On the derivation of the Kompaneets equation

TL;DR

This paper revisits the derivation of the Kompaneets equation, highlighting its structure as a diffusion approximation to the quantum Boltzmann equation and emphasizing the importance of the flux factor for deriving the continuity equation.

Contribution

It provides a new derivation of the Kompaneets equation emphasizing its structure as a Kramers-Moyal diffusion approximation, relaxing some original assumptions.

Findings

Reveals the importance of the flux or Møller factor in the derivation.

Shows the derivation does not require the Planck law to be stationary.

Highlights the potential for more general assumptions beyond original derivation.

Abstract

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 possible corrections to that Planck law in the cosmic microwave background, most notably from the Sunyaev-Zeldovich effect. 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.