Here are the slides of a short course on “Many-particle systems and kinetic theory” I gave at the spring school “Kinetic theory and fluid mechanics”:
Comments are most welcome.
Abstract: Many-particle systems (gas, plasmas, galaxy..) are in general too hard to study through microscopic models, which attempt at following all the trajectories in phase space. Hence the need for a many-particle limit and a statistical kinetic viewpoint. The first such limit was pionnered by Maxwell, Boltzmann, Grad in the case of collisional gases. Motivated by plasma physics, Vlasov proposed another limit in the case of long-range interactions, based on the self-induced mean-field of the particles. We shall first present an introduction to this discussion and its connexion with the problem of irreversibility. Second we shall focus on the mean-field approach and review some classical works of Hepp, Braun and Dobrushin on the Vlasov-Poisson equation. In the last part of the course, we shall present a program of research devised by Kac in 1956 for investigating the irreversibility and entropy structure of many-particle systems. It is also based on a mean-field viewpoint, however starting from a stochastic jump process. We shall conclude by outlining some recent results related to this program.