Stationary product measures for conservative particle systems and ergodicity criteria

TL;DR

This paper characterizes stationary product measures and ergodicity criteria for conservative particle systems on countable sets, covering exclusion, zero range, and misanthrope processes, with results on ergodicity linked to tail sigma algebras.

Contribution

It provides a complete characterization of stationary product measures and ergodicity conditions for a broad class of conservative particle systems, including new insights for infinite configurations.

Findings

Stationary product measures are explicitly characterized for various processes.

Ergodicity of a stationary measure is equivalent to triviality of the tail sigma algebra.

For finite W, the ergodic measures are fully classified.

Abstract

We study conservative particle systems on W^S, where S is countable and W = {0, ..., N} or the natural numbers. The rate of a particle moving from site x to site y is given by p(x,y) b(eta_x, eta_y), where eta_z is the number of particles at site z. Under assumptions on b and the assumption that p is finite range, which allow for the exclusion, zero range and misanthrope processes, we show exactly what the stationary product measures are. Furthermore we show that a stationary measure mu is ergodic if and only if the tail sigma algebra of the partial sums is trivial under mu. This is a consequence of a more general result on interacting particle systems that shows that a stationary measure is ergodic if and only if the sigma algebra of sets invariant under the transformations of the process is trivial. We apply this result combined with a coupling argument on the stationary product measures to determine which product measures are ergodic. For the case that W is finite this gives a complete characterisation. For the case that W is the set of natural numbers we show that for nearly all functions b a stationary product measure is ergodic if and only if it is supported by configurations with an infinite amount of particles. We show that this picture is not complete, we give an example of a system where b is such that there is a stationary product measure which is not ergodic, even though it concentrates on configurations with an infinite number of particles.