On the Dynamics of Large Particle Systems in the Mean Field Limit
By François Golse. 2013
250 citations recorded by the source corpus.
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 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.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Mechanics
View Paper: https://www.semanticscholar.org/paper/c9f34244e41be310a266c589c44ffc48fc719fcc
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Citations in the corpus
- Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “We refer to [41,55] for an introduction to the mean field question in this kinetic setting.”
- Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, background): “Then the empirical measure as defined by (1.9) satisfies that Theorem 2 implies that with probability 1, µ N will converge to f for the weak − ∗ topology of measures. presentation We refer to [23, 24, 31] for a more precise of this connection between the various concepts of Mean Field limit.”
- A review of the mean field limits for Vlasov equations (2014, methodology): “The best result so far requires F ∈ W 1,∞, see [147] for instance (and the nice presentation in [73]), and actually uses the system of particles (1.”
- Mean Field Limit for Stochastic Particle Systems (2017, background): “However we focus here on the stochastic case and refer to [35, 46] for a review of the mean field limit for deterministic systems.”