Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control

TL;DR

This paper analyzes the behavior of nonlocal Fokker-Planck equations with two small parameters, revealing that in the fast reaction regime, their dynamics can be described by a rate-independent model exhibiting hysteresis and phase transitions.

Contribution

It establishes a rigorous connection between nonlocal Fokker-Planck equations and rate-independent hysteresis models in the context of Kramers-type phase transitions.

Findings

Derives mass-dissipation estimates using Muckenhoupt constants

Characterizes mass flux between phases via moment estimates

Proves dynamical stability and compactness of macroscopic quantities

Abstract

The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a time- dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and char- acterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical sta- bility of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.