Math Courses

Derivation of effective equations in the mean-field regime

by Chiara Saffirio (University of Basel)


Lecturer: Chiara Saffirio (University of Basel)

Abstract: The derivation of effective macroscopic theories approximating microscopic systems of interacting particles in some scaling limit is a major question in non-equilibrium statistical mechanics. In this course we will be concerned with the dynamics of systems made of many interacting fermions. We will focus on the mean-field regime, i.e. weakly interacting particles 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 will obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we will 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. The structure of the initial data will play an important role at each step of the approximation.  


N. Benedikter, M. Porta, B. Schlein: Effective evolution equations from quantum dynamics. Springer Briefs in Mathematical Physics 7, 2016. 
C. Saffirio: The Vlasov equation as the mean-field and semiclassical limit of many interacting fermions. IAMP News Bulletin April 2022, 

Information. This minicourse is part of the school "Scaling Limits and Generalized Hydrodynamics", see