Simplified derivation of the Kompaneets equation

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 field 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.