From Hartree dynamics to the relativistic Vlasov equation

TL;DR

This paper derives the relativistic Vlasov equation from quantum Hartree dynamics for fermions, establishing a link between quantum and classical relativistic kinetic descriptions in the mean-field limit.

Contribution

It extends the derivation of the Vlasov equation to the relativistic setting, addressing challenges unique to relativistic quantum kinetic energy control.

Findings

Successfully derived the relativistic Vlasov equation from quantum Hartree dynamics.

Established a rigorous connection between quantum and classical relativistic kinetic theories.

Addressed and overcame the challenge of controlling relativistic quantum kinetic energy differences.

Abstract

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in [6]. The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation.