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). The equation is
mostly known from studies of astrophysical plasmas, for its convergence to the
Planck distribution and for explaining the Sunyaev-Zeldovich effect changing
the apparent brightness of the cosmic microwave background radiation. We
revisit its derivation emphasizing its structure as a Kramers-Moyal diffusion
approximation to the quantum Boltzmann equation or Master equation with
stimulated emission. We do not assume that the Planck law is stationary in
performing the continuum approximation but we emphasize the necessity of the
flux or M{\o}ller factor to arrive at a continuity equation. On the other hand,
the structure allows more general assumptions than originally envisioned by
Kompaneets.
