From particle systems to the Landau equation: a consistency result

TL;DR

This paper demonstrates that in a weak-coupling particle system, the smooth part of the dynamics converges to the Landau equation as the number of particles grows, establishing a link between microscopic and kinetic descriptions.

Contribution

It introduces a new hierarchy with a memory term for the smooth component and proves the convergence to the Landau equation, including propagation of chaos.

Findings

First order correction converges to Landau equation term

Propagation of chaos is established

Hierarchy with memory term accurately describes the limit

Abstract

We consider a system of N classical particles, interacting via a smooth, short- range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey to the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in term of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N \rightarrow \infty, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.