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

Benedikt Jahnel and Jonas Köppl
Vol. 6 (2025), No. 2, 479–503
DOI: 10.2140/pmp.2025.6.479
Abstract

By refining Holley’s free-energy technique, we show that, under quite general assumptions on the dynamics, a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions cannot exhibit time-periodic behaviour if the dynamics admits a reversible Gibbs measure. This is the first result that makes the physical intuition rigorous that time-periodic behaviour can only happen in driven, i.e., nonreversible systems.

Keywords
interacting particle systems, Gibbs measures, relative entropy, attractor, periodic behaviour
Mathematical Subject Classification
Primary: 82C22
Secondary: 60K35
Milestones
Received: 19 March 2023
Revised: 23 August 2024
Accepted: 21 December 2024
Published: 12 March 2025
Authors
Benedikt Jahnel
Institut für Mathematische Stochastik
Technische Universität Braunschweig
Braunschweig
Germany		 Weierstrass-Institut fur Angewandte Analysis und Stochastik
Berlin
Germany
Jonas Köppl
Weierstrass-Institut fur Angewandte Analysis und Stochastik
Berlin
Germany
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).