Attractor properties for irreversible and reversible interacting particle systems

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Date
2015
Volume
2145
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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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 also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

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Keywords
Interacting particle systems, non-equilibrium, non-reversibility, attractor property, relative entropy, Gibbs measures
Citation
Jahnel, B., & Külske, C. (2015). Attractor properties for irreversible and reversible interacting particle systems (Vol. 2145). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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