a schur horn theorem for symplectic eigenvalues abstract let x and y be positive n vectors we show that there exists a 2 n 2 n positive definite real matrix whose symplectic spectrum is y and the symplectic spectrum of whose diagonal is x if and only if x is weakly supermajorised by y

inequalities for quantum marginal problems with continuous variables we consider a mixed continuous variable bosonic quantum system and present inequalities which must be satisfied between principal values of the covariances of a complete set of observables of the whole system and the principal values of the covariances of a complete set of observables of a subsystem we use several classical results for the proof the courant fischer weyl min max theorem for hermitian operators and its consequence the cauchy interlacing theorem and prove their analogues in the symplectic setting for the case of passive transformations of gaussian mixed states we also prove that the obtained inequalities are in a sense the best possible the obtained mathematical results are applied to the system of n uncorrelated thermal modes of the electromagnetic field finally we present the results of numerical simulations of the problem suggesting avenues of further research

exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

gaussian quantum marginal problem the quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system this setting also referred to as the question of the compatibility of local spectra has several applications in quantum information theory here we introduce the analogue of this statement for gaussian states for any number of modes and solve it in generality for pure and mixed states both concerning necessary and sufficient conditions formally our result can be viewed as an analogue of the sing thompson theorem respectively horn s lemma characterizing the relationship between main diagonal elements and singular values of a complex matrix we find necessary and sufficient conditions for vectors d1 dn and c1 cn to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix respectively more physically speaking this result determines what local temperatures or entropies are consistent with a pure or mixed gaussian state of several modes we find that this result implies a solution to the problem of sharing of entanglement in pure gaussian states and allows for estimating the global entropy of non gaussian states based on local measurements implications to the actual preparation of multi mode continuous variable entangled states are discussed we compare the findings with the marginal problem for qubits the solution of which for pure states has a strikingly similar and in fact simple form

asymmetric simple exclusion process with open boundaries and askey wilson polynomials we study the one dimensional asymmetric simple exclusion process asep with open boundary conditions particles are injected and ejected at both boundaries it is clarified that the steady state of the model is intimately related to the askey wilson polynomials the partition function and the n point functions are obtained in the integral form with four boundary parameters the thermodynamic current is evaluated to confirm the conjectured phase diagram

continuity and nodal properties near infinity for solutions of 2 dimensional schrodinger equations on considere des solutions w 2 2 a valeurs reelles x de v e 0 pour x r r x r 2 x r r r 0 on suppose que e 0

exact solution of a reaction diffusion process with three site interactions the one dimensional reaction diffusion process aa a and a a aaa is exactly solvable through the empty interval method if the diffusion rate equals the coagulation rate independently of the particle production rate the model is always in the universality class of diffusion annihilation this allows us to check analytically the universality of finite size scaling in a non equilibrium critical point

phase transitions in an exactly soluble one dimensional exclusion process we consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one dimensional chain of sites the particles are allowed to hop onto unoccupied sites to the right only for the special case of 1 the model was solved previously by derridaet al here we extend the solution to general the phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation

exact results for one dimensional cellular automata with different types of updates we study two common types of time noncontinuous updates for one dimensional stochastic cellular automata with arbitrary nearest neighbor interactions and arbitrary open boundary conditions we first construct the stationary states using the matrix product formalism this construction then allows to prove a general connection between the stationary states which are produced by the two different types of updates using this connection we derive explicit relations between the densities and correlation functions for these different stationary states

directed percolation fractal roots and the lee yang theorem abstract in the directed percolation model we consider the probability p of having an open bond as a complex parameter we show that the roots of the survival probability p n p for a square lattice of n rows distribute themselves in a fractal manner in the complex p plane these roots have an accumulation point on the real axis which coincides with the critical probability p c 0 6447

the application of the yang lee theory to study a phase transition in a non equilibrium system we study a phase transition in a non equilibrium model first introduced in ref 5 using the yang lee description of equilibrium phase transitions in terms of both canonical and grand canonical partition function zeros the model consists of two different classes of particles hopping in opposite directions on a ring on the complex plane of the diffusion rate we find two regions of analyticity for the canonical partition function of this model which can be identified by two different phases the exact expressions for both distribution of the canonical partition function zeros and their density are obtained in the thermodynamic limit the fact that the model undergoes a second order phase transition at the critical point is confirmed we have also obtained the grand canonical partition function zeros of our model numerically the similarities between the phase transition in this model and the bose einstein condensation has also been studied

from log determinant inequalities to gaussian entanglement via recoverability theory many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities specialized to gaussian distributed vectors with prescribed covariances in particular strong subadditivity ssa yields ln det v ac ln det v bc ln det v abc ln det v c geq 0 for all 3 times 3 block matrices v abc where subscripts identify principal submatrices we shall refer to the above mentioned inequality as ssa of log det entropy in this paper we develop further insights on the properties of the above mentioned inequality and its applications to classical and quantum information theory in the first part of this paper we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs subsequently we discuss the role of the classical transpose channel also known as petz recovery map in this problem and find its action explicitly we then prove some extensions of the saturation theorem by finding faithful lower bounds on a log det conditional mutual information in the second part we focus on quantum gaussian states whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation for gaussian states the log det entropy is equivalent to the renyi entropy of order 2 we provide a strengthening of log det ssa for quantum covariance matrices that involves the so called gaussian renyi 2 entanglement of formation a well behaved entanglement measure defined via a gaussian convex roof construction we then employ this result to define a log det entropy equivalent of the squashed entanglement measure which is remarkably shown to coincide with the gaussian renyi 2 entanglement of formation this allows us to establish useful properties of such measure s such as monogamy faithfulness and additivity on gaussian states

exact solution of a cellular automaton for traffic we present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions e g cars can enter and leave a part of a highway with certain probabilities the model studied is the asymmetric exclusion process asep with simultaneous updating of all sites it is equivalent to a special case vmax 1 of the nagel schreckenberg model for highway traffic which has found many applications in real time traffic simulations the simultaneous updating induces additional strong short range correlations compared to other updating schemes the stationary state is written in terms of a matrix product solution the corresponding algebra which expresses a system size recursion relation for the weights of the configurations is quartic in contrast to previous cases in which the algebra is quadratic we derive the phase diagram and compute various properties such as density profiles two point functions and the fluctuations in the number of particles cars in the system the current and the density profiles can be mapped onto the asep with other time discrete updating procedures through use of this mapping our results also give new results for these models

exact stationary state for an asymmetric exclusion process with fully parallel dynamics the exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz we give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices we prove the stationarity via a cancellation mechanism and by making use of an explicit representation of the matrix algebra we easily find closed expressions for the correlation functions in the general probabilistic case asymptotic expressions obtained by making use of earlier results allow us to derive the exact phase diagram

exact density profiles for the fully asymmetric exclusion process with discrete time dynamics on semi infinite chains exact density profiles in the steady state of the one dimensional fully asymmetric simple exclusion process on a semi infinite chain are obtained in the case of forward ordered sequential dynamics by taking the thermodynamic limit in our recent exact results for a finite chain with open boundaries the corresponding results for sublattice parallel dynamics follow from the relationship obtained by rajewsky and schreckenberg physica a 245 139 1997 and for parallel dynamics from the mapping found by evans rajewsky and speer j stat phys 95 45 1999 our analytical expressions involve laplace type integrals rather than complicated combinatorial expressions which makes them convenient for taking the limit of a semi infinite chain and for deriving the asymptotic behavior of the density profiles at large distances from its end by comparing the asymptotic results appropriate for parallel update with those published in the above cited paper by evans rajewsky and speer we find complete agreement except in two cases in which we correct technical errors in the final results given there

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models are discussed

local inhomogeneity in asymmetric simple exclusion processes with extended objects totally asymmetric simple exclusion processes tasep with particles which occupy more than one lattice site and with a local inhomogeneity far away from the boundaries are investigated these non equilibrium processes are relevant for the understanding of many biological and chemical phenomena the steady state phase diagrams currents and bulk densities are calculated using a simple approximate theory and extensive monte carlo computer simulations it is found that the phase diagram for tasep with a local inhomogeneity is qualitatively similar to homogeneous models although the phase boundaries are significantly shifted the complex dynamics is discussed in terms of domain wall theory for driven lattice systems

partially asymmetric simple exclusion model in the presence of an impurity on a ring we study a generalized two species model on a ring the original model mallick k 1996 j phys a math gen 29 5375 describes ordinary particles hopping exclusively in one direction in the presence of an impurity the impurity hops with a rate different from that of ordinary particles and can be overtaken by them here we let the ordinary particles also hop backward with rate q using the matrix product ansatz we obtain the relevant quadratic algebra a finite dimensional representation of this algebra enables us to compute the stationary bulk density of the ordinary particles as well as the speed of impurity on a set of special surfaces of the parameter space we will obtain the phase structure of this model in the accessible region and show how the phase structure of the original model is modified in the infinite volume limit this model presents a shock in one of its phases

entropy of open lattice systems we investigate the behavior of the gibbs shannon entropy of the stationary nonequilibrium measure describing a one dimensional lattice gas of l sites with symmetric exclusion dynamics and in contact with particle reservoirs at different densities in the hydrodynamic scaling limit l the leading order o l behavior of this entropy has been shown by bahadoran to be that of a product measure corresponding to strict local equilibrium we compute the first correction which is o 1 the computation uses a formal expansion of the entropy in terms of truncated correlation functions for this system the k th such correlation is shown to be o l k 1 this entropy correction depends only on the scaled truncated pair correlation which describes the covariance of the density field it coincides in the large l limit with the corresponding correction obtained from a gaussian measure with the same covariance

symplectic spaces and pairs of symmetric and nonsingular skew symmetric matrices under congruence abstract let f be a field of characteristic not 2 and let a b be a pair of n n matrices over f in which a is symmetric and b is skew symmetric a canonical form of a b with respect to congruence transformations s t a s s t b s was given by sergeichuk 1988 25 up to classification of symmetric and hermitian forms over finite extensions of f we obtain a simpler canonical form of a b if b is nonsingular such a pair a b defines a quadratic form on a symplectic space that is on a vector space with scalar product given by a nonsingular skew symmetric form as an application we obtain known canonical matrices of quadratic forms and hamiltonian operators on real and complex symplectic spaces

exclusion process for particles of arbitrary extension hydrodynamic limit and algebraic properties the behaviour of extended particles with exclusion interaction on a one dimensional lattice is investigated the basic model is called l asep as a generalization of the asymmetric exclusion process asep to particles of arbitrary length l stationary and dynamical properties of the l asep with periodic boundary conditions are derived in the hydrodynamic limit from microscopic properties of the underlying stochastic many body system in particular the hydrodynamic equation for the local density evolution and the time dependent diffusion constant of a tracer particle are calculated as a fundamental algebraic property of the symmetric exclusion process the su 2 symmetry is generalized to the case of extended particles

two way traffic flow exactly solvable model of traffic jam we study completely asymmetric two channel exclusion processes in one dimension it describes a two way traffic flow with cars moving in opposite directions the interchannel interaction makes cars slow down in the vicinity of approaching cars in the other lane particularly we consider in detail the system with a finite density of cars on one lane and a single car on the other when the interchannel interaction reaches a critical value a traffic jam occurs which turns out to be of first order phase transition we derive exact expressions for the average velocities the current the density profile and the k point density correlation functions we also obtain the exact probability of two cars being in one lane of distance r apart provided there is a finite density of cars on the other lane and show that the two cars form a weakly bound state in the jammed phase

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

yang lee zeros for a nonequilibrium phase transition equilibrium systems which exhibit a phase transition can be studied by investigating the complex zeros of the partition function this method pioneered by yang and lee has been widely used in equilibrium statistical physics we show that an analogous treatment is possible for a nonequilibrium phase transition into an absorbing state by investigating the complex zeros of the survival probability of directed percolation processes we demonstrate that the zeros provide information about universal properties moreover we identify certain nontrivial points where the survival probability for bond percolation can be computed exactly

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

asymmetric exclusion process and extremal statistics of random sequences a mapping is established between sequence alignment one of the most commonly used tools of computational biology at a certain choice of scoring parameters and the asymmetric exclusion process one of the few exactly solvable models of nonequilibrium physics the statistical significance of sequence alignments is characterized through studying the total hopping current of the discrete time and space version of the asymmetric exclusion process

on the asymptotic decay of l2 solutions of one body schrodinger equations in unbounded domains the asymptotic decay of l 2 solutions of schrodinger equations v 0 in r x er n x r r is investigated where v x v 1 r v 2 x with v 1 for r and with some e 0 for large r under additional assumptions on the decay of v 1 pointwise upper bounds to and lower bounds to the spherical average of are given showing the same asymptotics for r for the case v const 0 for r investigated in 8 a simplified treatment is given

disordered totally asymmetric simple exclusion process exact results we study the effect of quenched spatial disorder on the current carrying steady states of a totally asymmetric simple exclusion process with spatially disordered jump rates expressions for the steady state weights and the current are derived for this model in one dimension these solutions are exploited to study analytically the exact symmetries of the system in particular we confirm the recent numerical observation of tripathy and barma phys rev e 58 1911 1998 that the magnitude of the steady state current is left invariant when the direction of all the allowed particle jumps are reversed

asymmetric exclusion model and weighted lattice paths we show that the known matrix representations of the stationary state algebra of the asymmetric simple exclusion process asep can be interpreted combinatorially as various weighted lattice paths this interpretation enables us to use the constant term method ctm and bijective combinatorial methods to express many forms of the asep normalization factor in terms of ballot numbers one particular lattice path representation shows that the coefficients in the recurrence relation for the asep correlation functions are also ballot numbers additionally the ctm has a strong combinatorial connection which leads to a new canonical lattice path representation and to the expansion which provides a uniform approach to computing the asymptotic behaviour in the various phases of the asep the path representations enable the asep normalization factor to be seen as the partition function of a more general polymer chain model having a two parameter interaction with a surface we show in the case 1 that the probability of finding a given number of particles in the stationary state can be expressed via non intersecting lattice paths and hence as a simple determinant

nonequilibrium stationary states and equilibrium models with long range interactions it was recently suggested by blythe and evans that a properly defined steady state normalization factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities in analogy with the lee yang description of phase transition of equilibrium systems they studied the zeros in the complex plane of the normalization factor in order to find phase transitions in nonequilibrium steady states we show that like for equilibrium systems the densities associated with the rates are nondecreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the densities we illustrate this phenomenon for the asymmetric exclusion process we actually show that its normalization factor coincides with an equilibrium partition function of a walk model in which the densities have a simple physical interpretation

yang lee zeros for an urn model for the separation of sand we apply the yang lee theory of phase transitions to an urn model for the separation of sand the effective partition function of this nonequilibrium system can be expressed as a polynomial of the size dependent effective fugacity z numerical calculations show that in the thermodynamic limit the zeros of the effective partition function are located on the unit circle in the complex z plane in the complex plane of the actual control parameter certain roots converge to the transition point of the model thus the yang lee theory can be applied to a wider class of nonequilibrium systems than those considered previously

exponential lower bounds to solutions of the schrodinger equation lower bounds for the spherical average for a large class of generalizedn body schrodinger operators h we show that ife inf ess h and is an eigenfunction ofh with eigenvaluee then begin array 20 c lim r to infty end array r 1 ln left int limits s n 1 psi r omega 2 d omega right 1 2 alpha 0 with 0 2 e a threshold similar results are given fore

the asymmetric exclusion process comparison of update procedures the asymmetric exclusion process asep has attracted a lot of interest not only because of its many applications e g in the context of the kinetics of biopolymerization and traffic flow theory but also because it is a paradigmatic model for nonequilibrium systems here we study the asep for different types of updates namely random sequential sequential sublattice parallel and parallel in order to compare the effects of the different update procedures on the properties of the stationary state we use large scale monte carlo simulations and analytical methods especially the so called matrix product ansatz mpa we present in detail the exact solution for the model with sublattice parallel and sequential updates using the mpa for the case of parallel update which is important for applications like traffic flow theory we determine the phase diagram the current and density profiles based on monte carlo simulations we furthermore suggest an mpa for that case and derive the corresponding matrix algebra

log majorizations for the symplectic eigenvalues of the cartan barycenter abstract in this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are lipschitz for the cartan hadamard riemannian metric and establish log majorizations for the symplectic eigenvalues of the cartan barycenter of integrable probability borel measures this leads a version of jensen s inequality for geometric integrals of matrix valued integrable random variables

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

exact solution of an exclusion process with three classes of particles and vacancies we present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies using a matrix product ansatz we find exact expressions for the weights of the configurations in the stationary state the solution involves tensor products of quadratic algebras

first order sensitivity analysis of symplectic eigenvalues abstract for every 2 n 2 n positive definite matrix a there are n positive numbers d 1 a d n a associated with a called the symplectic eigenvalues of a it is known that d m are continuous functions of a but are not differentiable in general in this paper we show that the directional derivative of d m exists and derive its expression we also discuss various subdifferential properties of d m such as clarke and michel penot subdifferentials

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

perturbation theory for the decay rate of eigenfunctions in the generalized n body problem simple examples are known where eigenfunctions decay faster than the usual upper bounds would lead one to believe we develop aspects of the perturbation theory of the decay rate of eigenfunctions as measured by radial exponential weights we show that generically in a baire category sense eigenfunction decay rates are governed by the lowest threshold

large deviation functional of the weakly asymmetric exclusion process we obtain the large deviation functional of a density profile for the asymmetric exclusion process of l sites with open boundary conditions when the asymmetry scales like l 1 we recover as limiting cases the expressions derived recently for the symmetric ssep and the asymmetric asep cases in the asep limit the nonlinear differential equation one needs to solve can be analysed by a method which resembles the wkb method

some general features of matrix product states in stochastic systems we will prove certain general relations in the matrix product ansatz mpa for one dimensional stochastic systems which are true in both random and sequential updates we will derive general mpa expressions for the currents and current correlators and find the conditions in the mpa formalism under which the correlators are site independent or completely vanishing

one dimensional partially asymmetric simple exclusion process on a ring with a defect particle the effect of a moving defect particle for the one dimensional partially asymmetric simple exclusion process on a ring is considered the current of the ordinary particles the speed of the defect particle and the density profile of the ordinary particles are calculated exactly the phase diagram for the correlation length is identified as a by product the average and the variance of the particle density of the one dimensional partially asymmetric simple exclusion process with open boundaries are also computed

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

the totally asymmetric exclusion process on a ring exact relaxation dynamics and associated model of clustering transition abstract the totally asymmetric simple exclusion process in discrete time is considered on finite rings with fixed number of particles a translation invariant version of the backward ordered sequential update is defined for periodic boundary conditions we prove that the so defined update leads to a stationary state in which all possible particle configurations have equal probabilities using the exact analytical expression for the propagator we find the generating function for the conditional probabilities average velocity and diffusion constant at all stages of evolution an exact and explicit expression for the stationary velocity of tasep on rings of arbitrary size and particle filling is derived the evolution of small systems towards a steady state is clearly demonstrated considering the generating function as a partition function of a thermodynamic system we study its zeros in planes of complex fugacities at long enough times the patterns of zeroes for rings with increasing size provide evidence for a transition of the associated two dimensional lattice paths model into a clustered phase at low fugacities

the asymmetric exclusion model with sequential update we present a solution for the stationary state of an asymmetric exclusion model with sequential update and open boundary conditions we solve the model exactly for random hopping in both directions by applying a matrix product formalism which was recently used to solve the model with sublattice parallel update hinrichsen h 1995 weizmann inst preprint cond mat 9512172 it is shown that the matrix algebra describing the sequential update and sublattice parallel update are identical and can be mapped onto the random sequential case treated by derrida et al derrida b evans m r hakim v and pasquier v 1993 j phys a math gen 26 1493

finite dimensional representations of the quadratic algebra applications to the exclusion process we study the one dimensional partially asymmetric simple exclusion process asep with open boundaries that describes a system of hard core particles hopping stochastically on a chain coupled to reservoirs at both ends derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra closely related to the deformed oscillator algebra we construct all finite dimensional irreducible representations of this algebra this enables us to compute the stationary bulk density as well as all correlation lengths for the asep on a set of special curves of the phase diagram

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

exponential bounds and absence of positive eigenvalues for n body schrodinger operators for a large class ofn body potentialsv we prove that if is an eigenfunction of v with eigenvaluee then sup 2 e 0 exp x l2 is either a threshold or consequences of this result are the absence of positive eigenvalues and optimal l2 exponential lower bounds

synchronous asymmetric exclusion processes for a fully synchronous asymmetric exclusion process with open or closed boundaries only partial analytic results are known owing to the appearance of strong short range correlations which invalidate simple mean field approximations here we present a new method for calculating basic properties of nonequilibrium steady states and calculate densities fluxes travel times spatial and temporal correlation functions phase diagrams profiles and widths of boundary layers and interfaces between phases in coexistence as well as their microstructures this paper is based on two new elements i a microscopic characterization of order parameters and local configurations in the relevant phases based on the microdynamics of the model and ii an improved mean field approximation which neglects certain four point and higher order correlation functions it is conjectured that the density profiles obtained here are exact up to terms that are exponentially small in the system size

spectral degeneracies in the totally asymmetric exclusion process we study the spectrum of the markov matrix of the totally asymmetric exclusion process tasep on a one dimensional periodic lattice at arbitrary filling although the system does not possess obvious symmetries except translation invariance the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules this behaviour is explained by a hidden symmetry property of the bethe ansatz assuming a one to one correspondence between the solutions of the bethe equations and the eigenmodes of the markov matrix we derive combinatorial formulae for the orders of degeneracy and the number of multiplets these results are confirmed by exact diagonalisations of small size systems this unexpected structure of the tasep spectrum suggests the existence of an underlying large invariance group

perturbation bounds for williamson s symplectic normal form abstract given a real valued positive semidefinite matrix williamson proved that it can be diagonalised using symplectic matrices the corresponding diagonal values are known as the symplectic spectrum this paper is concerned with the stability of williamson s decomposition under perturbations we provide norm bounds for the stability of the symplectic eigenvalues and prove that if s diagonalises a given matrix m to williamson form then s is stable if the symplectic spectrum is nondegenerate and s t s is always stable finally we sketch a few applications of the results in quantum information theory

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

perturbation of embedded eigenvalues in the generalized n body problem we discuss the perturbation of continuum eigenvalues without analyticity assumptions among our results we show that generally a small perturbation removes these eigenvalues in accordance with fermi s golden rule thus generically in a baire category sense the schrodinger operator has no embedded non threshold eigenvalues

matrix product states of three families of one dimensional interacting particle systems abstract the steady states of three families of one dimensional non equilibrium models with open boundaries first proposed in n j phys 5 2003 145 1 are studied using a matrix product formalism it is shown that their associated quadratic algebras have two dimensional representations provided that the transition rates lie on specific manifolds of parameters exact expressions for the correlation functions of each model have also been obtained we have also studied the steady state properties of one of these models first introduced in j phys a math gen a 36 2003 74 with more details by introducing a canonical ensemble we calculate the canonical partition function of this model exactly using the yang lee theory of phase transitions we spot a second order phase transition from a power law to a jammed phase the density profile of particles in each phase has also been studied a simple generalization of this model in which both the left and the right boundaries are open has also been introduced it is shown that double shock structures may evolve in the system under certain conditions

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

traffic of molecular motors through tube like compartments the traffic of molecular motors through open tube like compartments is studied using lattice models these models exhibit boundary induced phase transitions related to those of the asymmetric simple exclusion process asep in one dimension the location of the transition lines depends on the boundary conditions at the two ends of the tubes three types of boundary conditions are studied a periodic boundary conditions which correspond to a closed torus like tube b fixed motor densities at the two tube ends where radial equilibrium holds locally and c diffusive motor injection at one end and diffusive motor extraction at the other end in addition to the phase diagrams we also determine the profiles for the bound and unbound motor densities using mean field approximations and monte carlo simulations our theoretical predictions are accessible to experiments

on symplectic eigenvalues of positive definite matrices if a is a 2n 2n real positive definite matrix then there exists a symplectic matrix m such that mtam dood where d diag d1 a dn a is a diagonal matrix with positive diagonal entries which are called the symplectic eigenvalues of a in this paper we derive several fundamental inequalities about these numbers among them are relations between the symplectic eigenvalues of a and those of at between the symplectic eigenvalues of m matrices a1 am and of their riemannian mean a perturbation theorem some variational principles and some inequalities between the symplectic and ordinary eigenvalues

finite size scaling and universality for the totally asymmetric simple exclusion process the applicability of the concepts of finite size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one dimensional totally asymmetric simple exclusion process with open boundaries in the thermodynamic limit there are boundary induced transitions both of the first and second order between steady state phases of the model we derive finite size scaling expressions for the current near the continuous phase transition and for the local density near the first order transition under different stochastic dynamics and compare them to establish the existence of universal functions next we study numerically the finite size behavior of the lee yang zeros of the normalization factor for the different steady state probabilities

diagonalizability with respect to perplectic and pseudo unitary similarity transformations abstract let r n be the n by n backward identity matrix and let l n k i k i n k suppose a c n n is nonsingular we say that a is perplectic if r n a t r n a 1 and a is pseudo unitary if l n k a l n k a 1 we give necessary and sufficient conditions for a matrix to be diagonalizable via a perplectic or a pseudo unitary matrix

first order phase transition in a reaction diffusion model with open boundary the yang lee theory approach a coagulation decoagulation model is introduced on a chain of length l with open boundary the model consists of one species of particles which diffuse coagulate and decoagulate preferentially in the leftward direction they are also injected and extracted from the left boundary with different rates we will show that on a specific plane in the space of parameters the steady state weights can be calculated exactly using a matrix product method the model exhibits a first order phase transition between a low density and a high density phase the density profile of the particles in each phase is obtained both analytically and using the monte carlo simulation the two point density density correlation function in each phase has also been calculated by applying the yang lee theory we can predict the same phase diagram for the model this model is further evidence for the applicability of the yang lee theory in the non equilibrium statistical mechanics context

matrix product ground states for exclusion processes with parallel dynamics we show in the example of a one dimensional asymmetric exclusion process that stationary states of models with parallel dynamics may be written in a matrix product form the corresponding algebra is quadratic and involves three different matrices using this formalism we prove previous conjectures for the equal time correlation functions of the model

singular values doubly stochastic matrices and applications abstract the hadamard square of any square matrix a is bounded above and below by some doubly stochastic matrices times the square of the largest and the smallest singular values of a applications to graphs permanents and eigenvalue perturbations are discussed

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally

exact results for a fully asymmetric exclusion process with sequential dynamics and open boundaries an exact and rigorous calculation of the current and density profile in the steady state of the one dimensional fully asymmetric simple exclusion process with open boundaries and forward ordered sequential dynamics is presented the method is based on a matrix product representation of the steady state probability distribution the main idea is to choose a suitable representation in which the scalar products describing the current and local density profile for a chain of arbitrary finite size depend only on the elements in a finite number of rows and columns this makes possible the use of a truncated finite dimensional representation of the matrices and vectors involved after performing the calculations we lift the truncation by letting its dimensionality go to infinity in this limit the results become exact for any size of the chain by rescaling one of the infinitedimensional matrix representations found in the work of derrida et al j phys a 26 1493 1993 for their algebra we obtain a symmetric propagator matrix its truncated version is diagonalized by orthogonal transformation for easy calculation of the relevant scalar products an interpretation of the phase transitions between the different phases is given in terms of eigenvalue splitting from a bounded quasicontinuous spectrum a precise description of the local density profiles is given for all values of the parameters it is shown that the leading order asymptotic form of the position dependent terms in the local density changes within the low and high density phases signaling the presence of a higher order transition