Exact evaluation of collision integrals for the nonlinear Boltzmann equation

In this work, we evaluate analytically the collision integrals for the nonlinear isotropic Boltzmann equation for the hard spheres for several initial distribution functions. The initial distribution functions include combinations of Maxwellians with different temperatures and delta-functions. We do not find any preferential scattering into the high energy tail in the case when the initial distribution function is a sum of two Maxwellians with different temperatures. Our results may be used to devise a new method for the numerical integration of the nonlinear Boltzmann equation which avoids an approximate numerical evaluation of the collision integrals.