Microscopic derivation of Vlasov equation with compactly supported pair potentials

TL;DR

This paper provides a probabilistic proof that the microscopic N-particle dynamics with compactly supported pair potentials converges to the Vlasov equation, specifically deriving the Vlasov-Dirac-Benney system with delta-like interactions.

Contribution

It introduces a novel probabilistic approach to derive the Vlasov equation from microscopic dynamics with short-range forces for a range of parameters.

Findings

Convergence of Newtonian trajectories to Vlasov-Dirac-Benney characteristics.

Establishment of mean-field limit for compactly supported potentials.

Derivation of Vlasov-Dirac-Benney equation from microscopic N-particle system.

Abstract

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with compactly supported pair potentials of the form $N^{3\beta-1} \phi(N^{\beta}x)$ for $\beta\in\left[0,\frac{1}{7}\right)$ and $\phi\in L^{\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)$. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Dirac-Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Dirac-Benney equation from the microscopic $N$-particle dynamics with a strong short range force.