Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice

TL;DR

This paper investigates the attractor properties of Gibbs measures in non-reversible lattice dynamics, establishing conditions under which Gibbs measures are limit points of the dynamics, even with non-Gibbsian trajectories.

Contribution

It proves that Gibbs measures form attractors for a broad class of non-reversible lattice dynamics, including low-temperature and non-Gibbsian cases, under certain regularity conditions.

Findings

Weak limit points of invariant measures include Gibbs states.

Limits of sequences of Gibbs measures are Gibbs measures under uniform bounds.

Extension to non-Gibbsian trajectories with small discontinuities.

Abstract

We consider stochastic dynamics of lattice systems with finite local state space, possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: a) There is at least one stationary measure which is a Gibbs measure for an absolutely summable potential Phi. b) Zero loss of relative entropy density under dynamics implies the Gibbs property with the same Phi. We prove results on the attractor property of the set of Gibbs measures for Phi: 1. The set of weak limit points of any trajectory of translation-invariant measures contains at least one Gibbs state for Phi. 2. We show that if all elements of a weakly convergent sequence of measures are Gibbs measures for a sequence of some translation-invariant summable potentials with uniform bound, then the limiting measure must be a Gibbs measure for Phi. 3. We give an extension of the second result to trajectories which are allowed to be non-Gibbs, but have a property of asymptotic smallness of discontinuities. An example for this situation is the time evolution from a low temperature Ising measure by weakly dependent spin flips.