On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations

TL;DR

This paper proves the mean-field limit for particle systems governed by Vlasov-Poisson-Fokker-Planck equations, demonstrating convergence of the particle system to the effective PDE as the number of particles grows.

Contribution

It establishes a rigorous propagation of chaos result for systems with regularized Coulomb interactions, combining entropy methods and trajectory control techniques.

Findings

Proves strong convergence of particle marginals to the VPFP solution.

Handles systems with or without white noise in velocities.

Provides a unified approach for both VP and VPFP equations.

Abstract

The derivation of effective descriptions for interacting many-body systems is an important branch of applied mathematics. We prove a propagation of chaos result for a system of $N$ particles subject to Newtonian time evolution with or without additional white noise influencing the velocities of the particles. We assume that the particles interact according to a regularized Coulomb-interaction with a regularization parameter that vanishes in the $N\to\infty$ limit. The respective effective description is the so called Vlasov-Poisson-Fokker-Planck (VPFP), respectively the Vlasov-Poisson (VP) equation in the case of no or sub-dominant white noise. To obtain our result we combine the relative entropy method from \cite{jabinWang2016} with the control on the difference between the trajectories of the true and the effective description provided in \cite{HLP20} for the VPFP case respectively in \cite{LP} for the VP case. This allows us to prove strong convergence of the marginals, i.e. convergence in $L^1$.