Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case

Abstract

Application of the method of nonlinear moments to solve the Boltzmann equation generates the need to sum a series that is the expansion of the distribution function in basis functions. This series converged only if the Grad test is fulfilled. Such a limitation can be removed if the expansion of the distribution function is summed over the index related to only the expansion in velocity magnitude. In this case, the distribution function and the collision integral become expanded in only spherical harmonics and the expansion coefficients satisfy integro-differential equations. The kernels of these equations are the sums of the Sonine polynomials in the velocities of colliding and outgoing particles multiplied by matrix elements of the collision integral. For a number of arguments, the direct calculation of the kernels requires that a very large number of terms in the sum be taken into consideration. In this respect, an approach seems to be promising in which the asymptotics of the matrix elements and Sonine polynomials at large indices are used and summation over index is replaced by integration. In this paper, we apply this approach to calculate the linear kernel in the isotropic case, assuming that interaction between particles is described by a pseudopower law. With this approach, the collision integral kernel can be calculated with a high accuracy using as little as a few tens of series terms and the asymptotic estimate of the residue.