2023/07/15 by Chiara Saffirio, Saffirio, Chiara
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2307.07762
openalex publication_date 2023/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.