Weakly interacting Fermions: mean-field and semiclassical regimes

TL;DR

This paper reviews the derivation of macroscopic equations for weakly interacting fermions, focusing on mean-field and semiclassical regimes, and discusses techniques from second quantization to connect microscopic and effective descriptions.

Contribution

It presents a comprehensive review of deriving the Hartree-Fock and Vlasov equations for fermionic systems using mean-field and semiclassical approximations, highlighting recent methods.

Findings

Derivation of the time-dependent Hartree-Fock equation for fermions.

Approximation of many-body dynamics by the Vlasov equation at longer times.

Application of second quantization techniques in mean-field theory.

Abstract

The derivation of effective macroscopic theories approximating microscopic systems of interacting particles is a major question in non-equilibrium statistical mechanics. In these notes we present an approximation of systems made by many fermions interacting via inverse power law potentials in the mean-field and semiclassical regimes, reviewing the material presented at the 11th summer school "Methods and Models of Kinetic Theory" held in Pesaro in June 2022. More precisely, we focus on weakly interacting fermions whose collective effect can be approximated by an averaged potential in convolution form, and review recent mean-field techniques based on second quantization approaches. As a first step we obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we look at longer time scales where a semiclassical description starts to be relevant and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space.