Non-equilibrium Dynamics and Random Matrices

According to first principle quantum mechanics, the evolution of N fermions (particles with antisymmetric wave function) is governed by the many body Schroedinger equation. We are interested, in particular, in the evolution in the mean field regime, characterized by a large number of weak collisions among the particles. For fermions, the mean field regime is naturally linked with a semiclassical limit. Asymptotically, the many body Schroedinger evolution can therefore be described by the classical Vlasov equation. A better approximation, however, is given by the Hartree-Fock equation. In this talk, we will show precise bounds on the convergence of the Schroedinger dynamics towards the Hartree-Fock evolution, for initial data close to Slater determinants with the appropriate semiclassical structure.