On the derivation of the Kompaneets equation

The relaxation of a photon bath to thermal equilibrium via Compton scattering with electrons is described in the Kompaneets equation (1956). mostly known from studies astrophysical plasmas, for its convergence Planck distribution and possible corrections that law cosmic microwave background, most notably Sunyaev-Zeldovich effect. We revisit derivation emphasizing structure as Kramers-Moyal diffusion approximation quantum Boltzmann or Master stimulated emission. do not assume stationary performing continuum but we emphasize necessity flux M{\o}ller factor arrive at continuity equation. On other hand, allows more general assumptions than originally envisioned by Kompaneets.