Hartree-Fock dynamics for weakly interacting fermions

TL;DR

This paper reviews recent advances in understanding the evolution of weakly interacting fermionic systems in the mean field regime, demonstrating that their many-body dynamics can be effectively approximated by the Hartree-Fock equation with quantifiable convergence rates.

Contribution

It establishes rigorous bounds on the rate of convergence of many-body fermionic dynamics to the Hartree-Fock equation in the semiclassical limit.

Findings

Approximation of fermionic evolution by Hartree-Fock equation.

Quantitative bounds on convergence rates.

Applicability to initial states close to Slater determinants.

Abstract

We review recent results concerning the evolution of fermionic systems. We are interested in the mean field regime, where particles experience many weak collisions. For fermions, the mean field regime is naturally linked with a semiclassical limit. Assuming some regularity of the interaction potential we show that the many body evolution of initial states close to Slater determinants exhibiting the appropriate semiclassical structure can be approximated by the Hartree-Fock equation. Our method provides precise bounds on the rate of the convergence.