Séminaire Jeunes chercheurs (Côme Tabary, jeudi 13 novembre 2025)

La prochaine séance du séminaire Jeunes chercheurs du CEREMADE aura lieu le jeudi 13 novembre 2025 à 17h en salle A707. Nous aurons le plaisir d'écouter Côme Tabary (Paris Cité), qui nous parlera de

An introduction to propagation of chaos illustrated by the Boltzmann and Landau equations

Abstract

The Boltzmann equation provides a statistical description of dilute gas at a mesoscopic scale, intermediate to microscopic particle interactions and macroscopic fluid mechanics. A deep ansatz in its derivation is the assumption of molecular chaos, that has puzzled mathematicians and physicists for decades: It claims that any two gas particles are uncorrelated before interacting with each other, disregarding the previous interactions that should have made them correlated.

In 1956, Mark Kac proposed to justify this ansatz by starting from a simpler Markov process of N particles, and letting N go to infinity. He claimed that the law of the N particles should approach the N-fold tensor product of the solution of the Boltzmann equation, a limit now known as propagation of chaos: this would indeed show that the particles become independent of each other in the large N limit, despite constantly interacting with each other.

In this talk, I will introduce the Boltzmann equation as well as its close cousin the Landau equation, which is used to model plasmas, and showcase their key properties. I will motivate Kac's program for these equations, highlight the mathematical challenges it poses, and present a proof of the propagation of chaos for the Landau equation.