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

models of genetic drift as limiting forms of the lotka volterra competition model the relationship between the moran model and stochastic lotka volterra competition slvc model is explored via time scale separation arguments for neutral systems the two are found to be equivalent at long times for systems with selective pressure their behavior differs it is argued that the slvc is preferable to the moran model since in the slvc population size is regulated by competition rather than arbitrarily fixed as in the moran model as a consequence ambiguities found in the moran model associated with the introduction of more complex processes such as selection are avoided

phase separation and coarsening in one dimensional driven diffusive systems local dynamics leading to long range hamiltonians a driven system of three species of particles diffusing on a ring is studied in detail the dynamics is local and conserves the three densities a simple argument suggesting that the model should phase separate and break the translational symmetry is given we show that for the special case where the three densities are equal the model obeys detailed balance and the steady state distribution is governed by a hamiltonian with asymmetric long range interactions this provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension the steady state of finite size systems is studied using a generalized matrix product ansatz the coarsening process leading to phase separation is studied numerically and in a mean field model the system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size the typical domain size is shown to grow logarithmically in time generalizations to a larger number of species are discussed

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

stationary distributions of multi type totally asymmetric exclusion processes we consider totally asymmetric simple exclusion processes with n types of particle and holes n taseps on z and on the cycle z n angel recently gave an elegant construction of the stationary measures for the 2 tasep based on a pair of independent product measures we show that angel s construction can be interpreted in terms of the operation of a discrete time m m 1 queueing server the two product measures correspond to the arrival and service processes of the queue we extend this construction to represent the stationary measures of an n tasep in terms of a system of queues in tandem the proof of stationarity involves a system of n 1 taseps whose evolutions are coupled but whose distributions at any fixed time are independent using the queueing representation we give quantitative results for stationary probabilities of states of the n tasep on z n and simple proofs of various independence and regeneration properties for systems on z

matrix product representation of the stationary state of the open zero range process many one dimensional lattice particle models with open boundaries like the paradigmatic asymmetric simple exclusion process asep have their stationary states represented in the form of a matrix product with matrices that do not explicitly depend on the lattice site in contrast the stationary state of the open 1d zero range process zrp takes an inhomogeneous factorized form with site dependent probability weights we show that in spite of the absence of correlations the stationary state of the open zrp can also be represented in a matrix product form where the matrices are site independent non commuting and determined from algebraic relations resulting from the master equation we recover the known distribution of the open zrp in two different ways first using an explicit representation of the matrices and boundary vectors second from the sole knowledge of the algebraic relations satisfied by these matrices and vectors finally an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden markov chains

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

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

intrinsic noise in systems with switching environments we study individual based dynamics in finite populations subject to randomly switching environmental conditions these are inspired by models in which genes transition between on and off states regulating underlying protein dynamics similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread existing piecewise deterministic markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear noise approximation specifically we derive the stationary distributions of a number of model systems in good agreement with simulations this improves existing approaches which are limited to the regimes of fast and slow switching

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

multiband least squares fir filter design this paper gives two methods for the least squares approximation design of fir digital filters they both allow multiple pass and stop bands and arbitrary transition bands all with explicit control of band edges the first method is as simple to use as a window method this design method builds up an optimal multiband frequency response by sequentially adding and subtracting optimal lowpass filters with transition bands however it assumes the ideal response has spline transition functions and does not allow error weighting the second method allows different weights in the multiple passbands stopbands and transition bands but it requires numerical solution of simultaneous equations which may be ill conditioned these methods compliment the parks mcclellan algorithm which minimizes the chebyshev error

multispecies tasep and the tetrahedron equation we introduce a family of layer to layer transfer matrices in a three dimensional 3d lattice model which can be viewed as partition functions of the q oscillator valued six vertex model on an m n square lattice by invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions at q 0 and m n they ultimately yield a new proof of the steady state formula for the n species totally asymmetric simple exclusion process obtained recently by the authors revealing the 3d integrability in the matrix product construction

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

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

a deformation of affine hecke algebra and integrable stochastic particle system we introduce a deformation of the affine hecke algebra of type which describes the commutation relations of the divided difference operators found by lascoux and schutzenberger and the multiplication operators making use of its representation we construct an integrable stochastic particle system it is a generalization of the q boson system due to sasamoto and wadati we also construct eigenfunctions of its generator using the propagation operator as a result we get the same eigenfunctions for the boson process obtained by povolotsky

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

construction of a matrix product stationary state from solutions of a finite size system stationary states of stochastic models which have n states per site in matrix product form are considered first we give a necessary condition for the existence of a finite m dimensional matrix product stationary state for any n m second we give a method to construct the matrices from the stationary

evolutionary dynamics with fluctuating population sizes and strong mutualism game theory ideas provide a useful framework for studying evolutionary dynamics in a well mixed environment this approach however typically enforces a strictly fixed overall population size deemphasizing natural growth processes we study a competitive lotka volterra model with number fluctuations that accounts for natural population growth and encompasses interaction scenarios typical of evolutionary games we show that in an appropriate limit the model describes standard evolutionary games with both genetic drift and overall population size fluctuations however there are also regimes where a varying population size can strongly influence the evolutionary dynamics we focus on the strong mutualism scenario and demonstrate that standard evolutionary game theory fails to describe our simulation results we then analytically and numerically determine fixation probabilities as well as mean fixation times using matched asymptotic expansions taking into account the population size degree of freedom these results elucidate the interplay between population dynamics and evolutionary dynamics in well mixed systems

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

exact results for one dimensional totally asymmetric diffusion models several types of totally asymmetric diffusion models with and without exclusion are considered for some models conditional probabilities of finding n particles on lattice sites at time t with initial occupation at time t 0 are expressed in a determinant form on the other hand the q boson totally asymmetric diffusion model is introduced which interpolates the free boson model and the model with exclusion like interaction the effects of the interaction are compared for the case of two particle diffusion

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

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

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

population genetics with fluctuating population sizes standard neutral population genetics theory with a strictly fixed population size has important limitations an alternative model that allows independently fluctuating population sizes and reproduces the standard neutral evolution is reviewed we then study a situation such that the competing species are neutral at the equilibrium population size but population size fluctuations nevertheless favor fixation of one species over the other in this case a separation of timescales emerges naturally and allows adiabatic elimination of a fast population size variable to deduce the fluctuation induced selection dynamics near the equilibrium population size the results highlight the incompleteness of the standard population genetics with a strictly fixed population size

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

exact diffusion constant for the one dimensional partially asymmetric exclusion model we calculate exactly the diffusion constant associated with the fluctuations of the current for the partial asymmetric exclusion model on a ring with an arbitrary number of particles and holes we also give the diffusion constant of a tagged particle on that ring our approach extends using the deformed harmonic oscillator algebra a result already known for the fully asymmetric case in the limit of weak asymmetry we extract from our exact expression the crossover between the edwards wilkinson and the kardar parisi zhang equations in 1 1 dimensions

large deviations for a stochastic model of heat flow we investigate a one dimensional chain of 2n harmonic oscillators in which neighboring sites have their energies redistributed randomly the sites n and n are in contact with thermal reservoirs at different temperature and kipnis et al j statist phys 27 65 74 1982 proved that this model satisfies fourier s law and that in the hydrodynamical scaling limit when n the stationary state has a linear energy density profile bar theta u u 1 1 we derive the large deviation function s u for the probability of finding in the stationary state a profile u different from bar theta u the function s has striking similarities to but also large differences from the corresponding one of the symmetric exclusion process like the latter it is nonlocal and satisfies a variational equation unlike the latter it is not convex and the gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state we also briefly discuss more general models and find the features common in these two and other models whose s is known

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

exact solution of the totally asymmetric simple exclusion process shock profiles the microscopic structure of macroscopic shocks in the one dimensional totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles first and second class this nonequilibrium steady state factorizes about any second class particle which implies factorization in the one component system about the random shock position it also exhibits several other interesting features including long range correlations in the limit of zero density of the second class particles the solution also shows that a finite number of second class particles in a uniform background of first class particles form a weakly bound state

on the integrability of zero range chipping models with factorized steady states the conditions of the integrability of general zero range chipping models with factorized steady states which were proposed in evans et al 2004 j phys a math gen 37 l275 are examined we find a three parametric family of hopping probabilities for the models solvable by the bethe ansatz which includes most of known integrable stochastic particle models as limiting cases the solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations which is proved as a byproduct we use the bethe ansatz to solve an eigenproblem for the transition matrix of the markov process on its basis we conjecture an integral formula for the green function of the evolution operator for the model on an infinite lattice and derive the bethe equations for the spectrum of the model on a ring

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

a weighted least squares algorithm for quasi equiripple fir and iir digital filter design it has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response fir filter the weighted least squares solution is equiripple the crux of the problem lies in the determination of the necessary least squares frequency response weighting function a novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi equiripple design is presented the algorithm converges very rapidly it typically produces a design which is only about 1 db away from the minimax optimum solution in two iterations and converges to within 0 1 db in six iterations convergence speed is independent of the order of the filter it can be used to design filters with arbitrarily prescribed phase and amplitude response

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

exact diffusion constant for one dimensional asymmetric exclusion models the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions is considered on a ring of size n the steady state of this system is known all configurations have equal weight which allows for easy computation of the average velocity of a particle in the steady state here an exact expression for the diffusion constant of a particle is obtained for arbitrary number of particles and system size by using a matrix formulation two limits of infinite system size n are discussed firstly when the number of particles remains finite as n to infinity the diffusion constant remains dependent on the exact number of particles due to correlations between successive collisions secondly when the density p of particles is non zero i e when the number of particles is equal to n rho as n to infinity the diffusion constant scales as n 12 the exponent 1 2 is related to the dynamic exponent z 3 2 of the kpz equation in 1 1 dimensions

finite dimensional representations of a shock algebra the algebra describing a shock measure in the asymmetric simple exclusion model seen from a second class particle has finite dimensional representations if and only if the asymmetry parameterp of the model and the left and right asymptotic densitiesp of the shock satisfy 1 p p r p 1 p p 1 p for some integerr 1 the minimal dimension of the representation is then 2r these representations can be used to calculate correlation functions in the model

matrix product formula for u _ q a_ 2 1 zero range process the zero range processes introduced recently by mangazeev maruyama and the authors are integrable discrete and continuous time markov processes associated with the stochastic r matrix derived from the well known quantum r matrix by constructing a representation of the relevant zamolodchikov faddeev algebra we present for n 2 a matrix product formula for the steady state probabilities in terms of q boson operators

dynamical pair approximation for cellular automata with shuffle update the random shuffle update method for the asymmetric exclusion process asep is introduced and the dynamical pair technique is extended in order to analyse its dynamics a sequence of approximate models is introduced the first element of which corresponds to the classical parallel update rule whose pair dynamics is reviewed it is then shown how the argument may be extended inductively to solve for the two cell configuration probabilities for each element of the sequence of approximate models a formal limit is then taken and macroscopic velocities and flow rates are derived

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

pairwise balance and invariant measures for generalized exclusion processes we characterize the steady state of a driven diffusive lattice gas in which each site holds several particles and the dynamics is activated and asymmetric using a quantum hamiltonian formalism we show that for arbitrary transition rates the model has product invariant measure in the steady state a pairwise balance condition is shown to hold configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to pairwise balance is more general than the condition of detailed balance and holds in the non equilibrium steady state of a number of stochastic models

an l_1 method for the design of linear phase fir digital filters this paper considers the design of linear phase finite impulse response digital filters using an l1 optimality criterion the motivation for using such filters as well as a mathematical framework for their design is introduced it is shown that l1 filters possess flat passbands and stopbands while keeping the transition band comparable to that of least squares filters the uniqueness of l1 based filters is explored and an alternation type theorem for the optimal frequency response is derived an efficient algorithm for calculating the optimal filter coefficients is proposed which may be viewed as the analogue of the celebrated remez exchange method a comparison with other design techniques is made demonstrating that the l1 approach may be a good alternative in several applications

exact solutions of exactly integrable quantum chains by a matrix product ansatz most of the exact solutions of quantum one dimensional hamiltonians are obtained thanks to the success of the bethe ansatz on its several formulations according to this ansatz the amplitudes of the eigenfunctions of the hamiltonian are given by a sum of permutations of appropriate plane waves in this paper alternatively we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product the eigenvalue equation for the hamiltonian defines the algebraic properties of the matrices defining the amplitudes the consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model the matrix product ansatz we propose allows an unified and simple formulation of several exact integrable hamiltonians in order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the xxz chain spin 1 fateev zamolodchikov model izergin korepin model sutherland model t j model hubbard model etc formulation of the matrix product ansatz for quantum chains with open ends is also possible as an illustration we present the exact solution of an extended xxz chain with z magnetic fields at the surface and arbitrary hard core exclusion among the spins

the matrix product solution of the multispecies partially asymmetric exclusion process we find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles the solution is in the form of a matrix product representation where the matrices for a system of n species are defined recursively in terms of the matrices for a system of n 1 species a complete proof is given based on the quadratic relations verified by these matrices this matrix product construction is interpreted in terms of the action of a transfer matrix

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

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

on the ubiquity of matrix product states in one dimensional stochastic processes with boundary interactions abstract recently it has been shown that the zero energy eigenstate corresponding to the stationary state of a stochastic hamiltonian with nearest neighbour interaction in the bulk and single site boundary terms can generically be written in the form of a so called matrix product state we generalize this result to stochastic hamiltonians with arbitrary but finite interaction range as an application two different particle hopping models with three site bulk interaction are studied for these models which can be interpreted as cellular automata for traffic flow we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions

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

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

construction of diffusion algebras in isaev et al j phys a 34 5815 5834 2001 diffusion algebras were introduced in the context of one dimensional stochastic processes with exclusion in statistical mechanics while isaev et al is based on the needs of the physicist reader and thus states results without proofs and focuses on the discussion of lower dimensional examples it is the purpose of this paper to present a construction formalism for diffusion algebras and to use the latter to prove the results in isaev et al

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

fast singular value shrinkage with chebyshev polynomial approximation based on signal sparsity we propose an approximation method for thresholding of singular values using chebyshev polynomial approximation cpa many signal processing problems require iterative application of singular value decomposition svd for minimizing the rank of a given data matrix with other cost functions and or constraints which is called matrix rank minimization in matrix rank minimization singular values of a matrix are shrunk by hard thresholding soft thresholding or weighted soft thresholding however the computational cost of svd is generally too expensive to handle high dimensional signals such as images hence in this case matrix rank minimization requires enormous computation time in this paper we leverage cpa to approximately manipulate singular values without computing singular values and vectors the thresholding of singular values is expressed by a multiplication of certain matrices which is derived from a characteristic of cpa the multiplication is also efficiently computed using the sparsity of signals as a result the computational cost is significantly reduced experimental results suggest the effectiveness of our method through several image processing applications based on matrix rank minimization with nuclear norm relaxation in terms of computation time and approximation precision

fast and stable least squares approach for the design of linear phase fir filters we present a new numerical algorithm for solving the normal equations associated with the least squares design of linear phase fir filters the usual solution methods have a computational complexity of o n sup 3 moreover solving the normal equations with gaussian elimination commonly yields numerical errors especially when the filter is long here we convert a least squares method into the problem of constructing a system of orthonormal functions the proposed design algorithm needs only o n sup 2 computations and numerical errors can be reduced some examples are given to show the performance of the algorithm

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

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

phase transition in the abc model recent studies have shown that one dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules the mathrm abc model which comprises three particle species that diffuse asymmetrically around a ring shows anomalous coarsening into a phase separated steady state in the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one no phase separation occurs and the steady state of the system is disordered in the present work we consider the weak asymmetry regime q mathrm exp ensuremath ensuremath beta n where n is the system size and study how the disordered state is approached in the case of equal densities we find that the system exhibits a second order phase transition at some nonzero ensuremath beta c the value of ensuremath beta c 2 ensuremath pi sqrt 3 and the optimal profiles can be obtained by writing the exact large deviation functional for nonequal densities we write down mean field equations and analyze some of their predictions

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

matrix product solution of the stationary state of two species open zero range processes using the matrix product ansatz we obtain solutions of the steady state distribution of the two species open one dimensional zero range process our solution is based on a conventionally employed constraint on the hop rates which eventually allows us to simplify the constituent matrices of the ansatz it is shown that the matrix at each site is given by the tensor product of two sets of matrices and the steady state distribution assumes an inhomogeneous factorized form our method can be generalized to the cases of more than two species of particles

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

the noncommutative space of stochastic diffusion systems in the matrix product ground states approach to n species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process we show that the quadratic algebra defines a noncommutative space with a glq n quantum group action as its symmetry boundary processes account for the appearance of parameter dependent linear terms in the algebraic relations we argue that for systems with boundary conditions the diffusion algebras are also obtained either by a shift of basis in the n dimensional quantum plane or by an appropriate change of basis in a lower dimensional one which leads to a reduction of the glq n symmetry

least squared error fir filter design with transition bands the authors propose the use of transition bands and transition functions in the ideal amplitude frequency response to allow the analytical design of optimal least squared error fir digital filters with an explicit control of the transition band edges design formulas are derived for approximations to ideal frequency responses which use pth order spline transition functions a mixed analytical and numerical method for zero error weight in the transition bands and passband and stopband error weighting functions with an integral squared error approximation are derived a variable order spline transition function is developed and a method for choosing the optimal order to minimize the integral squared approximation error is given

exact joint density current probability function for the asymmetric exclusion process we study the asymmetric simple exclusion process with open boundaries and derive the exact form of the joint probability function for the occupation number and the current through the system we further consider the thermodynamic limit showing that the resulting distribution is non gaussian and that the density fluctuations have a discontinuity at the continuous phase transition while the current fluctuations are continuous the derivations are performed by using the standard operator algebraic approach and by the introduction of new operators satisfying a modified version of the original algebra

disordered asymmetric simple exclusion process mean field treatment we provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model the asymmetric simple exclusion process first a mean field steady state mapping is generalized to the disordered case where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady state current density plot for periodic boundary conditions this effect was earlier observed by phys rev e 58 1911 1998 but we provide a treatment for more general distributions of disorder including both numerical results and analytic expressions for the width 2 delta c of the flat section we then apply an argument based on moving shock fronts europhys lett 48 257 1999 to show how this leads to an increase in the high current region of the phase diagram for open boundary conditions second we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well known cole hopf mapping to linearize the equation within this approach we show that adding disorder induces a localization transformation verified by numerical scaling and delta c maps to an inverse localization length helping to give a physical interpretation to the problem

on matrix product ground states for reaction diffusion models we discuss a new mechanism leading to a matrix product form for the stationary state of one dimensional stochastic models the corresponding algebra is quadratic and involves four different matrices for the example of a coagulation decoagulation model explicit four dimensional representations are given and exact expressions for various physical quantities are recovered we also find the general structure of n point correlation functions at the phase transition

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