Convergence towards the Vlasov-Poisson Equation from the $N$-Fermionic Schr\"odinger Equation

TL;DR

This paper demonstrates how the quantum dynamics of a large number of interacting fermions converges to the classical Vlasov-Poisson system, using semiclassical and mean-field analysis of the Husimi measure.

Contribution

It establishes the derivation of the Vlasov-Poisson equation from the N-fermionic Schrödinger equation with regularized Coulomb interactions in the large N limit.

Findings

Convergence of quantum fermionic dynamics to the Vlasov-Poisson system.

Effective semiclassical and mean-field estimates.

Validation of the regularized Coulomb potential approach.

Abstract

We consider the quantum dynamics of $N$ interacting fermions in the large $N$ limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to $N$.From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.