A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions

Abstract

We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time-dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential. We extend the variational approach to this problem of time-dependent regularity in order to include Hamiltonian trajectories with a finite lifetime in closed domains with a boundary. This leads to new phenomena, such a recovery of smoothness. We hereby create a new and unifying approach for the study of mean-field Gibbs-non-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations.

arXiv commentThis version to appear in Trans. AMS. Minor changes in the introduction and a few minor corrections

References in corpus (3)

• Gibbs-non-Gibbs transitions via large deviations: computable examples
• Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom-Rowlinson models
• A large deviation perspective on exponential decay of entropy and lower bounds on the Ricci-curvature

Cited by in corpus (3)

• Dynamical Gibbs-non-Gibbs transitions in lattice Widom-Rowlinson models with hard-core and soft-core interactions
• Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3
• Gibbs-non Gibbs transitions in different geometries: The Widom-Rowlinson model under stochastic spin-flip dynamics