Properties of the Collision Integral in the Axisymmetric Boltzmann Equation

Abstract

We study a matrix representation of the Boltzmann collision integral, the distribution function being expanded in terms of spherical Hermite polynomials. The recurrence relationships between the matrix elements obtained by Ender and Ender (1999a, b) for an isotropic distribution function in velocity space follow from the condition of the collision integral invariancy relative to the choice of the basis. The capability of the momentum method to solve the Boltzmann equation is essentially enlarged. The new relationships between the matrix elements are found in the axisymmetric case. These recurrence relationships give an easy opportunity to determine all the nonlinear matrix elements, provided that the linear ones are known. It is shown that a Hecke theorem for the linear Boltzmann operator has a corollary. Namely, in the standard kinetic theory, which assumes that the particles have spherical symmetry, a generalized Hecke theorem can be proved, by virtue of which many nonlinear matrix elements turn out to be zero. Keywords: Boltzmann equationSpherical Hermite polynomialsMatrix elementsRecurrence relationsAxisymmetric caseGeneralized Hecke theorem Acknowledgment This work was supported by the Russian Foundation for Basic Research (grant no. 01‐02‐17903) and the Russian Federal Program “Integration” (grant no. B0029).