Kernels of the direct collision integral of the nonlinear Boltzmann equation

This paper reviews the properties of kernels of the direct collision integral describing the escape of particles from an element of the phase volume due to collisions, as well as properties of kernels of the integral operators arising in expansion of the direct collision integral in spherical harmonics. It is shown that, for isotropic scattering cross sections depending on the relative velocity by a power law, these kernels can be obtained in an analytical form. Since the analytical expressions for the kernels with large indices become too cumbersome, their asymptotics are found.