On the Dynamics of Large Particle Systems in the Mean Field Limit

TL;DR

This paper reviews rigorous derivations of mean field PDEs from microscopic particle systems, highlighting mathematical techniques like stability estimates and BBGKY hierarchies, and discusses the propagation of chaos as particle number grows.

Contribution

It provides a comprehensive overview of mathematical methods for deriving mean field equations from microscopic models, emphasizing the connections between different approaches.

Findings

Clarifies the role of Dobrushin's stability estimate in derivations.

Explains the relation between BBGKY hierarchies and mean field limits.

Highlights the importance of chaotic sequences and propagation of chaos.

Abstract

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics - can be rigorously derived from first principles, i.e. from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin's stability estimate in the Monge-Kantorovich distance for the empirical measures built on the solution of the N-particle motion equations in classical mechanics, or the theory of BBGKY hierarchies in the case of classical as well as quantum problems. We explain in detail how these different approaches are related; in particular we insist on the notion of chaotic sequences and on the propagation of chaos in the BBGKY hierarchy as the number of particles tends to infinity.