The Vlasov-Poisson dynamics as the mean field limit of extended charges

TL;DR

This paper demonstrates that the Vlasov-Poisson equation in multiple dimensions can be rigorously derived as a mean field limit from a system of extended charges, under specific initial conditions and convergence rates.

Contribution

It establishes the validity of the Vlasov-Poisson dynamics as a mean field limit of Coulomb systems with extended charges, including convergence conditions and typical initial configurations.

Findings

Vlasov-Poisson dynamics derived from Coulomb systems

Convergence depends on initial empirical distribution rates

Propagation of molecular chaos established

Abstract

The paper treats the validity problem of the nonrelativistic Vlasov-Poisson equation in $d\geq 2$ dimensions. It is shown that the Vlasov-Poisson dynamics can be derived as a combined mean field and point-particle limit of an N-particle Coulomb system of extended charges. This requires a sufficiently fast convergence of the initial empirical distributions. If the electron radius decreases slower than $N^{-\frac{1}{d(d+2)}}$, the corresponding initial configurations are typical. This result entails propagation of molecular chaos for the respective dynamics