Asymptotics of collision integral matrix elements in the isotropic case
É. A. TroppIoffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021L. A. BakaleĭnikovIoffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021A. Ya. ÉnderIoffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021I. A. ÉnderSt. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 198904, Russia
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The method of nonlinear moments, when used to solve the Boltzmann equation, necessitates the calculation of collision integral matrix elements. The matrix elements are hard to calculate numerically, especially at large indices. The asymptotics of the matrix elements are constructed. In terms of the model of pseudopower particle interaction, a formula free of summation is derived. This makes it possible to find the asymptotic behavior of linear and nonlinear elements when two indices are large. For an arbitrary interaction cross section, asymptotic expansions of linear and nonlinear matrix elements in one index are obtained. For Maxwellian molecules, asymptotic formulas are derived for three large indices.
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