Hartree-Fock dynamics of weakly interacting fermionic systems

In this talk I will discuss the dynamics of interacting fermionic systems in the mean-field regime. Compared to the bosonic case, fermionic mean-field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. From a physical point of view, as the number of particles grows one expects the quantum evolution of the system to be effectively described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation. I will consider initial data which are close to quasi-free states, both at zero and at positive temperature, with an appropriate semiclassical structure. Under some regularity assumptions on the interaction potential I will show that the time evolution of such initial data stays close to a quasi-free state, with reduced one-particle density matrix given by the solution of the time-dependent Hartree-Fock equation. The result holds for all (semiclassical) times, and gives effective bounds on the rate of convergence towards the Hartree-Fock dynamics as the number of particles goes to infinity