Derivation of the Vlasov-Maxwell system from the Maxwell-Schr\"odinger equations with extended charges

TL;DR

This paper rigorously derives the Vlasov-Maxwell system from Maxwell-Schrödinger equations with extended charges, establishing convergence and providing explicit error estimates in the semiclassical limit.

Contribution

It introduces a well-posedness and regularity framework for Maxwell-Schrödinger and Vlasov-Maxwell systems with extended charges, and proves their convergence in the semiclassical regime.

Findings

Convergence of Maxwell-Schrödinger equations to Vlasov-Maxwell dynamics.

Explicit error estimates for the approximation.

Well-posedness and regularity results for both systems.

Abstract

We consider the Maxwell-Schr\"odinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from non-relativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell-Schr\"{o}dinger equations for extended charges towards the non-relativistic Vlasov-Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell-Schr\"odinger equations and for the Vlasov-Maxwell system for extended charges.