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

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.

133 pages; lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School "Macroscopic and large scale phenomena", Universiteit Twente, Enschede (The Netherlands)

References in corpus (6)

• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• On the Mean Field and Classical Limits of Quantum Mechanics
• Empirical Measures and Vlasov Hierarchies
• The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics
• Mean-Field Approximation of Quantum Systems and Classical Limit

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• Cumulant expansions of operator groups of quantum many-particle systems