Kac's Program for the Landau Equation

TL;DR

This paper rigorously derives the Landau equation with Coulomb interactions from a many-particle system, extending Kac's program to soft potentials and establishing convergence in multiple senses.

Contribution

It provides the first proof of propagation of chaos for Coulomb Landau equation and extends Kac's approach to soft potentials, including new functional inequalities.

Findings

Proves convergence in weak, Wasserstein, and entropic senses.

Establishes strong $L^1$ convergence.

Handles singular soft potentials with extended duality and new estimates.

Abstract

We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative $N$-particle system, obtained by passing to the grazing limit on Kac's walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac's program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong $L^1$ convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch-Duerinckx-Jabin \cite{bresch2024duality} and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.