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

deterministic exclusion process with a stochastic defect matrix product ground states we study a one dimensional anisotropic exclusion model describing particles moving deterministically on a ring with a single defect across which they move with probability 0 q 1 we show that the stationary state of this model can be represented as a matrix product state

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 2d directed animals on a strip of finite width proves a conjecture giving the exact number of directed animals of s sites with any root on a strip of finite width of a square lattice the authors also rederive more simply some previous results concerning the connective constant and particular eigenvectors of the transfer matrix

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

exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics a one dimensional driven lattice gas with disorder in the particle hopping probabilities is considered it has previously been shown that in the version of the model with random sequential updating a phase transition occurs from a low density inhomogeneous phase to a high density congested phase here the steady states for both parallel fully synchronous updating and ordered sequential updating are solved exactly the phase transition is shown to persist in both cases with the critical densities being higher than that for random sequential dynamics the steady state velocities are related to the fugacity of a bose system suggesting a principle of minimization of velocity a generalization of the dynamics to the case where the hopping probabilities depend on the number of empty sites in front of the particles is also solved exactly in the case of parallel updating the models have natural interpretations as simplistic descriptions of traffic flow the relation to more sophisticated traffic flow models is discussed

comparison between simultaneous and sequential updating in 2n 1 1 cellular automata properties of cellular automata with packard wolfram code 2n 1 1 have been studied without and with noise on square n 4 and triangular n 6 lattices the outer totalistic rule used assign a central spin opposite to the majority spin of its neighbourhood and flips it in the case of equality starting from random configurations with a variable proportion r of up spins different relaxation behaviours and equilibrium structures are obtained a ferromagnetic f structure and an antiferromagnetic af structure for simultaneous and sequential updatings respectively on a square lattice when the spins are processed simultaneously in the presence of noise a first order transition f af is observed for a square lattice and a second order transition for a triangular lattice the results show similarities with ising models domains but also strong differences magnetization as a function of r

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

stochastic models on a ring and quadratic algebras the three species diffusion problem the stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations we consider only exclusion processes and restrict the type of algebras such that one has recurrence relations for traces of words of different lengths this is possible only if the rates satisfy certain compatibility conditions these conditions are derived and explicit representations of the generators of the quadratic algebras are given

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

partially asymmetric exclusion process with open boundaries exclusive diffusion on a one dimensional lattice is studied in the model particles hop stochastically in both directions but with different rates at the ends of the lattice particles are injected and removed the exact stationary probability measure is represented in the form of a matrix product as a generalization of the solution given by derrida et al j phys a 26 1493 1993 for the fully asymmetric process the phase diagram of the current on the infinite lattice is obtained analytic expressions for the current in the different phases are derived the model is equivalent to an xxz heisenberg chain with a certain type of boundary terms the ground state of which corresponds to the stationary solution of the master equation

global existence and blow up for the degenerate and singular nonlinear parabolic system with a nonlocal source abstract the existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate and singular parabolic system u t x u x x 0 a u m x t v n x t d x v t x v x x 0 a v p x t u q x t d x in 0 a 0 t where t a 0 are constants and m n p q are positive real numbers furthermore under certain conditions it is proved that the blow up set of the solution is the entire interval 0 a and we also obtain the blow up rate under the condition

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

car oriented mean field theory for traffic flow models we present a new analytical description of the cellular automaton model for single lane traffic in contrast to previous approaches we do not use the occupation number of sites as dynamical variable but rather the distance between consecutive cars therefore certain longer ranged correlations are taken into account and even a mean field approach yields non trivial results in fact for the model with the exact solution is reproduced for the fundamental diagram shows a good agreement with results from simulations

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

garden of eden states in traffic models we investigate the allowed configurations in the stationary state of the cellular automaton model for single lane traffic it is found that certain states in the configuration space cannot be reached if one uses parallel dynamics these so called garden of eden goe states do not exist for random sequential dynamics and are responsible for the strong short ranged correlations found in parallel dynamics by eliminating the goe states we obtain a simple and effective approximative description of the model for the exact solution is recovered for this elimination leads to much higher values of the flux compared to the mean field result which are in good agreement with monte carlo simulations

stochastic traffic model with random deceleration probabilities queueing and power law gap distribution we extend the nagel schreckenberg stochastic cellular automata model for single lane vehicular traffic to incorporate quenched random deceleration probabilities we show by computer simulations that at low densities this model displays queueing of cars with a power law probability distribution of gaps between the cars while at high densities the behaviour of the model is similar to the jammed phase of the standard nagel schreckenberg model the approach to the steady state is characterized by the same critical exponents as for the coarsening process in the simple exclusion processes with random rates recently investigated independently by krug and ferrari and evans the numerical values of the exponents for gap distributions are in agreement with the analytical conjecture of krug and ferrari which implies that the models belong to the same universality class

discrete stochastic models for traffic flow we investigate a probabilistic cellular automaton model which has been introduced recently this model describes single lane traffic flow on a ring and generalizes the asymmetric exclusion process models we study the equilibrium properties and calculate the so called fundamental diagrams flow versus density for parallel dynamics this is done numerically by computer simulations of the model and by means of an improved mean field approximation which takes into account short range correlations for cars with a maximum velocity of 1 the simplest nontrivial approximation gives the exact result for higher velocities the analytical results obtained by iterated application of the approximation scheme are in excellent agreement with the numerical simulations

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

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

jamming transition in a cellular automaton model for traffic flow the cellular automaton model for traffic flow exhibits a jamming transition from a free flow phase to a congested phase in the deterministic case this transition corresponds to a critical point with diverging correlation length in the presence of noise however no consistent picture has emerged up to now we present data from numerical simulations that suggest the absence of critical behavior the transition of the deterministic case is smeared out and one only observes the remnants of the critical point

cellular automation models and traffic flow a cellular automaton model for the description of traffic flow is investigated it generalizes asymmetric exclusion models which have attracted a lot of interest in the past the authors calculate the so called fundamental diagram flow versus density for parallel dynamics using an improved mean field approximation which takes into account short range correlations for maximum velocity they find that the simplest of these non trivial approximations gives the exact result for higher velocities their results are in excellent agreement with numerical data

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