The properties of a nonlinear collision integral of the Boltzmann equation are under investigation and a new approach to development of the nonlinear moment method for solving the Boltzmann equation is proposed. The approach is based on the invariance of the collision integral with respect to the choice of basis functions. It is shown that the matrix elements corresponding to the moments of the nonlinear collision integral are not independent, being related by simple recursion relations. In the isotropic case these relations were used to construct an effective numerical procedure for calculation of the matrix elements. Similar expressions reduce computer's time consumption by many orders of magnitude compared with using the common formulas. As a result, it was obtained the distribution function including up to ten thermal velocities. In the axially symmetric case, the nonlinear matrix elements of a collision operator are interrelated via the simple expressions, these relations being true for the arbitrary cross sections of the particle interactions even in the high electric and magnetic fields. The obtained recurrent relations give an opportunity to calculate the matrix elements for the large indices. In the standard gas kinetic theory, the nonoriented (isotropic) particles are under consideration. In this case, a Hecke theorem holds for the linear matrix elements. It has been shown that a general Hecke theorem holds for the nonlinear matrix elements limiting a representative domain of the nonzero matrix elements. In the case of the nonoriented particles the additional linearly independent relations between matrix elements are found. It is shown that any nonlinear matrix element is a linear combination of the linear isotropic diagonal matrix elements. A set of codes is created by means of which the computer derivation of an analytical formula for the linear nonisotropic matrix elements is carried out. In a general 3D case, an invariance of the collision integral relatively to rotations about two axes is used. If a space is isotropic then a general Hecke theorem is a condition of solvability of the algebraic equations' system and the matrix elements with are related with the axi-symmetrical ones via the Klebsch-Gordan coefficients. If a system has a principal direction some new relations of matrix elemenets can be deduced when describing the kinetic coefficients for the nonsymmetric oriented particles and determining what a dependence of the each matrix element on the angles of rotation can be. The results obtained are generalized concerning the arbitrary interaction potentials as well as the gaseous mixtures.

Page created by Alexander Potekhin (palex.astro@mail.ioffe.ru) on November 4, 2001, updated on November 6, 2001.