From a non-Markovian system to the Landau equation

TL;DR

This paper demonstrates that solutions to a truncated BBGKY hierarchy converge to the Landau equation in the weak coupling limit, bridging non-Markovian particle dynamics with macroscopic kinetic equations.

Contribution

It provides a rigorous proof of convergence from a non-Markovian hyperbolic equation to the Landau equation, clarifying the connection between particle systems and kinetic theory.

Findings

Convergence of truncated BBGKY solutions to the Landau equation

Reformulation of the problem as a non-Markovian hyperbolic equation

Validation of Bogolyubov's multiple time scale analysis approach

Abstract

In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics and can be reformulated as a non-Markovian hyperbolic equation, which converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov's derivation of the kinetic equation by means of a multiple time scale analysis of the BBGKY hierarchy.