Kompaneets equation for neutrinos: Application to neutrino heating in supernova explosions

Understand

We derive a `Kompaneets equation' for neutrinos, which describes how the distribution function of neutrinos interacting with matter deviates from a Fermi-Dirac distribution with zero chemical potential.

• To this end, we expand the collision integral in the Boltzmann equation of neutrinos up to the second order in energy transfer between matter and neutrinos.
• The distortion of the neutrino distribution function changes the rate at which neutrinos heat matter, as the rate is proportional to the mean square energy of neutrinos, $E_\nu^2$.
• For electron-type neutrinos the enhancement in $E_\nu^2$ over its thermal value is given approximately by $E_\nu^2/E_{\nu,\rm thermal}^2=1+0.086(V/0.1)^2$ where $V$ is the bulk velocity of nucleons, while for the other neutrino species the enhancement is $(1+\delta_v)^3$, where $\delta_v=mV^2/3k_BT$ is the kinetic energy of nucleons divided by the thermal energy.