Mean‐Field Evolution of Fermionic Mixed States

Abstract

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.

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

“…The strategy of [37] is based on a rigorous formulation of second order perturbation theory following [38,35,17,36], combined with methods developed in the context of many-body quantum dynamics [7,8,5,58]. For larger interaction potentials however, this method is limited to a lower bound of the right order in and N but not capturing the precise value.…”

Mean-field evolution of fermionic systems

“…Later, the same result has been obtained by Petrat-Pickl [22], with a different method. The result of [5] applies to initial data describing pure states (e.g., approximate Slater determinants); very recently a similar result has been proven for mixed states (describing positive temperature states) by Benedikter-Jaksic-Porta-Saffirio-Schlein [7].…”

“…Here we present the main result of [7], in a simplified form. We refer the reader to [7] for more details.…”

Semi-classical limit of large fermionic systems at positive temperature

“…Let β, a, ρ > 0, let A, V satisfy the assumptions of Theorem 1, let w = aδ 0 for some a > 0. If m ∈ L 1 (R 2d ) satisfies the non-linear equation (5), then ρ m ∈ L 1+d/2 R d .…”

“…For atoms the positive Thomas-Fermi model was derived for the first time in [43]. There are several mathematical works on the time-dependent setting [42,52,3,14,17,8,5,1,44,7,13], in which the Schrödinger dynamics has been proved to converge to the Vlasov time-dependent equation in the limit N → ∞. Finally, the first two terms in the expansion of the (free) energy of a Fermi gas with spin in the limit ρ → 0 was provided in [32] at T = 0 and in [50] at T > 0.…”