Attractor Properties for Irreversible and Reversible Interacting Particle Systems

Abstract

We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a timestationary 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 timestationary 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.

Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees

“…More recently, Jahnel and Külske [JK19] have extended the free energy density approach to non-reversible dynamics on integer-lattice systems.…”

Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3

“…Let us mention here time-evolved discrete lattice spins [19,30], continuous lattice spins [24,34], time-evolved models of point particles in Euclidean space [17], and models on trees [32]. For a discussion of non-Gibbsian behavior of timeevolved lattice measures in regard to the approach to a (possibly non-unique) invariant state under dynamics, see [16], for relevance of non-Gibbsianness to the infinite-volume Gibbs variational principle (and its possible failure) see [22,25]. For recent developments for one-dimensional long-range systems, and the relation between continuity of one-sided (vs. two-sided) conditional probabilities see [2][3][4]31].…”

Gibbs-Non Gibbs Transitions in Different Geometries: The Widom-Rowlinson Model Under Stochastic Spin-Flip Dynamics

“…In our example above P t is the symmetric independent spin-flip dynamics, and does not involve a randomization of the spatial degrees of freedom. However it is clear that one would like to study more generally also dependent dynamics, and also possibly irreversible dynamics, compare [31,19]. Such studies have been performed at first for the Ising model on the lattice, for work on this and related work see [8,30,28,11,12,9,13,26].…”