Attractor properties for irreversible and reversible interacting particle systems

TL;DR

This paper proves that under certain conditions, the long-term behavior of translation-invariant interacting particle systems converges to Gibbs measures, extending previous results to include irreversible and non-ergodic dynamics.

Contribution

It introduces a method to show that limit points of particle system trajectories are Gibbs measures, even for irreversible and non-ergodic dynamics, under mild conditions.

Findings

Weak limit points are Gibbs states for the same specification.

The method applies to both reversible and irreversible systems.

Convergence results are extended using relative entropy techniques.

Abstract

We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification. We show how to prove the non-nullness for a large number of cases, and also give an alternate version of the last condition such that the non-nullness requirement can be dropped. As an application we obtain the attractor property if there is a reversible Gibbs measure. Our method generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones.