Kompaneets equation
= Kompaneets equation
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The <Kompaneets equation> describes frequency diffusion of a mean <photon> <occupation number> $n$ through repeated nonrelativistic <Compton scattering>. For thermal <Electrons>, with $x_e=h\nu/(k_BT_e)$ and $dy=(k_BT_e/m_ec^2)n_e\sigma_Tc\,dt$, its full homogeneous form is $\partial_y n=x_e^{-2}\partial_{x_e}[x_e^4(\partial_{x_e}n+n+n^2)]$. The drift and stimulated terms permit a thermal stationary spectrum. In the hot-electron limit $T_e\gg T_\gamma$, with low photon energies, they can be neglected relative to the diffusion term; any fixed linear frequency variable $x$ then gives $\partial_y n=x^{-2}\partial_x(x^4\partial_xn)$. The quantity $n$ is not the frequency-integrated <photon number density>.