# Mean field evolution of fermonic systems

# Mean field evolution of fermonic systems

Please note different day (Wednesday), time and location (Jadwin 303). 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. As the number of particles grows, the quantum evolution of the system is expected 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, at zero (pure states) or at positive temperature (mixed states), with an appropriate semiclassical structure. Under mild 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 can be extended to Coulomb interactions, under the assumption that the solution of the time-dependent Hartree-Fock equation preserves the semiclassical structure of the initial data. If time permits, the convergence from the time-dependent Hartree-Fock equation to the Vlasov equation will also be discussed. The results hold for all semiclassical times, and give effective bounds on the rate of convergence towards the effective dynamics as the number of particles goes to infinity.