our measurements last only hours , an effect can not completely excluded . a _ fourth _ reason may be that instead of the shear viscosity the extensional viscosity has to be taken into account in eq.([eq : disprel ] ) . indeed , besides a small viscous sublayer , the flow profile of surface waves can `` be described by a potential and is rotational free and purely elongational '' @xcite . most recently a capillary - break - up - extensional - rheometer was subjected to magnetic fields oriented along the direction of the capillary @xcite . for increasing fields an enlarged elongational viscosity was observed . this effect was also attributed to chain formation . however , to measure the elongational viscosity of ferrofluids is a difficult task , and sensitive devices have still to be developed . using a highly viscous magnetic fluid , the dynamics of the formation of the rosensweig instability can be slowed down to the order of minutes . therefore , it is possible to measure the dynamics using a two - dimensional imaging technique , in contrast to previous work @xcite , where only a one - dimensional cut through the two - dimensional pattern was accomplished . by means of a specific measurement protocol we were able to seed regular patterns of small amplitude , suitable for a comparison with linear theory . from the evolution of their amplitudes we could estimate the linear growth and decay rates , respectively . our experiment confirmed for the very first time a _ linear _ scaling of the growth rate with the magnetic inductions , as predicted @xcite for the immediate vicinity of the bifurcation point . thus the scaling behavior of the growth rate is now confirmed for supercritical magnetizations not only above @xcite but also below the boundary of the two scaling regimes at @xmath12 . additionally , we uncovered , that the rates of growth and decay are slightly different , a phenomenon not predicted by the theory . a possible origin of this discrepancy is the formation of chains of magnetic particles . their presence in our ferrofluid is indicated by the magnetically enhanced shear thinning . the build up of chains in the static spikes , and their subsequent destruction during the decay may change the effective viscosity of the structured ferrofluid , and thus explain the deviations . so far our theory is based on newtonian liquids . an extension to shear thinning and structured ferrofluids is referred to future investigations . it may be able to reproduce the scaling of the effective viscosity as described phenomenologically by eq.([eq : scaling_visc ] ) . we thank m. mrkl for measuring the surface tension of the used magnetic fluid . the temperature - controlled container was made with the help of klaus oetter and the mechanical and electronic workshop the university of bayreuth . moreover discussions with thomas friedrich , werner khler , konstantin morozov and christian wagner are gratefully acknowledged . r.r . is deeply indebted to the emil - warburg foundation for financially supporting repair and upgrade of the magnetorheometer . the coefficients for the forth and fifth order of @xmath127 in the scaling laws ( [ eq : hat_omega2_hatm ] , [ eq : hat_k_hatm ] ) are @xmath128}{4\bar\nu^5}\\ \label{eq : iota } & + \frac{407(2+a_\chi)^5}{64\bar\nu^9}\ ; , \\ \nonumber o = & \frac{3a_\chi(2a_\chi+1)+6b_\chi(3a_\chi+2+b_\chi)}{\bar\nu^2 } + \frac{3(2+a_\chi)^3(23a_\chi^2 + 158a_\chi+33b_\chi+125)}{\bar\nu^6}\\ \nonumber & - \frac{3(2+a_\chi)\left[36a_\chi^3+a_\chi^2(18b_\chi+217)+a_\chi(138b_\chi+282)+b_\chi(138 + 11b_\chi)+83\right]}{4\bar\nu^4}\\ \label{eq : o } & - \frac{491(2+a_\chi)^5}{32\bar\nu^8 } \ ; .\end{aligned}\ ] ] 45ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop @noop * * , ( ) @noop `` , '' ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1017/jfm.2015.565 [ * * , ( ) ] @noop * * , ( ) in @noop _ _ , vol . , , ( , , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) , @noop * * , ( ) @noop * * , ( ) , , @noop _ _ , , vol . ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) \doibase http://dx.doi.org/10.1122/1.4902356 [ * * , ( ) ] | using a highly viscous magnetic fluid , the dynamics in the aftermath of the rosensweig instability can be slowed down by more than 2000 times . in this way
we expand the regime where the growth rate is predicted to scale linearly with the bifurcation parameter by six orders of magnitude , while this regime is tiny for standard ferrofluids and can not be resolved experimentally there .
we measure the growth of the pattern by means of a two - dimensional imaging technique , and find that the slopes of the growth and decay rates are not the same - a qualitative discrepancy to the theoretical predictions .
we solve this discrepancy by taking into account a viscosity which is assumed to be different for the growth and decay .
this may be a consequence of the measured shear thinning of the ferrofluid .
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in physics , the study of colloidal brownian motion has a long history beginning with einstein s famous paper @xcite in 1905 , and the understanding of its mechanism has been systematically developed in molecular kinetic theory @xcite . recently , experimental developments have enabled researchers to observe a single trajectory itself of brownian motion @xcite , which engages physicists in quantitative modelling of sub - micrometer systems , such as molecular motors @xcite . likewise , recent breakthroughs for computational technologies have enabled physicists to study microstructure of financial brownian motion in detail . they have applied their knowledge beyond material science and into studying in particular price movements in financial markets for about a quarter century . there consequently appeared new approaches as mentioned below , in contrast to conventional mathematical finance where a priori theoretical dynamics is assumed @xcite . there are three physical approaches to financial brownian motions as shown in figure [ fig : micro - meso - macro ] . the microscopic approach focuses on the dynamics of traders in the financial markets ( figure [ fig : micro - meso - macro]a ) . traders correspond to molecules in the modelling of materials , which enables both a numerical and a theoretical analysis of the macroscopic motion of prices @xcite . another approach is based on macroscopic empirical analyses of price time series ( figure [ fig : micro - meso - macro]c ) and the direct empirical modelling of price dynamics focusing on fat - tailed distributions and long - time correlations in volatility @xcite . the third approach focuses on mesoscopic dynamics concerning order - books ( figure [ fig : micro - meso - macro]b ) , which are accumulated buy sell orders initiated by traders in the price axis where deals occur either at the best bid ( buy ) or ask ( sell ) price defining the market prices . numerical simulations of markets become available by introducing models of order - book dynamics @xcite . recently , the mesoscopic approach developed considerably with the analysis of high frequency financial market data in which the whole history of orders is tractable using a direct analogy between order - books and colloids @xcite . from the order - book data @xcite , the importance of inertia was reported for market price , implying the existence of market trends . the langevin equation was then found to hold most of time showing that the fluctuation dissipation relation can be extended to order - book dynamics . however , microscopic mechanisms behind market trends have not been clarified so far because direct information is required on individual traders strategies . in the present paper , we analyse more informative order - book data in which traders can be identified in an anonymised way for each order so that we can estimate each trader s dynamics directly from the data . here we present a minimal model validated by direct observation of individual traders trajectories for financial brownian motion . we first report a novel statistical law on the trend - following behaviour of foreign exchange ( fx ) traders by tracking the trajectories of all individuals . we next introduce a corresponding microscopic model incorporated with the empirical law . we reveal the mesoscopic and macroscopic behaviour of our model by developing a parallel theory to that for molecular kinetics : boltzmann - like and langevin - like equations are derived for the order - book and the price dynamics , respectively . a quantitative agreement with empirical findings is finally presented without further assumptions . we analysed the high - frequency fx data between the us dollar ( usd ) and the japanese yen ( jpy ) from the 5th 16.00 to the 10th 20.00 gmt september 2016 on electronic broking services , one of the biggest fx platforms in the world . all trader activities were recorded for our dataset with anonymised trader ids with one - millisecond time - precision . the minimum price - precision was 0.005 yen for the usd / jpy pair at that time , and the currency unit in this paper is 0.001 yen , called the tenth pip ( tpip ) . the minimum volume unit for transaction was one million usd , and the total monetary flow was about @xmath0 billion usd during this week . the market adopts the double - auction system , where traders quote bid or ask prices . in this paper , we particularly focused on the dynamics of high - frequency traders ( hfts ) , who frequently submit or cancel orders according to algorithms ( see appendix [ app : def_hft ] for the definition ) . the presence of hfts has rapidly grown recently @xcite and @xmath1 of the total orders were submitted by the hfts in our dataset . we first illustrate the trajectories of bid and ask prices quoted for the top 3 hfts in figure . [ fig : trajectory_traders]a c . we observed that with the two - sided quotes typical hfts tend to play the role of liquidity providers ( called market - makers @xcite ) . we also observed that buy sell spreads ( i.e. , the difference between the bid and ask prices for a single market - maker ) fluctuated around certain time - constants , showing a strong coupling between these prices . indeed , the buy sell spread distributions exhibit sharp peaks for individual hfts as shown in the insets in figure [ fig : trajectory_traders]a c ( see also appendix [ app : buy - sell ] ) . we next report the empirical microscopic law for the trend - following strategy of individual traders . let us denote the bid and ask prices of the top @xmath2th hft by @xmath3 and @xmath4 ( see appendix [ app : trend - following ] for the definitions ) . we investigated the average movement of the mid - price @xmath5 between transactions conditional on the previous market price movement @xmath6 ( figure [ fig : trend_follow]a ) . for the top 20 hfts ( figure [ fig : trend_follow]b and c ) , we find that the average movement is described by @xmath7 where the conditional average @xmath8 is taken when the previous price movement is @xmath6 and @xmath9 ( see appendix [ app : trend - following ] for the detail ) . @xmath10 and @xmath11 are constants characterizing the price movement and the saturation threshold against the market trend . here , typical values are given using @xmath12 [ tpip ] and @xmath13 [ tpip ] . the empirical law ( [ eq : trendfollow ] ) implies that the reaction of traders is linear for small market trends but saturates for large market trends . remarkably , a similar behaviour was reported from a full macroscopic analysis of market price data at one - month precision @xcite . here we introduce a minimal microscopic model incorporating the empirical law ( [ eq : trendfollow ] ) . we make four assumptions : ( i ) the number of traders is sufficiently large . ( ii ) they always quote both bid and ask prices ( for the @xmath2th trader , @xmath3 and @xmath4 ) simultaneously with a unit volume as market - makers . ( iii ) buy sell spreads are time - constants unique to traders with distribution @xmath14 . the trader dynamics is then characterized by the mid - price @xmath15 . ( iv ) trend - following random walks are assumed in the microscopic dynamics ( see figure [ fig : dealermodel]a c ) : @xmath16 with a constant strength for trend - following @xmath17 , white gaussian noise @xmath18 with constant variance @xmath19 , and a requotation jump @xmath20 after transactions ( figure [ fig : dealermodel]c ) . @xmath20 is defined by @xmath21 with jump size @xmath22 and the @xmath23th transaction time @xmath24 between traders @xmath2 and @xmath25 . the transaction condition at time @xmath26 is given by @xmath27 ( figure [ fig : dealermodel]b ) . this model can be assessed using parallel tools employed in molecular kinetic theory . in this theory @xcite , the boltzmann equation is first derived for the one - body velocity distribution from the hamiltonian dynamics by the method of bogoliubov , born , green , kirkwood , and yvon assuming molecular chaos . the langevin equation is derived in turn from the boltzmann equation for massive brownian particles @xcite . here we have followed the same mathematical procedure to elucidate the dynamics behind order - book profiles and financial brownian motion . for relative position @xmath28 from the centre of mass @xmath29 , the boltzmann - like equation is first derived for the one - body probability distribution density @xmath30 conditional for a trader with a buy sell spread @xmath31 from the multi - agent dynamics ( [ eq : dealermodel ] ) : @xmath32\label{eq : financialboltzmann}\ ] ] with @xmath33 and @xmath34 . here , @xmath35 ( @xmath36 ) represents transactions as bid ( ask ) orders . the average order - book profile is given by @xmath37 for the ask side . the integral term in equation ( [ eq : financialboltzmann ] ) corresponds to the collision integral in the conventional boltzmann equation . as the langevin equation is derived from it @xcite , then similarly the langevin - like equation is derived from the boltzmann - like equation ( [ eq : financialboltzmann ] ) , @xmath38 where @xmath39 and @xmath40 are the market price movement and transaction time interval at the @xmath41th tick time . the tick time is an integer time incremented by every transaction and the mean time interval between transactions is @xmath42 seconds in this week . the first and second terms on the right - hand side of eq . ( [ eq : financiallangevin ] ) describe trend - following and random noise , respectively ; the trend - following term corresponds to the momentum inertia in the conventional langevin equation . equations ( [ eq : financialboltzmann ] ) and ( [ eq : financiallangevin ] ) can be solved for @xmath43 under an appropriate boundary condition ( see appendix [ app : boundaryconditions ] ) . we first set the buy sell spread distribution as @xmath44 with decay length @xmath45}$ ] , empirically validated in our dataset ( figure [ fig : pricediff]a and appendix [ app : buy - sell ] ) . the average order - book profile @xmath46 is given for @xmath47 by @xmath48 . \label{eq : orderbook}\ ] ] the tail of the price movement is approximately given by @xmath49 with decay length @xmath50 , average movement from trend - following @xmath51 , average transaction interval @xmath52 , and complementary cumulative distribution @xmath53 . further technical details are to be published in a forthcoming paper . we next investigated the consistency between our microscopic model and our dataset . the empirical daily profile was first studied for the average order - book @xmath46 in figure [ fig : pricediff]b ( see appendix [ app : order - book ] for the detail ) . surprisingly , we found an agreement with our theoretical lines ( [ eq : orderbook ] ) without fitting parameter that strongly supports the validity of our description . we also empirically evaluated the two - hourly segmented cumulative distribution for the price movement in one - tick precision @xmath54 ( figure [ fig : pricediff]c ) , which obeys an exponential law that is consistent with our theoretical prediction ( [ eq : theory_pricemove ] ) . in our dataset , the decay length @xmath55 is approximately constant over a two - hour period but varies over time during the week . to remove this non - stationary feature , we introduced the two - hourly scaled cumulative distribution @xmath56 with scaling parameters @xmath55 and @xmath57 ( figure [ fig : pricediff]d ) , thereby incorporating the two - hourly exponential - law for the whole week . the price movements obey an exponential law for short periods but simultaneously obeys a power - law over long periods with exponent @xmath58 ( figure [ fig : pricediff]e ) . this apparent discrepancy is explained by the power - law nature of the decay length @xmath55 . because @xmath55 approximately obeys a power - law cumulative distribution @xmath59 over the week with @xmath60 ( figure [ fig : pricediff]f ) , the one - week distribution @xmath61 obeys the power - law as a superposition of the two - hourly segmented exponential distribution , @xmath62 with @xmath63 . remarkably , @xmath55 tends to be long when the market is inactive ( figure [ fig : pricediff]g ) . we therefore obtain a consistent result with the previously reported power - law @xcite as a non - stationary property of @xmath55 . we note that our model can exhibit super - diffusion under an appropriate parameter set as @xmath64 with @xmath65 for short periods ( figure [ fig : pricediff]h ) , which is consistent with previous reports @xcite . here the mean squared displacement is defined by @xmath66 as a function of tick time @xmath41 with the ensemble average @xmath67 . we also note that our model asymptotically shows ballistic behaviour @xmath68 when trend - following is sufficiently large , which implies that it plays the role of a momentum inertia " in financial markets . we further note that our model can show sub - diffusion ( @xmath69 ) when trend - following is sufficiently small . in this article , we have presented an intensive data analysis of anonymised traders in a foreign exchange market to directly observe strategies of hfts . we first report a simple empirical law characterizing trend - following behaviour of individual traders against market trends . a trend - following random walk model is correspondingly introduced as a microscopic dynamics of the financial market . the mesoscopic and macroscopic behaviours of this model are systematically analysed in a parallel calculation to molecular kinetic theory . our theoretical model reproduces the average order - book profile and the price movement distributions empirically . this work would be an important step toward unified description of financial markets from individual traders dynamics . we discuss here a possible reason behind the success of our kinetic - like theory for our model . in material physics , mean - field approximations are invalid for low - dimensional systems because the low - dimensional geometry does not allow two - body correlations to disappear after collision . in contrast , in our model , traders are separated compulsorily after transactions , and there is little possibility for the same pair to enter successive transactions . the two - body correlation then quickly decays after collision assuming molecular chaos " is valid . this scenario implies that the kinetic - like description may work well in various socio - economic systems , in addition to the previously studied examples , such as opinion formation and wealth distribution @xcite . our report dealt mainly with short - duration trends ; their correlation with long - duration trends @xcite is a topic to future studies . of interest is the study of traders behaviour in unstable markets triggered by external shocks , as those that occur in financial crises and flash crashes . the economics reason behind the hyperbolicity ( [ eq : trendfollow ] ) in trend - following needs to be pursed further . we greatly appreciate icap for their provision of the ebs data . we also appreciate m. katori , h. hayakawa , s. ichiki , k. yamada , s. ogawa , f. van wijland , d. sornette , t. sano , and t. ito for fruitful discussions . this work was supported by jsps kakenhi ( grand no . 16k16016 ) and jst , strategic international collaborative research program ( sicorp ) on the topic of ict for a resilient society " by japan and israel . 99 einstein , a. ber die von der molekularkinetischen theorie der wrme geforderte bewegung von in ruhenden flssigkeiten suspendierten teilchen . _ _ 322 , 549 - 560 ( 1905 ) . chapman , s. & cowling , t. g. _ the mathematical theory of non - uniform gases _ ( cambridge univ . press , cambridge , 1970 ) . van kampen , n. g. _ stochastic processes in physics and chemistry _ ( amsterdam , north - holland , 2007 ) . van den broeck , c. , kawai , r. & meurs , p. microscopic analysis of a thermal brownian motor . _ 93 , 090601 - 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book fluid and fluctuation - dissipation relations . _ phys . _ 112 , 098703 - 098707 ( 2014 ) . hendershott , t. , jones , c. m. & menkveld , a. j. does algorithmic trading improve liquidity ? _ the journal of finance _ 66 , 1 - 33 ( 2011 ) . menkveld , a. j. high frequency trading and the new market makers . _ journal of financial markets _ 16 , 712 - 740 ( 2013 ) . ebs dealing rules appendix ebs market ( at the time of june 2016 ) . lemprire , y. , deremble , c. , seager , p. , potters , m. & bouchaud , j .- two centuries of trend following . _ journal of investment strategies . _ 3 , 41 - 61 ( 2014 ) . slanina , f. _ essentials of econophysics modelling . _ ( oxford university , oxford , 2014 ) . pareschi , l. & toscani , g. _ interacting multiagent systems : kinetic equations and monte carlo methods . _ ( oxford university press , oxford , 2013 ) . for this paper , we define a high frequent trader ( hft ) as a trader who submits more than 500 times a day on average ( i.e. , more than 2500 times for the week ) . as several traders are unwilling to transact and often interrupt orders at the instant of submission ( called flashing ) , we excluded traders with live orders of less than @xmath70 percent of the transaction time . with this definition , the number of hfts was 134 during this week , whereas the total number of traders was 1015 . we note that the total number of traders who submitted limit orders was 922 ; the other 93 traders submitted only market orders . we calculated the percentage of two - sided quotes as follows : when a bid ( ask ) order is submitted by a trader , we check whether corresponding ask ( bid ) orders exist . we then count the number of two - sided quotes for all traders at every order submission and finally divide it by the total number of submissions . the difference in the median bid and ask prices was studied as a buy sell spread for an hft . samples where only both bid and ask prices exist are taken in the one - second time - precision for figure [ fig : trajectory_traders ] and [ fig : pricediff]a . we plotted standard deviations of the averages as error bars for each point . we remark on the precise definition of the bid ( ask ) price of individual hfts for the analysis of trend - following . if a trader quotes both single - bid and single - ask orders at any time , the bid and ask prices are defined literally . in the presence of multiple bid or ask orders , we use the value of the median for the bid or ask orders as @xmath3 or @xmath4 . in the absence of any bid or ask orders , we use the previous bid or ask price as @xmath3 or @xmath4 for interpolation for figure . [ fig : trend_follow]b and c. we also remark that exceptional samples where the bid or ask price is far from the market price by 10 yen ( 0.0659% of the total ) are excluded from the calculation of the conditional ensemble average @xmath71 . we note that the standard deviations of the conditional averages are plotted for each point as error bars . also , median values in the top 20 hfts are given using @xmath12 [ tpip ] and @xmath72 [ tpip ] . for the boltzmann - like equation ( [ eq : financialboltzmann ] ) , we first introduce sufficiently large cutoffs at @xmath73 . the limit @xmath43 is taken with reflecting boundaries assumed at @xmath73 . a large cutoff limit @xmath74 is taken finally . for the langevin - like equation ( [ eq : financiallangevin ] ) , we make the following two assumptions . ( i ) trend - following has the same order as random noise : @xmath75 . ( ii ) saturation in trend - following ( [ eq : trendfollow ] ) is dominant : @xmath76 . we note that equation ( [ eq : trendfollow ] ) can be approximated for large fluctuations as @xmath77 under these conditions with the signature function @xmath78 defined by @xmath79 for @xmath80 and @xmath81 for @xmath82 . the daily average order - book profile is calculated for the hfts . we took snapshots of the order - book every second and its ensemble average every day . we also plotted standard deviations of the averages as error bars for each point . we take snapshots of the order - book after every transaction and count the total number of different trader ids for both bid and ask sides . the counting weight for an hft quoting both sides is set to 1 and that for an hft quoting one side is 1/2 . we then plot the average of the number of trader ids for both bid and ask sides every two hours in figure [ fig : pricediff]g . the typical number of hft ids was about @xmath83 in our dataset with this definition . the number of total volumes quoted by hfts is typically about @xmath84 . admittedly , there is room for debate on which number is appropriate for the calibration of the total number of traders in our model ; it remains a topic for future study . | brownian motion has been a pillar of statistical physics for more than a century , and recent high - frequency trading data have shed new light on microstructure of brownian motion in financial markets .
though evidences of trend - following behaviour of traders were indirectly shown in such trading data , the microscopic model has not been established so far by direct observation of trajectories for individual traders . in this paper
, we present a minimal microscopic model for financial brownian motion through an intensive analysis of trajectory data for all individuals in a foreign exchange market .
this model includes a novel empirical law quantifying traders trend - following behaviour that can create the inertial motion in market prices over short durations .
we present a systematic solution paralleling molecular kinetic theory to reveal mesoscopic and macroscopic dynamics of our model .
our model exhibits quantitative agreements with empirical results strongly supporting our analysis . |
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this paper presents data for the last two patches ( c and d ) of the sky observed by the public eso imaging survey ( eis ) , being carried out in preparation for the first year of regular operation of vlt . the i - band data reported here covers a total area of 12 square degrees , down to @xmath6 , corresponding to two patches probing separated regions of the sky , 6 square degrees each . the present work complements earlier papers in the series ( nonino 1998 ; paper i , prandoni 1998 ; paper iii ) and completes the presentation of the data accumulated by the eis observations carried out in the period july 1997-march 1998 as part of the wide - angle imaging survey originally described by renzini and da costa ( 1997 ) and in paper i. the primary science goal for surveying patches c and d was to search for and produce a list of distant galaxy cluster candidates that would complement those of the other two patches ( a and b ) reported earlier ( olsen 1998a , b : paper ii and v ) , providing vlt targets nearly year - round . patches c and d were also selected to overlap with the ongoing 92 cm westerbork survey in the southern hemisphere ( wish ) being carried out in the region @xmath7 and @xmath8 . originally , the eis observations were expected to be carried out in two passbands ( v and i ) . however , because of time constraints and the prospect of supplementing the eis observations at the ntt with the new wide - field imager for the 2.2 m eso / mpia telescope , preference was given to increase the area covered by the i - band observations , more suitable for identifying distant clusters with @xmath9 ( see paper v ) . this decision allowed the full coverage of the selected patches , yielding a total coverage of 12 square degrees . combined with the data for patches a and b the eis i - band data covers a total area of about 17 square degrees , currently the largest available survey of its kind in the southern hemisphere . the goal of the present paper is to describe the characteristics of the i - band observations of patches c and d. in section 2 , the observations , calibration and the quality of the data are described . in section 3 , the object catalogs extracted from the images are examined and compared with data from the other patches and other data sets to comparable depth . concluding remarks are presented in section 4 . the observations of patches c and d were carried out over several months in the period november ( december for patch d ) 1997 to march 1998 , using the red channel of the emmi camera on the 3.5 m new technology telescope ( ntt ) at la silla . the red channel of emmi is equipped with a tektronix 2046 @xmath10 2046 chip with a pixel size of 0.266 arcsec and a useful field - of - view of about @xmath11 . the observations were carried out as a series of overlapping 150 sec exposures , with each position on the sky being sampled at least twice , using the wide - band filter wb829#797 described in paper i , and for which the color term relative to the cousins system is small . the data for patches c and d consist of 1348 frames but only 1203 were accepted for final analysis , discarding 145 frames obtained in poor seeing condition ( @xmath12 arcsec ) . the frames actually accepted have a seeing in the range 0.5 to 1.6 arcsec , considerably better than the data available for patches a and b obtained at the peak of el nio . figure [ fig : seeing ] shows the seeing distribution of all observed frames in each patch . for comparison the figure also shows the seeing distribution of the accepted frames , with the vertical lines in each panel indicating the median seeing and the quartiles of the distribution . from the figure one finds that the median seeing for both patches is sub - arcsec ( @xmath13 arcsec ) with only 25% of the area covered by frames with a seeing larger than 1 arcsec . the good quality of the observations can also be seen from figure [ fig : limiso ] which shows the @xmath14 limiting isophote within 1 arcsec for each patch . apart from one subrow in patch c , in both cases the limiting isophote is typically @xmath15 25.3 @xmath16 mag arcsec@xmath17 . the two - dimensional distributions of the seeing and limiting isophote are shown in figures [ fig : seeing_cont ] and [ fig : limiso_cont ] . comparison with similar distributions presented in earlier papers ( paper i and iii ) shows that the data for patches c and d are significantly better . note that for each patch tables are available listing the position of each accepted frame , its seeing , limiting isophote and photometric zero - point and can be found at `` http://www.eso.org/eis '' . in late february 1998 , a realignment of the secondary mirror was carried out by the ntt team in an attempt to minimize the image distortions seen in the upper part , especially the upper - right corner , of the emmi frames . some frames for patch c and most of the frames in patch d were observed with the new setup of the ntt . examination of the point spread function for these frames showed no significant improvement in the quality of the images . this points out the need to introduce a position - dependent estimator for the point - spread function to assure uniformity in the star / galaxy separation across the frame . this is particularly important for images observed under good seeing conditions . in fact , examining the uniformity of the classification as a function of position on the chip it is found that there is a 10% increase in the density of galaxies at the upper edge of the chip , due to misclassifications , significantly larger than that seen in paper i. in the last three runs ( january - march ) it was also noticed faint ( at the @xmath18 level of the background noise ) linear features aligned along the east - west direction ( perpendicular to the readout axis ) associated with moderately bright stars located in the lower half of the ccd not previously seen . the cause for the these features are at the present time unclear but are probably due to the electronic of the old - generation ccd controller of emmi , when used in a dual - port readout mode . these affects two - thirds of the patch c frames and essentially all the patch d frames . these light trails occur randomly in the patch and there is no obvious way of correcting for them a priori . an important consequence of this problem is that it leads to a localized increase in the detection of low - surface brightness objects over a range of magnitudes ( typically @xmath19 ) which can have a significant impact in the cluster detection algorithm ( scodeggio 1998 , paper vii ) . this is unfortunate because both patches c and d are located at lower galactic latitudes ( @xmath20 ) with almost an order of magnitude larger density of stars than the previous patches . the photometric calibration of the patch was carried out , as described in papers i and iii , by determining a common zero - point for all frames from the solution of a global least - squares fit to all the relative zero - points , constraining their sum to be equal to zero . the absolute zero - point was determined by a simple zero - point offset determined from the common zero - point of all frames observed in photometric conditions . there are 340 and 290 such frames , covering about 80% and 60% of the surveyed area , in patches c and d , respectively ( see figure [ fig : overlaps ] ) . the zero - points for these frames were determined using a total of 10 fields containing of the order of 45 standard stars taken from landolt ( 1992 a , b ) , observed in 10 nights for patch c and in 11 nights for patch d. altogether 215 independent measurements of standards in the three passbands were used in the calibration . comparison with external data suggests that a zero - point offset provides an adequate photometric calibration for the entire patch . in order to check the photometric calibration and the uniformity of the zero - points , strips from the denis survey ( epchtein 1996 ) crossing the surveyed area the regions of overlap of these data are shown in figure [ fig : overlaps ] , which shows that there are five strips crossing patch c and two strips crossing patch d. in the figure the regions observed under photometric conditions are also indicated . comparison of this figure with their counterparts presented in papers i and iii , clearly shows that the data for patches c and d are of superior quality , with a much larger fraction of frames taken under photometric conditions . in order to investigate possible systematic errors in the photometric zero - point over the scale of the patch , the eis catalogs were compared with object catalogs extracted from the denis strips that cross the survey regions ( see figure [ fig : overlaps ] ) . comparison of the catalogs allows one to investigate the variation of the zero - point over the patch . the results are shown in figure [ fig : denis ] . the domain in which the comparison can be made is relatively small because of saturation of objects in eis at the bright end ( @xmath21 ) and the shallow magnitude limit of denis ( @xmath22 ) . still , within the two magnitudes where comparison is possible one finds a roughly constant zero - point offset of less than 0.02 mag for both strips and a scatter of @xmath23 mag that can be attributed to the errors in the denis magnitudes ( deul 1998 ) . in order to evaluate the quality of the data simple statistics computed from the object catalogs extracted from the images are compared in this section with model predictions and other data sets . the catalogs derived from individual frames are used to generate the even , odd and best seeing catalogs , described in earlier papers . the spatial distribution of stars and galaxies , defined using similar star / galaxy classification criteria as in previous papers of the series , are shown in figures [ fig : pc_visu ] and [ fig : pd_visu ] down to @xmath24 and @xmath25 for stars and galaxies , respectively . the latter corresponds roughly to the completeness limit of the object catalog . this limit was established using the object catalog extracted from the co - addition of images of a reference frame taken periodically during the observations of a patch . note that because of the much better seeing star / galaxy classification is possible down to @xmath26 and the completeness is about 0.5 mag deeper . some improvement in the classification is expected from a new estimator being implemented in sextractor based on a position - dependent psf fitting scheme currently being tested . this new version should also improve the uniformity of the classification across the chip . the distribution of the stars and galaxies shown in figures [ fig : pc_visu ] and [ fig : pd_visu ] is remarkably homogeneous and considerably better than those seen in the previous eis patches due to the much better observing conditions . this is true except for a small region of about 0.2 square degrees in patch c which has been removed , as indicated in figure [ fig : pc_visu ] . the only problem seen with the galaxy catalogs in these patches is the presence of several relatively thin linear features clearly seen at high resolution ( see eis release page ) . these features are a consequence of the electronic problem mentioned above and are not easily corrected for at the image level . in order to evaluate the data the general properties of the extracted object catalogs are investigated and compared with model predictions and other data sets . note that patches c and d are located at lower galactic latitude and the number of stars is considerably larger . in addition , the seeing is considerably better than in previous patches . therefore , it is of interest to re - evaluate the overall performance of the eis pipeline reduction under these new conditions . figure [ fig : star_counts ] , shows the comparison of the star counts for patches c and d derived using the stellar sample extracted from the object catalogs , with the predicted counts based on a galactic model composed of an old - disk , a thick disk and a halo . the star - counts have been computed using the model described by mndez and van altena ( 1996 ) , using the standard parameters described in their table 1 and an @xmath27 of 0.015 and 0.010 for patches c and d , respectively . it is important to emphasize that no attempt has been made to fit any of the model parameters to the observed counts . the model is used solely as a guide to evaluate the data . as can be seen there is a good agreement at bright magnitudes ( @xmath28 ) , but the observed counts show an excess at fainter magnitudes ( @xmath29 ) . even though it is unlikely that this excess is due to misclassified galaxies at these relatively bright magnitudes , a better agreement can be achieved if a higher stellarity index is assumed . on the other hand , it is also possible that the model underestimates the contribution of the thick - disk which makes a significant contribution in this magnitude range . the steep drop in the stellar counts beyond @xmath30 is partially due to the relatively high stellarity index adopted , which was chosen to minimize the losses of galaxies . by adopting a stellarity index of 0.5 the drop in the counts may be avoided down to @xmath31 . however , at these magnitudes and this value of the stellarity index contamination by galaxies may be significant . another potential problem at these faint magnitudes is the misclassification of stars as a consequence of the distortion effects in emmi , that can have some impact for images taken in good seeing conditions . in order to evaluate the depth of the galaxy samples , galaxy counts in patches c and d are compared with those of previous patches in figure [ fig : gal_counts ] . there is a remarkable agreement among the counts derived for the different patches , indicating that the identification of galaxies has not been affected by the observations at lower galactic latitudes . the galaxy counts obtained from the different patches have been combined to compute the mean galaxy counts and the variance . this is also shown in figure [ fig : gal_counts ] where it is compared to other ground - based counts ( postman 1998 ) and those from hdf ( williams 1996 ) , appropriately converted to the cousins system ( see paper iii ) . as can be seen the eis galaxy counts agree extremely well with the ground - based data covering comparable area over the entire magnitude range down to @xmath32 and with the bright end of the hdf counts . the excellent internal and external agreement of the i - band galaxy counts serves as a confirmation of the reliability of the eis galaxy catalogs . extraction from co - added images should allow reaching about 0.5 mag deeper . one way of examining the overall uniformity of the galaxy catalogs is to use the two - point angular correlation function , @xmath33 , as departures from uniformity should affect the correlation function especially at faint magnitudes . the latter should be sensitive to artificial patterns , especially to the imprint of the individual frames , or possible gradients in the density over the field , which could result from large - scale gradients of the photometric zero - point . note that any residual effect due to the improper association of objects in the border of overlapping frames would lead to a grid pattern ( see the weight map in the eis release page ) that could impact the angular correlation function . figure [ fig : w ] shows @xmath33 obtained for different magnitude intervals for both patches , using the estimator proposed by landy & szalay ( 1993 ) . the calculation has been done over the entire area of patch d and most of the area of patch c , with only one subrow ( 10 consecutive frames ) removed according to the discussion above ( see section [ obs ] ) . for comparison , @xmath33 computed for the other patches are also shown ( papers i and iii ) from which the cosmic variance can be evaluated directly from the data . as can be seen there is a remarkable agreement for all the magnitude intervals considered . moreover , the larger contiguous area of patches c and d allows to estimate the angular correlation function out to @xmath34 degree . in all cases @xmath33 is well described by a power law @xmath35 with @xmath36 in the range 0.7 - 0.8 . note that for patch b the results refer to the galaxy sample obtained after removing the foreground cluster ( see paper iii ) . in particular , there is no evidence for any underlying pattern associated with the overlap of different frames . the effect on @xmath33 was evaluated by carrying out simulations by adding to the observed galaxy distribution a grid pattern with different density contrast . it was found that for high contrast this would lead to local depressions in the angular correlation function on scales of half the size of the diagonal of the grid and its multiples , with the depth of depression depending on the relative density . none such features are seen further indicating the uniformity of the derived galaxy catalogs . finally , note that the good agreement of @xmath33 for the different patches confirms that the observed small - scale linear features associated with the faint light trails , mentioned in section [ obs ] , have very little impact in the angular correlation function . as shown in paper iii the dependence of the amplitude of the correlation function on the limiting magnitude of the sample is consistent with earlier estimates based on significantly smaller areas and the recent results reported by postman ( 1998 ) . these results show that the eis galaxy catalogs are spatially uniform and form a homogeneous data set independent of the patch , yielding reproducible results . finally , note that even though a single power - law with a slope between 0.7 - 0.8 gives a reasonable fit for the correlation computed in all magnitude bins , there is some indication that for fainter samples ( @xmath37 ) the angular correlation function may be better represented by two distinct power - laws . on small scales ( @xmath38 ) the slope remains the same while on larger scales it becomes gradually flatter . a similar behavior is seen in the @xmath33 computed for all four patches . this flattening seems to be consistent with earlier claims by campos ( 1995 ) and neuschaefer and windhorst ( 1995 ) using significantly smaller samples , and more recently by postman ( 1998 ) with a sample of similar size to eis but covering a single contiguous area . one year after the first observations , the full data set accumulated by eis is being made public in the form of astrometrically and photometrically calibrated pixel maps and object catalogs extracted from individual images . in addition , separate papers have presented derived catalogs listing candidate targets for follow - up work . the eis data set consists of about 6000 science and calibration frames , totaling 96 gb of raw data and over 200 gb of reduced images and derived products . all the information regarding these frames are maintained in a continuously growing database . together with the science archive group a comprehensive interface has been built to provide users with a broad range of products and information regarding the survey . from the verification of the object catalogs and their comparison against model predictions and other observations , it has been found that the extracted catalogs are reliable and uniform . when all patches are included , the combined eis galaxy catalog contains about one million galaxies and it is by far the largest data set of faint galaxies currently available in the southern hemisphere . the star counts show a good agreement with current galactic models , especially at high - galactic latitudes , and the galaxy counts agree remarkably well with other ground - based observations as well as with the counts derived from hdf . the data from the different patches seem to be rather homogeneous , as strongly suggested from measurements of the angular two - point correlation function which should be sensitive to large - scale gradients in a patch or to relative offsets of the photometric zero - points for the different patches . as expected eis - wide has provided large samples ( 50 to over 200 candidates ) of distant clusters of galaxies ( olsen 1998a , b , scodeggio 1998 ) and of potentially interesting point sources ( zaggia 1998 ) , more than adequate for the first year of observations with vlt , the main goal of eis . some of the targets can also be observed nearly year round . in order to expedite the delivery of the products all the results refer to single exposure frames as discussed in the previous papers of the series . even though co - addition has been done for all the patches some problems have been uncovered during the verification of the object catalogs extracted from them and require further work . however , the samples already public are sufficiently deep and large for programs to be conducted in the first year of operation of the vlt . the results obtained from the co - added images will become available before the vlt proposal deadline . this paper completes the first phase of eis which will now focus on the deep observations of the hdf - south ( @xmath39 ) and axaf deep ( @xmath40 ) fields . the results presented so far show the value of a public survey providing the community at large with the basic data and tools required to prepare follow - up observations at 8-m class telescopes . the experience acquired by eis in pipeline processing , data archiving and mining will now be transferred to the pilot survey , a deep wide - angle imaging survey to be conducted with the wide - field camera mounted on the eso / mpia 2.2 m telescope . we thank all the people directly or indirectly involved in the eso imaging survey effort . in particular , all the members of the eis working group for the innumerable suggestions and constructive criticisms , the eso archive group and the st - ecf for their support . we also thank the denis consortium for making available some of their survey data . the denis project development was made possible thanks to the contributions of a number of researchers , engineers and technicians in various institutes . the denis project is supported by the science and human capital and mobility plans of the european commission under the grants ct920791 and ct940627 , by the french institut national des sciences de lunivers , the education ministry and the centre national de la recherche scientifique , in germany by the state of baden - wurttemberg , in spain by the dgicyt , in italy by the consiglio nazionale delle richerche , by the austrian fonds zur frderung der wissenschaftlichen forschung und bundesministerium fr wissenschaft und forschung , in brazil by the fundation for the development of scientific research of the state of so paulo ( fapesp ) , and by the hungarian otka grants f-4239 and f-013990 and the eso c & ee grant a-04 - 046 . our special thanks to the efforts of a. renzini , vlt programme scientist , for his scientific input , support and dedication in making this project a success . finally , we would like to thank eso s director general riccardo giacconi for making this effort possible in the short time available . | this paper presents the i - band data obtained by the eso imaging survey ( eis ) over two patches of the sky , 6 square degrees each , centered at @xmath0 , @xmath1 , and @xmath2 , @xmath3 .
the data are being made public in the form of object catalogs and , photometrically and astrometrically calibrated pixel maps .
these products together with other useful information can be found at `` http://www.eso.org/eis '' .
the overall quality of the data in the two fields is significantly better than the other two patches released earlier and cover a much larger contiguous area .
the total number of objects in the catalogs extracted from these frames is over 700,000 down to @xmath4 , where the galaxy catalogs are 80% complete .
the star counts are consistent with model predictions computed at the position of the patches considered .
the galaxy counts and the angular two - point correlation functions are also consistent with those of the other patches showing that the eis data set is homogeneous and that the galaxy catalogs are uniform .
1@xmath5 # 1 |
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last time the interest has sharply increased for searching the conditions for realization supersolidity phenomenon in solid @xmath1he @xcite , when the crystalline order combines with superfluidity . in spite of the great number of experimental and theoretical investigations in this area , the consensus has not been attained yet . for the present , it has been determined well that observing effects strongly depend on the growing conditions and annealing degree of helium crystals . the special modeling which was conducted from the first principles by monte - carlo method , showed that in the perfect hcp @xmath1he crystal the supersolidity effects can not appear @xcite . the most authors connect such effects in solid @xmath1he at low temperatures with the disorder in helium samples . possible kinds of the disorder may be the defects , grain boundaries @xcite , glass phase , or liquid inclusions @xcite . also , the possible interpretation @xcite of the experiments on flow the superfluid helium through the solid helium @xcite show the essential role of the liquid channels , which may exist in the solid helium up to the ultralow temperatures . in this connection , the experiments which allow to identify the kind of the disorder , for example , in rapidly grown helium crystals , interesting . these data can be obtained by nuclear magnetic resonance ( nmr ) . whereas for its realization the nuclei of @xmath0he are necessary , we deal hereafter with the samples of not pure @xmath1he but with dilute @xmath0he-@xmath1he mixture . since nmr technique allows to measure diffusion coefficient in different coexisting phases and difference of diffusion coefficients in liquid and solid helium are several orders of the magnitude then such an experiment may answer the question whether liquid inclusions are formed in solid helium under very rapid crystal growing . the aim of present work is to elucidate this problem . we detect , by nmr technique , the presence of liquid phase in solid helium samples grown in different conditions and also establish the influence of annealing effect on character of diffusion processes . the crystals were grown by the capillary blocking method from initial helium gas mixture with a 1% of @xmath0he concentration . the copper cell of cylindrical form with inner diameter of 8 mm and length of 18 mm has the nmr coil glued to the inner surface of the cell . the pressure and temperature variations of the sample in the cell were controlled by two capacitive pressure gauges fixed to the both cylinder ends and by two resistance thermometers attached to the cold finger of the cell with sensitivities about 1 mbar and 1 mk , respectively . two series of crystals under the pressure above 33 bar were studied . the first one ( `` low quality crystals '' ) was prepared by quick step - wise cooling from the melting curve down to the lowest temperature ( 1.27 k ) without any special thermal treatment . to improve the crystal quality of the second series ( `` high quality crystals '' ) a special three - stage thermal treatment was used : annealing at the melting curve , thermocycling in single phase regions and annealing in the hcp single phase region near the melting curve @xcite . the criterions of crystal quality are , first , constancy of the pressure with time under constant temperature which is closed to melting and , second , reaching the pressure minimum under thermal cycling . the spin diffusion coefficient was determined with the help of the pulsed nmr technique at a frequency of @xmath2 mhz . the carr - purcell ( @xmath3 ) spin - echo method @xcite was used with a 90@xmath4-@xmath5 - 180@xmath4 sequence of probe pulses as well as the method of stimulated echo ( @xmath6 ) with the sequence of three probes pulses 90@xmath4-@xmath7 - 90@xmath4-@xmath8 - 90@xmath4 were applied to the nuclear system of the sample . generally , if a few phases do coexist in the sample , the echo amplitude @xmath9 for @xmath3 is given by @xmath10 and for @xmath6 @xmath11 \label{2}\ ] ] where @xmath12 is the maximal amplitude of a echo amplitude at @xmath13 , @xmath14 is the magnetic field gradient , @xmath15 is a gyromagnetic ratio , index @xmath16 numerates coexisting phases with the diffusion coefficients @xmath17 , @xmath18 is the relative content of the @xmath16-th phase in the sample . one can choose duration parameters @xmath5 , @xmath7 , and @xmath8 in order to get the strongest @xmath19 dependence and to single out @xmath17 fitting parameter . it should be emphasized that spin - diffusion coefficient @xmath20 measurement was just the method to identify a thermodynamical phases by their typical @xmath20 value . neither contribution of @xmath0he atoms in a phase transition processes nor even the dynamics of different phase s ratio could be tracking because of too long spin - lattice relaxation times . the typical results of nmr measurements for diffusion coefficients in two - phase sample on the melting curve are presented in fig . [ fig_mc ] in @xmath19 scale . there are two slopes for the data obtained which correspond to two different diffusion coefficients . experimental data analysis according to eq . ( [ 1 ] ) gives for curve piece with sharp slope @xmath21 @xmath22/s which corresponds to diffusion in liquid phase @xcite and for curve piece with mildly slope @xmath23 @xmath22/s which corresponds to diffusion in hcp phase @xcite . the phase ratio is @xmath24 . then this sample was rapidly cooled down to 1.3 k in the hcp region . the results of nmr measurements are shown in fig . [ fig_quenched ] . the presence of significant contribution ( @xmath25 ) of phase with fast diffusion coefficient ( @xmath26 @xmath22/s ) was unexpected . this fact can be interpreted as existence of liquid - like inclusions in hcp matrix which were apparently quenched from the melting curve . such a situation was visually observed in pure @xmath1he in refs . [ 1,4,15,16].the liquid droplets formation was also observed by nmr technique in 1% @xmath0he-@xmath1he mixture under bcc and hcp phases coexistence @xcite . note that this effect was observed in all three low - quality samples studied . after that this crystal was heated up to melting curve and , after annealing procedure described above ( sec . [ method ] ) , to avoid a thermal shock , was slowly cooled down to 1.3 k ( the hcp region ) . the results are presented in fig . [ fig_good ] . both the absence of visible @xmath19 functional dependence ( see eq . ( [ 1 ] ) ) which should be characteristic feature for @xmath27 @xmath22/s under @xmath28 ms at @xmath29 gs/@xmath22 and the position ( 0 ; 0 ) of the intersection point of @xmath3 and @xmath6 data curves are the evidences of the liquid - like diffusion absence in the crystal . it also should be noted that monotonous pressure decrease was observed in low - quality samples with fast diffusion coefficient . the typical pressure relaxation times were about @xmath30 hour . after annealing of such samples along with fast diffusion process disappearing , monotonous pressure decreasing was also stopped . this relaxation indirectly confirms our speculation about liquid - like inclusions quenched from the melting curve in the samples without any annealing . detailed study of pressure relaxation in quenched samples is projected . it is shown that under rapidly cooling from the melting curve ( without annealing ) solid helium samples contain liquid - like inclusions identified by additional fast diffusion decay of echo - signal . subsequent annealing of these samples leads to fast diffusion disappearing which is connected with crystallization of liquid - like inclusions . coming out of these defects is accompanied by pressure relaxation in the system . we thank b.cowan for useful consultations and for applying of his nmr spectrometer . this work has also been partially supported by grant stcu # 3718 , program of cooperation in research and education in science and technology for the 2008 ukrainian junior scientist research collaboration , and the ministry of education and science of ukraine ( project m/386 - 2009 ) . | the study of phase structure of dilute @xmath0he - @xmath1he solid mixture of different quality is performed by spin echo nmr technique .
the diffusion coefficient is determined for each coexistent phase .
two diffusion processes are observed in rapidly quenched ( non - equilibrium ) hcp samples : the first process has a diffusion coefficient corresponding to hcp phase , the second one has huge diffusion coefficient corresponding to liquid phase .
that is evidence of liquid - like inclusions formation during fast crystal growing .
it is established that these inclusions disappear in equilibrium crystals after careful annealing .
pacs numbers : 61.72.cc , 66.30.ma , 61.50.-f , 64.70.d- keywords : nmr , @xmath0he-@xmath1he solid mixture , diffusion , defects * * + _ ye.o .
vekhov , a.p .
birchenko , n.p .
mikhin , and e.ya .
rudavskii _ + _ _ |
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the wheeler - feynman@xciteelectrodynamics developed from the schwarzschild - tetrode - fokker@xcite direct - interaction functional . equations of motion are derived from hamilton s principle for the action integral @xmath5 where the four - vector @xmath6 represents the four - position of particle @xmath7 parametrized by arc - length @xmath8 , double bars indicate quadri - vector modulus @xmath9 and the dot indicates the usual minkowski relativistic scalar product of four - vectors . ( integration is to be carried over the hole particle trajectories , at least formally ) . the above action integral describes an interaction at the advanced and retarded light - cones with an electromagnetic potential given by half the sum of the advanced and retarded linard - wierchert potentials @xcite . wheeler and feynman showed that electromagnetic phenomena can be described by this direct action - at - a - distance theory in complete agreement with maxwell s theory as far as the classical experimental consequences@xcite . this direct - interaction formulation of electrodynamics was developed to avoid the complications of divergent self - interaction , as there is no self - interaction in this theory , and also to eliminate the infinite number of field degrees of freedom of maxwell s theory @xcite . it was a great inspiration of wheeler and feynman in 1945 , that followed a lead of tetrode @xcite and showed that with the extra hypothesis that the electron interacts with a completely absorbing universe , the advanced response of this universe to the electron s retarded field arrives _ at the present time of the electron _ and is equivalent to the local instantaneous self - interaction of the lorentz - dirac theory@xcite . the action - at - a - distance theory is also symmetric under time reversal , as the fokker action includes both advanced and retarded interactions . dissipation in this time - reversible theory becomes a matter of statistical mechanics of absorption@xcite . the area of wheeler - feynman electrodynamics has been progressing slowly but steadily since 1945 : quantization was achieved by use of the feynman path integral technique and the effect of spontaneous emission was successfully described in terms of interaction with the future absorber , in agreement with quantum electrodynamics@xcite . it was also shown that it is possible to avoid the usual divergencies associated with quantum electrodynamics by use of proper cosmological boundary conditions@xcite . as far as understanding of the dynamics governed by the equations of motion , the state of the art is as follows : the exact circular orbit solution to the attractive two - body problem was proposed in 1946@xcite and rediscovered by schild in 1962@xcite . the 1-dimensional symmetric two - electron scattering is a special case where the equations of motion simplify a lot and it has been studied by many authors , both analytically and numerically @xcite . in this very special case the initial value functional problem surprisingly requires much less than an arbitrary initial function to determine a solution manifold with the extra condition of bounded manifold for all times . it was shown that the solution is uniquely determined by the interelectronic distance at the turning point if this distance is large enough ( this minimum distance curiously evaluates to 0.49 bohr radii by the action - at - a - distance theory@xcite , much larger than about one classical electronic radius that one would naively guess ) . as a result of this theorem , there is a single continuous parameter ( the positive energy ) describing the unique non - runaway symmetric orbit at that given positive energy . the noether s four - constant of motion derived from the fokker lagrangian involves an integral over the past history@xcite . for example in the case of a hydrogen atom this four - momentum constant evaluates to @xcite @xmath10 where @xmath11 represents the derivative of the delta function@xcite . notice that because of this delta function , only finite portions of the trajectory are involved : actually an extent of length @xmath12 approximately . this non - local constant will behave very differently from the local coulombian energy , that is known to confine orbits of a negative energy within a maximum separation distance . in the case where the particles acquire a large separation ( unbound state ) , the hole past history is involved ( @xmath13 ) in the determination of the non - local energy constant . as regards the mathematical structure of the equations of motion , for the case of a two - electron atom the acceleration of electron 1 is given by@xcite @xmath14 where @xmath15 and @xmath16 are the electronic charge and mass , @xmath17 @xmath18and @xmath19 and @xmath20 are the total electric and magnetic fields produced by electron 2 and the nucleus . in the action - at - a - distance theory these fields are given by the average of the retarded and advanced linard - wiechert fields , calculated with the instantaneous position of the stationary nucleus and the retarded and advanced positions of electron 2 at the times @xmath21 , which is defined by the implicit condition @xmath22 where the minus and plus signs are the conditions for the retarded and advanced times respectively . the partial electric fields of electron 2 acting on electron 1 at time @xmath23are @xcite @xmath24 , \end{aligned}\ ] ] where @xmath25 , @xmath26 and @xmath27 is the speed of light . the advanced field @xmath28 is obtained from the above expression by replacing @xmath29 by @xmath30 and @xmath27 by @xmath31 the partial magnetic fields of electron 2 are @xmath32 where the @xmath33 is to ensure an outgoing poynting vector @xmath34 for the retarded fields and an incoming poynting vector for the advanced fields . the total electric field in equation ( [ motion1 ] ) must include also the instantaneous coulomb electric field of the stationary nucleus . equation ( [ motion1 ] ) can suggest a paradox about causality , as the force depends on the future of particle 2 . in the following , and to finish this introduction , we show that equation ( [ motion1 ] ) , when written properly , becomes a functional differential equation with _ delayed argument only _ , as first observed in@xcite . to outline the essentials of the explanation , let us first ignore the field of the nucleus and take the nonrelativistic limit of ( [ motion1 ] ) ( @xmath35 ) in this approximation the electric field @xmath19 entering in equation ( [ motion1 ] ) evaluates to @xmath36 . then we note that one can use equation ( [ motion1 ] ) as an equation of motion for _ particle 2 , _ by solving the rearranged form of ( [ motion1 ] ) , @xmath37 for the most advanced acceleration of particle 2 , @xmath38 in the above form it is clear that the right hand side involves only functions evaluated at times prior to the most advanced time , defined by @xmath39 , and no further advanced information is necessary , eliminating the ghost of dependence on the future . in the same way , the causal equation of motion of particle 1 is to be produced from the equation for particle 2 by solving for the most advanced acceleration of particle 1 . for the special case of 1-dimensional motion of two electrons , @xmath40 depends only on the advanced velocity , and ( [ causal ] ) can easily be solved for this advanced velocity as a function of the past history . in the 3-dimensional case there is an extra complexity , as the acceleration appears in the linard - wiechert partial field @xmath28 in the form @xmath41 . the bad news is that the component of the acceleration along the advanced normal can not be solved for from the value of the double - vector - product only . because of this degeneracy , equation ( [ causal ] ) is an algebraic - differential equation , and the null direction of the left hand side of ( [ causal ] ) is a constraint to be satisfied by the right hand side ( the scalar product with @xmath42 must vanish ) . the numerically correct way to integrate this type of equation is by use of the modern integrators for algebraic - differential equations like dassl @xcite adapted for retarded equations ( which has never been done yet ) or by dealing directly with the algebraic constraint @xcite . according to the standard classification of g. a. kamenskii@xcite , equation ( [ causal ] ) belongs to the class of differential - difference equations of neutral type . even though more complex , the motion is still causally determined by the past trajectory , as we wanted to demonstrate , the price being an algebraic neutral delay equation . as far as initial conditions go , the general theory on delay equations @xcite tells us that we need to provide an initial @xmath43 function describing the position of particle 2 from @xmath44 up to the initial instant @xmath45 . the information on particle 1 needed is also to be provided over twice the retardation lag seen by particle 1 . this is a short piece of trajectory for bound nonrelativistic atomic orbits , but for a ionized state or a runaway orbit this can be the whole past history ! unless further simplifications or conditions are added , this is the generic problem at hand . the 3-dimensional cases of atomic interest ( e.g. helium ) have never been studied , and they are more complex than the 1-d scattering because one can have negative energy bound states for example . most relevant for physics is the question of the conditions for the existence of a bounded manifold solution , which still needs to be understood in the general case ( it would be very curious if they turned out to be a discrete set of negative energies ) . the only existing analytical result in the 3-dimensional case is the linear stability of the schonberg - schild circular orbits@xcite , resulting in an infinite number of unstable solutions to the characteristic equation . the numerical treatment ofthe exact neutral equations displays instabilities and is generally difficult . in the following we resort to the darwin approximation not as much as a mathematical approximation to the action - at - a - distance electrodynamics , but as a physical approximation of lorentz - invariant dynamics in the atomic ( shallow ) energy range . to introduce our numerical calculations , we start from the scale - invariant coulomb limit of the tetrode - fokker - wheeler - feynman interaction : let @xmath15 and @xmath16 be the electronic charge and mass respectively and @xmath46 the nuclear charge of our two - electron atom , which in this work is assumed to have an infinite mass . all our numerical work uses a scaling which exploits the scale invariance of the coulomb dynamics : given a negative energy , there is a unique circular orbit at that energy with frequency @xmath47 and radius@xmath48 related by @xmath49 . we scale distance , momentum , time and energy as @xmath50 , @xmath51 , @xmath52 and @xmath53 , respectively . in these scaled units , the coulomb dynamics of the two - electron atom is described by the scaled hamiltonian @xmath54 where @xmath55(single bars represent euclidean modulus ) and @xmath56 . for a generic non - circular orbit , @xmath57plays the role of a scale parameter , and we recover the value of the energy in ergs through @xmath58 . notice that @xmath57 does not appear in the scaled hamiltonian , which is the scale invariance property . from the scaled frequency @xmath59 and scaled angular momentum @xmath60 we can recover the actual values in cgs units by the formulas @xmath61 the only other analytic constant of the coulomb dynamics , besides the energy ( [ hamitwo ] ) is the total angular momentum , and this dynamics in chaotic and displays arnold diffusion , as proved in @xcite for a similar three - body system . the numerical calculations were performed using a 9th - order runge - kutta embedded integrator pair@xcite . we chose the embedded error per step to be @xmath62 , and after ten million time units of integration the percentage changes in energy and total angular momentum were less than @xmath63 . as a numerical precaution we performed the numerical calculations using the double kustanheimo coordinate transformation to regularize single collisions with the nucleus@xcite . as these alone are not enough for faithful integration , we checked that there was never a triple collision , as the minimum inter - electronic distance was about @xmath64 units while the minimum distance to the nucleus was @xmath65 units for all the orbits considered in this work . we also checked that along stable non - ionizing orbits we can integrate forward up to fifty thousand time units , reverse the integration , go backwards another fifty thousand units and recover the initial condition with a percentile error of @xmath66 . for longer times this precision of back and forth integration degenerates rapidly , which is due to the combined effect of numerical truncation and stochasticity . the question of how far in time the numerical trajectories approximate shadowing trajectories in the present system is far from trivial @xcite , but we assume it to be a time at least of the order of these one hundred thousand units . ( energy conservation of one part in a million is achieved for much longer times , even one billion time units ) . the study of orbits of a two - electron atom was greatly stimulated by the recent interest in semiclassical quantization , and these studies discovered two types of stable zero - angular - momentum periodic orbits for helium ( @xmath67 ) : the langmuir orbit and the frozen - planet orbit @xcite . a detailed study of the non - ionizing orbits of coulombian helium was initiated in reference @xcite for plane orbits , and we describe some of their results below . there are basically two types of non - ionizing orbits : symmetric if @xmath68 for all times and asymmetric if @xmath69 generically . symmetric orbits are produced by symmetric initial conditions like for example @xmath70 and @xmath71 or @xmath70 and @xmath72 with @xmath73 @xcite because ( [ hamitwo ] ) is symmetric under particle exchange , these orbits satisfy @xmath68 at all times , and therefore can not ionize if @xmath74 ( both electrons would have to ionize at the same time , which is impossible at negative energies ) . for example the double - elliptical orbits ( two equal ellipses symmetrically displaced along the x - axis ) discussed in @xcite are in this class . double - elliptical orbits are known to be unstable @xcite and we find that they ionize in about one hundred turns because of the numerical truncation error . most symmetric plane orbits are very unstable to asymmetric perturbations , with the exception of the langmuir orbit for a small range of @xmath75 values around @xmath67 @xcite@xmath76 the simplest way to produce an asymmetric non - ionizing plane orbit is from the initial condition @xmath77 , @xmath78 , @xmath79 , @xmath80 , as suggested in @xcite . in figure 1 we show the electronic trajectories for the first three hundred scaled time units along a two - dimensional non - ionizing orbit of @xmath81 with @xmath82 and @xmath83 in the above defined condition . we used a numerical refining procedure to finely adjust @xmath84as to maximize the non - ionizing time and this condition of figure 1 does not ionize for one million time units . the orbit survives that far only for a very sharp band of values of @xmath85 , other neighboring values producing quick ionization . this orbit was named double - ring torus in @xcite . the other possible type of non - ionizing orbit resulting from the above initial condition , depending on @xmath86 is what was named braiding torus in reference @xcite , with both electrons orbiting within the same region . a search over @xmath87 was conducted in @xcite , and it was found that most values of @xmath87 produce quick ionization except for a zero - measure set of @xmath87 values where braiding tori or double ring orbits are found . this suggests the general result that non - ionizing orbits are rare in phase space . to search for general tridimensional non - ionizing orbits in phase space , it is convenient to introduce canonical coordinates @xmath88 and @xmath89 @xmath90 initial conditions with @xmath91 @xmath92 describe double - elliptical orbits ( and circular as a special case ) . to generate an elliptical initial condition , we exploit the scale invariance and set the energy to minus one . it is easy to check that elliptical orbits of the hamiltonian ( [ hamitwo ] ) with an energy of minus one must have a total angular momentum of magnitude ranging from zero to two . to exploit the rotational invariance of ( [ hamitwo ] ) , we can choose the plane defined at @xmath91 @xmath92 by the angular momentum @xmath93 @xmath94 to be the @xmath95 plane . on this @xmath95 plane a single number @xmath96 ( the angular momentum ) , determines completely the elliptical orbit . the next step in producing a generic orbit is to add all possible perturbations along @xmath97 and @xmath98 to the chosen elliptical orbit . these are six directions and once we are looking for bound oscillatory orbits , we can choose @xmath99 , once @xmath100 has to cross the @xmath95 plane at some point . these are five numbers to vary and plus the angular momentum of the elliptical orbit it totals six parameters . our numerical search procedure consists in varying these six parameters over a fine grid , integrating every single initial condition until the distance from one electron to the nucleus is greater than twenty units , which is our ionization criterion . this criterion fails if the orbit has a very low angular momentum because these can go far away from the nucleus and come back , and therefore our search possibly misses low - angular - momentum non - ionizing orbits . as the majority of the initial conditions ionize very quickly , this search procedure is reasonably fast . we first perform a coarse search for ionization times above one thousand units and then refine in the neighborhood of each surviving condition to get conditions that do not ionize after one million time units . using the above numerical search procedure we found the tridimensional non - ionizing initial condition of figure 2 for helium , a tridimensional double - ring orbit generated by the initial condition @xmath101 which does not ionize before ten million turns . ( after the search and refinement , we scaled this orbit s energy to minus one , for later convenience ) . we also found the non - ionizing orbit orbit of figure 3 for h - minus ( @xmath102 ) , a tridimensional orbit generated by the condition @xmath103 which does not ionize before one million turns ( coulombian energy of this condition is also minus one ) . this last orbit is fragile and numerically harder to find : as the first electron has an orbit very close to the positive @xmath102 charge , there remains only a dipole field to bind the second electron . as the outer electron is much slower in the scaled units , we had to plot the first @xmath104 time units of evolution to display the generic features of the trajectory . non - ionizing orbits of @xmath105 are very rare in phase space , which is reminiscent of the quantum counterpart , as the @xmath106 ion is known to have only one quantum bound state at @xmath107 , very close to the ionization threshold @xmath108@xcite . one remarkable fact about these non - ionizing orbits is that they all have a very sharp fourier transform . this property makes them approximately quasi - periodic orbits . for example in figure 4 we plot the fast fourier transform of the orbit of figure 2 , performed using @xmath109 points . ( it seems that there are at least two basic frequencies in the resonance structure of figure 4 ) . even though these orbits look like quasi - periodic tori , there seems to be a thin stochastic tube surrounding each orbit , as evidenced by a small positive maximum lyapunov exponent . we calculated numerically this maximum lyapunov exponent by doubling the integration times up to @xmath110 and found that the exponent initially decreases but then saturates to a value of about @xmath111 for the orbits of figures 1 , 2 and 3 . the gravitational three - body problem has recently been proved to display arnold diffusion@xcite , and this numeriacally calculated positive lyapunov exponent suggests that the same is true for the two - electron coulombian atom . the numerical integrations in this section are performed using the darwin approximation . the darwin equations of motion are a @xmath112 perturbation of the coulomb dynamics , of size @xmath113 for atomic energies . in the scaled units of section ii the darwin hamiltonian is the following @xmath112 perturbation of hamiltonian ( [ hamitwo ] ) @xmath114 \nonumber \\ & & -\frac{\beta ^{2}}{8}[|\vec{p}_{1}|^{4}+|\vec{p}_{2}|^{4 } ] , \label{darwin}\end{aligned}\ ] ] where @xmath115 . the second line represents the biot - savart magnetic interaction plus the first relativistic correction to the static electric field and the last line describes the relativistic mass correction . notice that these are both proportional to the small parameter @xmath112 , which makes them a small scale - dependent perturbation on the scale invariant coulomb hamiltonian ( first line ) . it is possible to regularize the darwin equations with the same double - kustanheimo transformation@xcite , only that here one needs to define the regularized time using the higher powers @xmath116 , instead of the lower powers @xmath117 used to regularize the coulomb equations@xcite . the main question we address numerically in this section is the dependence of the stability of a non - ionizing orbit with the energy scale of the orbit . here we use the word stability to mean ionization - stability : we call an initial condition ionization - stable if any small perturbation of it produces another non - ionizing orbit . the scale - dependent darwin terms ( of size @xmath118 produce significant deviations from the coulomb dynamics only in a time - scale of order @xmath119 which we find numerically to be the typical time for a non - ionizing coulombian initial condition to ionize along the darwin vector field . this poses a numerical difficulty if @xmath57 is too small because one has to integrate the orbit for very long times to investigate the stability . it turns out that ionization - stable orbits can be found at larger values of @xmath57 for larger values of @xmath120 . here the dynamical stability mechanism is reminiscent of quantum atomic physics , where the values of @xmath57 vary with the nuclear charge as @xmath121 . large values of @xmath120 facilitate the numerical procedure and in the following we present the numerical investigation of the stability of non - ionizing orbits starting from the large @xmath120 case . let us start with the @xmath122 calcium ion two - electron system along the non - ionizing orbit of figure 1 by fixing @xmath82 and @xmath123 in the condition defined in section ii . to test the stability of the orbit at each value of @xmath57 we add a random perturbation of average size @xmath124 to the initial condition and integrate the darwin dynamics until either we find ionization or the time of integration is greater than @xmath125 time units we repeat this for at least twelve randomly chosen perturbations ( because of the twelve degrees of freedom ) and the minimum time to ionization is plotted in figure 5 as a function of @xmath57 . it can be seen that only for a narrow set of values around @xmath126 this minimum time to ionization was greater than @xmath127 . or the other values it decreases rapidly to a value of about @xmath128 . one could argue that for the other values of @xmath57 the non - ionizing initial condition has shifted away from the @xmath83 initial condition and this being the reason that our orbit ionized . to test this , we fixed @xmath57 at a bad value for example @xmath129 and varied the plane initial condition in the neighborhood of this condition of figure 1 . we found that the minimum time to ionization was always about @xmath128 ( also the maximum time before ionization was about @xmath128 ) . we also searched in a bigger neighborhood , of size proportional to @xmath130 . this suggests the interpretation that for the special resonant value of @xmath131 the net diffusive effect of the scale - dependent term vanishes , allowing a non - ionizing perturbed manifold . in order to have a direct interpretation ( in atomic units ) of the scale parameter @xmath130 , it is convenient to scale to minus one the energy of the initial condition of figure 1 ( by exployting the coulombian scale invariance ) . after this , the energy of the orbit in ergs evaluates to @xmath132 , and for @xmath131 this is approximately @xmath133 atomic units . the total angular momentum of this orbit is @xmath134 . this orbit s energy is above the ionization continuum of the ion , @xmath135 atomic units , but it is still in the quantum range . it serves nevertheless to demonstrate that this dynamical system might exhibit non - ionizing stable orbits only at very sharply defined energy values . for the orbits of figures 2 and 3 , the above procedure becomes prohibitively slow , as the value of @xmath130 are much smaller and one must integrate for very long times , much beyond the estimated shadowing time . to partially overcome this we used a larger amplitude random perturbation ( of average size @xmath136 ) , to produce faster ionization . the drawback with this is that the minimum ionization time does not show pronounced peaks , only the average ionization time still showing a signature of scale dependence . in figure 6 we show this average time for the orbit of figure 3 . this property of sharply defined energies can possibly be found for the lower - lying energies below the ionization threshold as well . these orbits would involve configurations where the electrons come very close to the nucleus and acquire a large velocity . even though our integrator is regularized , the correct physical electronic repulsion is greatly amplified when one electron has a relativistic velocity and the darwin approximation can not describe the physics then . actually , it is known that the darwin interaction can produce unphysical effects when pushed to relativistic energies@xcite . we therefore do not expect to find these low - lying atomic energy scales with the present darwin approximation and shall be contempt with these interesting result already . for the same reason given above , we do not study here the frozen - planet periodic orbit ( the two electrons performing one - dimensional periodic motion on the same side of the nucleus , with the inner electron rebounding from the origin , an artifact of regularization ) . the main problem being the failure of the darwin approximation , as the inner particle goes to the speed of light@xcite . the correct relativistic dynamics can actually produce a new _ physical _ inner turning point very close to the origin but not _ at _ the origin as the regularized motion , and we discuss elsewhere@xcite . last , we consider the non - ionizing symmetric periodic orbit called the langmuir orbit , where the two electrons perform symmetric bending motion shaped approximately like a semi - circle@xcite . for the coulomb two - electron atom with @xmath67 this orbit was found to have a zero maximum lyapunov exponent@xcite . the orbit is therefore neutrally stable , which is the best one can expect from a periodic orbit of a hamiltonian vector field . ( absolute stability violates the symplectic symmetry , which says that to every stability exponent @xmath137 one should have a @xmath138 exponent ) . it is a simple matter to obtain the langmuir - like orbit for the darwin hamiltonian at any given value of @xmath57 : all it takes is a little adjusting in the neighborhood of the coulombian langmuir condition . we attempted to investigate numerically any scale - dependent diffusion away from this darwin - langmuir condition for @xmath57 in the atomic range , but again the numerics is prohibitively slow at the time of writing this work . the simplified dynamical mechanism behind resonant non - ionization seems to go intuitively as follows : the peculiar scale - invariant coulomb dynamics determines the non - ionizing orbits within narrow stochastic tubes . the next step is the action of the small scale - dependent relativistic corrections that produce a slow diffusion of the orbit out of the thin tube in a time of the order of @xmath139 . after this , quick ionization follows . only at very special resonant values of @xmath57 the relativistic terms leave the orbit within the tube , a resonant effect that depends on @xmath140 , fixing the energy scale . in the literature , the escape to infinity from simpler to understand two - degree - of - freedom systems has been attributed to cantori , which , as is well known , can trap chaotic orbits near regular regions for extremely long times@xcite . in the present larger dimensional case it appears that resonances are also controlling the escape to infinity of one electron by the existence of extra resonant constants of motion@xcite . this seems to be in agreement with the numerical results of very sharp peaks for the minimum ionization time . we have tried to concentrate on the physics described by this combination of chaotic dynamics on a two - electron atom with inclusion of relativistic correction , while discussing this highly nontrivial result of nonlinear dynamics . in references @xcite we noticed that a simple resonant normal form criterion gives a surprisingly good prediction for the discrete atomic energy levels of helium . the resonant structure was calculated using the darwin interaction ( [ darwin ] ) , which is the low - velocity approximation to both maxwell s @xcite and wheeler - feynmans@xcite electrodynamics . as we saw in section ii , the coulombian non - ionizing orbits are far from circular , and these orbits would radiate even in dipole according to the time - irreversible maxwell s electrodynamics ( circular orbits radiate only in quadrupole but are linearly unstable ) . it becomes then clear that the heuristic results of @xcite can only have a physical meaning in the context of a time - reversible theory ( as the action - at - a - distance electrodynamics for example ) . the combination of chaotic dynamics with relativistic invariance has never been explored numerically , and most known lorentz - invariant dynamical systems are for one particle and possess trivially integrable dynamics . the situation gets unexpectedly much more complicated for more than one particle ( apart from the trivial non - interacting many - particle system ) : due to the famous no - interaction theorem@xcite , the relativistic description of two directly interacting particles is impossible within the hamiltonian formalism and its set of ten canonical generators for the poincare group @xcite . description of interacting particles is possible only in the context of constraint dynamics , with eleven canonical generators and with the dirac bracket replacing the poisson bracket . for example the relativistic action - at - a - distance equations for two interacting electrons are non - local and possess only infinite - dimensional constrained hamiltonian representations@xcite . the interested reader should consult some recently found two - body direct - interaction relativistic lagrangian dynamical systems@xcite as well as the constraint - dynamics direct - interaction models recently used in chromodynamics and two - body dirac equations@xcite . the nonlinear dynamics of these models could display interesting and so far inexplored dynamical behaviour . it would be natural to wonder if one can find an analogous scale - dependent dynamics for a dynamical system describing the hydrogen atom , apparently the simplest example of lorentz - invariant two - body relativistic dynamics of atomic interest . it turns out that hydrogen is not simpler than helium at all , but it appears to us that there is an essential difference which has actually made the interesting dynamics of a two - electron atom amenable to study already within the darwin approximation : in a two - electron atom orbits with a negative energy can ionize , while in hydrogen this might be possible only if one includes all orders of the relativistic action - at - a - distance interaction . ( as we saw in section i , the noether s energy constant involves a segment of the past trajectory , and a negative value does not forbid ionization ) . ionization with a negative energy would be impossible for hydrogen within the darwin approximation ( unless the electron goes to the speed of light ) . this is indication that in hydrogen the essential physics described by the action - at - a - distance electrodynamics is of non - perturbative character . the paradoxical result of the infinite linear instability of circular orbits in atomic hydrogen@xcite is another warning of this non - perturbative dynamics . the author acknowledges the support of fapesp , proc . 96/06479 - 9 and cnpq , proc . 301243/94 - 8(nv ) . s. bleher , c. grebogi , e. ott and r. brown , _ phys . a _ , * 38 * , 930 ( 1988 ) , f. t. arecchi , r. badii and a. politi,_phys . rev . a_. * 32 * , 402 , ( 1985 ) , f. c. moon and g .- x.li , _ lett . _ * 55 * , 1439 ( 1985 ) , e. g. gwinn and r. m. westervelt , _ phys lett . _ * 54 * , 1613 ( 1985 ) , c. grebogi , s. mcdonald , e. ott and j. yorke , _ phys . _ * 99a * , 415 , ( 1983 ) . r. n. hill , _ relativistic action at a distance : classical and quantum aspects _ , proceedings , barcelona , spain 1981 , edited by j. llosa , lecture notes in physics * 162 * , 104 , ( springer , new york 1982 ) . k. richter and d. wintgen , _ j. phys . b * 23 * , _ l197 ( 1990 ) , k. richter and d. wintgen _ phys . lett . _ * 65 * , 1965 ( 1990 ) , k. richter , g. tanner and d. wintgen , _ phys . a _ * 48 * , 4182 ( 1993 ) , d. wintgen , a. brgers , k. richter and g. tanner,._progress of theoretical physics supplement * 116 * , 121 ( 1994 ) . _ d. g. currie , t. f. jordan and e. c. g. sudarshan , _ rev . of mod phys _ , * 35 * , 350 ( 1963 ) , also in _ the theory of action - at - a - distance in relativistic particle dynamics , _ edited by edward h. kerner , ( gordon and breach , new york 1972 ) . | we study numerically the dynamical system of a two - electron atom with the darwin interaction as a model to investigate scale - dependent effects of the relativistic action - at - a - distance electrodynamics .
this dynamical system consists of a small perturbation of the coulomb dynamics for energies in the atomic range .
the key properties of the coulomb dynamics are : ( i ) a peculiar mixed - type phase space with sparse families of stable non - ionizing orbits and ( ii ) scale - invariance symmetry , with all orbits defined by an arbitrary scale parameter .
the combination of this peculiar chaotic dynamics ( ( i ) and ( ii ) ) , with the scale - dependent relativistic corrections ( darwin interaction ) generates the phenomenon of scale - dependent stability : we find numerical evidence that stable non - ionizing orbits can exist only for a discrete set of resonant energies .
the fourier transform of these non - ionizing orbits is a set of sharp frequencies .
the energies and sharp frequencies of the non - ionizing orbits we study are in the quantum atomic range .
the coulomb dynamical system of the helium atom is a very peculiar chaotic system that exhibits arnold diffusion@xcite , and with a typical trajectory having an infinity of possible time - asymptotic final states .
for example , almost all negative - energy trajectories of coulombian helium display the generic phenomenon of ionization , namely , the ejection of one electron@xcite . several nonlinear dynamical systems share this property of having more than one time - asymptotic final state , with the respective basins for each outcome having a complicated structure in initial condition space@xcite . the numerical work on this paper
is based on stable coulombian orbits of a two - electron atom that do not ionize for several millions of turns of one electron around the nucleus .
it is a property of the coulomb dynamics of a two - electron atom that most initial conditions with a negative energy ionize very quickly in about 20 turns@xcite .
then there are the very special initial conditions that do not ionize due to a precise phase balance between the two electrons .
these rare non - ionizing orbits are defined very sharply in phase space and were first studied in reference @xcite for plane orbits . here
we also develop a numerical procedure to search for non - ionizing orbits among a large number of possible tridimensional initial conditions
. the coulomb hamiltonian exhibits the scale invariance degeneracy : if we scale time and space as @xmath0 , @xmath1 , for @xmath2 , the equations of motion are left invariant . for this reason ,
the behavior of the coulomb dynamics is the same in all scales , a degeneracy which is broken by the relativistic effects of electrodynamics .
the phenomenon of breaking the scale invariance in electrodynamics was explored analytically in @xcite for the darwin interaction , which is the low - velocity approximation to the wheeler - feynman action - at - a - distance electrodynamics @xcite .
it was found in @xcite that a simple resonant normal form approximation theory predicts a discrete set of quantized scales very close to the quantum atomic energies . using these preliminary findings as guide
, we present a numerical investigation of the stability of non - ionizing orbits for the darwin dynamics and its dependence on the energy scale .
it turns out that for energies of atomic interest , the darwin equations of motion approximate the coulomb equations plus a perturbation of size @xmath3 with @xmath4 .
therefore , non - ionizing stable orbits of the darwin dynamics should exist in the neighborhood of non - ionizing stable coulombian orbits if the perturbation does not force ionization . for these ,
our numerical results with the darwin dynamics indicate that the non - ionizing property plus stabilty require sharply defined discrete energies .
the darwin interaction is not exactly a lorentz invariant interaction@xcite , so we study it as an approximation to the relativistic action - at - a - distance electrodynamics , for the sake of including the present approach into an underlying physical theory .
maxwell s theory would seem to be the natural candidate for the comprehensive physical theory , but it lacks time - reversibility and dipolar dissipation would forbid the orbits studied in this paper .
there is also the choice of other more recent lorentz - invariant lagrangian@xcite systems and constrained hamiltonian dynamical systems@xcite , whose exact forms are actually more amenable to numerical treatment than the wheeler - feynman electrodynamics , but we shall not consider them here . the interested reader should consult reference @xcite , where a covariant approximation to wheeler - feynman electrodynamics is attempted by the two - body todorov equation of constraint dynamics .
this paper is organized as follows : in section i we review the state of the art of the time - reversible action - at - a - distance electrodynamics , and if the reader wants to skip this part the rest of the paper makes full sense as a nonlinear dynamics study , except for the discussion at the end . in section
ii we describe the numerical calculations with the coulomb limit of the darwin interaction , find some non - ionizing orbits and their fourier transforms . in section iii
we include the scale dependent darwin terms and investigate the possibility of stable non - ionizing orbits . in section
iv we put the conclusions and discussion . |
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rrrrrrr & & & & + & & & & & & + 104 & 0.05 & 0.06 & -0.71 & -0.70 & -0.78 & 14.05 @xmath12 0.05 + 288 & 0.03 & 0.04 & -1.40 & -1.07 & -1.14 & 15.40 @xmath12 0.05 + 362 & 0.05 & 0.06 & -1.33 & -1.15 & -1.09 & 15.51 @xmath12 0.05 + 1261 & 0.01 & 0.01 & -1.32 & & -1.08 & 16.68 @xmath12 0.05 + 1851 & 0.02 & 0.03 & -1.23 & & -1.03 & 16.18 @xmath12 0.05 + 1904 & 0.01 & 0.01 & -1.67 & -1.37 & -1.37 & 16.15 @xmath12 0.05 + 3201 & 0.21 & 0.27 & -1.53 & -1.23 & -1.24 & 14.75 @xmath12 0.05 + 4590 & 0.04 & 0.05 & -2.11 & -1.99 & -2.00 & 15.75 @xmath12 0.10 + 4833 & 0.33 & 0.42 & -1.92 & -1.58 & -1.71 & 15.70 @xmath12 0.10 + 5272 & 0.01 & 0.01 & -1.66 & & -1.33 & 15.58 @xmath12 0.05 + 5466 & 0.00 & 0.00 & -2.22 & & -2.13 & 16.60 @xmath12 0.05 + 5897 & 0.08 & 0.10 & -1.93 & -1.59 & -1.73 & 16.30 @xmath12 0.10 + 5904 & 0.03 & 0.04 & -1.38 & -1.11 & -1.12 & 15.00 @xmath12 0.05 + 6093 & 0.18 & 0.23 & -1.75 & & -1.47 & 16.25 @xmath12 0.05 + 6171 & 0.33 & 0.42 & -1.09 & & -0.95 & 15.65 @xmath12 0.05 + 6205 & 0.02 & 0.03 & -1.63 & -1.39 & -1.33 & 14.95 @xmath12 0.10 + 6218 & 0.19 & 0.24 & -1.40 & & -1.14 & 14.70 @xmath12 0.10 + 6254 & 0.28 & 0.36 & -1.55 & -1.41 & -1.25 & 15.05 @xmath12 0.10 + 6341 & 0.02 & 0.03 & -2.24 & & -2.10 & 15.20 @xmath12 0.10 + 6352 & 0.21 & 0.27 & -0.50 & -0.64 & -0.70 & 15.25 @xmath12 0.05 + 6362 & 0.09 & 0.12 & -1.18 & -0.96 & -0.99 & 15.35 @xmath12 0.05 + 6397 & 0.18 & 0.23 & -1.94 & -1.82 & -1.76 & 12.95 @xmath12 0.10 + 6541 & 0.12 & 0.15 & -1.79 & & -1.53 & 15.40 @xmath12 0.10 + 6637 & 0.17 & 0.22 & -0.72 & & -0.78 & 15.95 @xmath12 0.05 + 6656 & 0.34 & 0.44 & -1.75 & & -1.41 & 14.25 @xmath12 0.10 + 6681 & 0.07 & 0.09 & -1.64 & & -1.35 & 15.70 @xmath12 0.05 + 6723 & 0.05 & 0.06 & -1.12 & & -0.96 & 15.45 @xmath12 0.05 + 6752 & 0.04 & 0.05 & -1.54 & -1.42 & -1.24 & 13.80 @xmath12 0.10 + 6779 & 0.20 & 0.26 & -1.94 & & -1.61 & 16.30 @xmath12 0.05 + 6809 & 0.07 & 0.09 & -1.80 & & -1.54 & 14.45 @xmath12 0.10 + 7078 & 0.09 & 0.12 & -2.13 & -2.12 & -2.02 & 15.90 @xmath12 0.05 + in very recent times , new determinations of galactic globular cluster ( ggc ) metallicities have provided us with new homogeneous @xmath13\textrm { } $ ] scales . in particular , carretta & gratton ( @xcite ; cg ) obtained metallicities from high resolution spectroscopy for 24 ggcs , with an internal uncertainty of 0.06 dex . for an even larger sample of 71 ggcs , metallicities have been obtained by rutledge et al . ( @xcite ; rhs97 ) based on spectroscopy of the caii infrared triplet . the equivalent widths of the caii triplet have been calibrated by rhs97 on both the cg scale and the older zinn & west ( @xcite ; zw ) scale . the compilation by rhs97 is by far the most homogeneous one which is currently available . in the same period , we have been building the largest homogeneous @xmath0 photometric sample of galactic globular clusters ( ggc ) based on ccd imaging carried out both with northern ( isaac newton group , ing ) and southern ( eso ) telescopes ( rosenberg et al . @xcite , @xcite ) . the main purpose of the project is to establish the relative age ranking of the clusters , based on the methods outlined in saviane et al . ( @xcite , @xcite ; srp97 , srp99 ) and buonanno et al . ( @xcite ; b98 ) . the results of this investigation are presented in rosenberg et al . ( @xcite ; rspa99 ) . here suffice it to say that for a set of clusters we obtained @xmath14 vs. @xmath15 color - magnitude diagrams ( cmd ) , which cover a magnitude range that goes from a few mags below the turnoff ( to ) up to the tip of the red giant branch ( rgb ) . at this point both a spectroscopic and photometric homogeneous databases are available : the purpose of this study is to exploit them to perform a thorough analysis of the morphology of the rgb as a function of the cluster s metallicity . as a first step , we want to obtain a new improved calibration of a few classical photometric metallicity indices . secondly , we want to provide to the community a self - consistent , * analytic , * family of giant branches , which can be used in the analysis of old stellar populations in external galaxies . photometric indices have been widely used in the past to estimate the mean metallicities of those stellar systems where direct determinations of their metal content are not feasible . in particular , they are used to obtain @xmath13\textrm { } $ ] values for the farthest globulars and for those resolved galaxies of the local group where a significant pop ii is present ( e.g. the dwarf spheroidal galaxies ) . the calibration of @xmath0 indices is particularly important , since with comparable exposure times , deeper and more accurate photometry can be obtained for the cool , low - mass stars in these broad bands than in @xmath16 . moreover , our huge cmd database allows a test of the new cg scale on a large basis : we are able to compare the relations obtained for both the old zw and new scale , and check which one allows to rank ggcs in the most accurate way . indeed , the most recent calibration of the @xmath0 indices ( carretta & bragaglia @xcite ) is based on just 8 clusters . a reliable metallicity ranking of ggc giant branches also allows studies that go beyond a simple determination of the _ mean _ metallicity of a stellar population . as an illustration , we may recall the recent investigation of the halo metallicity distribution function ( mdf ) of ngc 5128 ( harris et al . @xcite ) , which was based on the fiducial gc lines obtained by da costa & armandroff ( @xcite , hereafter da90 ) . these studies can be made more straightforward by providing a suitable analytic representation of the rgb family of ggcs . indeed , assuming that most of the ggcs share a common age ( e.g. rosenberg et al . @xcite ) , one expects that there should exist a `` universal '' function of @xmath17\ } $ ] able to map any @xmath18 $ ] coordinate pair into the corresponding metallicity ( provided that an independent estimate of the distance and extinction of the star are available ) . we will show here that such relatively simple mono - parametric function can actually be obtained , and that this progress is made possible thanks to the homogeneity of both our data set and analysis . in order to enforce a proper use of our calibrations , we must clearly state that , in principle , the present relations are valid only for rigorously old stellar populations ( i.e. for stars as old as the bulk of galactic globulars ) . at fixed abundance , giant branches are somewhat bluer for younger ages ( e.g. bertelli et al . moreover , in real stellar systems agb stars are also present on the blue side of the rgb ( cf . [ f : pars - partb ] ) . both effects must be taken into account when dealing with lg galaxies , since they could lead to systematic effects in both the mean abundances and the abundance distributions ( e.g. saviane et al . @xcite ) . the observational sample , on which this investigation is based , is presented in sect . [ s : sample ] . [ s : indices ] is devoted to the set of indices which are to be calibrated . they are defined in sect [ s : defindices ] . the reliability of our sample is tested in sect . [ s : checks ] , where we demonstrate that our methodology produces a set of well - correlated indices . in sect . [ s : newda90 ] we show that , once a distance scale is assumed for the ggcs , our whole set of rgbs can be approximated by a _ single _ analytic function , which depends on the metallicity alone . this finding allows a new and easier way to determine the distances and mean metallicities of the galaxies of the local group , extending the methods of da costa & armandroff ( @xcite ) , and lee et al . ( @xcite ) . the metallicity indices are calibrated in sect . [ s : calibrations ] , where analytic relations are provided both for the zw and for the cg scales . using these indices , we are able to test our analytic rgb family in sect [ s : testfits ] . our conclusions are in sect . [ s : conclusioni ] . clusters have been observed with the eso / dutch 0.9 m telescope at la silla , and at the rgo / jkt 1 m telescope in la palma . this database comprises @xmath19 of the ggc whose distance modulus is @xmath20 . the zero - point uncertainties of our calibrations are @xmath21 mag for each band . three clusters were observed both with the southern and the northern telescopes , thus providing a consistency check of the calibrations : no systematic differences were found , at the level of accuracy of the zero - points . a detailed description of the observations and reduction procedures will be given in forthcoming papers ( rosenberg et al . @xcite , @xcite ) presenting the single clusters . a subsample of this database was used for the present investigation . we retained those clusters whose cmd satisfied a few criteria : ( a ) the hb level could be well determined ; ( b ) the rgb was not heavily contaminated by foreground / background contamination ; and ( c ) the rgb was well defined up to the tip . this subsample largely overlaps that used for the age investigation , but a few clusters whose to position could not be measured , are nevertheless useful for the metallicity indices definition . conversely , in a few cases the lower rgb could be used for the color measurements , while the upper branch was too scarcely defined for a reliable definition of the fiducial line . two of the cmds that were used are shown in figs . [ f : pars - partea ] ( ngc 1851 ) and [ f : pars - partb ] ( ngc 104 ) , and they illustrate the good quality of the data . the dataset of clusters used in this paper is listed in table [ t : the - sample ] . from left to right , the columns contain the ngc number , the reddening both in @xmath22 and @xmath15 , the metallicity according to three different scales , and the apparent magnitude of the horizontal branch ( hb ) . the @xmath23 values were taken from the harris ( @xcite ) on - line table . the @xmath15 reddenings were obtained by assuming that @xmath24 ( dean et al . the values of the metallicity were taken from rhs97 : they represent the equivalent widths of the caii infrared triplet , calibrated either onto the zinn & west ( @xcite ) scale ( zw column ) or the carretta & gratton ( @xcite ) scale ( rhs97 column ) . moreover , the original carretta & gratton metallicities ( cg column ) are also given for the clusters comprised in their sample . the hb level was found in different ways for clusters of different metallicity . for the the metal rich and metal intermediate clusters , a magnitude distribution of the hb stars was obtained , and the mode of the distribution was taken . where the hb was too scarcely populated , a horizontal line was fitted through the data . the blue tail of the metal poorest clusters does not reach the horizontal part of the branch : in that case , a fiducial hb was fitted to the tail , and the magnitude of the horizontal part was taken as the reference level . the fiducial branch was defined by taking a cluster having a bimodal hb color distribution ( ngc 1851 , cf . [ f : pars - partea ] ) and then extending its hb both to the red and to the blue by `` appending '' clusters being more and more metal rich and metal poor , respectively . the details of this procedure , as well as the errors associated to the @xmath25 in table [ t : the - sample ] , are discussed in rspa99 . for ngc 1851 , @xmath26 was adopted ( dashed line in fig . [ f : pars - partea ] ) , and this value is just @xmath27 mag brighter than the value found by walker ( @xcite ) and saviane et al . ( @xcite ) . based on this observational sample , a set of metallicity indices were measured on the rgbs of the clusters . in the next section , the indices are defined and the measurement procedures are described . consistency checks are also performed . the metallicity indices calibrated in this study are represented and defined in fig . [ f : pars - partea ] and fig . [ f : pars - partb ] . the figures represent the cmd of ngc 1851 and ngc 104 in different color - magnitude planes , and the crosses mark the position of the rgb points used in the measurement of the indices . the left panel of fig . [ f : pars - partea ] shows the apparent colors and magnitudes for ngc 1851 : the inclined line helps to identify the first index , @xmath1 . this was defined , in the @xmath28 plane , by hartwick ( @xcite ) as the slope of the line connecting two points on the rgb : the first one at the level of the hb , and the second one 2.5 mag brighter . we use the same definition for the @xmath4 plane here ; however , in order to be able to use our metal richest clusters , we redefined @xmath1 by measuring the second rgb point mag brighter than the hb . since @xmath1 is measured on the apparent cmd , it is independent both from the reddening and the distance modulus . the right panel of the same figure , shows the apparent @xmath14 magnitude vs. the de - reddened @xmath29 color . in this panel , four other indices are identified , i.e. @xmath30 , @xmath2 , @xmath6 , and @xmath3 . the first one is the rgb color at the level of the hb , and the other three measure the magnitude difference between the hb and the rgb at a fixed color @xmath31 , 1.2 and 1.4 mag . the former index was originally defined by sandage & smith ( @xcite ) and the latter one by sandage & wallerstein ( @xcite ) , in the @xmath32 plane . the other two indices , @xmath2 and @xmath6 , are introduced later to measure the metal richest gcs . these indices require an independent color excess determination . finally , fig . [ f : pars - partb ] shows the cmd of ngc 104 ( 47 tuc ) in the absolute @xmath33 plane : the adopted distance modulus , @xmath34 , was obtained by correcting the apparent luminosity of the hb according to lee et al . ( @xcite ; cf . [ s : calibrations ] ) . by comparison , harris catalog reports @xmath35 . two other indices are represented in the figure : @xmath7 and @xmath8 . they are defined as the rgb color at a fixed absolute @xmath36 magnitude of @xmath37 ( da costa & armandroff @xcite ) or @xmath38 ( lee et al . the latter index was also discussed by armandroff et al . ( @xcite ) , and a calibration formula was given in caldwell et al . ( @xcite ) . this is based on the da90 clusters plus m5 and ngc 362 from lloyd evans ( @xcite ) . since these two indices are defined on the bright part of the rgb , they can be measured even for the farthest objects of the local group ( lg ) . due to the fast luminosity evolution of the stars on the upper rgb , this part of the branch was typically under - sampled by the early small - size ccds , so no wide application of these indices has been made for galactic globulars . however , this is of no concern for galaxy - size stellar systems . it will be shown in sect . [ s : calibrations ] that good accuracies can be obtained even for gcs , provided that the analytic function of eq . ( [ e : iperbole ] ) is used . lrrrrrrr & & & & + & & & & & & & + 104 & 0.99 & 4.13 & 0.78 & 1.27 & 1.87 & 1.94 & 1.57 + 288 & 0.95 & 6.39 & 1.25 & 1.75 & 2.36 & 1.51 & 1.35 + 362 & 0.90 & 7.28 & 1.67 & 2.09 & 2.57 & 1.45 & 1.28 + 1261 & 0.91 & 7.77 & 1.62 & 2.13 & 2.73 & 1.39 & 1.25 + 1851 & 0.97 & 7.41 & 1.23 & 1.82 & 2.55 & 1.45 & 1.31 + 1904 & 0.94 & 8.56 & 1.58 & 2.14 & 2.83 & 1.35 & 1.24 + 3201 & 0.99 & 8.72 & 1.19 & 1.91 & 2.71 & 1.39 & 1.27 + 4590 & 0.91 & 9.98 & 1.90 & 2.52 & 3.25 & 1.24 & 1.16 + 4833 & 0.92 & 9.25 & 1.80 & 2.36 & 3.12 & 1.28 & 1.19 + 5272 & 0.91 & 7.60 & 1.66 & 2.13 & 2.81 & 1.36 & 1.24 + 5466 & 0.91 & 9.85 & 1.93 & 2.50 & 3.18 & 1.24 & 1.16 + 5897 & 0.97 & 8.73 & 1.34 & 2.00 & 2.79 & 1.35 & 1.25 + 5904 & 0.93 & 6.91 & 1.41 & 1.91 & 2.55 & 1.44 & 1.30 + 6093 & 0.93 & 8.02 & 1.58 & 2.12 & 2.91 & 1.34 & 1.24 + 6171 & 1.07 & 5.66 & 0.31 & 1.09 & 1.93 & 1.67 & 1.49 + 6205 & 0.89 & 7.70 & 1.75 & 2.20 & 2.75 & 1.37 & 1.23 + 6218 & 0.95 & 7.09 & 1.34 & 1.88 & 2.51 & 1.46 & 1.31 + 6254 & 0.90 & 8.25 & 1.75 & 2.29 & 3.17 & 1.30 & 1.21 + 6341 & 0.88 & 9.92 & 2.15 & 2.69 & 3.40 & 1.21 & 1.13 + 6352 & 1.12 & 3.11 & -0.16 & 0.52 & 1.30 & 1.99 & 1.75 + 6362 & 0.93 & 5.84 & 1.31 & 1.76 & 2.32 & 1.55 & 1.37 + 6397 & 0.89 & 9.45 & 1.98 & 2.49 & 3.12 & 1.26 & 1.16 + 6541 & 1.01 & 8.59 & 1.03 & 1.77 & 2.67 & 1.39 & 1.29 + 6637 & 0.96 & 4.39 & 0.96 & 1.41 & 1.97 & 1.82 & 1.53 + 6656 & 0.86 & 10.32 & 2.27 & 2.69 & 2.96 & 1.24 & 1.12 + 6681 & 0.95 & 7.54 & 1.35 & 1.92 & 2.76 & 1.37 & 1.27 + 6723 & 1.01 & 6.02 & 0.76 & 1.38 & 2.18 & 1.55 & 1.41 + 6752 & 0.99 & 7.16 & 1.08 & 1.69 & 2.46 & 1.45 & 1.33 + 6779 & 0.94 & 8.74 & 1.60 & 2.18 & 2.94 & 1.32 & 1.22 + 6809 & 0.93 & 9.38 & 1.72 & 2.29 & 2.87 & 1.32 & 1.20 + 7078 & 0.88 & 9.82 & 2.10 & 2.62 & 3.27 & 1.23 & 1.14 + colors and magnitudes were measured on a fiducial rgb , which has been found by least - square fitting an analytic function to the observed branch . after some experimenting , it was found that the best solution is to use the following relation : @xmath39 where @xmath40 and @xmath41 represent the color and the magnitude , respectively . one can see from figs . [ f : pars - partea ] and [ f : pars - partb ] that the function is indeed able to represent the giant branch over the typical metallicity range of globular clusters . moreover , it is shown in sect . [ s : newda90 ] that , when the cmds are corrected for distance and reddening , the four coefficients can be parametrized as a function of [ fe / h ] , so that one is able to reproduce the rgb of each cluster , using just one parameter : the metallicity . at any rate , the indices were measured on the original loci , so that an independent check of the goodness of the generalized hyperbolae can be made , by comparison of the measured vs. predicted indices . all the indices values that have been measured are reported in table [ t : the - indices ] . in this table , the cluster ngc number is given in column 1 ; the following columns list , from left to right , @xmath5 , @xmath1 , @xmath2 , @xmath6 , @xmath3 , and finally the rgb color measured at @xmath42 and @xmath43 . the lee et al . ( 1990 ) distance scale was used to compute the last two indices ( cf . [ s : calibrations ] ) . before discussing the indices as metallicity indicators , we checked their internal consistency . we will show in sect . [ s : calibrations ] that the index @xmath1 is the most accurate one , as expected , since it does not require reddening and distance corrections . the rest of the indices are therefore plotted vs. @xmath1 in figs . [ f : intcheck2 ] and [ f : intcheck1 ] , and we expect that most of the scatter will be in the vertical direction . second order polynomials were fitted to the distributions , and the _ rms _ of the fit was computed for each index . in order to intercompare the different indices , a relative uncertainty has been computed by dividing the _ rms _ by the central value of each parameter ( this value is identified by a dotted line in each figure ) . in this way , the scatter of the metal index @xmath44 is @xmath45 , 0.02 , 0.04 , 0.06 , 0.12 , and 0.26 , for the indices @xmath46 , @xmath8 , @xmath5 , @xmath3 , @xmath6 , and @xmath2 , respectively . these values confirm the visual impression of the figures , that @xmath47 and @xmath48 are the lowest dispersion indices , followed by @xmath5 and @xmath3 . the indices will be calibrated in terms of [ fe / h ] in sect . [ s : calibrations ] ; however , before moving to this section , we want to present a new way to provide `` standard '' ggc branches in the @xmath10 plane , along the lines of the classical da costa & armandroff ( @xcite ) study . using this family of rgb branches , we are able to make predictions on the trend of the already defined indices with metallicity ; these trends can thus be compared to the observed ones , and therefore provide a further test of the reliability of our rgb family ( cf . sect [ s : testfits ] ) . lcccccc cluster & @xmath49 & @xmath50 & @xmath51 & [ fe / h]@xmath52 & [ fe / h]@xmath53 & [ fe / h]@xmath54 + ngc 104 & 14.05 & 0.050 & 0.064 & @xmath55 & @xmath56 & @xmath57 + ngc 5904 & 15.00 & 0.023 & 0.029 & @xmath58 & @xmath59 & @xmath60 + ngc 288 & 15.40 & 0.036 & 0.046 & @xmath61 & @xmath62 & @xmath63 + ngc 6205 & 14.95 & 0.000 & 0.000 & @xmath64 & @xmath65 & @xmath66 + ngc 5272 & 15.58 & 0.002 & 0.003 & @xmath67 & @xmath65 & + ngc 6341 & 15.20 & 0.010 & 0.013 & @xmath68 & @xmath69 & + rrrrrrrrrrrr & & & & & + @xmath70 & @xmath71 & @xmath70 & @xmath71 & @xmath70 & @xmath71 & @xmath70 & @xmath71 & @xmath70 & @xmath71 & @xmath70 & @xmath71 + 13.782 & 0.978 & 15.359 & 0.914 & 15.492 & 0.852 & 14.725 & 0.926 & 14.645 & 0.867 & 15.060 & 0.852 + 13.604 & 0.994 & 15.107 & 0.939 & 15.151 & 0.874 & 14.457 & 0.942 & 14.322 & 0.890 & 14.720 & 0.872 + 13.443 & 1.008 & 14.849 & 0.960 & 14.789 & 0.892 & 14.221 & 0.961 & 14.033 & 0.909 & 14.395 & 0.894 + 13.317 & 1.021 & 14.593 & 0.984 & 14.597 & 0.910 & 14.040 & 0.978 & 13.788 & 0.929 & 14.079 & 0.916 + 13.075 & 1.045 & 14.342 & 0.999 & 14.359 & 0.929 & 13.878 & 0.994 & 13.595 & 0.944 & 13.789 & 0.937 + 12.862 & 1.070 & 14.109 & 1.018 & 14.143 & 0.955 & 13.700 & 1.009 & 13.381 & 0.966 & 13.533 & 0.953 + 12.619 & 1.101 & 13.881 & 1.036 & 13.796 & 0.990 & 13.456 & 1.032 & 13.170 & 0.984 & 13.303 & 0.974 + 12.346 & 1.136 & 13.649 & 1.062 & 13.517 & 1.021 & 13.190 & 1.061 & 12.984 & 1.005 & 13.082 & 0.994 + 12.035 & 1.185 & 13.376 & 1.090 & 13.265 & 1.046 & 12.916 & 1.091 & 12.832 & 1.019 & 12.850 & 1.020 + 11.761 & 1.231 & 13.058 & 1.132 & 13.005 & 1.076 & 12.655 & 1.122 & 12.631 & 1.045 & 12.611 & 1.039 + 11.461 & 1.281 & 12.766 & 1.173 & 12.759 & 1.110 & 12.419 & 1.154 & 12.363 & 1.077 & 12.351 & 1.067 + 11.101 & 1.362 & 12.534 & 1.210 & 12.519 & 1.148 & 12.231 & 1.183 & 12.118 & 1.111 & 12.075 & 1.102 + 10.696 & 1.459 & 12.380 & 1.233 & 12.302 & 1.187 & 12.073 & 1.212 & 11.945 & 1.138 & 11.771 & 1.148 + 10.330 & 1.600 & 12.163 & 1.268 & 12.109 & 1.227 & 11.868 & 1.254 & 11.844 & 1.156 & 11.492 & 1.195 + 10.062 & 1.720 & 11.928 & 1.317 & 11.878 & 1.275 & 11.615 & 1.305 & 11.707 & 1.178 & 11.284 & 1.233 + 9.877 & 1.856 & 11.617 & 1.411 & 11.741 & 1.310 & 11.335 & 1.371 & 11.571 & 1.204 & 11.154 & 1.265 + 9.706 & 2.019 & 11.427 & 1.483 & 11.575 & 1.344 & 11.116 & 1.422 & 11.395 & 1.252 & 11.008 & 1.295 + 9.602 & 2.148 & & & 11.494 & 1.377 & 10.902 & 1.489 & 11.141 & 1.312 & 10.854 & 1.320 + 9.524 & 2.315 & & & 11.330 & 1.406 & 10.652 & 1.585 & 10.870 & 1.376 & 10.709 & 1.351 + 9.573 & 2.576 & & & 11.240 & 1.447 & 10.457 & 1.680 & 10.643 & 1.444 & & + 9.619 & 2.768 & & & 11.112 & 1.488 & 10.343 & 1.742 & 10.552 & 1.492 & & + & & & & 11.078 & 1.528 & & & & & & + & & & & 11.047 & 1.546 & & & & & & + da costa & armandroff ( @xcite ) presented in tabular form the fiducial ggc branches of 6 globulars , covering the metallicity range @xmath72\leq -0.71 $ ] . the rgbs were corrected to the absolute @xmath10 plane using the apparent @xmath14 magnitude of the hb , and adopting the lee et al . ( @xcite ) theoretical hb luminosity . since the da90 study , these branches have been widely used for stellar population studies in the local group . based on these rgbs , in particular , a method to determine both the distance and mean metallicity of an old stellar population was presented by lee et al . ( @xcite ) . both da90 and lee et al . ( 1993 ) provided a relation between the metallicity [ fe / h ] and the color of the rgb at a fixed absolute @xmath36 magnitude ( @xmath42 and @xmath43 , respectively ) , and recently a new relation for @xmath73 has also been obtained by caldwell et al . ( @xcite ) . once the distance of the population is known ( e.g. via the luminosity of the rgb tip ) , then an estimate of its _ mean _ metallicity can be obtained using one of the calibrations . it is assumed that the age of the population is comparable to that of the ggcs , and that the age spread is negligible compared to the metallicity spread ( rspa99 ) . in such case , one expects that any rgb star s position in the absolute cmd is determined just by its metallicity , and that a better statistical determination of the population s metal content would be obtained by converting the color of _ each _ star into a [ fe / h ] value . with this idea in mind , in the following sections we will show that this is indeed possible , at least for the bright / most sensitive part of the giant branch . we found that a relatively simple _ continuous _ function can be defined in the @xmath9 $ ] space , and that this function can be used to transform the rgb from the @xmath10 plane to the @xmath74},m_{i } $ ] plane . in order to obtain this function , we first selected a subsample of clusters with suitable characteristics , so that a reference rgb grid can be constructed . the fiducial branches for each cluster were then determined in an objective way , and they were corrected to the absolute @xmath75 plane . in this plane , the analytic function was fitted to the rgb grid . these operations are described in the following sections . the clusters that were used for the definition of the fiducial rgbs are listed in table [ t : fiducialgc ] , in order of increasing metallicity . the table reports the cluster name , and some of the parameters listed in table [ t : the - sample ] are repeated here for ease of use . the values of the reddening were in some cases changed by a few thousandth magnitudes ( i.e. well within the typical uncertainties on @xmath23 ) , to obtain a sequence of fiducial lines that move from bluer to redder colors as [ fe / h ] increases , and again the corresponding @xmath76 values were obtained assuming that @xmath24 ( dean et al . @xcite ) . indeed , due to the homogeneity of our sample , we expect that if a monotonic color / metallicity sequence is not obtained , then only the uncertainties on the extinction values must be taken into account . in order to single out these clusters from the total sample , some key characteristics were taken into account . in particular , we considered clusters whose rgbs are all well - defined by a statistically significant number of stars ; they have low reddening values ( @xmath77 ) ; and they cover a metallicity range that includes most of our ggcs ( @xmath78\leq -0.7 $ ] on the zw scale ) . the da90 fiducial clusters were ngc 104 , ngc 1851 , ngc 6752 , ngc 6397 , ngc 7078 and ngc 7089 ( m2 ) . ngc 104 is the only cluster in common with the previous study , and m2 is not present in our dataset . the other objects have been excluded from our fiducial sample since they have too large reddening values ( @xmath79 for ngc 6397 and ngc 7078 ) , or their rgbs are too scarcely populated in our cmds ( ngc 1851 and ngc 6752 ) . nevertheless , the calibrations that we obtain for the @xmath7 and @xmath8 are in fairly good agreement with those obtained by da90 ( for the small discrepancies at the high metallicity end , cf . [ s : da90indices1 ] and [ s : da90indices2 ] ) , and in particular with the recent caldwell et al . ( @xcite ) calibration for the @xmath73 index . the ridge lines of our fiducial rgbs were defined according to the following procedure . the rgb region was selected from the calibrated photometry , by excluding both hb and agb stars . all stars bluer than the color of the rr lyr gap were removed ; agb stars were also removed by tracing a reference straight line in the cmd , and by excluding all stars blue - side of this line . this operation was carried out in the @xmath80 plane , where the rgb curvature is less pronounced , and a straight line turns out to be adequate . the fiducial loci were then extracted from the selected rgb samples . the @xmath15 and @xmath36 vectors were sorted in magnitude , and bins were created containing a given number of stars . within each bin , the median color of the stars and the mean magnitude were used as estimators of the bin central color and brightness . the number of stars within the bins was exponentially increased going from brighter to fainter magnitudes . in this way , ( a ) one can use a small number of stars for the upper rgb , so that the color of the bin is not affected by the rgb slope , and ( b ) it is possible to take advantage of the better statistics of the rgb base . finally , the brightest two stars of the rgb were not binned , and were left as representatives of the top branch . after some experimenting , we found that a good rgb sampling can be obtained by taking for each bin a number of stars which is proportional to @xmath81 , where @xmath44 is an integer number . the resulting fiducial vectors were smoothed using an average filter with a box size of 3 . the rgb regions of the 6 clusters are shown in fig . [ f : zooms ] , together with the fiducial lines : it can be seen that in all cases the agbs are easily disentangled from the rgbs . the values of the fiducial points corresponding to the solid lines in fig . [ f : zooms ] , are listed in table [ t : fidtable ] . the fiducial branches defined in sect . [ s : fiducials ] were fitted with a parametrized family of hyperbolae . first , the rgbs were moved into the absolute @xmath10 plane . the distance modulus was computed from the apparent magnitude of the hb ( cf . table [ t : fiducialgc ] ) and by assuming the common law @xmath82+b $ ] ; in order to compare our results with those of da90 , @xmath83 and @xmath84 were used , but we also obtained the same fits using more recent values as in carretta et al . ( @xcite ) , i.e. @xmath85 and @xmath86 . the rgb was modeled with an hyperbola as in rosenberg et al . ( @xcite ) , but in this case the coefficients were taken as second order polynomials in [ fe / h ] . in other words , we parametrized the whole family of rgbs in the following way : @xmath87\ ] ] where @xmath88^{2}+k_{2}[{\rm fe / h}]+k_{3}\ ] ] @xmath89^{2}+k_{5}[{\rm fe / h}]+k_{6}\ ] ] @xmath90^{2}+k_{8}[{\rm fe / h}]+k_{9}\ ] ] @xmath91 the list of the parameters of the fits in magnitude is reported in table [ t : coeffs ] , together with the _ rms _ of the residuals around the fitting curves . the table shows that the parameter @xmath92 does not depend on the choice of the distance scale , as expected . even the other coefficients are little dependent on the distance scale , apart from @xmath93 . it is affected by the zero - point of the hb luminosity - metallicity relation , and indeed there is the expected @xmath94 mag difference going from the ldz to the c99 distance scale . one could question the choice of a constant @xmath92 , but after some training on the theoretical isochrones , we found that even allowing for a varying parameter , its value indeed scattered very little around some mean value . this empirical result is a good one , in the sense that it allows to apply a robust linear least - square fitting method for any choice of @xmath95 , and then to search for the best value of this constant by a simple _ rms _ minimization . we chose to fit the @xmath96\ } $ ] function , and not the @xmath97 ) $ ] function , since the latter one would be double - valued for the brightest part of the metal rich clusters rgbs . this choice implies that our fits are not well - constrained for the vertical part of the giant branch , i.e. for magnitudes fainter than @xmath98 . however , we show in the next section that our analytic function is good enough for the intended purpose , i.e. to obtain the [ fe / h ] of the rgb stars in far local group populations , and thus to analyze how they are distributed in metallicity . our synthetic rgb families are plotted in figs . [ f : fitszw ] and [ f : fitscg ] , for the ldz distance scale . in the former figure , the zw metallicity scale is used , while the cg scale is used in the latter one . the figures show that the chosen functional form represents a very good approximation to the true metallicity `` distribution '' of the rg branches . the _ rms _ values are smaller than the typical uncertainties in the distance moduli within the local group . we further stress the excellent consistency of the empirical fiducial branches for clusters of similar metallicity . we have two pairs of clusters whose metallicities differ by at most 0.03 dex ( depending on the scale ) : ngc 288 and ngc 5904 on the one side , and ngc 5272 and ngc 6205 on the other side . the figures show that the fiducial line of ngc 288 is similar to that of ngc 5904 , and the ngc 5272 fiducial resembles that of ngc 6205 , further demonstrating both the homogeneity of our photometry and the reliability of the procedure that is used in defining the cluster ridge lines . if the coefficients of the hyperbolae are taken as third order polynomials , the resulting fits are apparently better ( the _ rms _ is @xmath99 mag ) ; however , the trends of the metallicity indices show an unphysical behavior , which is a sign that further clusters , having metallicities not covered by the present set , would be needed in order to robustly constrain the analytic function . in the following section , the indices are calibrated in terms of metallicity , so that in sect . [ s : testfits ] they will be used to check the reliability of our generalized fits . .the coefficients that define the functions used to interpolate our rgbs ( see text ) ; the top header line identifies the two distance scales used , while the two metallicities are identified in the second line of the header[t : coeffs ] [ cols= " > , > , > , > , > " , ] cccccccc & d.sc . & metallicity & @xmath100 & @xmath101 & @xmath102 & _ rms _ & fit + & & cg & -0.03 & 0.23 & -1.19 & 0.13 & 2 + & & zw & -0.004 & -0.18 & 0.08 & 0.12 & 2 + & & zw & -0.24 & 0.28 & & 0.12 & 1 + @xmath8 & ldz & cg & 0.00487 & -0.0057 & & 0.13 & @xmath103 + & & zw & -2.12 & 8.81 & -9.75 & 0.13 & 2 + & c99 & cg & 0.0045 & -0.0053 & & 0.15 & @xmath103 + & & zw & -2.05 & 8.57 & -9.61 & 0.12 & 2 + @xmath7 & ldz & cg & 0.0068 & -0.0076 & & 0.15 & @xmath103 + & & zw & -3.34 & 12.37 & -11.91 & 0.14 & 2 + & c99 & cg & 0.0065 & -0.0073 & & 0.15 & @xmath103 + & & zw & -3.233 & 12.23 & -11.96 & 0.14 & 2 + @xmath3 & & cg & -0.34 & 0.93 & -1.37 & 0.16 & 2 + & & zw & -0.063 & -0.56 & 0.41 & 0.16 & 2 + & & zw & -0.87 & 0.77 & & 0.16 & 1 + @xmath6 & & cg & -0.36 & 0.55 & -0.97 & 0.19 & 2 + & & cg & -0.69 & 0.0007 & & 0.22 & 1 + & & zw & -0.13 & -0.38 & -0.28 & 0.20 & 2 + & & zw & -0.82 & 0.06 & & 0.20 & 1 + @xmath2 & & cg & -0.30 & 0.09 & -0.81 & 0.23 & 2 + & & cg & -0.59 & -0.52 & & 0.25 & 1 + & & zw & -0.13 & -0.42 & -0.68 & 0.25 & 2 + & & zw & -0.70 & -0.56 & & 0.25 & 1 + @xmath5 & & cg & 4.25 & -5.37 & & 0.32 & 1 + & & zw & 5.25 & -6.52 & & 0.33 & 1 + in order to obtain analytic relations between the indices and the actual metallicity , our photometric parameters were compared both with the zw and the cg values . a summary of the resulting equations is given in table [ t : rms ] . for each index ( first column ) both linear and quadratic fits were tried , of the form : @xmath104=\alpha \cdot index + \beta $ ] and @xmath104=\alpha \cdot index^{2}+\beta \cdot index + \gamma $ ] . the coefficients of the calibrating relation are given in the columns labelled @xmath105 , @xmath106 , and @xmath107 ; in column 7 , the _ rms _ of the residuals is also given . in the case of the @xmath7 and @xmath8 indices , neither the linear nor the quadratic fits give satisfactory results , when the cg scale is considered . instead , a good fit is obtained if a change of variables is performed , setting @xmath108 } $ ] , and linearly interpolating in the index ( i.e. setting @xmath109 ) . the column 8 of table [ t : rms ] identifies the kind of fitting function that is used for each parameter / metallicity combination : the symbols `` 1 '' , `` 2 '' and `` @xmath103 '' refer to the linear , quadratic , and linear in @xmath103 fits , respectively . relations on both the cg and zw metallicity scales are given , and column 3 flags the [ fe / h ] scale that is used . in order to measure the @xmath46 and @xmath8 indices ( cf . [ s : indices ] ) a distance scale must be adopted . the most straightforward way is to use the observed @xmath25 ( cf . table [ t : the - sample ] ) coupled with a suitable law for the hb absolute magnitude . it has become customary to parameterize this magnitude as @xmath110+b $ ] , although there is no consensus on the value of the two parameters @xmath111 and @xmath112 . the current calibrations of these two metallicity indices were obtained by da costa & armandroff ( @xcite ) and lee et al . ( @xcite ) , and they are based on the lee et al . ( @xcite ; ldz ) theoretical luminosities of the hb . ldz gave a relation @xmath113 + 0.82 $ ] valid for @xmath114 . as discussed in sect [ s : newda90 ] , since many current determinations of population ii distances within the local group are based on the lee et al . ( @xcite ) distance scale , and for the purpose of comparison with previous studies , we provide a calibration using the latter hb luminosity - metallicity relation . however , in the last ten years revisions of this relation have been discussed by many authors , so we also calibrated the two indices using @xmath115 + 0.90 $ ] ( carretta et al . @xcite ) , which is one of the most recent hb - based distance scales . we must stress that _ metallicities on the zw scale must be used in the @xmath116 vs. _ [ fe / h ] _ relation_. indeed , cg showed that their scale is not linearly correlated to that of zw , so not even the _ @xmath116 vs. _ [ fe / h ] relation will be linear : if one wishes to use the new scale , then _ the absolute magnitude of the hb must be re - calibrated _ in a more complicated way . the best calibrating relations are shown in figs . [ f : cals ] to [ f : caldv14 ] . in the following sections , for each index a few remarks on the accuracy of the calibrations and comparisons with past studies are given . on the cg scale , the second - order fit has a residual _ rms _ of 0.12 dex in [ fe / h ] . on the zw scale , the linear fit is obtained with a _ rms _ of 0.12 dex . this index can therefore be calibrated on both scales , with a comparable level of accuracy . a parabolic fit does not improve the relation on the zw scale , since the coefficient of the quadratic term is very small ( -0.004 ) and the _ rms _ is the same . these relations are shown in fig . [ f : cals ] as solid lines , where the upper panel is for the zw scale , and the lower panel for the cg scale ( this layout is reproduced in all the following figures ) . the cluster ngc 6656 ( m22 ) was excluded from the fits , and is plotted as an open circle in fig . [ f : cals ] . it is well - known that m22 is a cluster that shows a metallicity spread , and indeed it falls outside the general trend in most of the present calibrations . the first definition of the @xmath7 index was given in da costa & armandroff ( @xcite ) , where a calibration in terms of the zw scale was also given : @xmath11=-15.16 + 17.0\ , ( v - i)_{-3}-4.9\ , ( v - i)_{-3}^{2 } $ ] . the same index ( measured on the _ absolute _ rgbs corrected with the ldz hb luminosity - metallicity relation ) is plotted , in fig . [ f : calvi-3 ] , as a function of the metallicity on both scales , and the solid lines represent our calibrations . the top panel shows the quadratic relation on the zw scale , whose _ rms _ is 0.14 dex . the bottom panel of fig . [ f : calvi-3 ] shows the relation on the cg scale . in this case , a quadratic fit is not able to reproduce the trend of the observational data . a better result can be obtained by making a variable change , i.e. using the variable @xmath119 } $ ] ; in this case , a linear relation is found , and its _ rms _ is 0.15 dex . this measure of the residual scatter has been computed after transforming back to metallicity , so the reliability of the index can be compared to that of the other ones . again , the index can be calibrated on both scales with a comparable accuracy . the dashed curve in the upper panel of fig . [ f : calvi-3 ] shows the original relation obtained by da90 : there is a small discrepancy at the high - metallicity end , which can be explained by the different 47 tuc fiducial line that was adopted by da90 ( cf . below the discussion on @xmath8 ) . as already recalled , we checked the effect of adopting another distance scale , by repeating our measurements and fits , and adopting the c99 distance scale . for the zw metallicity scale , we obtain the quadratic relation whose coefficients are listed in table [ t : rms ] , and whose _ rms _ is 0.15 dex . the bottom panel of fig . [ f : calvi-3 ] shows the relation on the cg scale . again , a quadratic fit is not able to reproduce the trend of the observational data . making the already discussed variable substitution , the linear relation in @xmath103 has an _ rms _ of 0.16 dex , so the two metallicity scales yield almost comparable results . using the same `` standard '' gc branches of da90 , lee et al . ( @xcite ) defined a new index , @xmath8 , to be used for the farthest population ii objects . it was also calibrated in terms of the zw scale : @xmath11=-12.64 + 12.6\ , ( v - i)_{-3.5}-3.3\ , ( v - i)_{-3.5}^{2 } $ ] . a new calibration was also given recently in caldwell et al . ( @xcite ) : [ fe / h]@xmath121 , where @xmath122 $ ] . the index and our calibrations ( solid lines ) are plotted , in fig . [ f : calvi-3.5 ] , on both metallicity scales . again , the measurements were made in the absolute cmd , assuming the ldz distance scale . our quadratic calibration vs. the zw scale has a residual _ rms _ scatter of 0.13 dex , which is the same of the linear relation on the cg metallicity vs. @xmath103 . the lee et al . relation ( dashed line ) predicts slightly too larger metallicities on the zw scale , for @xmath104>-1 $ ] . this can also be interpreted as if the da90 47 tuc branch were @xmath123 mag bluer than ours . indeed , if one looks at fig . 5 of da90 , one can easily see that some weight is given to the brightest rgb star , which is brighter than the trend defined by the previous ones . the result is a steeper branch , which also justifies the da90 slightly bluer rgb fiducial . since our metal richest point is defined by two clusters , and since the two measured parameters agree very well , we are confident that our calibration is reliable . in any case , the discrepancy between the two scales is no larger than @xmath94 dex . it is also reassuring that the caldwell et al . ( @xcite ) relation ( pluses ) is closer to the present calibration , since the former is based on a larger set of clusters . this might be an indication that the lee et al . relation is actually inaccurate at the metal rich end , due to the small set of calibrating clusters . as before , we obtained a further calibration also using the c99 @xmath116 vs. [ fe / h ] relation ; the quadratic fit on the zw scale has a residual _ rms _ scatter of 0.13 dex , while the @xmath103 variable can be fitted with a straight line , with an _ rms _ of 0.14 dex . for any @xmath126 index , the quadratic relations vs. the zw metallicity do not improve the _ rms _ and they are not plotted in the figures . the coefficients are listed in table [ t : rms ] . the best metallicity estimates of the `` @xmath126 family '' are obtained with the @xmath3 index . the errors on @xmath11 $ ] are just slightly larger than the standard uncertainties of the spectroscopic determinations . the solid lines of fig . [ f : caldv14 ] show the calibrations that we obtain . the quadratic equation on the cg scale , and the linear one on the zw scale , are obtained with residual scatters of 0.16 dex . the rest of the indices in this family , and @xmath127 , lack the precision of the other abundance indicators . this is due to the fact that the error on any @xmath126 index is proportional to the uncertainty on the color of the rgb ( which depends on the reddening ) , times its local slope where the reference point is measured . since the rgb slope increases going away from the tip ( i.e. towards bluer colors ) , we expect that the scatter on the @xmath126 indices will also increase as the color of the reference point gets bluer . indeed , table [ t : rms ] shows that in most cases the _ rms _ uncertainties are @xmath128 dex for these indices . the residual scatter is largest for the @xmath127 index , which is the most affected by the uncertainties on the reddening . the @xmath6 and @xmath127 parameters have been earlier calibrated , on the cg scale , by carretta & bragaglia ( @xcite ) . using their quadratic relation for @xmath6 , and both their linear and quadratic relations for @xmath127 , the corresponding _ rms _ of the residuals in metallicity are 0.21 dex and @xmath129 dex , respectively . our new and the old calibrations are therefore compatible , within the ( albeit large ) uncertainties . a straightforward test of our new analytic rgbs can be made by generating the same metallicity indices that have been measured on the observed rgbs , and then checking the consistency of the predicted vs. measured quantities . to this aim , for a set of discrete [ fe / h ] values a @xmath130 vector was generated , and the combination of the two was used to compute the @xmath131 vector of the giant branch , using eqs . ( [ e : general]-[e : d ] ) . then for each branch the metallicity indices were measured as it was done for the clusters fiducials . in figs . [ f : cals ] to [ f : caldv14 ] , the predicted indices are identified by the small open squares ( spaced by 0.1 dex ) connected by a solid line . the best predictions are for those indices that rely on the brightest part of the rgb ( i.e. @xmath7 , @xmath8 and @xmath132 ) , while the computations are partially discrepant for those indices that rely on a point that is measured on the faint rgb . this is easily explained by the nature of our fit : since the best match is searched for along the ordinates ( for the reasons discussed in sect . [ s : newda90 ] ) , then it is better constrained in the upper part of the rgb , where its curvature becomes more sensitive to metallicity . we must also stress that the metal richest cluster in the reference grid is 47 tuc ( [ fe / h]@xmath133 on the zw scale ) , whereas ngc 6352 ( [ fe / h]@xmath134 on the same scale ) is the metal richest cluster for which metallicity indices have been measured . some of the discrepancies that are seen at the highest metallicities are therefore due to the lack of low - reddening clusters that can be used to extend the reference grid to the larger [ fe / h ] values . the mean differences between the predicted and fitted indices are , on the zw scale , around 0.03 dex for the @xmath7 and @xmath8 indices . they are around 0.08 dex for the @xmath6 , @xmath3 , and @xmath1 indices . they rise to @xmath94 and @xmath135 dex for the @xmath2 and @xmath5 indices . a similar trend is seen for the comparison on the cg scale . in this case , the mean differences are @xmath99 dex for @xmath7 , @xmath8 , and @xmath1 ; they are @xmath94 dex for @xmath6 and @xmath3 ; and they are 0.12 and 0.27 for the @xmath2 and @xmath5 indices . we can therefore conclude that , apart from the @xmath2 and @xmath5 indices , our mono - parametric rgb family gives a satisfactory reproduction of the actual changes of the rgb morphology and location , as a function of metallicity . it is then expected that , using this approach , one can exploit the brightest @xmath136 mags of the rgb to determine the mean metallicity , and even more important , the metallicity _ distribution _ of the old stellar population of any local group galaxy . in a forthcoming paper , we will demonstrate such possibility by re - analyzing our old photometric studies of the dwarf spheroidal galaxies tucana ( saviane et al . @xcite ) , phoenix ( held et al . @xcite ; martnez - delgado et al . @xcite ) , fornax ( saviane et al . @xcite ) , lgs 3 ( aparicio et al . @xcite ) , leo i ( gallart et al . @xcite ; held et al . @xcite ) and ngc 185 ( martnez - delgado et al . in this work , we have provided the first calibration of a few metallicity indices in the @xmath137 plane , namely the indices @xmath1 , @xmath2 and @xmath3 . calibrations on both the zinn & west ( 1984 ) and carretta & gratton ( 1997 ) scales have been obtained . the metallicity indices @xmath5 , @xmath6 , @xmath7 and @xmath8 have been also calibrated on both scales , and we have shown that our new relations are consistent with existing ones . in the case of the latter two indices , we have obtained the first calibration on the cg scale ; for both scales , we have also obtained the first calibration that takes into account new results on the rr lyr distances . the accuracy of the calibrations is generally better than 0.2 dex , regardless of the metallicity scale that is used . our results are an improvement over previous calibrations , since a new approach in the definition of the rgb is used , and since our formulae are based on the largest homogeneous photometric database of galactic globular clusters . the availability of such database also allowed us a progress towards the definition of a standard description of the rgb morphology and location . we were able to obtain a function in the @xmath9 $ ] space which is able to reproduce the whole set of ggc giant branches in terms of a single parameter ( the metallicity ) . we suggest that the usage of this function will improve the current determinations of metallicity and distances within the local group , extending the methods of lee et al . ( 1993 ) . we thank the referee , gary da costa , for helpful suggestions that improved the final presentation of the manuscript . i.s . acknowledges the financial support of italian and spanish foreign ministries , through an ` azioni integrate / acciones integradas ' grant . aparicio a. , gallart c. , bertelli g. , 1997 , aj 114 , 680 armandroff t.e . , da costa g.s . , caldwell n. , seitzer p. , 1993 , aj 106 , 986 bertelli g. , bressan a. , chiosi c. , fagotto f. , nasi e. , 1994 , a&as 106 , 275 buonanno r. , corsi c.e . , pulone l. , fusi pecci f. , bellazzini m. , 1998 , a&a 333 , 505 ( b98 ) caldwell n. , armandroff t.e . , da costa g.s . , seitzer p. , 1998 , aj 115 , 535 carretta e. , bragaglia a. , 1998 , a&a 329 , 937 carretta e. , gratton r. , 1997 , a&as 121 , 95 ( cg ) carretta e. , gratton r.g . , clementini g. , fusi pecci f. , 1999 , apj , in press ( c99 ) da costa g.s . , armandroff t.e . , 1990 , aj 100 , 162 ( da90 ) dean j.f . , warren p.r . , cousins a.w.j . , 1978 , mnras 183 , 569 gallart c. , freedman w. , aparicio a. , bertelli g. , chiosi c. , 1999 , aj , , 118 , 2245 harris g.l.h . , harris w.e . , poole g.b . , 1999 , aj 117 , 855 harris w.e , 1996 , aj 112 , 1487 hartwick f.d.a . , 1968 , apj 154 , 475 held e.v . , saviane i. , momany y. , 1999a , a&a 345 , 747 held e.v . , saviane i. , momany y. , carraro g. , 1999b , apj , in press lee m.g . , freedman w.l . , madore b.f . , 1993 , apj 417 , 553 lee y.w . , demarque p. , zinn r. , 1990 , apj 350 , 155 ( ldz ) lloyd evans t. , 1983 , s. afr . circ . 7 , 86 martnez - delgado d. , aparicio . a. , gallart c. , 1999a , aj , 118 , 2229 martnez - delgado d. , gallart c. , aparicio a. , 1999b , aj , 118 , 862 rosenberg a. , saviane i. , piotto g. , aparicio a. , 1999a , aj , 118 , 2306 ( rspa99 ) rosenberg a. , piotto g. , saviane i. , aparicio a. , 1999b , a&as , in press rosenberg a. , aparicio a. , saviane i. , piotto g. , 1999c , a&as , submitted rutledge a.g . , hesser j.e . , stetson p.b . , 1997 , pasp 109 , 907 ( rhs97 ) sandage a. , smith l.l . , 1966 , apj 144 , 886 sandage a. , wallerstein g. , 1960 , apj 131 , 598 saviane i. , held e.v . , bertelli g. , 1999a , a&a , in press saviane i. , held e.v . , piotto g. , 1996 , a&a 315 , 40 saviane i. , rosenberg a. , piotto g. , 1997 . in : rood , a.renzini ( eds . ) advances in stellar evolution , cambridge university press , cambridge , p. 65 ( srp97 ) saviane i. , piotto g. , fagotto f. , et al . , 1998 , a&a 333 , 479 saviane i. , rosenberg a. , piotto g. , 1999b . in : b. k. gibson , t. s. axelrod , m. e. putman ( eds . ) , `` the third stromlo symposium : the galactic halo '' ( srp99 ) walker a. , 1992 , pasp 104 , 1063 zinn r. , west m. , 1984 , apjs 55 , 45 ( zw ) | the purpose of this study is to carry out a thorough investigation of the changes in morphology of the red giant branch ( rgb ) of galactic globular clusters ( ggc ) as a function of metallicity , in the @xmath0 bands . to this aim ,
two key points are developed in the course of the analysis . *
( a ) * using our photometric @xmath0 database for galactic globular clusters ( the largest homogeneous data sample to date ; rosenberg et al .
@xcite ) _ we measure a complete set of metallicity indices _ , based on the morphology and position of the red - giant branch . in particular , we provide here the first calibration of the @xmath1 , @xmath2 and @xmath3 indices in the @xmath4 plane .
we show that our indices are internally consistent , and we calibrate each index in terms of metallicity , both on the zinn & west ( 1984 ) and the carretta & gratton ( 1997 ) scales .
our new calibrations of the @xmath5 , @xmath6 , @xmath7 and @xmath8 indices are consistent with existing relations . *
( b ) * using a grid of selected rgb fiducial points , _ we define a function in the @xmath9 $ ] space which is able to reproduce the whole set of ggc giant branches in terms of a single parameter _ ( the metallicity ) . as a first test , we show that the function is able to predict the correct trend of our observed indices with metallicity
. the usage of this function will improve the current determinations of metallicity and distances within the local group , since it allows to easily map @xmath10 coordinates into @xmath11,m_{i } $ ] ones . to this
aim the `` synthetic '' rgb distribution is generated both for the currently used lee et al .
( 1990 ) distance scale , and for the most recent results on the rr lyr distance scale . |
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et al_. [ 1 ] , introduced the first hierarchical clustering algorithm for wsns , called leach . it is one of the most popular protocols in wsns . the main idea is to form clusters of the sensor nodes . leach outperforms classical clustering algorithm by using adaptive clustering and rotating chs . this saves energy as transmission will only be performed on that specific ch rather than all the nodes . leach performs well in homogeneous environment however , it s performance deteriorates in heterogeneous environment . threshold - sensitive energy efficient network ( teen ) [ 2 ] is a reactive protocol for time critical applications . the ch selection and cluster formation of nodes is same as that of leach . in this scheme , ch broadcasts two threshold values i.e. hard threshold ( ht ) and soft threshold ( st ) . ht is the absolute value of an attribute to trigger a sensor node . ht allows nodes to transmit the event , if the event occurs in the range of interest . therefore , this not only reduces transmission to significant numbers but also increases network lifetime . georgios _ et al_. [ 3 ] , proposed a two level heterogeneous aware protocol , consisting of normal and advance ( high energy ) nodes . it is based on the weighted election probabilities of each node according to their respective energy to become a ch . intuitively , advance nodes have more probability to become a ch than normal nodes , which seems logical according to their energy consumption . stable election protocol ( sep ) does not require any global knowledge of the network . the drawback of sep is that it does not consider the changing residual energy of the node hence , the probability of advanced nodes to become ch remains high irrespective of the residual energy left in the node . moreover , sep performs below par if the network is more than two levels . in [ 4 ] , authors proposed distributed energy efficient clustering ( deec ) protocol for wsns . deec is a clustering protocol for two and multilevel heterogeneous networks . the probability for a node to become ch is based on residual energy of the nodes and average energy of network . the epoch for nodes to become chs is set according to the residual energy of a node and average energy of the network . the node with higher initial and residual energy has more chances to become a ch than the low energy node . a number of routing protocols have been proposed in the area of wsns . most of them are based on ch selection . however , not much attention has been devoted towards time critical applications . most of the routing protocols are for proactive networks . deec , being a proactive heterogeneous network protocol is not well suited for time critical applications . teen is a reactive protocol which guarantees that in homogeneous environment , unstable region will be short . after the death of the first node , all the remaining nodes are expected to die on average within a small number of rounds as a consequence of the uniform remaining energy due to the well distributed energy consumption . on the other hand teen in the presence of high energy nodes yields a large unstable region . the reason being , all high energy nodes are equipped with almost the same energy however , the ch selection process is unstable and as a result most of the time these nodes are idle , as there is no ch to transmit . hence , in our research paper we focus on developing a protocol that gives us better results for time critical applications in both environments i.e. homogeneous and heterogeneous environment . we define following parameters which evaluate and compare the performance of clustering protocols . it is the time interval when the network starts its operation till the death of the first node . it is also referred to as stable region or steady state . it is the time interval from the death of the first node until the death of the last node or till the time the network is dead . it is referred as unstable region . it is the time interval from the start of operation of the network till the death of last alive node : @xmath0 this instantaneous measure shows the total number of nodes ( advanced , normal ) alive i.e. the nodes having energy greater than zero . it is the total rate of data sent over the network from nodes to their respective chs and from chs to base station . in this section , we describe the functionality and characteristics of our model of a wsns . we particularly present the setting , the ch selection and how transmissions occur . teen is the first reactive protocol . in this scheme , closer nodes form clusters with a ch to transmit the collected data to one upper layer . this is same as leach protocol however , at every cluster change time , the ch broadcasts two threshold values i.e hard and st . ht is the absolute value of an attribute to trigger on its transmitter and report to its respective ch . ht allows nodes to transmit data , if the data occurs in the range of interest . therefore , a significant reduction of the transmission delay occurs . moreover , st is the small change in the value of the sensed attribute . next transmission occurs when there is a small change in the sensed attribute once it reaches the ht . so , it further reduces the number of transmissions . deec is a proactive protocol designed for two and multi level heterogeneous networks . all the nodes use the initial and residual energies to select a ch . the node with higher initial and residual energy has greater probability to become a ch . in a two - level heterogeneous network , we have two categories of nodes , m.n advanced nodes with initial energy equal to @xmath1 and ( 1-m).n normal nodes , where @xmath2 is the initial energy . moreover , @xmath3 and @xmath4 are two variables which control the nodes ( advanced or normal ) percentage types and @xmath5 is the total energy in the network . the value of @xmath5 is given as : @xmath6 the value of the total initial energy of the multi - level heterogeneous networks is given as : @xmath7 the probability of a node to become ch in a two level heterogeneous network is : @xmath8 for normal nodes @xmath8 for advanced nodes this model can be easily extended to multi - level heterogeneous networks : @xmath9 as we are assuming uniformly distributed node , so distance of cluster members from ch is : @xmath10 average distance between base station and ch is : @xmath11 in this section , we describe heer , which improves the stable region for clustering hierarchy process for a reactive network in homogeneous and heterogeneous environment . we use the initial and residual energies of the nodes to become ch similar to that of deec . it does not require any global knowledge of energy at any election round.when cluster formation is done , the ch transmits two threshold values , i.e @xmath12 and @xmath13 . the nodes sense their environment repeatedly and if a parameter from the attributes set reaches its @xmath12 value , the node switches on its transmitter and transmits data . the current value ( @xmath14 ) on which first transmission occurs , is stored in an internal variable in the node called sensed value ( @xmath15 ) . this reduces the number of transmissions . now the nodes will again transmit the data in same cluster period when @xmath16 . that is , if @xmath14 differs from @xmath15 by an amount equal to or greater than @xmath13 , then it further reduces the number of transmissions . figure . 1 shows different states of a cluster i.e. from data sensing to data transmitting . every node selects itself as a ch on the basis of its initial energy and residual energy . in state ( 1 ) a cluster is formed the node senses its environment continuously until the parameter ( cv ) reaches its ht value . when cv reaches ht value , the nodes become green as shown in the figure in state ( 2 ) . the node then switches on its transmitter and sends the data to the ch . the ch aggregates and transmits data to base station . the cv on which first transmission occurs is stored in sv . the node , then again starts sensing its environment as shown in state ( 3 ) until the cv differs from sv by an amount equal to or greater than st . when this condition becomes true , the node again switches on its transmitter and sends data to ch.the ch then transmits data to base station as shown in state ( 4 ) of figure . 1 . * heer performs best for time critical applications in both homogeneous and heterogeneous environment . * it reduces the number of transmissions resulting in the reduction of energy consumption . * it increase the stability period and network lifetime . proactive protocols sense their environment and transmit data periodically . they consume energy continuously with time due to periodic transmission . our main focus in proactive protocols is on increasing lifetime , throughput and to decrease energy consumption . contrary to proactive protocol , reactive protocol is application dependent . it senses the environment periodically but transmits data only when its @xmath14 reaches to absolute value of the attribute . as data transmission consumes more energy than data sensing , so , in reactive network the throughput can be minimized or maximized as per its application . the throughput in reactive networks is inversely proportional to the network lifetime or its stability period . if transmissions are less the stability period and network lifetime will be prolonged as @xmath14 does not reach the absolute value . however , if the @xmath14 reaches @xmath12 value ( absolute value ) repeatedly then maximum number of transmissions will occur and nodes will die quickly . in this section , we simulate an environment with varying temperature in different regions . our field has dimensions of @xmath17 square units . the number of nodes in the field is @xmath18 . we assume that base station is in the center of sensing nodes . to evaluate the performance of heer , we simulate it with teen and deec . the parameters used in our simulation are listed in table 1 . our goals in conducting the simulation are as follows . * we examine the performance of teen and heer for the prolonging of stability period and network life time . * we also observe the throughput of both the protocols . we observe the performance of teen , deec and heer under homogeneous environments . we also examine the sensitivity of our protocol to the degree of heterogeneity in the sensor network . .parameters used in our simulations [ cols="<,<",options="header " , ] = 100 and @xmath13=2,width=340,height=264 ] = 70 and @xmath13=10,width=340,height=264 ] = 70 and @xmath13=10,width=340,height=264 ] in this section , we compare heer , teen and deec protocols in homogeneous environment . we observe from figure . 3 that in teen after the death of first node , all the remaining nodes die within a small number of rounds . this is due to the reason that all the nodes have same probability to become a ch . deec prolongs the stability period and network life time due to the election of ch on the basis of residual energy . high residual energy nodes have greater probability of becoming a ch . the stability period of heer ( hard and soft ) is much longer than that of teen and deec . heer introduces ht value which decreases the number of transmissions to base station . this increases the stability period and network lifetime . st further reduces the number of transmissions resulting in the reduction of energy consumption . this prolongs the network life time . heer outperforms deec in terms of stability period and network life time by a factor of 1.78 and 1.60 respectively . moreover , both stability period and network life time of heer also outperforms teen by a factor of 2.0 . from figure . 5 , we can observe that by changing the values of thresholds , the stability period and network life time of heer ( hard ) changes noticeably . the stability period and network lifetime decreases if we decrease the value of @xmath19 as shown in figure . moreover , figure.6 shows that the throughput difference between heer ( hard ) and deec decreases by decreasing the difference between the two thresholds . st also effects the network lifetime . if number of transmissions increase , we observe a decrease in network lifetime and vice versa . in this paper , we present a hybrid reactive protocol of teen and deec for homogeneous environment . heer minimizes the energy consumption by first distributing load to all high energy nodes and then on to low energy nodes . like teen , it is well suited for time critical applications and is more efficient than teen and deec . w.heinzelman , a. chandrakasan , and h. balakrishnan , `` energy - efficient communication protocol for wireless sensor networks , '' in the proceeding of the hawaii international conference system sciences , hawaii , january 2000 . a. manjeshwar and d. p. agarwal , `` teen : a routing protocol for enhanced efficiency in wireless sensor networks , '' in 1st international workshop on parallel and distributed computing issues in wireless networks and mobile computing , april 2001 . g. smaragdakis , i. matta , a. bestavros , sep : a stable election protocol for clustered heterogeneous wireless sensor networks , in : second international workshop on sensor and actor network protocols and applications ( sanpa 2004 ) , 2004 . | wireless sensor networks ( wsns ) consist of numerous sensors which send sensed data to base station .
energy conservation is an important issue for sensor nodes as they have limited power .
many routing protocols have been proposed earlier for energy efficiency of both homogeneous and heterogeneous environments .
we can prolong our stability and network lifetime by reducing our energy consumption . in this research paper
, we propose a protocol designed for the characteristics of a reactive homogeneous wsns , heer ( hybrid energy efficient reactive ) protocol . in heer
, cluster head(ch ) selection is based on the ratio of residual energy of node and average energy of network .
moreover , to conserve more energy , we introduce hard threshold ( ht ) and soft threshold ( st ) .
finally , simulations show that our protocol has not only prolonged the network lifetime but also significantly increased stability period .
wireless , sensor , networks , energy , hybrid , cluster , reactive |
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medical imaging including x - rays , magnetic resonance imaging ( mri ) , computer tomography ( ct ) , ultrasound etc . are susceptible to noise @xcite . reasons vary from use of different image acquisition techniques to attempts at decreasing patients exposure to radiation . as the amount of radiation is decreased , noise increases @xcite . denoising is often required for proper image analysis , both by humans and machines . image denoising , being a classical problem in computer vision has been studied in detail . various methods exist , ranging from models based on partial differential equations ( pdes ) @xcite , domain transformations such as wavelets @xcite , dct @xcite , bls - gsm @xcite etc . , non local techniques including nl - means @xcite , combination of non local means and domain transformations such as bm3d @xcite and a family of models exploiting sparse coding techniques @xcite . all methods share a common goal , expressed as @xmath0 where @xmath1 is the noisy image produced as a sum of original image @xmath2 and some noise @xmath3 . most methods try to approximate @xmath2 using @xmath1 as close as possible . in most cases , @xmath3 is assumed to be generated from a well defined process . with recent developments in deep learning @xcite , results from models based on deep architectures have been promising . autoencoders have been used for image denoising @xcite . they easily outperform conventional denoising methods and are less restrictive for specification of noise generative processes . denoising autoencoders constructed using convolutional layers have better image denoising performance for their ability to exploit strong spatial correlations . in this paper we present empirical evidence that stacked denoising autoencoders built using convolutional layers work well for small sample sizes , typical of medical image databases . which is in contrary to the belief that for optimal performance , very large training datasets are needed for models based on deep architectures we also show that these methods can recover signal even when noise levels are very high , at the point where most other denoising methods would fail . rest of this paper is organized as following , next section discusses related work in image denoising using deep architectures . section iii introduces autoencoders and their variants . section iv explains our experimental set - up and details our empirical evaluation and section v presents our conclusions and directions for future work . although bm3d @xcite is considered state - of - the - art in image denoising and is a very well engineered method , burger et al . @xcite showed that a plain multi layer perceptron ( mlp ) can achieve similar denoising performance . denoising autoencoders are a recent addition to image denoising literature . used as a building block for deep networks , they were introduced by vincent et al . @xcite as an extension to classic autoencoders . it was shown that denoising autoencoders can be stacked @xcite to form a deep network by feeding the output of one denoising autoencoder to the one below it . jain et al . @xcite proposed image denoising using convolutional neural networks . it was observed that using a small sample of training images , performance at par or better than state - of - the - art based on wavelets and markov random fields can be achieved . xie et al . @xcite used stacked sparse autoencoders for image denoising and inpainting , it performed at par with k - svd . agostenelli et al . @xcite experimented with adaptive multi column deep neural networks for image denoising , built using combination of stacked sparse autoencoders . this system was shown to be robust for different noise types . an autoencoder is a type of neural network that tries to learn an approximation to identity function using backpropagation , i.e. given a set of unlabeled training inputs @xmath4 , it uses @xmath5 an autoencoder first takes an input @xmath6^d$ ] and maps(encode ) it to a hidden representation @xmath7^{d'}$ ] using deterministic mapping , such as @xmath8 where @xmath9 can be any non linear function . latent representation @xmath3 is then mapped back(decode ) into a reconstruction @xmath1 , which is of same shape as @xmath2 using similar mapping . @xmath10 in , prime symbol is not a matrix transpose . model parameters ( @xmath11 ) are optimized to minimize reconstruction error , which can be assessed using different loss functions such as squared error or cross - entropy . basic architecture of an autoencoder is shown in fig . [ autoencoder_fig ] @xcite here layer @xmath12 is input layer which is encoded in layer @xmath13 using latent representation and input is reconstructed at @xmath14 . using number of hidden units lower than inputs forces autoencoder to learn a compressed approximation . mostly an autoencoder learns low dimensional representation very similar to principal component analysis ( pca ) . having hidden units larger than number of inputs can still discover useful insights by imposing certain sparsity constraints . denoising autoencoder is a stochastic extension to classic autoencoder @xcite , that is we force the model to learn reconstruction of input given its noisy version . a stochastic corruption process randomly sets some of the inputs to zero , forcing denoising autoencoder to predict missing(corrupted ) values for randomly selected subsets of missing patterns . basic architecture of a denoising autoencoder is shown in fig . [ denautoencoder_fig ] denoising autoencoders can be stacked to create a deep network ( stacked denoising autoencoder ) @xcite shown in fig . [ sautoencoder_fig ] @xcite . output from the layer below is fed to the current layer and training is done layer wise . convolutional autoencoders @xcite are based on standard autoencoder architecture with convolutional _ encoding _ and _ decoding _ layers . compared to classic autoencoders , convolutional autoencoders are better suited for image processing as they utilize full capability of convolutional neural networks to exploit image structure . in convolutional autoencoders , weights are shared among all input locations which helps preserve local spatiality . representation of @xmath15th feature map is given as @xmath16 where bias is broadcasted to whole map , @xmath17 denotes convolution ( 2d ) and @xmath9 is an activation . single bias per latent map is used and reconstruction is obtained as @xmath18 where @xmath19 is bias per input channel , @xmath20 is group of latent feature maps , @xmath21 is flip operation over both weight dimensions . backpropogation is used for computation of gradient of the error function with respect to the parameters . we used two datasets , mini - mias database of mammograms(mmm ) @xcite and a dental radiography database(dx ) @xcite . mmm has 322 images of 1024 @xmath22 1024 resolution and dx has 400 cephalometric x - ray images collected from 400 patients with a resolution of 1935 @xmath22 2400 . random images from both datasets are shown in fig . [ random_real ] . all images were processed prior to modelling . pre - processing consisted of resizing all images to 64 @xmath22 64 for computational resource reasons . different parameters detailed in table [ datasets ] were used for corruption . .dataset perturbations [ cols="<,<",options="header " , ] + @xmath23 represents 50% corrupted images with @xmath24 , @xmath25 are images corrupted with @xmath26 , @xmath27 are corrupted with @xmath28 and @xmath29 are corrupted with a poisson noise using @xmath30 also , as the noise level is increased the network has trouble converging . [ troublecon ] shows the loss curves for gaussian noise with @xmath31 . even using 100 epochs , model has not converged . we have shown that denoising autoencoder constructed using convolutional layers can be used for efficient denoising of medical images . in contrary to the belief , we have shown that good denoising performance can be achieved using small training datasets , training samples as few as 300 are enough for good performance . our future work would focus on finding an optimal architecture for small sample denoising . we would like to investigate similar architectures on high resolution images and the use of other image denoising methods such as singular value decomposition ( svd ) and median filters for image pre - processing before using cnn dae , in hope of boosting denoising performance . it would also be of interest , if given only a few images can we combine them with other readily available images from datasets such as imagenet @xcite for better denoising performance by increasing training sample size . agostinelli , forest , michael r. anderson , and honglak lee . `` adaptive multi - column deep neural networks with application to robust image denoising . '' _ advances in neural information processing systems_. 2013 . burger , harold c. , christian j. schuler , and stefan harmeling . `` image denoising : can plain neural networks compete with bm3d ? . '' _ computer vision and pattern recognition ( cvpr ) _ , 2012 ieee conference on . ieee , 2012 . rudin , leonid i. , and stanley osher . `` total variation based image restoration with free local constraints . '' image processing , 1994 . proceedings . _ icip-94_. , ieee international conference . 1 . ieee , 1994 . vincent , pascal , et al . `` stacked denoising autoencoders : learning useful representations in a deep network with a local denoising criterion . '' _ journal of machine learning research _ 11.dec ( 2010 ) : 3371 - 3408 . yaroslavsky , leonid p. , karen o. egiazarian , and jaakko t. astola . `` transform domain image restoration methods : review , comparison , and interpretation . '' _ photonics west 2001-electronic imaging_. international society for optics and photonics , 2001 . | image denoising is an important pre - processing step in medical image analysis .
different algorithms have been proposed in past three decades with varying denoising performances . more recently ,
having outperformed all conventional methods , deep learning based models have shown a great promise .
these methods are however limited for requirement of large training sample size and high computational costs . in this paper
we show that using small sample size , denoising autoencoders constructed using convolutional layers can be used for efficient denoising of medical images .
heterogeneous images can be combined to boost sample size for increased denoising performance .
simplest of networks can reconstruct images with corruption levels so high that noise and signal are not differentiable to human eye .
image denoising , denoising autoencoder , convolutional autoencoder |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this article , the term cdw is used to indicate a periodic modulation of the charge density @xmath2 , irrespective of the process behind its generation . this modulation is described , in the unperturbed system and for the simplest form of cdw , as @xmath3 , where @xmath4 is the intensity and @xmath5 the characteristic wavelength . perturbations to cdws have been studied extensively @xcite , but most studies are concerned either with uniform perturbations ( e.g. an external electric field ) or point - like perturbations ( e.g. pinning by defects ) , and often consider one - dimensional models , appropriate for quasi - one - dimensional materials , where the coherence length in the perpendicular directions is smaller than the atomic distance . what is considered here , instead , is the effect of a localized but extended perturbation of typical length scale similar to the cdw wavelength , acting on a material where the coherence length is macroscopic in more than one dimension . starting from the standard fukuyama - lee - rice model @xcite for cdw , the charge modulation is described through a ginzburg - landau theory as a classical elastic medium . the complex order parameter @xmath6 is considered , taking into account the amplitude degree of freedom @xmath7 , as well as the phase @xmath8 . the charge density can be expressed as @xmath9 , so for the unperturbed system @xmath10 and @xmath11 are constant ( at least locally ) , and the free energy reads : @xmath12=\int\left[-2f_0{\left|\psi({\mathbf{r}})\right|}^2+f_0{\left|\psi({\mathbf{r}})\right|}^4+\kappa{\left|\nabla\psi({\mathbf{r}})\right|}^2\right]\mathrm{d}{\mathbf{r}},\ ] ] where @xmath13 sets the energy scale and @xmath14 represents the elastic energy contribution . considering the external perturbation by the afm tip , generally described as a potential @xmath15 coupling to the charge density , adds an interaction term to the free energy : @xmath16=\int\left[v({\mathbf{r}})\mathrm{re}(\psi({\mathbf{r}})e^{i{\mathbf{q}}\cdot{\mathbf{r}}})\right]\mathrm{d}{\mathbf{r}}.\ ] ] a well studied case is the impurity one @xcite , where @xmath17 . in that case phase oscillations are often enough to describe the ground state of the system , which comes from the balance of elastic and potential energy . these point - like perturbations , however , only impose a likewise point - like constraint on the order parameter , and therefore can not lead to a phase slip in the absence of an external driver @xcite . i will therefore consider the case where @xmath15 has a finite width of the order of the wavelength @xmath5 , and minimize the total free energy @xmath18 given a specific shape of @xmath15 . following in the steps of the impurity model and considering only phase perturbations , the functional would be @xmath19=\int\left[\kappa{\left|\nabla\phi({\mathbf{r}})\right|}^2+v({\mathbf{r}})\rho_0\cos({\mathbf{q}}\cdot{\mathbf{r}}+\phi({\mathbf{r}}))\right]\mathrm{d}{\mathbf{r}}.\ ] ] this model , however , is inadequate to describe this specific effect . in fact , considering a purely one - dimensional model would lead to a linear behavior of @xmath8 where the potential is zero . since we expect a decay far from the perturbation , this is a clearly unphysical result . moreover , due to the nature of the phase , which is defined modulo @xmath20 , given some boundary conditions the solution is not univocally defined unless the total variation of @xmath8 is also specified . assuming the phase to have the unperturbed value @xmath21 far from the perturbation , we can define the integer _ winding number _ @xmath22 of a solution as the integral @xmath23 taken along the cdw direction @xmath24 ( with @xmath25 typically representing the unperturbed case ) . since any change in the winding number along the @xmath24 direction would extend to the whole sample and unnaturally raise the energy of such a solution , to recover a physical result one needs to take into account the amplitude degree of freedom , which will allow for the presence of dislocations and local changes in the winding number . for these reasons , the minimization will be performed in two dimensions , with the complete complex order parameter and in a subspace with a defined winding number . the final result is expected to be similar to what previously considered in the wider context of phase slip @xcite and more specifically in the case of localized phase slip centers @xcite . namely , the local strain induced by the perturbation on the phase will reduce the order parameter amplitude , to the point where a local phase slip event becomes possible . in more than one dimension , the boundary between areas with different winding number will be marked by structures such as _ vortices_. from this preliminary analysis , the mechanism responsible for the dissipation peaks can be understood : as the tip approaches the surface , it encounters points where the energies of solutions with different winding number undergo a crossover . at these points the transition between manifolds is not straightforward , due to the mechanism required to create the vortices , therefore the oscillations lead to jumps between different manifolds , resulting in hysteresis for the tip and ultimately dissipation . to asses the validity of the proposed mechanism , numerical simulations of the tip - surface interaction were performed . a two - dimensional model is considered , since it takes into account the relevant effects while keeping the simulation simpler ; moreover , the experimental substrate has a quasi - two - dimensional structure , so that volume effects can be expected to be negligible . differently from the experimental system @xcite , the cdw is characterized by a single wave vector @xmath24 , leading to a simpler order parameter and a clearer effect . to represent the effect of the tip , the shape of a van der waals potential @xmath26 is integrated over a conical tip at distance @xmath27 from the surface . the result can be reasonably approximated in the main area under the tip by a lorentzian curve @xmath28 where @xmath29 is the distance in the plane from the point right below the tip and the parameters are found to scale like @xmath30 and @xmath31 . free energy @xmath32 as a function of tip distance @xmath27 for subspaces with different winding number @xmath22 . results from simulations on a 201@xmath33201 grid with parameters ( see text ) @xmath34 ev / nm , @xmath35 ev , @xmath36 nm@xmath37 , @xmath38 ev@xmath39 nm , @xmath40 nm@xmath37 and boundary conditions @xmath41 . insets : charge density @xmath2 ( full lines ) and phase @xmath8 ( dashed lines ) along the line passing right below the tip ( indicated by the vertical dashed line ) for different winding number @xmath22 , at the positions indicated by the dots on the energy curves . ] knowing the shape of the perturbation , the total free energy @xmath18 is minimized numerically on a square grid of points with spacing much smaller than the characteristic wavelength of the cdw , imposing a constant boundary condition @xmath42 on the sides perpendicular to @xmath24 , while setting periodic boundary conditions in the other direction to allow for possible phase jumps . the minimization is carried out through a standard conjugated gradients algorithm @xcite . the parameters employed are order of magnitude estimates of the real parameters , which reproduce the relevant experimental effects in a qualitative fashion . the charge density @xmath2 and phase @xmath8 profile along the line passing right below the tip is shown in the insets of fig . [ fig : encurves ] for solutions with different winding number @xmath22 . the position of the attractive tip is indicated by the vertical dashed line , and the phase slip occurred under it , leading to and increase in charge density , is clearly visible . parallel lines far from the tip always revert to the @xmath25 manifold , through the creation of vortices . the main curves in fig . [ fig : encurves ] represent the minimum energy at defined @xmath22 as a function of the distance @xmath27 . since the solution with a given winding number lies in a local minimum , it is possible to use the minimization algorithm to find solutions in a certain subspace , even when this is not the global minimum for a specific distance , by starting from a reasonable configuration ( namely , the minimum at close distance ) . two different crossing points can be seen , which would give rise to two peaks in the experimental dissipation trace . of course a more complex cdw configuration or different parameters would give rise to more peaks . force as a function of distance for evolutions with @xmath43 nm , @xmath44 nm and different values of @xmath45 with @xmath46 ev@xmath39s . inset : total work @xmath47 as a function of oscillation frequency @xmath45 . ] to completely justify the validity of the dissipation mechanism , we need to look into the dynamics of the cdw , to guarantee that the evolution through a crossing point does not lead to immediate relaxation between different manifolds . to do this , the time evolution of the system was simulated , following the time - dependent ginzburg - landau equation @xcite @xmath48 this equation can be interpreted as an overdamped relaxation , with a coefficient @xmath49 , of the order parameter towards the equilibrium position . integrating this equation ( through a standard runge - kutta algorithm @xcite ) , the force as a function of the distance can be computed for a tip performing a full oscillation perpendicular to the surface according to the law @xmath50 . [ fig : fcurves ] shows the force evolution during such oscillations at different frequencies . as we can see the tip suffers a hysteresis even at low frequencies , since the decay from one manifold to the other happens far from the crossing point . the area of the loops represents directly the dissipated energy per cycle @xmath47 , as reported in the inset . based on these simulations , the theory hereby presented is shown to be compatible with the experimental findings : it accounts for the existence of multiple peaks , their appearance far away from the surface and their nature being related to the cdw structure of the sample . it is interesting to notice that the vortices appearing in our simulations have been described before in the context of cdw conduction noise @xcite , where their creation and movement justifies the phase slip near the cdw boundaries . in this sense , this theory lies in between these macroscopic effect and the simple one - dimensional model of defect pinning and phase slip @xcite , as is appropriate for a localized but extended perturbation . to conclude , i have presented a mechanism to explain peaks in the dissipation of a tip oscillating at specific distances above a cdw surface : these occur around instability points corresponding to the crossing of energy levels characterized by different winding numbers . numerical simulations in a system of reduced complexity support the validity of this mechanism . it would be interesting to investigate the same effect in other systems displaying cdw or even spin density waves , as well as systems where the origin of charge modulation is related to the fermi surface and not to other effects . _ acknowledgements _ the author thanks his collaborators g.e . santoro and e. tosatti . he acknowledges research support by snsf , through sinergia project crsii2 136287/1 , by erc advanced research grant n. 320796 modphysfrict , and by miur , through prin-2010llkjbx_001 vanossi a. , manini n. , urbakh m. , zapperi s. , and tosatti e. , rev . phys . * 85 * 529 ( 2013 ) . langer m. , kisiel m. , pawlak r. , pellegrini f. , santoro g.e . , buzio r. , gerbi a. , balakrishnan g. , baratoff a. , tosatti e. , and meyer e. , nature mater . * 13 * 173177 ( 2014 ) . grner g. , rev . phys . * 60 * 1129 ( 1988 ) . fukuyama h. , and lee p.a . b * 17 * 535 ( 1978 ) . lee p.a . , and rice t.m . , b * 19 * 3970 ( 1979 ) . tucker j.r . b * 40 * 5447 ( 1989 ) . ttt i. , and zawadowski a. , phys . b * 32 * 2449 ( 1985 ) . maki k. , phys . a * 202 * 313 ( 1995 ) . gorkov l.p . , zh . eksp . 86 * 1818 ( 1984 ) . mcmillan w.l . b * 12 * 1187 ( 1975 ) . press w.h . , teukolsky s.a . , vetterling w.t . , and flannery b.p . _ numerical recipes : the art of scientific computing ( 3rd ed . ) _ , cambridge university press ( 2007 ) . , verma g. , and maki k. , phys . lett . * 52 * 663 ( 1984 ) . | a mechanism is proposed to describe the occurrence of distance - dependent dissipation peaks in the dynamics of an atomic force microscope tip oscillating over a surface characterized by a charge density wave state .
the dissipation has its origin in the hysteretic behavior of the tip oscillations occurring at positions compatible with a localized phase slip of the charge density wave .
this model is supported through static and dynamic numerical simulations of the tip surface interaction and is in good qualitative agreement with recently performed experiments on a nbse@xmath0 sample .
the study of the microscopic mechanisms leading to energy dissipation and friction has very important theoretical and practical implications . in recent years
, experiments have started to single out the effects of microscopic probes in contact or near contact with different surfaces , and much theoretical effort has been devoted to the full understanding of such experiments @xcite . in particular ,
the minimally invasive non - contact experiments offer a chance to investigate delicate surface properties and promise to bring new insight on localized effects and their interaction with the bulk .
recently , a non - contact atomic force microscopy ( afm ) experiment @xcite on a nbse@xmath0 sample has shown dissipation peaks appearing at specific heights from the surface and extending up to @xmath1 nm far from it .
these peaks were obtained with tips oscillating both parallel and perpendicular to the surface , and in a range of temperatures compatible with the surface charge density wave ( cdw ) phase of the sample . in this paper ,
a model is proposed explaining in detail the mechanism responsible for these peaks : the tip oscillations induce a charge perturbation in the surface right under the tip , but , due to the nature of the cdw order parameter , multiple stable charge configurations exist characterized by different `` topological '' properties .
when the tip oscillates at distances corresponding to the crossover of this different manifolds , the system is not allowed to follow the energy minimum configuration , even at the low experimental frequencies of oscillation , and this gives rise to a hysteresis loop for the tip , leading to an increase in the dissipation .
while the idea behind this dissipation mechanism has been proposed by the author and collaborators in the original paper @xcite , this article expands on the technical aspects of the model , highlighting details to appear in a future publication . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
dielectric resonators have applications in microwave and optical frequency ranges , including antennas @xcite and as building blocks of impedance - matched huygen s metasurfaces @xcite . approximate methods for finding the modes of dielectric resonators are known @xcite , which usually assume that @xmath0 . these methods are inaccurate for the moderate values of permittivity available at optical frequencies , and more sophisticated methods are needed to account for radiation effects . open nanophotonic resonators such as meta - atoms , nano - antennas and oligomers are typically strongly radiative systems , where loss can not be treated as a perturbation . furthermore , in many nanophotonic systems , material dispersion and losses can not be neglected , further complicating the problem of finding their modes . in radiating and dissipative systems the modes have complex frequencies @xmath1 , corresponding to damped oscillations of the form @xmath2 , with @xmath3 . the corresponding modal fields @xmath4 do not possess the orthogonality usually found in the modes of closed systems , and they are commonly referred to as quasi - normal modes @xcite . they are particularly useful for solving dipole emission problems @xcite , since they allow a mode volume to be defined for open cavities @xcite . a significant practical difficulty is the requirement to normalise a mode with diverging far - fields @xcite . a different perspective on this problem can be found within the microwave engineering literature @xcite , originally motivated by time - domain radar scattering problems . by using integral methods to solve maxwell s equation , only currents on the scatterer need to be solved for , avoiding having to explicitly handle the diverging far - fields . as it is based on finding the singularities of a scattering operator , this approach is referred to as the singularity expansion method ( sem ) . the field distributions corresponding to these singularities are identical to the quasi - normal modes at the complex frequencies of the singularities @xmath1 . when solving scattering problems on the @xmath5 axis , the fields in the sem approach are reconstructed from the dyadic green s function , which remains finite in the far - field . thus the sem avoids the most significant practical disadvantage of quasi - normal modes based on fields . recently it has been shown that the singularity expansion method can be applied to meta - atoms and plasmonic resonators @xcite , clearly identifying the modes which contribute to scattering and coupling problems . however , finding all modes within a region of the complex frequency plane requires an iterative procedure with multiple contour integrations@xcite . this greatly increases the computational burden , and it remains unclear how robust this procedure is . in addition , it has not yet been demonstrated whether all spectral features can be explained by such a model , particularly the interference between non - orthogonal modes in the extinction spectrum and suppression of back - scattering corresponding to the huygens condition@xcite . in this work , a robust integral approach to finding modes of open resonators is demonstrated for an all - dielectric meta - atom , based on the singularity expansion method . it is shown how this leads to a clear decomposition of the extinction spectrum of a silicon disk , automatically accounting for interference between the non - orthogonal modes . by performing a vector spherical harmonic decomposition of each mode , the unidirectional scattering behaviour of the disk is explained . it is shown that higher - order modes can also interfere to supress back - scattering , corresponding to the previously reported generalized huygens condition @xcite . a brief overview of integral equation methods to solve maxwell s equations is given , followed by the robust approach to find the modes . here dielectric objects are considered , and treated through a surface equivalent problem , with surface equivalent electric and magnetic currents , @xmath6 and @xmath7 , where @xmath8 is the surface normal . these surface currents can be excited by the incident electric or magnetic field , yielding the electric field integral equation and magnetic field integral equation respectively . to yield a stable solution , both of these equations must be combined using some chosen weighting coefficients@xcite . in this work the combined - tangential form is used , as detailed in ref . . this gives us an operator equation relating equivalent surface currents to the tangential components of the incident fields @xmath9 in this work the time convention @xmath10 is used , with @xmath11 . note that in contrast to other conventions , the imaginary part gives the oscillation rate , and the real part gives the decay rate . we could find the corresponding time - domain function @xmath12 of a frequency - domain function @xmath13 through the inverse laplace transform @xmath14 . physically observable quantities must be represented by a real function in the time domain , thus they must satisfy the constraint @xmath15 in the frequency domain . equation is solved using the boundary element method ( also known as the method of moments@xcite ) . after choosing sets of basis functions @xmath16 and testing functions @xmath17 ( both are loop - star functions @xcite in this work ) , the operator equation becomes a finite - dimensional matrix equation @xmath18 where @xmath19 is the vector containing the weighted equivalent surface currents @xmath20^t,\ ] ] and @xmath21 is the source vector containing the projected incident fields @xmath22^t.\ ] ] the impedance matrix @xmath23 is dense and frequency dependent , and contains all information regarding the response of the scatterer to arbitrary incident fields . the unknown current vector @xmath19 is solved for a given incident field vector @xmath21 , with the solution is given by @xmath24 . the singularities of @xmath25 will dominate the spectrum of the response , and by mittag - leffler s theorem the response may be expanded in terms of these singularities @xcite . they correspond to solutions which can exist in the absence of a source , and hence they can be used to model the response to an arbitrary incident field . the most important singularities of the impedance matrix are its poles , corresponding to the quasi - normal modes of the system . in practice it may usually be assumed that all poles are of first order @xcite . the poles correspond to frequencies @xmath26 , satisfying @xmath27 and @xmath28 for non - zero @xmath29 and @xmath30 . these are the left and right eigenvectors of the system respectively , with @xmath29 being the surface current distribution of the mode . the singularities of the impedance matrix are found by the contour integration procedure outlined in ref . . first a pair of matrix integrals @xmath31 and @xmath32 is evaluated about a contour containing all modes of interest , as shown schematically in fig . [ fig : contour ] . the contour is offset slightly from the @xmath5 axis to eliminate any modes which do not couple to incident radiation , hence have @xmath33 . the desired radiating modes are shown by green crosses , and have @xmath3 . since currents must be real functions in the time domain , for each pole there is a corresponding complex conjugate pole at @xmath34 , shown in orange . as the poles and residues are just complex conjugates of those with positive @xmath35 , they can be found by symmetry , and do not need to be included within the contour . note that some poles are over - damped , with @xmath36 , and these poles do not appear in conjugate pairs . the contour incorporates the @xmath37 axis in order to capture these poles . an arc is used to eliminate spurious numerical poles which cluster near the origin when using integral operators of the first kind @xcite . and their residues with only a single integration . green crosses : physical modes with finite radiation damping . red crosses : spurious internal solutions with no damping . orange crosses : conjugate modes which can be found by symmetry.[fig : contour ] ] the mode frequencies and currents are eigenvalues and eigenvectors of @xmath38 . a singular value decomposition is used to determine the number of valid solutions to this equation @xcite and solving for the corresponding left eigenvalue problem yields the projectors @xmath30 . this procedure can yield solutions lying both inside and outside the contour , and those falling outside the contour are discarded . the poles and currents are further improved by newton iteration , then normalised so that @xmath39 . this ensures that the dyadic product of the eigenvectors matches the pole residue , i.e. @xmath40 in the vicinity of @xmath41 , simplifying the pole expansion . note that in general no orthogonality relation exists between these mode currents . as is discussed in appendix [ sec : orthogonality ] , orthogonality is not required for this approach . it will be shown how this non - orthogonality leads to physically meaningful interference effects . due to geometric symmetry , many modes are degenerate , with several different eigenvectors having the same pole location @xmath41 . when solving the structure numerically , the imperfect symmetry of the mesh usually results in some frequency splitting of these degenerate modes , so a thresholding procedure is used to group closely spaced poles . the contour integration and iterative search procedure were found to cope with these nearly degenerate poles without requiring any special handling . note that it is not necessary to orthogonalise degenerate modes , since the method is intrinsically able to account for non - orthogonality , as long as the modes span the full eigenspace . once the modes have been found , currents can be solved for arbitrary incident fields , @xmath42 where we consider excitation at physically realisable frequencies on the @xmath5 axis . the projector @xmath30 operates on the incident field @xmath21 to give its overlap with the mode . the bracketed term accounts for close the excitation frequency is to the mode s resonant frequency . note that this polynomial has the correct asymptotic behaviour , thus improving the convergence and removing the need to include an entire function contribution@xcite . the important result obtained from eq . is a scalar weighting of each mode s current vector @xmath29 . regardless of whether it is calculated directly from eq . or as a superposition of modes from eq . , the current vector @xmath19 can be used to find the currents at any point on the surface . the dyadic green s function can be then be used to find the fields anywhere in space . in practice this is not usually necessary , since many quantities of physical interest such as scattering , radiation forces and torques can be calculated directly from the currents @xcite . the quantity of most interest is the extinction cross - section @xmath43 giving the total work done by the incident fields on the currents . here @xmath44 is the electric field of the incident plane - wave . this quantity can be defined for each mode by substituting the mode s current and its weighting from eq . . , summed over all values of azimuthal index @xmath45.[fig : extinction_multipole ] ] the techniques outlined in section [ sec : modelling ] are now applied to study the scattering behaviour of a silicon disk meta - atom . as a first step , the structure is modelled directly without considering the modes , using eq . . the radius is taken as 242 nm , height 220 nm and edges are rounded with radius 50 nm . the material properties of silicon were obtained by fitting an 8 pole model to the experimental data from ref . . in fig . [ fig : extinction_multipole ] the extinction cross - section of the disk is plotted . the incident wave - vector is parallel to the axis of the disk . as a first attempt to explain the spectral features , a multipole expansion is also shown in fig . [ fig : extinction_multipole ] . details of the expansion are given in appendix [ sec : multipole ] . solid lines show the electric multipole moments @xmath46 , and dashed curves show the magnetic moments @xmath47 . although the multipoles accurately reproduces the total extinction , there is no direct correspondence between modes and multipoles , with each peak exhibiting contributions from many multipole moments . furthermore , several multipole moments show peaks and dips at similar locations , but it is unclear if these moments are linked to each other . therefore _ the multipole decomposition is unable to resolve the internal dynamics _ which are observed in the extinction spectrum . it will be demonstrated that the model based on eq . can resolve these internal dynamics , showing which modes correspond to each of the spectral features . ] the modes of the silicon disk are found by the procedure outlined in section [ sec : finding - modes ] . figure [ fig : extinction_modes](a ) shows the location of the poles in the complex frequency plane , with many of them being doubly degenerate . since currents decay in time as @xmath48 , more highly damped modes have more negative values of @xmath49 . the schematic of the incident field orientation is shown in the inset . the modes which most strongly couple to this incident field are marked with coloured dots . the equivalent surface current @xmath50 of the first 5 of these modes is shown in fig . [ fig : mode - currents ] . note that these currents are complex , hence the plotted vectors give a snapshot of the oscillating current distribution . the divergence @xmath51 is proportional to the equivalent surface charge ( and hence to the normal component of the electric field ) and is indicated by the shading of the surface . the colors of the markers next to each current distribution correspond to the poles shown in fig . [ fig : extinction_modes](a ) . each mode is also given an arbitrary label in roman numerals for reference purposes . . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents],title="fig : " ] + ' '' '' . right : spherical multipoles of each mode , scaled normalised according to the total scattered power.[fig : mode - currents ] ] to understand the nature of these modes , we compare the disk with rounded edges to the sphere , since the two are topologically equivalent . in ref . it is shown that the poles of a sphere can be found from the roots of the denominators of the coefficients from mie theory , involving spherical bessel and spherical hankel functions . the field corresponding to each of these poles is a pure vector spherical harmonic , with poles corresponding to radial index @xmath52 having degeneracy of @xmath53 . note that for each vector spherical harmonic there is an infinite number of poles , corresponding to different number of radial oscillations inside the sphere @xcite . we can consider the dielectric disk to be a sphere which has been transformed in a continuous manner , breaking the spherical symmetry . this means that the total number of poles is the same for both geometries , but the degeneracy is reduced by poles splitting to different locations . as a result , the corresponding current for each pole of the disk is not a pure vector spherical harmonic . by performing a multipole decomposition of the current for each mode of the disk , we can see which mode of the sphere it is most closely related to . this is shown in the right column of fig . [ fig : mode - currents ] , where each mode s multipole moments are normalised to the total scattered power , as outlined in appendix [ sec : multipole ] . in all cases there is a single dominant multipole moment , although for higher order modes the influence of higher moments becomes more significant . in the following sections this multipole expansion of the modes will be used to explain their contributions to extinction and scattering . figure [ fig : extinction_modes](b ) shows the extinction contribution from each of the modes , obtained from each term in eq . . the extinction from degenerate pairs of modes has been combined , along with the contribution of their conjugate modes at @xmath34 . it can be seen that all features in the extinction spectrum can be clearly attributed to the modal contributions . the extinction spectrum for each mode exhibits only a single feature , being a peak and/or dip in the vicinity of its pole frequency @xmath54 . there is a very clear correspondence between the damping rate @xmath49 and the sharpness of the features in the corresponding extinction curve . note that for more highly damped modes , there is some shift between the peak and pole frequencies . this is because such modes couple strongly to the incident field , and therefore the overlap term in eq . can shift the spectral features away from the natural frequency @xmath35 . the accuracy and convergence of this model of extinction is shown in appendix [ sec : accuracy ] . one of the most striking features of fig . [ fig : extinction_modes](b ) is that several modes show negative contributions to extinction . this is due to the non - orthogonality of the modes , which means that even if the incident field matches the profile of one mode , it may still excite others . it can be seen that the dip in extinction at around 260thz can be attributed to a strong negative contribution from mode iii , emitting radiation in the forward direction that is in - phase with the incident field . in ref . it was shown how extinction can be decomposed into direct terms from each mode , plus inteference terms between every pair of modes . the current obtained from eq . does not explicitly show separate direct and interference terms . energy conservation dictates that the total extinction must be positive , so a sufficient set of modes must be included to have a physically meaningful result . to quantify the interference between modes , their overlap @xmath55 is plotted in fig . [ fig : mode - overlap ] , normalised such that @xmath56 . this indicates how strongly an incident field having the shape of mode @xmath45 excites mode @xmath57 . for example , we see strong overlap between modes i and iii , showing that the field which excites mode iii also strongly excites mode i. this leads to the strong negative extinction observed for mode iii . the strong overlap of modes i and iii is is consistent with their multipole decomposition shown in fig . [ fig : mode - currents ] , where both are dominated by the @xmath58 electric dipole term . the strongest mode overlap observable in fig . [ fig : mode - overlap ] is between modes ii and iv , consistent with both having strong magnetic dipole moment @xmath59 . however this does not lead to strong interference in fig . [ fig : extinction_modes](b ) . the low damping rates of these modes seen in fig . [ fig : extinction_modes](a ) results in narrow resonant peaks which have little overlap . a notable feature of fig . [ fig : mode - overlap ] is that mutual overlap terms can be greater than self terms , a consequence of the unconjugated inner product which appears in the formalism . it should also be noted that similarities in the multipole decomposition of modes is not always a good predictor of their overlap . for example , modes ii and v have both have dominant magnetic dipole moments @xmath59 , but nonetheless have relatively weak overlap observable in fig . [ fig : mode - overlap ] . ] ] to calculate the total scattering cross section , vector spherical harmonics are used , since the total scattering is the incoherent sum of all multipole contributions , given by eq . . figure [ fig : multipole - scattering ] shows the contribution of each multipole coefficient to the scattering cross - section . as with the multipole extinction spectrum shown in fig . [ fig : extinction_multipole ] , the features of the multipole scattering spectra are rather complex , but can be explained by considering the contributions of different modes . in the wavelength range above 1000 nm , corresponding to measured range in ref . , it can be seen that the scattering is dominated by the electric dipole and magnetic dipole moments @xmath60 @xmath61 . the magnetic dipole moment can be attributed to the resonance of mode ii , which has negligible contributions from other moments . the electric dipole moment @xmath58 appears to have two distinct maxima in fig . [ fig : multipole - scattering ] . from the coefficients shown in fig . [ fig : mode - currents ] , it is clear that only modes i and iii contribute to this dipolar scattering . from fig . [ fig : extinction_modes](b ) , we can see that mode i has a very broad resonance , while mode iii has a much narrower resonance , with a negative contribution to extinction . this results in cancellation of electric dipole radiation , corresponding to an anapole distribution@xcite . this effect is typically explained in terms of a quasi - static electric dipole ( a linear current distribution ) interfering with a toroidal dipole ( a poloidal current distribution ) . the surface currents shown in fig . [ fig : mode - currents ] are consistent with this explanation , however the explanation in terms of modes is more general , and does rely on any low frequency approximation . indeed , in ref . it was shown that for spheres , this condition occurs when the contributions from the first and second @xmath58 modes cancel . the situation for the disk is similar , the difference being that the interfering modes i and iii have additional contributions from other multipole moments . ] for applications in huygens metasurfaces , the most important attribute of a meta - atom is to have suppressed back scattering and strong forward scattering . this is typically achieved by overlapping electric and magnetic dipole type resonances . [ fig : directional - scattering ] shows the forward and backward scattering amplitudes , with peaks labelled according to the corresponding resonant modes . the first peak of forward scattering corresponds to the overlap of modes i and iii , with almost purely electric dipole radiation , and mode ii , with almost purely magnetic dipole radiation . it can also be seen that at the resonances of modes iv and v there are additional highly directional scattering features , as these modes also overlap with the electric - dipole type modes i and iii . examining the multipole decompositions in fig . [ fig : mode - currents ] , it can be seen that mode iv is dominated by its electric quadrupole response , with a significant contribution from its magnetic dipole response . in contrast , mode v is dominated by its magnetic dipole response , with lesser contributions from electric quadrupole and magnetic octupole moments . it is significant that all of these multipole moments radiate anti - symmetric electric fields into the forward and backward directions . thus all of these moments are able to cancel the electric dipole and magnetic quadrupole moments of modes i and iii , which radiate with symmetric electric fields in the forward and backward direction . considering the contribution of modes to this directional scattering process , the generalised huygens condition introduced in ref . can be re - interpreted as interference between modes of different symmetry . this suggests that to optimise this generalised huygens effect , the meta - atoms should be placed within a homogeneous dielectric environment , as has been done for all - dielectric huygens metasurface @xcite . a dielectric substrate without a compensating superstrate introduces coupling between modes of opposite symmetry@xcite , greatly complicating the design process and degrading the directionality of scattering a robust technique based on the singularity expansion method was presented to find the modes of a meta - atom , fully accounting for radiative losses . by solving maxwell s equations using integral techniques , the normalisation of diverging fields typically required when using quasi - normal modes is avoided . the technique was applied a silicon disk , a building block which enables optical metasurfaces having low loss , and full manipulation of the transmitted phase . it was demonstrated that the complicated features of the extinction spectrum can be readily explained in terms of contributions from the modes . interference between non - orthogonal modes was shown to play a key role . when considering far - field scattering properties , a vector spherical harmonic expansion yields an accurate , if somewhat opaque , description . by combining it with the modal analysis , the nature and origin of all scattering features can be elucidated . in the case of the silicon disk , there are several bands of strong forward scattering and suppressed backscattering , corresponding to the generalised huygens condition . it was shown that each band corresponds to the overlap of modes with odd and even radiation symmetry . the techniques used to find modes and construct models of scatterers are implemented in an open - source code openmodes@xcite , along with notebooks to reproduce all results in this paper@xcite . the author acknowledges useful discussions with andrey miroshnichenko , sarah kostinski , mingkai liu , and yuri kivshar . this research was funded by the australian research council . the electric multipole coefficients @xmath62 and magnetic multipoles coefficients @xmath63 were computed directly from the surface currents using the formulas from ref . . duality allows these formulas to be generalised to include the equivalent magnetic currents through the substitution @xmath64 . the normalisation of multipole coefficients from ref . is used , as this simplifies the expression for scattering cross - section , which is given by @xmath65 where the coefficients include contributions from all values of azimuthal index @xmath45 : @xmath66 in fig . [ fig : mode - currents ] @xmath67 and @xmath68 are normalised to their sum , and their square root is plotted since it more clearly shows the smaller contributions . in fig . [ fig : multipole - scattering ] these terms are plotted including the pre - factor from eq . to give them dimensions of scattering cross - section . for a plane wave propagating in the @xmath69 direction , with incident electric field along the @xmath70 direction , the extinction cross - section is given by @xcite @xmath71\right.\nonumber\\ * + \left.\left[\sum_{m=-1,1}m\mathrm{im}\{b_{lm}\}\right]\right ) . \label{eq : multipole_extinction}\end{aligned}\ ] ] the quantities in square brackets are plotted in fig . [ fig : extinction_multipole ] , including all common pre - factors in eq . . for 3 terms of the multipole expansion , the extinction plotted in fig . [ fig : extinction_multipole ] agrees with the direct calculation to a relative error below 2% for frequencies below 350thz . by adapting the formulas from mie theory @xcite , forward scattering can be found as @xmath72 while back - scattering is given by @xmath73 as losses are low in this system , the total extinction and scattering are approximately equal , due to the optical theorem . however , this still allows each multipole s contribution to extinction shown in fig . [ fig : extinction_multipole ] to be different from its contribution to scattering shown in fig . [ fig : multipole - scattering ] . as discussed in ref . , the electric fields of quasi - normal modes do not obey the usual orthogonality relationship based on a conjugated inner product , i.e. @xmath74 . however , they do obey an unconjugated orthgonality relationship , which is required for normalisation of modes@xcite , and projection of external fields onto modal fields . in contrast , the current vectors on the scatterer obtained from the singularity expansion method do not exhibit any form of orthogonality . however , such orthogonality is not required when working with modal currents , since they are normalised by weighting them to match the residue of the pole , as shown in eq . . in addition to providing the current vector @xmath29 , this approach also yields the correctly normalised projector @xmath30 , which gives the projection of an arbitrary field onto each mode by a simple scalar product , as used in eq . . it is noted that in the literature a number of orthogonal decompositions of the impedance matrix @xmath75 have been presented , most prominently the characteristic mode analysis @xcite . as these mode vectors are real , they exhibit the conventional conjugated orthogonality . however , such decompositions suffer from a number of problems which make them unsuited for physically modelling open resonators . first , the eigenvalue problem must be solved at each frequency , yielding a different set of current vectors at each frequency . this requires some algorithm to track modes with frequency @xcite , and effectively prevents their use in time - domain problems . more significantly , the enforcement of mode orthogonality on an inherently non - hermitian system results in an artificial set of basis vectors which contain a complex mixture of underlying eigenvectors . this manifests itself in unphysical avoided crossings , whereby the nature of a pair of modes is swapped in some frequency region @xcite . the author has observed similar behaviour when utilising other orthogonal decompositions of the impedance matrix , such as the singular value decomposition . in order to reproduce the interference phenomena observed in fig . [ fig : extinction_modes ] , it is essential to use the non - orthgonal modes obtained from singularity expansion method , or quasi - normal modes approaches . ] to confirm the accuracy of the mode expansion , the directly calculated extinction curve is plotted in fig . [ fig : extinction - accuracy ] ( solid line ) , as well as the sum of all contributions plotted in fig . [ fig : extinction_modes](b ) ( red dashed line ) . it can be seen that the agreement is good for frequencies below 250thz , however at high frequencies it becomes poorer . by increasing the number of poles considered from 28 ( i.e. 7 modes , each doubly degenerate and with conjugate poles ) to 145 , much better agreement is achieved , as shown by the blue dashed curve . clearly a model involving so many parameters is less useful as a design tool , thus there is an inevitable trade - off between accuracy and the level of insight provided . however , in contrast to simpler approaches based on point dipole or equivalent circuit models , it is possible to control the level of detail which is included within the model by choosing to include or exclude poles . as this work includes materials with dispersion and dissipative losses , the impedance matrix @xmath23 may exhibit branch point singularities , in addition to poles . the green s function used to calculate elements of the impedance matrix has terms proportional to @xmath76 . the complex wave - number @xmath77 has branch points at the poles and zeros of the permittivity , connected by branch cuts@xcite . for the material data used in this work , all such branch points occur at frequencies above 800thz , thus their contribution is neglected in eq . . the accuracy of the results shown in fig . [ fig : extinction - accuracy ] confirms that no significant contribution from branch points is missing from the result . the lack of branch points in the frequency range of interest also ensures that the integration contour illustrated in fig . [ fig : contour ] does not intersect any of the branch cuts . applying the contour integration in a frequency range of high dispersion would require choosing the contour so that it encircles branch points in pairs to avoid crossing branch - cuts . 39ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty , ed . , @noop _ _ , no . ( , , ) link:\doibase 10.1002/adom.201400584 [ * * , ( ) ] link:\doibase 10.1109/tmtt.1975.1128528 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.70.1545 [ * * , ( ) ] link:\doibase 10.1021/ph400114e [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.237401 [ * * , ( ) ] link:\doibase 10.1103/physreva.92.053810 [ * * , ( ) ] link:\doibase 10.1109/proc.1976.10379 [ * * , ( ) ] link:\doibase 10.1109/jstqe.2012.2227684 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.075108 [ * * , ( ) ] link:\doibase 10.1103/physrevb.89.165429 [ * * , ( ) ] link:\doibase 10.1109/jphot.2014.2331236 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.197401 [ * * , ( ) ] link:\doibase 10.1063/1.4949007 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1029/2004rs003169 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1109/8.761074 [ * * , ( ) ] link:\doibase 10.1080/02726348108915144 [ * * , ( ) ] link:\doibase 10.1080/02726348108915141 [ * * , ( ) ] link:\doibase 10.1109/jlt.2012.2234723 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1109/tap.2015.2438393 [ * * , ( ) ] link:\doibase 10.1002/pip.4670030303 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1103/physreva.93.053837 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.053819 [ * * , ( ) ] link:\doibase 10.1038/ncomms9069 [ * * ( ) , 10.1038/ncomms9069 ] link:\doibase 10.1063/1.3486480 [ * * , ( ) ] @noop `` , '' @noop , link:\doibase 10.1088/1367 - 2630/14/9/093033 [ * * , ( ) ] @noop _ _ , ed . ( , ) @noop _ _ ( , , ) link:\doibase 10.1103/physreva.49.3057 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2579668 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2556698 [ * * , ( ) ] link:\doibase 10.1109/tap.2016.2550098 [ * * , ( ) ] ( ) pp . | the modes of silicon disk meta - atoms are investigated , motivated by their use as a building block of huygens metasurfaces .
a model based on these modes gives a clear physical explanation of all features in the extinction spectrum , in particular due to the interference between non - orthogonal modes . by performing a vector spherical harmonic expansion of each mode ,
the complex features of the far - field scattering spectrum are also readily explained .
it is shown that in general each mode has contributions from many multipole moments .
higher order modes with appropriate symmetry are also able to satisfy the huygens condition , leading to multiple bands of strong forward scattering and suppressed back scattering .
these results demonstrate a robust approach to find the modes of nano - photonic scatterers , commonly referred to as quasi - normal modes . by utilising an integral formulation of maxwell s equations , the problem of normalising diverging far - fields
is avoided .
the approach is implemented in an open - source code . |
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.comparison of recent wide - area low - redshift galaxy surveys [ tab : othersurveys ] [ cols="<,^,^,^",options="header " , ] @xmath3 objects removed from the initial target list ( due to changes in + the 2mass source catalogue after 6dfgs was underway ) . + i d @xmath4 or 9999 in these cases . + * columns : * + ( 1 ) i d : programme i d ( progid in the database ; see sec . [ sec : release ] ) . + ( 2 ) survey sample : first sample ( in order of progid ) in which + object is found . + ( 3 ) 6dfgs spectra : number of spectra obtained for this sample . + note that some objects were observed more than once . the + numbers include spectra of all qualities and galactic sources . + ( 4 ) good @xmath5 : number of robust extragalactic 6dfgs redshifts , + ( those with @xmath6 or 4 ) . reflects contents of database . + ( 5 ) lit . @xmath5 : additional literature extragalactic redshifts ( ignoring + repeats and overlaps ) . + ( 6 ) total @xmath5 : total number of extragalactic redshifts for objects + in this sample . + only @xmath7 redshifts should be used in any galaxy analysis . ( the distinction between @xmath8 and @xmath9 is less important than that between @xmath10 and @xmath8 , since the former represent a successful redshift in either case . ) galaxies with repeat observations have all spectra retained in the database , and the final catalogued redshift is a weighted mean of the measurements with @xmath7 , excluding redshift blunders . descriptions of the redshift quality scheme in its previous forms can be found in sec . 2.1 of @xcite and sec . 4.4 of @xcite . unreliable ( @xmath10 ) or unusable ( @xmath11 ) galaxy redshifts together comprise around 8 percent of the redshift sample . galactic sources ( @xmath12 ) represent another 4 percent . the remaining 110256 sources with @xmath7 are the robust extragalactic 6dfgs redshifts that should be used ( alongside the 14815 literature redshifts ) in any analysis or other application . tables [ tab : distribution ] and [ tab : breakdown ] give the breakdown of these numbers across various 6dfgs sub - samples . redshift uncertainties and blunder rates were estimated from the sample of 6dfgs galaxies with repeat redshift measurements . there were 8028 such redshift pairs , 43percent of which were first - year ( pre-2002.5 ) data . most repeat measurements were made because of a low - quality initial measurement , or because of a change in the field tiling strategy after the first year of observations . we define a blunder as a redshift mismatch of more than 330 ( @xmath13 ) between a pair of redshift measurements that we would expect to agree . the blunder rate on individual 6dfgs redshifts is 1.6percent , the same as reported for the first data release @xcite . in late 2002 , a new transmissive volume - phase holographic ( vph ) gratings ( 580v and 425r ) replaced the existing reflection gratings ( 600v and 316r ) , resulting in improved throughput , uniformity , and data quality . excluding first - year repeats reduces the individual blunder rate to 1.2percent . while the first - year data represent nearly half of all repeat measurements , they represent less than a fifth of the overall survey . table [ tab : blunder ] summarises the blunder rates and other statistics for both the full and post - first - year data . figure [ fig:6dfvs6df ] shows repeat redshift measurements for 6dfgs observations with the vph gratings , representative of the great majority of survey spectra ( around 80 percent : 4570 sources spanning 2002.5 to 2006 ) . blunder measurements ( 106 of them ) have been circled , and the scatter in redshift offset , @xmath14 , as a function of redshift is also shown . not surprisingly , redshifts becoming increasingly difficult to secure as one moves to higher values . the inset in figure [ fig:6dfvs6df ] displays the distribution in @xmath15 for measurement pairs grouped by their redshift quality @xmath8 or 4 classifications . there are 3611 pairs in the non - blunder sample with both measurements of quality @xmath9 , with scatter implying a redshift uncertainty in an individual @xmath9 measurement of @xmath16 . likewise , the scatter in the much smaller @xmath8 sample ( 33 pairs ) suggests @xmath17 . if we include the pre-2002.5 non - vph data , the implied redshift uncertainties are unchanged for @xmath9 and increase slightly in the case of @xmath8 ( @xmath18 ) . note that these redshift uncertainties are less than those estimated in @xcite from the first data release , demonstrating the improved integrity of the 6dfgs data since the early releases . llr + * 6dfgs ( full sample ) : * & + total repeat measurements ( @xmath19 ) : & 8028 + rms scatter of all redshift measurement pairs@xmath20 & @xmath21 + @xmath9 redshift uncertainty ( 6051 sources ) & @xmath22 + @xmath8 redshift uncertainty ( 104 sources ) & @xmath23 + & + number of blunders@xmath24 ( @xmath25 ) : & 260 + 6dfgs pair - wise blunder rate : & 3.2% + 6dfgs single - measurement blunder rate : & 1.6% + + * 6dfgs ( vph grating only , 2002.5 2006 ) : * & + total repeat measurements ( @xmath19 ) : & 4570 + rms scatter of all redshift measurement pairs@xmath20 & @xmath23 + @xmath9 redshift uncertainty ( 3611 sources ) & @xmath26 + @xmath8 redshift uncertainty ( 33 sources ) & @xmath27 + & + number of blunders@xmath24 ( @xmath25 ) : & 106 + 6dfgs pair - wise blunder rate : & 2.3% + 6dfgs single - measurement blunder rate : & 1.2% + + * 6dfgs ( vph only ) vs. sdss dr7 : * & + number of comparison sources ( @xmath28 ) : & 2459 + number of blunders@xmath20 ( @xmath28 ) : & 95 + pair - wise blunder rate : & 3.9% + implied blunder rate for sdss : & 2.7% + + @xmath3 clipping the most extreme @xmath29% of outliers ( @xmath30% either side ) . + @xmath31 a blunder is defined as having @xmath32(@xmath13 ) . + an external comparison of 6dfgs redshifts to those overlapping the seventh data release ( dr7 ) of the sloan digital sky survey ( sdss ; * ? ? ? * ) was also made and is shown in fig . [ fig : litvs6df ] . although the full sdss dr7 contains over a million classified extragalactic spectra , almost all are too northerly to overlap significantly with the southern 6dfgs or are too faint . however , the 2459 sources in common to both catalogues provide a valuable test of redshift success rates . the pair - wise blunder fraction in this case is 3.9percent . splitting this with the 6dfgs blunder rate of 1.2percent implies an sdss blunder rate of 2.7percent , although the 6dfgs blunder rate at the fainter sdss magnitudes is likely to be somewhat higher than the 1.2percent measured overall . the 6dfgs online database is hosted at the wide field astronomy unit of the institute for astronomy at the university of edinburgh . data are grouped into 15 inter - linked tables consisting of the master target list , all input catalogues , and their photometry . users can obtain fits and jpeg files of 6dfgs spectra as well as 2mass and supercosmos postage stamp images in and where available , and a plethora of tabulated values for observational quantities and derived photometric and spectroscopic properties . the database can be queried in either its native structured query language ( sql ) or via an html web - form interface . more complete descriptions are given elsewhere @xcite , although several new aspects of the database are discussed below . figure [ fig : example ] shows two examples of the way data are presented in the database . table [ tab : paramcontents ] shows the full parameter listing for the 6dfgs database . individual database parameters are grouped into lists of related data called _ tables_. parameter definitions are given in documentation on the database web site . the target table contains the original target list for 6dfgs , and so contains both observed and unobserved objects . individual entries in this table are celestial sources , and the targetid parameters are their unique integer identifiers . note that the original target list _ can not _ be used to estimate completeness , due to magnitude revisions in both the 2mass xsc and supercosmos magnitudes subsequent to its compilation . item ( iv ) below discusses this important issue in more detail . the spectra table holds the redshift and other spectroscopic data obtained by the 6df instrument through the course of the 6dfgs . many new parameters have been introduced to this table for this release ( indicated in table [ tab : paramcontents ] by the @xmath3 symbol ) . individual entries in this table are spectroscopic observations , meaning that there can be multiple entries for a given object . the specid parameter is the unique integer identifier for 6dfgs observations . most 6dfgs spectra consist of two halves , observed separately through different gratings , and subsequently spliced together : a v portion ( @xmath335600 ) and an r portion ( @xmath347500 ) . or @xmath35 passbands . ] ( data taken prior to october 2002 used different gratings , spanning 40005600 and 55008400 . ) various parameters in spectra belonging to the individual v or r observations carry a _ v or _ r suffix , and are listed in table [ tab : paramcontents ] for v ( with slanted font to indicated that there is a matching set of r parameters ) . the twomass and supercos tables hold relevant 2mass xsc and supercosmos photometric and spatial information . likewise , the remaining eleven tables contain related observables from the input lists contributing additional 6dfgs targets to target . while some of the parameter names have been duplicated between tables ( e.g. mag_1 , mag_2 ) their meaning changes from one table to the next , as indicated in table [ tab : paramcontents ] . database tables can be queried individually or in pairs . alternatively , positional cross - matching ( r.a . and dec . ) can be done between database sources and those in a user - supplied list uploaded to the site . search results can be returned as html - formatted tables , with each entry linking to individual gif frames showing the 6dfgs spectrum alongside its @xmath36 postage stamp images , as shown in fig . [ fig : example ] . individual object fits files of the same data can also be accessed in this way . long database returns can also be emailed to the user as an ascii comma - separated variable ( csv ) text file . alternatively , the fits files of all objects found through a search can be emailed to the user as a single tar file under a _ tar saveset _ option . llcl + table name & description & progid & parameters + + target & the master target list & @xmath37 & targetid , targetname , htmid , ra , dec , cx , cy , cz , gl , gb , + & & & a_v , progid , bmag , rmag , sg , zcatvel , zcaterr , zcatref , + & & & bmagsel , rmagsel , templatecode@xmath3 , framename + spectra & redshifts and observational data & @xmath37 & specid , targetid , targetname , obsra , obsdec , match_dr , + & & & htmid , cx , cy , cz , z_origin , z , z_helio , quality , abemma , + & & & nmbest , ngood , z_emi , q_z_emi , kbestr , r_crcor , z_abs , + & & & q_z_abs , q_final , ialter , z_comm , zemibesterr , zabsbesterr , + & & & zfinalerr , _ title_v , cenra_v , gratslot_v , cendec_v , _ + & & & _ appra_v , appdec_v , actmjd_v , conmjd_v , progid_v , _ + & & & _ label_v , obsid_v , run_v , exp_v , ncomb_v , gratid_v , _ + & & & _ gratset_v , gratblaz_v , source_v , focus_v , tfocus_v , _ + & & & _ gain_v , noise_v , ccd_v , utdate_v , utstrt_v , _ + & & & _ mjdobs_v , name_v , thput_v , ra_v , dec_v , x_v , y_v , _ + & & & _ xerr_v , yerr_v , theta_v , fibre_v , pivot_v , recmag_v , _ + & & & _ pid_v , _ framename , _ axisstart_v , axisend_v , _ matchspecid , + & & & z_initial@xmath3 , z_helio_initial@xmath3 , z_update_flag@xmath3 , z_update_comm@xmath3 , + & & & slit_vane_corr@xmath3 , quality_initial , xtalkflag@xmath3 , xtalkscore@xmath3 , + & & & xtalkveloff@xmath3 , xtalkcomm@xmath3 , quality_update_comm , deprecated@xmath3 + & & & revtemplate@xmath3 , revcomment@xmath3 , z_comm_inital@xmath3 + twomass & 2mass input catalogues & 1 ( @xmath38 ) , & objid , catname , targetname , targetid , ra , dec , priority , + & & 3 ( @xmath39 ) , & mag_1 , progid , mag_2 , j_m_k20fe , h_m_k20fe , + & & 4 ( @xmath40 ) & k_m_k20fe , radius , a_b , muk20fe , corr , j , h , kext , k , + & & & kext_k , prevcatname@xmath3 , rtot@xmath3 , jtot@xmath3 , htot@xmath3 , ktot@xmath3 + supercos & supercosmos input catalogues & 8 ( ) , & objid , catname , targetname , targetid , ra , dec , priority , + & & 7 ( ) & mag_1 ( old ) , progid , mag_2 ( old ) , comment + fsc & iras faint source catalogue sources & 126 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 , comment + rass & rosat all - sky survey candidate agn & 113 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 , comment + hipass & sources from the hipass hi survey & 119 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + durukst & durham / ukst galaxy survey extension & 78 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + shapley & shapley supercluster galaxies & 90 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + denisi & denis survey galaxies , @xmath41 & 6 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 , comment + denisj & denis survey galaxies , @xmath42 & 5 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 , comment + agn2mass & 2mass red agn survey candidates & 116 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 , mag_3 + he s & hamburg / eso survey candidate qsos & 129 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + nvss & candidate qsos from nvss & 130 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + sumss & bright radio sources from sumss & 125 & objid , catname , targetname , targetid , ra , dec , priority , + & & & mag_1 , progid , mag_2 + + + @xmath3 new parameters created for the final data release . + _ slanted font _ v - spectrum parameters ( _ _ v _ ) have matching r - spectrum ( _ _ r _ ) parameters . + additional downloads in the form of ascii files are also available from the database web site . these include a master catalogue compilation of all redshifts ( from both 6dfgs and the literature ) , as well as a comma - separated file of the spectral observations . the latter contains an entry for every 6dfgs observation held by the database ( including repeats ) , regardless of redshift quality . the master catalogue attempts to assign the best available redshift to those sources determined to be extragalactic . in the case of repeats , a combined 6dfgs redshift is obtained by error - weighting ( @xmath43 ) those @xmath7 redshifts within @xmath13 ( 330 ) of an initial @xmath7 median , thereby excluding blunders . where literature redshifts exist and are consistent with the 6dfgs redshift , the latter is used in the catalogue . in cases of disagreement ( @xmath44 difference ) , the 6dfgs redshift is taken and the mismatch is flagged . literature redshifts are use , where they exist , for objects that 6dfgs failed to secure . the master catalogue includes the targetid for each object and the specid references for each 6dfgs observation contributing to the final redshift , to facilitate cross - referencing with the 6dfgs database . completeness maps ( calculated from the revised target lists , after 2mass and supercos magnitude changes ) will be made available at a future date . table [ tab : parameters ] lists a subset of the more commonly - used database parameters , along with detailed descriptions . new parameters for this final release are indicated . users should pay particular attention to the important differences between parameters which have similar - sounding names but which are significantly different in purpose . examples to note are : ( i ) z , z_origin , z_helio , z_initial , and z_helio_initial , ( ii ) quality and q_final , ( iii ) ( jtot , htot , ktot ) , ( j , h , k ) , and ( mag1 , mag2 ) ( from the twomass table ) , and , ( iv ) ( bmag , rmag ) , ( bmagsel , rmagsel ) and ( bmag , rmag ) ( from the supercos table ) . table [ tab : parameters ] details the differences between them . lcl + parameter & associated table(s ) & notes + + & & + targetid & all & unique source i d ( integer ) , used to link tables . + targetname & all & source name , ` g@xmath45 ' . ( sources observed but not in the original + & & target list have the form ` c@xmath45 ' ) . + progid & all & programme i d ( integer ) , identifying the origin of targets . @xmath46 for main samples . + objid & all except target & unique object i d ( integer ) , assigned to each object in all input catalogues . + & and spectra & + & & + bmag , rmag & target & new supercosmos magnitudes following the revision for 2dfgrs by peacock , + & & hambly and read . first introduced for dr2 . the most reliable 6dfgs magnitudes . + zcatvel , zcaterr & target & existing redshifts and errors ( ) from zcat @xcite where available . + zcatref & target & code indicating source of zcat redshift : ` @xmath47 ' for earlier 6dfgs redshifts ( subsequently + & & ingested by zcat ) , ` @xmath48 ' for zcat - ingested 2dfgrs redshifts . the ` @xmath49 ' in both cases + & & holds redshift quality ( see quality below ) . zcatref@xmath50 for other zcat surveys . + bmagsel , rmagsel & target & old supercosmos magnitudes compiled by w. saunders . never used for selection + & & and not intended for science . previously under bmag and rmag in pre - dr2 releases . + templatecode & target & code indicating cross - correlation template : ` n ' = no redshift , ` z ' = zcat redshift + & & ( no template used ) , ` t ' = 2dfgrs ( no template used ) , 1 @xmath51 9 = 6dfgs template code . + & & + specid & spectra & unique spectral i d ( integer ) . different for repeat observations of the same object . + z_origin & spectra & is ` c ' for most spectra , which come from ( c)ombined ( spliced ) v and r spectral frames . + & & is ` v ' or ` r ' for unpaired ( orphan ) data , as applicable . + kbestr & spectra & template spectrum i d ( integer ) used for redshift cross - correlation . + z_helio & spectra & heliocentric redshift . corrected by @xmath52 for template offset if kbestr@xmath53 or 7 . + & & the redshift intended for science use . + z & spectra & raw measured redshift . not intended for science use . also template offset corrected . + z_initial@xmath3 & spectra & initial copy of redshift z , uncorrected ( e.g. for slit vane shifts ) . not for scientific use . + z_update_flag@xmath3 & spectra & z_helio corrections : ` 1 ' if slit - vane corrected , ` 2 ' if template corrected , ` 3 ' for both . + z_helio_initial@xmath3 & spectra & initial version of z_helio , uncorrected ( e.g. for slit vane shifts ) . not for science use . + quality & spectra & redshift quality , @xmath54 ( integer ) : ` 1 ' for unusable measurements , ` 2 ' for possible but unlikely + & & redshifts , ` 3 ' for a reliable redshift , ` 4 ' for high - quality redshifts , and ` 6 ' for + & & confirmed galactic sources . only quality@xmath55 or 4 should be used for science . ( quality + & & does _ not _ measure spectral quality . ) + q_final & spectra & final redshift quality assigned by software . not intended for general use . . + quality_initial@xmath3 & spectra & quality value at initial ingest , before database revision . not for general use . + quality_update_comm@xmath3 & spectra & explanation of quality value changes during database revision . + title_v , title_r & spectra & observation title from sds configuration file ( consisting of field name and plate number ) . + xtalkflag@xmath3 & spectra & fibre number of a nearby object suspected of spectral cross - talk contamination . ` @xmath56 ' if + & & object is a contaminator itself . ` 0 ' if neither a contaminator nor contaminee . + xtalkscore@xmath3 & spectra & score from ` 0 ' ( none ) to ` 5 ' ( high ) assessing the likelihood of spectral cross - contamination . + xtalkveloff@xmath3 & spectra & velocity offset ( ) between contaminator and contaminee in cross - contamination . + xtalkcomm@xmath3 & spectra & comment about cross - talk likelihood . + slit_vane_corr@xmath3 & spectra & correction ( ) made to a redshift affected by slit vane shifts during observing . + revtemplate@xmath3 & spectra & code of any spectral template used during the database revision of redshifts . + revcomment@xmath3 & spectra & explanation of any redshift changes resulting from the database revision . + & & + catname & twomass & 2mass name . ( prior to this release , catname held the old names now in prevcatname ) . + prevcatname@xmath3 & twomass & old 2mass name ( as at 2001 ) . + rtot@xmath3 & twomass & 2mass xsc extrapolated / total radius ( 2mass r_ext parameter ) . + jtot , htot , ktot@xmath3 & twomass & revised 2mass xsc total magnitudes ( 2mass j_m_ext , _ etc . _ ) . for science use . + mag_1,mag_2 & twomass & input catalogue magnitudes . not used in twomass table and so default non - value is 99.99 . + & & superseded by jtot , htot , and ktot . + corr & twomass & magnitude correction ( based on average surface brightness ) used to calculate kext_k . + j , h , k & twomass & old 2mass xsc total magnitudes . @xmath57 used for selection . superseded by jtot , _ etc_. + kext & twomass & redundant 2mass extrapolated @xmath38 magnitudes , previously used to obtain kext_k . + kext_k & twomass & old total @xmath38 magnitude estimated from kext and corr . used in original 6dfgs @xmath38-band + & & selection ( see @xcite for a discussion ) . now redundant . + & & + mag_1,mag_2 & supercos & old supercosmos magnitudes compiled by saunders , parker and read for target + & & selection . now superseded by the revised magnitudes bmag and rmag in the target table . + & & + + @xmath3 new parameters created for the final data release . + all of the changes previously implemented for dr2 @xcite have been retained , with some modifications . in particular , some fields rejected from earlier data releases on technical grounds have been fixed and included in the final release . the final data span observations from 2001 may to 2006 january inclusive . new changes are as follows : 1 . * revised 2mass names : * between the creation of the initial 6dfgs target list in 2001 and the final 2mass xsc data release in 2004 , the 2mass source designations changed in the last two digits in both the r.a . and dec . components of the source name . the original 2mass names ( previously held in the 6dfgs database twomass table under the attribute catname ) have been retained but re - badged under a new attribute prevcatname . the revised 2mass names are stored in catname and are consistent with the final data release of the 2mass xsc . original 6dfgs sources that were subsequently omitted from the final 2mass data release have catname= ` ' . * revised 2mass photometry : * the total magnitudes used to select 6dfgs sources were also revised by 2mass between 2001 and 2004 . these new values are held in the newly - created jtot , htot , ktot . the revisions amount to less than 0.03mag , except in the case of corrected blunders . the old magnitudes used for target selection continue to be held in j , h and kext_k , the latter being derived from surface brightness - corrected 2mass extrapolated magnitudes ( see @xcite for a full discussion ) . * revised supercosmos photometry : * as discussed in @xcite for dr2 , the supercosmos magnitudes were also revised between 2001 and 2004 . as was the case for dr2 , bmag and rmag are the revised magnitudes , which should be used for science purposes . however , some of the values in bmag and rmag have changed because of an improvement in the algorithm we have used to match 6dfgs objects with new supercosmos magnitudes . this has removed much more of the deblending discussed in sec . 2.3 of @xcite . the historical magnitudes held in bmagsel , rmagsel ( in the target table ) and mag_1 , mag_2 ( in the supercos table ) retain their dr2 definitions and values . * redshift completeness : * the 2mass and supercosmos magnitude revisions have imparted a small but non - negligible scatter between the old and new versions of , particularly @xmath38 . they have a non - negligible impact on estimates of 6dfgs redshift completeness at the faint end ( faintest @xmath58 mag ) of each distribution . in this regime , the new magnitudes cause increasing numbers of original 6dfgs targets to lie beyond the cut - off and increasing numbers of sources that were not original targets to fall inside the cut - off . consequently , new target lists were compiled using the revised magnitudes , the completeness estimates were recalculated , and the results are presented along with the luminosity and mass functions in jones et al . ( in prep . ) . 5 . * fibre cross - talk : * instances of fibre cross - talk , in which bright spectral features from one spectrum overlap with an adjacent one , have been reviewed and are now flagged in the database through three new parameters : xtalkflag , xtalkscore , and xtalkveloff , defined in table [ tab : parameters ] . the flags are not definitive and are only meant to reflect the _ likelihood _ that a redshift has been affected thus . specifically , users are urged to use extreme caution with redshifts from sources having @xmath59 , @xmath60 and @xmath61 . cases of @xmath62 are weak candidates where cross - talk is possible but not fully convincing ( e.g. only the v or the r spectra are affected , but not both ) . @xmath63 are good candidates , but which carry the previous caveat . @xmath64 are likely cross - talk pairs which are usually confirmed through visual inspection of the spectra . cross - talk is an uncommon occurrence ( about @xmath65 percent of all spectra ) , and it only affects the redshifts for spectra with fewer real features than false ones . an algorithm was used to search for coincident emission lines in adjacent spectra and a cross - talk severity value assigned from 1 to 5 . users are urged to exercise caution with spectra and redshifts having cross - talk values of 3 or greater . a detailed discussion of the cross - talk phenomenon can be found in the database documentation on the website . 6 . * highest redshift sources : * very occasionally , spurious features due to cross - talk or poor sky - subtraction led to erroneously high redshifts . this is particularly the case with the additional target samples ( @xmath66 ) , whose selection criteria do not necessarily ensure reliable detections at the optical wavelengths of 6dfgs spectra . special care should be taken with the high redshift sources reported for these targets . all sources ( across all programmes ) with @xmath67 were re - examined and re - classified where necessary . in addition , those sources from the primary and secondary samples ( @xmath68 ) with redshifts in the range @xmath69 were re - examined . there are 318 6dfgs sources with @xmath70 , mostly qsos , and a further 7 possible cases . the highest of these is the @xmath71 qso g2037567@xmath37243832 . other notable examples are the candidate double qso sources g0114547@xmath37181903 ( @xmath72 ) shown in fig . [ fig : example](@xmath73 ) and g2052000@xmath37500523 ( @xmath74 ) . deep follow - up imaging in search of a foreground source is necessary to decide whether these sources are individual gravitationally lensed qsos or genuine qso pairs . even with such data in hand , the distinction is quite often equivocal ( e.g. * ? ? ? * ; * ? ? ? * and references therein ) . 7 . * orphan fields : * the final data release includes ( for the first time ) data from 29 orphan fields . these are fields that , for various reasons , are missing either the v or r half of the spectrum . these fields have a reduced redshift yield because of the restricted access to redshifted spectral features , particularly in the case of missing r spectra . orphan field data are flagged in the database through the z_origin parameter ( see table [ tab : parameters ] ) . * re - examination of q=1 and q=2 spectra : * all sources originally classified as either being extragalactic and @xmath10 , or non-2mass - selected ( @xmath75 ) and @xmath11 , have been re - examined . this was done primarily to improve the identification of faint high - redshift qsos . many qsos were poorly identified in the early stages of the survey due to the absence of suitable qso templates for redshifting . redshift data for 4506 @xmath10 and 3687 @xmath11 sources were checked , and the database updated where necessary . * image examination of all q=6 sources and re - redshifting : * in the initial redshifting effort , 6212 sources were classified as @xmath12 ( i.e. confirmed galactic sources with @xmath76 ) on the basis of their spectra and redshifts alone . once spectral and imaging data were assembled side - by - side in the 6dfgs database , it was straightforward to examine the postage - stamp images of these sources , given their spectral classification . most were confirmed as being true galactic sources ( stars , hii regions , planetary nebulae , ysos ) , or galactic objects in close proximity to an extragalactic source . a small number were also found to be 2mass imaging artefacts , or parts of larger objects . however , a significant number ( 847 ) were found to be galaxies with near - zero redshifts , which were subsequently re - redshifted and re - classified , and updated in the database . in some cases , even though the source was clearly a galaxy on the basis of its imaging , its true redshift could not be obtained . the most common causes were scattered light from a nearby star , or contamination from a foreground screen of galactic emission . 10 . * anomalous @xmath38@xmath5 sources with q=3,4 : * the @xmath38@xmath5 magnitude - redshift relation was used to identify anomalous redshifts ( @xmath7 ) outside the envelope normally spanned by this relation at typical 6dfgs redshifts . the postage - stamp images of these sources were compared to their spectra and redshifts to decide if the initial redshift was incorrect . there were 120 objects deemed to have an anomalous @xmath38@xmath5 ; 94 were found to have incorrect redshifts , which were re - examined and re - incorporated into the database * correction of slit - vane shifted fields : * midway through the survey it became apparent that the magnetically - held vane supporting the spectrograph slit was shifting occasionally between exposures . this problem was discovered prior to dr2 but the affected redshifts were withheld ; they have been corrected and provided in the final release . the resulting spectra from affected fields show a small wavelength offset ( greater than @xmath77 and up to a few ) , dependent on fibre number . the v and r spectral halves were sometimes affected individually , and at other times in unison . instances of shifting were isolated by comparing the wavelength of the [ oi]@xmath785577.4 sky line , as measured from the 6dfgs spectra , to its true value . a search found 125 affected fields able to be satisfactorily fit ( measured [ oi ] against fibre number ) and redshift corrected . in all , 18438 galaxies were corrected in this way ( approximately 14 percent of the entire sample of _ all _ spectra ) , with corrections @xmath79 . redshift template values kbestr were used to determine whether to apply a correction . if an object used @xmath80 ( corresponding to early - type galaxy templates ) , the redshift was deemed to be due to absorption - lines , which occur predominantly in the v half . if the corresponding v frame was indeed slit - vane affected , a correction was applied to the redshift for this galaxy based on the fit to the v frame _ alone_. alternatively , if @xmath81 ( corresponding to late - type galaxy templates ) , then the redshift was deemed to be emission - line dependent , and the corresponding r frame correction was made where necessary . users can find those galaxies with slit - vane corrected redshifts through the new slitvanecorr parameter , which gives the size ( in ) of any corrections applied . unaffected galaxies have @xmath82 . the corrected redshifts are the heliocentric redshifts held by z_helio correction for template offset values : * various tests comparing 6dfgs redshifts to independent measurements found small systematic offsets in the case of a couple of templates . the discrepancy is almost certainly due to a zero - point error in the velocity calibration of the template spectra . this effect was discovered prior to dr2 and is discussed in @xcite , although no corrections were applied to the affected redshifts in that release . for this final release , corrections of @xmath83 have been applied to redshifts derived from templates @xmath84 . the corrected redshifts are both the raw ( z ) and heliocentric ( z_helio ) redshifts . the redshift offsets were found to be consistent between a 2004 comparison of 16127 6dfgs and zcat redshifts , and a 2007 comparison of 443 redshifts from various peculiar velocity surveys @xcite . 13 . * telluric sky line subtraction : * the redshifting software used by 6dfgs automatically removed telluric absorption lines from spectra , but the database spectra have hitherto retained their imprint . for the final release we have re - spliced spectra and incorporated telluric line removal . an example spectrum is shown in fig . [ fig : example ] . a small number of spectra which failed to re - splice successfully have had their old telluric - affected versions retained . * spurious clustering : * the entire sample of reliable redshifts ( @xmath7 ) was tested for spurious clusters , caused by any systematic effect that produces noticeable numbers of objects from the same field with nearly identical redshifts . possible causes include poor sky subtraction and/or splicing of spectra , and the fibre cross - talk effect discussed in item ( v ) . fields containing at least 16 cases of galaxy groups ( 3 or more members ) with redshift differences of less than 30 had their redshifts re - examined : 171 galaxies from 7 fields . no prior knowledge of real galaxy clustering was used for the re - redshifting , and the database was updated with new redshifts and quality assignments . the field 0058m30 was particularly prominent with 48 galaxies at or near an apparent redshift of 0.1590 . this was due to the over - subtraction and subsequent mis - identification of the 7600 telluric absorption band with redshifted h@xmath85 . a further 134 objects with redshifts in the range @xmath86 were reexamined for this effect , and 118 given corrected @xmath5 or @xmath54 values . almost all of the affected spectra are among the earliest observations of survey data ( 2001 ) , prior to the switch to vph gratings * rass sources : * all sources in the rosat all - sky survey ( rass ) additional target sample ( @xmath87 ; 1850 sources ) were re - examined using the full qso template set . the database was updated with new redshifts and quality assignments . @xcite describe the selection and characteristics of this sample in more detail . the wide sky coverage of the 6df galaxy survey affords the most detailed view yet of southern large - scale structures out to @xmath8830000 . the 6dfgs extends the sky coverage of the 2dfgrs @xcite by an order of magnitude , and likewise improves by an order of magnitude on the sampling density of the all - sky pscz survey @xcite . prominent southern structures such as shapley , hydra - centaurus and horologium - reticulum have received much special attention in their own right over recent years @xcite . however , a detailed large - scale mapping of all intervening structures ( and the voids between them ) with a purpose - built instrument has remained unavailable until now . the complementary 2mass redshift survey ( 2mrs ; huchra et al . , in prep ) uses the 6dfgs in the south to provide an all - sky redshift survey of some 23000 galaxies to @xmath89 ( @xmath90 ) . it is hoped it will one day be extended to reach an equivalent depth to 6dfgs in the north in those areas not already covered by sdss . figures [ fig : colplot1 ] and [ fig : colplot2 ] show the @xmath91 universe as seen by 6dfgs in the plane of the sky , projected in galactic coordinates . the two figures show the northern and southern galactic hemispheres , respectively . familiar large - scale concentrations such as shapley are obvious , and several of the key structures have been labelled . at @xmath92 , filamentary structures such as the centaurus , fornax and sculptor walls @xcite interconnect their namesake clusters in a manner typical of large structures generally . at @xmath93 to 0.01 the centaurus wall crosses the galactic plane zone of avoidance ( zoa ) and meets the hydra wall at the centaurus cluster . the hydra wall then extends roughly parallel to the zoa before separating into two distinct filaments at the adjacent hydra / antlia clusters , both of which extend into the zoa . behind these , at @xmath94 to 0.02 , a separate filament incorporates the norma and centaurus - crux clusters , and encompasses the putative great attractor region ( * ? ? ? * ; * ? ? ? * and references therein ) . beyond these , at @xmath95 to 0.05 , lies the shapley supercluster complex , a massive concentration of clusters thought to be responsible for 10 percent of the local group motion @xcite or even more @xcite . figures [ fig : pieinner ] and [ fig : pieouter ] show an alternative projection of these structures , as conventional radial redshift maps , cross - sectioned in declination . the two figures show the same data on two different scales , out to limiting redshifts of @xmath96 and 0.1 respectively . the empty sectors in our maps correspond to the zoa region . these declination - slice sky views can also be cross - referenced with the aitoff - projected sky redshift maps presented in @xcite for the 6dfgs data available up to 2004 , as well as figs [ fig : colplot1 ] and [ fig : colplot2 ] . figure [ fig : pieouter ] similarly displays the local universe out to @xmath97 with hitherto unseen detail and sky coverage . while it extends and confirms the now familiar labyrinth of filaments and voids , it also reveals evidence of inhomogeneity on a still larger scale the plot for @xmath98 ( middle right panel ) is a good example . a large under - dense region ( @xmath99 ) at @xmath100hr to 5hr separates regions of compact high - density filaments ; similar inhomogeneities are visible in the other plots . an extraordinarily large void ( @xmath101 by 0.07 ) is apparent in the plot for @xmath102 , towards @xmath103hr . other voids of this size are apparent when the data are examined in cartesian coordinates . the most extreme inhomogeneity , however , is the over - dense shapley region , which is unique within the sample volume . @xcite have used spherical harmonics and wiener filtering to decompose the density and velocity field of the shallower 2mrs . the correspondence between the largest - scale superclusters and voids seen in both surveys at @xmath104 is clear . our southernmost projection ( @xmath105 ) confirms the most distant ( pavo ) of the three tentative superclusters of @xcite while indicating that the other two are not major overdensities . we point out that this southern region is where 6dfgs coverage is generally lowest , with below - average completeness between 0 hr and 6 hr and around the pole ( poor sky coverage ) , and at 11 hr to 17 hr ( zoa ) . azimuthal stretching effects are also evident , due to the wide r.a . span of single fields at polar declinations . work is currently underway cataloguing new clusters and groups from 6dfgs ( merson et al , in prep . ) using a percolation - inferred friends - of - friends algorithm @xcite . at the same time , a preliminary list of @xmath106 void regions has been compiled as a reference for future work on under - dense regions . a power spectrum analysis of the clustering of 6dfgs galaxies will be published elsewhere . the 6df galaxy survey ( 6dfgs ) is a combined redshift and peculiar velocity survey over most of the southern sky . here we present the final redshift catalogue for the survey ( version 1.0 ) , consisting of 125071 extragalactic redshifts over the whole southern sky with @xmath0 . of these , 110256 are new redshifts from 136304 spectra obtained with the united kingdom schmidt telescope ( ukst ) between 2001 may and 2006 january . with a median redshift of @xmath107 , 6dfgs is the deepest hemispheric redshift survey to date . redshifts and associated spectra are available through a fully - searchable online sql database , interlinked with photometric and imaging data from the 2mass xsc , supercosmos and a dozen other input catalogues . peculiar velocities and distances for the brightest 10 percent of the sample will be made available in a separate future release . in this paper we have mapped the large - scale structures of the local ( @xmath108 ) southern universe in unprecedented detail . in addition to encompassing well - known superclusters such as shapley and hydra - centaurus , the 6dfgs data reveal a wealth of new intervening structures . the greater depth and sampling density of 6dfgs compared to earlier surveys of equivalent sky coverage has confirmed hundreds of voids and furnished first redshifts for around 400 southern abell clusters @xcite . more detailed quantitative analyses of 6dfgs large - scale structure will be the subject of future publications . the unprecedented combination of angular coverage and depth in 6dfgs offers the best chance yet to minimise systematics in the determination of the luminosity and stellar mass functions of low - redshift galaxies , both in the near - infrared and optical ( e.g. * ? ? ? while surveys containing @xmath109-galaxy redshifts ( such as 6dfgs ) have now reduced random errors to comparable levels of high precision , systematic errors remain the dominant source of the differences between surveys . for example , the evolutionary corrections that initially beset comparisons between 2dfgrs and sdss ( cf . * ? ? ? * ; * ? ? ? * ) are negligible for 6dfgs , which spans lookback times of only 0.2 to 0.7 gyr across @xmath110 $ ] ( compared to 0.5 to 1.3 gyr for sdss and 2dfgrs ) . the minimisation of such systematics are a feature of the 6dfgs stellar mass and luminosity functions derived for the final redshift set ( jones et al . , in prep ) . in addition to these studies , 6dfgs redshift data have already been used to support a variety of extragalactic samples selected from across the electromagnetic spectrum . deep hi surveys planned for next - generation radio telescopes @xcite will also benefit from this redshift information as they probe the gas content of the local southern universe over comparable volumes . dhj acknowledges support from australian research council discovery projects grant ( dp-0208876 ) , administered by the australian national university . jph acknowledges support from the us national science foundation under grant ast0406906 . we dedicate this paper to two colleagues who made important contributions to the 6df galaxy survey before their passing : john dawe ( 1942 2004 ) , observer and long - time proponent of wide - field fibre spectroscopy on the ukst from its earliest days , and tony fairall ( 1943 2008 ) , whose unique insights from a career - long dedication to mapping the southern universe underpin much of the interpretation contained herein . , h. , tago , e. , einasto , m. , einasto , j. , jaaniste , j. , 2005 , in nearby large - scale structures and the zone of avoidance , astronomical society of the pacific conference series vol . 329 , a. p. fairall and p. a. woudt eds . , p283 , f. g. , parker , q. a. , bogatu , g. , farrell , t. j. , hingley , b. e. , miziarski , s. , 2000 , in iye m. , moorwood a. f. , eds . spie vol . 4008 , optical and ir telescope instrumentation and detectors , spie , bellingham , wa , p. 123 | we report the final redshift release of the 6df galaxy survey , a combined redshift and peculiar velocity survey over the southern sky ( @xmath0 ) .
its 136304 spectra have yielded 110256 new extragalactic redshifts and a new catalogue of 125071 galaxies making near - complete samples with @xmath1 .
the median redshift of the survey is 0.053 .
survey data , including images , spectra , photometry and redshifts , are available through an online database .
we describe changes to the information in the database since earlier interim data releases .
future releases will include velocity dispersions , distances and peculiar velocities for the brightest early - type galaxies , comprising about 10% of the sample .
here we provide redshift maps of the southern local universe with @xmath2 , showing nearby large - scale structures in hitherto unseen detail .
a number of regions known previously to have a paucity of galaxies are confirmed as significantly underdense regions .
the url of the 6dfgs database is http://www-wfau.roe.ac.uk/6dfgs
. surveys galaxies : clustering galaxies : distances and redshifts cosmology : observations cosmology : large scale structure of universe |