A note on a Vlasov-Fokker-Planck equation with non-symmetric interaction

TL;DR

This paper investigates a Vlasov-Fokker-Planck equation with non-symmetric interaction, analyzing its stability, free energy properties, and the related particle system, revealing conditions for global convergence and non-equilibrium behavior.

Contribution

It extends the analysis of VFP equations to non-symmetric interactions, establishing free energy as a Lyapunov function and proving global convergence under small Lipschitz interaction forces.

Findings

Free energy acts as a Lyapunov function with a suitable definition.

Global convergence occurs when the interaction force is sufficiently small.

The particle system is a non-equilibrium Langevin process with contractive Fisher information.

Abstract

In the recent [3], Cesbron and Herda study a Vlasov-Fokker-Planck (VFP) equation with non-symmetric interaction, introduced in physics to model the distribution of electrons in a synchrotron particle accelerator. We make four remarks in view of their work: first, it is noticed in [3] that the free energy classically considered for the (symmetric) VFP equation is not a Lyapunov function in the non-symmetric case, and we will show however that this is still the case for a suitable definition of the free energy (with no explicit expression in general). Second, when the interaction is sufficiently small (in $W^{1,\infty}$), it is proven in [3] that the equation has a unique stationary solution which is locally attractive; in this spirit, we will see that, when the interaction force is Lipschitz with a sufficiently small constant, the convergence is global. Third, we also briefly discuss the mean-field interacting particle system corresponding to the VFP equation which, interestingly, is a non-equilibrium Langevin process. Finally, we will see that, in the small interaction regime, a suitable (explicit) non-linear Fisher information is contracted at constant rate, similarly to the situation of Wasserstein gradient flows for convex functionals (although here the dynamics is not a gradient flow).