WIAS Preprint No. 3004, (2023)

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


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

2020 Mathematics Subject Classification

  • 82C22 60K35


  • Interacting particle systems, Gibbs measures, relative entropy, attractor




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.

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