Radiative Transport Limit of Dirac Equations with Random Electromagnetic Field

TL;DR

This paper rigorously derives the radiative transport limit for the Dirac equation with a random electromagnetic field, revealing how solutions behave in the kinetic limit using advanced mathematical tools.

Contribution

It provides the first detailed mathematical analysis of the kinetic limit for the Dirac equation with random electromagnetic fields, including convergence of cross-modes.

Findings

Cross-modes converge weakly to zero in space and almost surely in probability.

Propagating modes converge strongly to their deterministic limit.

The analysis employs a matrix-valued Wigner transform and martingale methods.

Abstract

This paper concerns the kinetic limit of the Dirac equation with random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared to the scalar (Schr\"odinger) case is the proof of convergence of cross-modes to 0 weakly in space and almost surely in probability. The propagating modes are shown to converge in an appropriate strong sense to their deterministic limit.