Mean-field Evolution of Fermionic Systems

TL;DR

This paper analyzes the mean-field limit of fermionic systems, demonstrating that initial states close to a Slater determinant evolve near Hartree-Fock dynamics with explicit convergence rates.

Contribution

It establishes the persistence of Slater determinant structure under fermionic mean-field evolution and provides effective bounds on convergence to Hartree-Fock equations.

Findings

Evolution remains close to Slater determinant for all times

Convergence to Hartree-Fock dynamics with explicit rates

Applicable under regular interaction potentials

Abstract

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of initial data close to a Slater determinant, whose reduced one-particle density is an orthogonal projection $\omega_N$ with the appropriate semiclassical structure. Assuming some regularity of the interaction potential, we show that the evolution of such an initial data remains close to a Slater determinant, with reduced one-particle density given by the solution of the Hartree-Fock equation with initial data $\omega_N$. Our result holds for all (semiclassical) times, and gives effective bounds on the rate of the convergence towards the Hartree-Fock dynamics.