Abstract
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.
Acknowledgments
The authors would like to thank the three anonymous referees for their insightful feedback that helped to substantially improve this manuscript. Benedikt Jahnel acknowledges the financial support of the Leibniz Association within the Leibniz Junior Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz Competition. Furthermore, Benedikt Jahnel is also affiliated with the Institute for mathematical stochastics at the Technical University of Braunschweig.
Citation
Benedikt Jahnel. Jonas Köppl. "Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties." Ann. Appl. Probab. 33 (6A) 4570 - 4607, December 2023. https://doi.org/10.1214/22-AAP1926
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