Comparison of numerical methods for computing the repeated Compton
  scattering of photons in isotropic media
  Repeated Compton scattering of photons with thermal electrons is one of the
fundamental processes at work in many astrophysical plasma. Solving the exact
evolution equations is hard and one common simplification is based on
Fokker-Planck (FP) approximations of the Compton collision term. Here we carry
out a detailed numerical comparison of several FP approaches with the exact
scattering kernel solution for a range of test problems assuming isotropic
media and thermal electrons at various temperatures. The Kompaneets equation,
being one of the most widely used FP approximations, fails to account for
Klein-Nishina corrections and enhanced Doppler boosts and recoil at high
energies. These can be accounted for with an alternative FP approach based on
the exact first and second moments of the scattering kernel. As demonstrated
here, the latter approach works very well in dilute media, but inherently fails
to reproduce the correct equilibrium solution in the limit of many scattering.
Conditions for the applicability of the FP approximations are clarified,
overall showing that the Kompaneets equation provides the most robust
approximation to the full problem, even if inaccurate in many cases. We close
our numerical analysis by briefly illustrating the solutions for the spectral
distortions of the cosmic microwave background (CMB) after photon injection at
redshift $z\lesssim 10^5$, when double Compton and Bremsstrahlung emission can
be omitted. We demonstrate that the exact treatment using the scattering kernel
computed with {\tt CSpack} is often needed. This work should provide an
important step towards accurate computations of the CMB spectral distortions
from high-energy particle cascades.
