Isotropic radiative transfer as a phase space process: Lorentz covariant Green's functions and first-passage times

TL;DR

This paper reformulates radiative transfer as a phase space process, deriving Lorentz covariant Green's functions and analyzing first-passage times, thus addressing fundamental properties of radiative transfer solutions.

Contribution

It introduces a phase space approach to radiative transfer, providing exact solutions and demonstrating Lorentz covariance, which was not previously established.

Findings

Green's functions satisfy Chapman-Kolmogorov relation in phase space

Exact solutions as probability densities of persistent random walks

First-passage time distributions reveal effective delays in pulse measurements

Abstract

The solutions of the radiative transfer equation, known for the energy density, do not satisfy the fundamental transitivity property for Green's functions expressed by Chapman-Kolmogorov's relation. I show that this property is retrieved by considering the radiance distribution in phase space. Exact solutions are obtained in one and two dimensions as probability density functions of continous-time persistent random walks, the Fokker-Planck equation of which is the radiative transfer equation. The expected property of Lorentz covariance is verified. I also discuss the measured signal from a pulse source in one dimension, which is a first-passage time distribution, and unveil an effective random delay when the pulse is emitted away from the observer.