Kompaneets equation for neutrinos: Application to neutrino heating in
  supernova explosions
  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. This enhancement has
a significant implication for supernova explosions, as it would aid
neutrino-driven explosions.
