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

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.

revised version with enlarged introduction, additional overview on the proof strategy, and minor corrections in the proofs; 51 pages, several figures

References in corpus (2)

• Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
• Kramers and non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint

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