WIAS Preprint No. 2935, (2022)

Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties


  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Köppl, Jonas
    ORCID: 0000-0001-9188-1883

2020 Mathematics Subject Classification

  • 82C20 60K35


  • Gibbs measures, interacting Markov jump processes, Gibbs variational principle, time-reversed dynamics, relative-entropy density, relative-entropy production, omega-limit set, non-reversibility




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.

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