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arXiv · 1212.3128

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

Michael Herrmann·Barbara Niethammer·Juan J. L. Velázquez
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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.

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Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. 2014-07-05. Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control. https://doi.org/10.1007/s00205-014-0782-5

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