Mean-Field Limits in Statistical Dynamics

Abstract

These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum N -particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul.

A rigorous derivation of the Hamiltonian structure for the Vlasov equation

“…In other directions, mean-field convergence has been shown for Coulombtype potentials which are regularized at some small scale 𝜖 𝑁 vanishing as 𝑁 → ∞ [BP16, Laz16, LP17, Gra21] or when the initial data are of so-called monokinetic type [Ser20]. For reviews of Vlasov mean-field limits, the reader may consult [Jab14,Gol16] and, in particular, the recent lecture notes [Gol22].…”

Hydrodynamic and kinetic representation of the microscopic classic dynamics at the transition on the macroscopic scale

“…Some mathematical concepts regarding the mean-field limit of an N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system are presented by Golse (2012). Some mathematical aspects related to the microscopic derivation of Vlasov equations with singular potentials are discussed by Golse (2022) and Grass (2021), where the transition to the mean-field in the quantum theory is discussed as well. These works do not include the concept of construction of the material fields based of motion of classical particles suggested by Drofa & Kuz'menkov (1996) and developed in this paper.…”