https://hal-polytechnique.archives-ouvertes.fr/hal-03514290Golse, FrançoisFrançoisGolseCMLS - Centre de Mathématiques Laurent Schwartz - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueMEAN-FIELD LIMITS IN STATISTICAL DYNAMICSHAL CCSD2022Mean-field limitClassical limit of quantum mechanicsSchrödinger equationHartree equationVlasov equationEmpirical measureWasserstein distance[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]Golse, François2022-01-06 11:26:092022-01-12 03:46:092022-01-11 10:07:33enPreprints, Working Papers, ...application/pdf1These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum N-particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul.