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Attractor Properties for Irreversible and Reversible Interacting Particle Systems

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Abstract

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

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Acknowledgements

The authors thank the editor and anonymous referees for comments and suggestions that helped to improve the presentation of the material. This research was supported by the Leibniz program Probabilistic Methods for Mobile Ad-Hoc Networks.

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Authors and Affiliations

  1. Weierstrass Institute Berlin, Mohrenstr. 39, 10117, Berlin, Germany

    Benedikt Jahnel

  2. Fakultät für Mathematik, Ruhr-Universität Bochum, 44801, Bochum, Germany

    Christof Külske

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  1. Benedikt Jahnel
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  2. Christof Külske
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Correspondence to Christof Külske.

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Communicated by H. Spohn

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Jahnel, B., Külske, C. Attractor Properties for Irreversible and Reversible Interacting Particle Systems. Commun. Math. Phys. 366, 139–172 (2019). https://doi.org/10.1007/s00220-019-03352-4

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