Dynamical Gibbs Variational Principles for Irreversible Interacting Particle Systems with Applications to Attractor Properties

TL;DR

This paper establishes that for certain irreversible particle systems, measures with zero relative entropy loss are Gibbs measures, leading to an attractor property where limits of trajectories are Gibbs measures, extending previous results.

Contribution

It proves that zero entropy loss implies Gibbsianity for a broad class of irreversible systems, and demonstrates the attractor property for these systems.

Findings

Zero entropy loss measures coincide with Gibbs measures.

Weak limit points of trajectories are Gibbs measures.

Results extend to general irreversible interacting particle systems.

Abstract

We consider irreversible translation-invariant interacting particle systems on the $d$-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.