Vlasov limit for a chain of oscillators with Kac potentials

TL;DR

This paper proves the convergence of a chain of oscillators with long-range interactions to a Vlasov-type equation and demonstrates energy diffusion in certain anharmonic potentials, advancing understanding of long-range interacting systems.

Contribution

It establishes the Vlasov limit for a chain of oscillators with Kac potentials and proves energy diffusion for specific anharmonic potentials, despite non-exchangeability.

Findings

Convergence of empirical measure to Vlasov-type equation

Energy diffusion in a class of anharmonic potentials

Extension of mean field limits to non-exchangeable particles

Abstract

We consider a chain of anharmonic oscillators with local mean field interaction and long-range stochastic exchanges of velocity. Even if the particles are not exchangeable, we prove the convergence of the empirical measure associated with this chain to a solution of a Vlasov-type equation. We then use this convergence to prove energy diffusion for a restricted class of anharmonic potentials.