On the long-time behaviour of reversible interacting particle systems in one and two dimensions

Abstract

By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett.

arXiv comment28 pages; final version

References in corpus (5)

• Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators
• Rhythmic behavior in a two-population mean field Ising model
• Rotating states in driven clock- and XY-models
• A class of nonergodic interacting particle systems with unique invariant measure
• Dynamical Gibbs Variational Principles for Irreversible Interacting Particle Systems with Applications to Attractor Properties

Cited by in corpus (1)

• Time-periodic behaviour in one- and two-dimensional interacting particle systems