Two-particle kinetic equation and its self-similar exact solutions

Abstract

In the paper [AIP Conf. Proc., Vol. 1333, Part I, p. 134-139 (2011)], the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. By now, only a few exact explicit solutions of the Boltzmann kinetic equation are discovered. They play an important role in the kinetic theory of rarefied gases. A well known example of such solution is the BKW solution for Maxwell molecules. In the report we presented a new exact solution for the two-particle kinetic equation that is intermediate asymptotic for space homogeneous relaxation problems. After reducing the two-particle distribution function to one-particle distribution function the solution is reduced to BKW solution.In the paper [AIP Conf. Proc., Vol. 1333, Part I, p. 134-139 (2011)], the kinetic equation for two-particle distribution function was obtained by making use of exactly the same physical assumptions as Ludwig Boltzmann did. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. The Boltzmann equation follows from this two-particle equation without any additional assumptions after a simple integration. By now, only a few exact explicit solutions of the Boltzmann kinetic equation are discovered. They play an important role in the kinetic theory of rarefied gases. A well known example of such solution is the BKW solution for Maxwell molecules. In the report we presented a new exact solution for the two-particle kinetic equation that is intermediate asymptotic for space homogeneous relaxation problems. After reducing the two-particle distribution function to one-particle distribution function the solution is reduced to BKW solution.