Chiara Saffirio[a]
Weakly interacting Fermions: mean-field and semiclassical regimes
Pages: 307-332
Received: 17 July 2023
Accepted in revised form: 14 January 2025
Mathematics Subject Classification: 82C10, 35Q41, 35Q55, 82C05, 35Q83.
Keywords: Mean-field limit, semiclassical limit, Hartree-Fock equation, many-body Schrödinger equation, Vlasov equation, singular interaction.
Authors address:
[a]: University of Basel, Department of Mathematics and Computer Science, Basel, Switzerland
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.