Mean-field Evolution of Fermionic Mixed States

TL;DR

This paper analyzes the mean-field dynamics of fermionic mixed states, showing that their evolution can be approximated by the time-dependent Hartree-Fock equation with explicit convergence rates.

Contribution

It proves the convergence of fermionic many-body dynamics to the Hartree-Fock equation for initial states close to quasi-free states, with effective convergence estimates.

Findings

Convergence of reduced density matrices to Hartree-Fock solutions

Validity of Hartree-Fock approximation for all times

Explicit rate of convergence in the mean-field limit

Abstract

In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital 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 one.