Martin R. Evans

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.

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.

Nonequilibrium statistical mechanics of the zero-range process and related models

We review recent progress on the zero-range process, a model of interacting particles which hop between the sites of a lattice with rates that depend on the occupancy of the departure site. We discuss several applications which have stimulated interest in the model such as shaken granular gases and network dynamics; we also discuss how the model may be used as a coarse-grained description of driven phase-separating systems. A useful property of the zero-range process is that the steady state has a factorized form. We show how this form enables one to analyse in detail condensation transitions, wherein a finite fraction of particles accumulate at a single site. We review condensation transitions in homogeneous and heterogeneous systems and also summarize recent progress in understanding the dynamics of condensation. We then turn to several generalizations which also, under certain specified conditions, share the property of a factorized steady state. These include several species of particles; hop rates which depend on both the departure and the destination sites; continuous masses; parallel discrete-time updating; non-conservation of particles and sites.

Phase transitions in one-dimensional nonequilibrium systems

The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transitions which have been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is known exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. It is also shown that even when the conditions for condensation are not fulfilled one can still observe very sharp crossover behaviour and apparent condensation in a finite system. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.

Bose-Einstein condensation in disordered exclusion models and relation to traffic flow

A disordered version of the one-dimensional asymmetric exclusion model where the particle hopping rates are quenched random variables is studied. The steady state is solved exactly by use of a matrix product. It is shown how the phenomenon of Bose condensation whereby a finite fraction of the empty sites are condensed in front of the slowest particle may occur. Above a critical density of particles a phase transition occurs out of the low-density phase (Bose condensate) to a high-density phase. An exponent describing the decrease of the steady-state velocity as the density of particles goes above the critical value is calculated analytically and shown to depend on the distribution of hopping rates. The relation to traffic flow models is discussed.

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.

Shock formation in an exclusion process with creation and annihilation

We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.

Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra

We study the partially asymmetric exclusion process with open boundaries. We generalize the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q . Due to the relationship between the matrix algebra and the q -deformed quantum harmonic oscillator algebra we find that q -Hermite polynomials, along with their orthogonality properties and generating functions, are of great utility. We employ two distinct sets of q -Hermite polynomials, one for q 1. It turns out that these correspond to two distinct regimes: the previously studied case of forward bias (q 1) where the boundaries support a current opposite in direction to the bulk bias. For the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.

Phase Separation in One-Dimensional Driven Diffusive Systems

Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel(February 7, 2008)A driven diffusive model of three types of particles that exhibits phase separation on a ring isintroduced. The dynamics is local and comprises nearest neighbor exchanges that conserve each ofthe three species. For the case in which the three densities are equal, it is shown that the modelobeys detailed balance. The Hamiltonian governing the steady state distribution in this case is givenand is found to have long range asymmetric interactions. The partition sum and bounds on somecorrelation functions are calculated analytically demonstrating phase separation.PACS numbers: 02.50.Ey; 05.20.-y; 64.75.+g

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.