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

TL;DR

This paper introduces a Hamilton-Jacobi framework to analyze the regularity and transitions of large deviation rate functions in mean-field models, revealing new phenomena like smoothness recovery.

Contribution

It develops a unifying Hamiltonian dynamics approach to study Gibbs-non-Gibbs transitions and extends variational methods to include finite-lifetime Hamiltonian trajectories.

Findings

Hamiltonian flow describes the evolution of the rate function's gradient.

New phenomena such as recovery of smoothness are identified.

A unifying approach based on Hamilton-Jacobi equations is proposed.

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.