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.
Keywords
Cite
@article{arxiv.1212.3128,
title = {Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control},
author = {Michael Herrmann and Barbara Niethammer and Juan J. L. Velázquez},
journal= {arXiv preprint arXiv:1212.3128},
year = {2015}
}
Comments
revised version with enlarged introduction, additional overview on the proof strategy, and minor corrections in the proofs; 51 pages, several figures