  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.
