A brief introduction to the scaling limits and effective equations in kinetic theory

TL;DR

This paper introduces the fundamental equations in kinetic theory, focusing on their derivation from particle systems, the associated mathematical challenges, and the current open problems in establishing rigorous limits.

Contribution

It provides an overview of the derivation of Boltzmann and Landau equations from particle systems and discusses open mathematical problems in proving these limits rigorously.

Findings

Formal derivations of kinetic equations from particle systems

Discussion of mathematical challenges in rigorous proofs

Identification of open problems in kinetic theory

Abstract

These lecture notes provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context.