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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 $x_e=h\nu /(k_B{T}_e)$ and $dy=(k_B{T}_e/m_e{c}^2)n_e{\sigma }_{T}cdt$, 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 ${\partial }_{y}n=x^{-2}{\partial }_x(x^4{\partial }_{x}n)$. The quantity $n$ is not the frequency-integrated photon number density.

In the Maxwell-Boltzmann velocity distribution, each Cartesian component has mean-square velocity $k_B{T}_e/m_e$. Thus $⟨v^2⟩=3k_B{T}_e/m_e$. The original PDF calls this quantity an rms velocity, but the rms speed is its square root. Also, its $n_{\gamma }$ must mean the dimensionless mean occupation number $n$, since its equilibrium value is a Planck photon distribution; it is not the frequency-integrated photon number density.

Take $x=h\nu /(k_B{T}_{\gamma })$ at a fixed local radiation temperature. Because $x$ is proportional to frequency,The hot-electron Kompaneets equation therefore becomesFor traversal length $d\ell =cdt$, the Compton y parameter isIt measures electron pressure integrated along the line of sight, or the Thomson scattering optical depth weighted by $k_B{T}_e/(m_e{c}^2)$. In this hot-electron limit the mean fractional energy gain is approximately $4y$. The derivation assumes nonrelativistic electrons, Thomson-limit photon energies and $T_e\gg T_{\gamma }$. The original PDF uses $h\nu$, as required for ordinary frequency; the converted TeX incorrectly substitutes $\hslash \nu$.

For a Planck photon distribution, the hot-electron Kompaneets equation gives $\Delta n=yx{e}^x(e^x-1{)}^{-2}[x\coth (x/2)-4]$, where $x=h\nu /(k_B{T}_{\gamma })$. The equivalent thermodynamic temperature perturbation has the displayed form, while the Rayleigh-Jeans law gives $\Delta T/T=-2y$. The distortion changes sign at $x\simeq 3.830$. This first-order expression assumes a small Compton y parameter and nonrelativistic thermal Electrons.