TEMPERATURE INTEGRAL BRACKETS FOR A ONE-COMPONENT SYSTEM WITH SMALL INTERACTION

Abstract

The problem of description of the hydrodynamic stage of system evolution and the corresponding calculation of the system kinetic coefficients is urgent for statistical physics. The Chapman-Enskog method is widely applied to the corresponding problem for different systems, and the Sonine polynomials are widely used for the calculation of approximate solutions for the system distribution function. The standard hydrodynamic theory leads to Fredholm integral equations of the first kind, for which the solutions based on Sonine polynomials are considered to be convergent. It should be stressed that analytical calculations are often restricted to the one- or two-polynomial approximations because of the cumbersomeness of such calculations and the fact that the convergence of the solutions with increasing number of polynomials is considered to be rather fast.

However, the numerical investigation of the corresponding convergence is of interest. For example, a numerical investigation of the corresponding convergence for the simple and rigid-sphere gas approximations up to as many as 150 polynomials was made by S.K. Loyalka, R.V. Tompson, and E.L. Tipton on the basis of the Boltzmann kinetic equation. However, we do not know any works where such investigations would be made for systems described by the Landau kinetic equation.

As is known, the so-called systems with small interaction are described by the Landau kinetic equation, which contains the Landau collision integral. For example, systems with Coulomb interaction are described by this mathematical apparatus. In particular, some previous investigations were devoted to a completely ionized two-component plasma, and in most cases the one- or two-polynomial approximations were used. In this paper we investigate the corresponding integral brackets for a one-component system with small interaction, but the calculation of the integral brackets is made up to the thirteen-polynomial approximation. Here we restrict ourselves only to the integral brackets necessary for the calculation of the temperature part of the first-order-in-gradients distribution function. Exact analytical results for the integral brackets under consideration are obtained. The obtained results are important for further numerical investigation of the convergence of the results for the system thermal conductivity with increasing number of polynomials in corresponding approximations. The brackets that are necessary for the calculation of the velocity part of the distribution function may be the subject matter of another paper.

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