Electron kinetic equations for comptonization by isotropic photons

We derive the kinetic equations that describe the time evolution of the distribution function of nondegenerate nonrelativistic electrons interacting with isotropic thermal photons via Compton scattering. Two cases, when the electrons are isotropic and when the electron motion is restricted to one spatial dimension due to a strong magnetic field, are considered. These equations are relevant in the limit when the photon energy density dominates the electron energy density, and they are complementary to the Kompaneets equation, which obtains in the opposite limit. Our result is compared with previous results, and the applicability of these equations is examined. Finally, by explicitly calculating the entropy generation, we show that these kinetic equations satisfy Boltzmann's H-theorem for the coupled electron-photon system as a whole