Comparison of the Bhatnagar-Gross-Krook approximation with the exact Coulomb collision operator

The collisional relaxation of a bi-Maxwellian and a double-beam distribution is studied in order to assess the reliability of the Bhatnagar-Gross-Krook (BGK) operator in comparison with the full Fokker-Planck Coulomb collision operator. If a velocity-dependent relaxation rate is employed, qualitative agreement can be achieved between both models. Using the friction rate ${\ensuremath{\nu}}_{F}$ in the BGK model yields numerical results most similar to those of the exact Fokker-Planck model. However, in quantitative terms the collisional relaxation of higher moments is only poorly described. In particular, for the skewness or heat flux the numerical differences are intolerable. For this and other reasons detailed in the paper, the BGK model cannot serve as a reliable calculation of transport properties. Nevertheless, it may have merits in qualitatively assessing collisional effects in inhomogeneous systems which otherwise have to be treated as entirely collision-free.