Instabilities in the mean field limit

TL;DR

This paper investigates how systems of particles interacting via Coulomb or smooth potentials exhibit instabilities in the mean field limit, especially around certain unstable equilibria, over timescales logarithmic in the number of particles.

Contribution

It demonstrates that in the mean field limit, particle systems can become unstable in logarithmic time for configurations near unstable equilibria, highlighting limitations of mean field approximations.

Findings

Instabilities occur in the mean field limit at times of order log N.

Unstable homogeneous equilibria lead to system instabilities.

Results apply to Coulomb and smooth potentials in various dimensions.

Abstract

Consider a system of $N$ particles interacting through Newton's second law with Coulomb interaction potential in one spatial dimension or a $\mathcal{C}^2$ smooth potential in any dimension. We prove that in the mean field limit $N \to + \infty$, the $N$ particles system displays instabilities in times of order $\log N$ for some configurations approximately distributed according to unstable homogeneous equilibria.