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 timedependent 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 characterize 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 stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.

Population dynamics of driven autocatalytic reactive mixtures

“…As mentioned above, Herrmann et al [10,18,19] performed extensive analysis on the various regimes of the population dynamics at different currents for the simple kinetics R = R 0 ∆µ. Our more general model approaches it asymptotically as R → 0.…”

“…The point of exit from phase separation is oscillatory with respect to total reaction rate. This is due to the varying frequency observed in the regime of oscillatory phase transition, which is addressed extensively previously and is beyond the scope of this work [10,18,19].…”

“…Emergent behavior arises in population dynamics with nonlinear dynamics. Herrmann et al [18,19] studied phase transition based on Dreyer's model in the case of constant total current. The coupling of particles through the constraint of total current gives rise to a variety of regimes ranging from Kramers type transition, oscillatory bimodal distribution, to unimodal distribution with increasing total current.…”

Gradient flow formulation and longtime behaviour of a constrained Fokker–Planck equation

“…Then, equation (1.1) is obtained by passing to the limit in the time step of the discrete scheme. In addition, this provides an alternative well-posedness result to [14,Lemma 1] that is based on a fixed point argument. Let us also note, that well-posedness in the case of compact state space is also obtained in [8].…”

Mass transport in Fokker–Planck equations with tilted periodic potential

“…More precisely, we show in §2.3 for given t that almost all mass is in fact contained in the vicinity of the local minima p = P j provided that D(t) from (1.16) is sufficiently small. Similar massdissipation estimates have been used in [HNV14].…”