Nonequilibrium stationary states with Gibbs measure for two or three species of interacting particles

TL;DR

This paper constructs explicit one-dimensional driven diffusive models with two or three particle species that have nonequilibrium dynamics but stationary states matching specific Gibbs measures, highlighting constraints on asymmetric stochastic processes.

Contribution

It provides explicit constructions of driven diffusive systems with prescribed Gibbs stationary measures, analyzing parameter constraints and balance conditions for multiple species.

Findings

Models with two species have five free parameters, generally not obeying pairwise balance.

Totally asymmetric dynamics with two species obey pairwise balance.

Three species models with totally asymmetric dynamics have two free parameters and obey pairwise balance.

Abstract

We construct explicit examples of one-dimensional driven diffusive systems for two and three species of interacting particles, defined by asymmetric dynamical rules which do not obey detailed balance, but whose nonequilibrium stationary-state measure coincides with a prescribed equilibrium Gibbs measure. For simplicity, the measures considered in this construction only involve nearest-neighbor interactions. For two species, the dynamics thus obtained generically has five free parameters, and does not obey pairwise balance in general. The latter property is satisfied only by the totally asymmetric dynamics and the partially asymmetric dynamics with uniform bias, i.e., the cases originally considered by Katz, Lebowitz, and Spohn. For three species of interacting particles, with nearest-neighbor interactions between particles of the same species, the totally asymmetric dynamics thus obtained has two free parameters, and obeys pairwise balance. These models are put in perspective with other examples of driven diffusive systems. The emerging picture is that asymmetric (nonequilibrium) stochastic dynamics leading to a given stationary-state measure are far more constrained (in terms of numbers of free parameters) than the corresponding symmetric (equilibrium) dynamics.