Combinatorial mappings of exclusion processes

Abstract

We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties.Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state-such as the entropy-can be interpreted in terms of how this larger state space has been partitioned.These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.

Integrability of two-species partially asymmetric exclusion processes

“…The fluctuations are controlled by the defect, which creates low density "holes" at rate αp, that travel backwards, and high density waves at rate px/α, that travel forwards. This creates fluctuations, of magnitude L(ρ 1 − ρ 2 )(1 − u) and L(ρ 1 − ρ 2 )u respectively, whose squares have to be added weighted by their rate of creation to obtain the total variance (13).…”

“…Cases to which such models have been applied include the motion of RNA polymerase during DNA translation [1] and ribosome dynamics in mRNA translation [2], traffic flow on a busy street [3,4], and driven colloids in a narrow channel [5][6][7]. Moreover, these models have been shown to have links to many other problems in statistical physics, including disordered polymers in random media [8], surface growth models [9] (notably, some of the models are known to lie within the KPZ universality class [10]), diffusion in strongly anisotropic materials [11], equations in fluid dynamics, such as the Burgers equation [12], and certain combinatorial problems [13].…”

Matrix product solution for a partially asymmetric 1D lattice gas with a free defect

“…The steady state of this system is of the matrix product type (see [12,26] for reviews of the models that have been solved and the combinatorial mappings of the solutions). In other words, there exist matrices X 0 , X 1 , X 2 , such that the weight of any configuration is given by the corresponding matrix product:…”

Steady state distributions in generalized exclusion processes

“…In addition to classical transitions such as entry, exit and hopping, the Boolean functions for the following dynamics can also be written using Identities (5) and (6). We leave this task to the reader.…”

“…Determining the steady state distribution is of particular interest in studying the ASEP and a wide array of methods are used, from matrix ansatzes [5] to combinatorial enumeration [6]. Zhao and Krishnan [7] introduced an approach using probabilistic Boolean networks (PBN).…”