From the Hartree dynamics to the Vlasov equation

TL;DR

This paper investigates how the quantum Hartree dynamics for many fermions approximates the classical Vlasov equation as the number of particles grows large, providing explicit convergence rates for regular potentials.

Contribution

It establishes rigorous bounds on the rate at which quantum fermionic dynamics converge to the classical Vlasov equation in the mean field limit for regular interaction potentials.

Findings

Proves convergence of Hartree to Vlasov as N increases

Provides explicit convergence rate bounds for regular potentials

Bridges quantum and classical descriptions of fermionic systems

Abstract

We consider the evolution of quasi-free states describing $N$ fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large $N$, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the rate of convergence.