Mean field limit for the one dimensional Vlasov-Poisson equation

TL;DR

This paper provides a simplified proof of the mean field limit for the one-dimensional Vlasov-Poisson equation, establishing convergence of particle systems to the PDE and demonstrating global solutions for both particle and continuum models.

Contribution

It offers a simpler proof of the mean field limit and propagation of chaos for the 1D Vlasov-Poisson system, along with existence results for solutions.

Findings

Simplified proof of mean field convergence

Establishment of propagation of molecular chaos

Existence of global solutions for particle and PDE models

Abstract

We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in \cite{Tro86}. Here we shall give a simpler proof of this result, and explain why it implies the so-called "Propagation of molecular chaos". More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.