ABSTRACT. We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann-Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (appeared after the well known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states that are close to quasi-free states and prove that, under suitable assumptions on the initial data and on the many-body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi-free state. In particular, we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many-body dynamics towards the Hartree-Fock evolution.

We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N , we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the rate of convergence. 2 HS ≤ C , (1.6)where the full energy is of order N (here we used that the Hilbert-Schmidt norm 2 of ω N is bounded by N 1/2 ). Because of the smallness of the exchange term, instead of considering the Hartree-Fock 1 In general quasi-free states are characterized by two operators on L 2 (R 3 ), a one-particle reduced density ωN and a pairing density α. Here we restrict our attention to states with α = 0; this is expected to be a very good approximation for equilibrium states of fermions in the mean field regime considered here.2 The Hilbert-Schmidt norm of a compact operator A is defined as A 2 HS = trA * A.

We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally.