A mean field limit for the Vlasov-Poisson system

TL;DR

This paper provides a probabilistic proof demonstrating that as the number of particles increases, their empirical distribution converges to the Vlasov-Poisson system, capturing Coulomb and Newton interactions with a specific cutoff scaling.

Contribution

It introduces a novel probabilistic approach to establish the mean field limit and propagation of chaos for particle systems with Coulomb or Newton potentials, including a specific cutoff scaling.

Findings

Convergence of empirical distributions to Vlasov-Poisson solutions

Applicable to systems with Coulomb and Newton interactions

Uses a probabilistic proof technique

Abstract

We present a probabilistic proof of the mean field limit and propagation of chaos $N$-particle systems in three dimensions with positive (Coulomb) or negative (Newton) $1/r$ potentials scaling like $1/N$ and an $N$-dependent cut-off which scales like $N^{-1/3+ \epsilon}$. In particular, for typical initial data, we show convergence of the empirical distributions to solutions of the Vlasov-Poisson system with either repulsive electrical or attractive gravitational interactions.