Simplified derivation of the Kompaneets equation

TL;DR

This paper presents a simplified derivation of the Kompaneets equation, describing how photon energy distribution evolves in a plasma due to Compton scattering and drag forces, connecting it to fundamental thermodynamic principles.

Contribution

It offers a new, simplified derivation of the Kompaneets equation, linking photon diffusion to classical thermodynamics and Einstein's recoil concept.

Findings

Derivation of the Kompaneets equation from basic principles

Connection between photon diffusion and Bose-Einstein distribution

Insight into the thermodynamic basis of photon energy evolution

Abstract

An isotropic electromagnetic field in a plasma of thermalized electrons undergoes changes in energy as a result of Compton scattering and an Einstein-Hopf drag force on the electrons, eventually approaching a Bose-Einstein photon distribution at the electron temperature. The rate of change of feld energy due to the combined effects of Compton scattering and the drag force is shown to be described by the Kompaneets equation for photon diffusion in frequency space. A similarity is noted between this approach and Einstein's derivation of the Planck spectrum based on the recoil of atoms as they absorb and emit radiation.