Dynamical Gibbs variational principles and applications

Koeppl Jonas, Weierstrass Institute

We consider irreversible translation-invariant interacting particle systems on the $d$-dimensio\-nal hypercubic 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 the relative entropy is non-increasing as a function of time and that a vanishing relative entropy loss for 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. We first explain the general idea behind the method on the simple example of a continuous time Markov chain on a finite state space and then discuss how one can extend it to irreversible interacting particle systems in infinite volumes.

Area: CS17 - Interacting systems in statistical physics I (Alexander Zass)

Keywords: interacting particle system, Gibbs Measure, entropy, variational principle, limit theorem

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