Kompaneets equation

Remarks

Example: the Bose–Einstein distribution $n(x) = 1/(e^{x + \mu} - 1)$ with a constant $\mu \ge 0$ is a stationary solution, since $\partial n/\partial x = -e^{x+\mu}/(e^{x+\mu} - 1)^2 = -(n + n^2)$, so the bracket vanishes. Non-example: a Planck distribution $n = 1/(e^{x'} - 1)$ at a temperature different from $T_e$, with $x' = h\nu / (k_B T_\gamma)$ and $T_\gamma \ne T_e$, is not stationary: the bracket does not vanish and the spectrum evolves. The three terms in the bracket describe diffusion in energy, recoil and induced scattering. The equation conserves the photon number $\int_0^\infty x^2 n \, dx$, since the right side is $x^{-2}$ times a derivative and the flux vanishes at $x = 0$ and $x = \infty$ for regular solutions. One often writes it in terms of the Compton parameter $y$ with $dy = \frac{k_B T_e}{m_e c^2} n_e \sigma_T c \, dt$.