Stationary solutions of the Vlasov-Fokker-Planck equation: existence, characterization and phase-transition

TL;DR

This paper investigates the stationary solutions of the Vlasov-Fokker-Planck equation, revealing non-uniqueness and phase transitions in particle distributions influenced by potential, interaction, friction, and stochastic forces.

Contribution

It establishes the existence of multiple stationary solutions and phase transitions for the Vlasov-Fokker-Planck equation under certain conditions, extending recent McKean-Vlasov results.

Findings

Non-uniqueness of stationary solutions

Existence of phase transitions

Dependence on potential and interaction parameters

Abstract

In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean-Vlasov equation.