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

Kraaij, Richard; Redig, Frank; van Zuijlen, Willem

doi:10.1090/tran/8408

A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions
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by Richard C. Kraaij, Frank Redig and Willem B. van Zuijlen PDF

Trans. Amer. Math. Soc. 374 (2021), 5287-5329 Request permission

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.

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