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

François Golse (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Keywords: Monge-Kantorovich distance; Propagation of chaos; BBGKY hierarchy; Trace-class operators; Time-dependent Hartree equation; Time-dependent Hartree-Fock equations; Cubic nonlinear Schrödinger equation; Mean field limit; Vlasov-Poisson system; Vorticity formulation of the Euler equation; Empirical measure

Domains: Mathematics [math]; Analysis of PDEs [math.AP]; Mathematics [math]; Mathematical Physics [math-ph]

François Golse. On the Dynamics of Large Particle Systems in the Mean Field Limit. Adrian Muntean, Jens Rademacher, Antonios Zagaris. Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, 3, Springer, Cham, pp.1-144, 2016, Lecture Notes in Applied Mathematics and Mechanics, ⟨10.1007/978-3-319-26883-5_1⟩. ⟨hal-00776304⟩