Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

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.

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional \({\mathscr {E}}_\varepsilon \) and the dissipation potential \({\mathscr {R}}_\varepsilon \) or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions \(u_\varepsilon \). We investigate the question under which conditions the limits u of (subsequences of) the solutions \(u_\varepsilon \) are solutions of the gradient system generated by the \(\varGamma \)-limits \({\mathscr {E}}_0\) and \({\mathscr {R}}_0\). Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where \({\mathscr {R}}_\varepsilon \) is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.