Boltzmann equation without molecular chaos hypothesis

S. V. Bogomolov ; T. V. Zakharova

Matematičeskoe modelirovanie, Tome 33 (2021) no. 1, pp. 3-24

Résumé

A physically clear probabilistic model of a hard sphere gas is considered both with the help of the theory of random processes and in terms of the classical kinetic theory for the densities of distribution functions in the phase space: from the system of nonlinear stochastic differential equations (SDE), first the generalized, and then — random and non-random integro-differential Boltzmann equation taking into account correlations and fluctuations are derived. The main feature of the original model is the random nature of the intensity of the jump measure and its dependence on the process itself. For the sake of completeness, we briefly recall the transition to more and more rough approximations in accordance with a decrease in the dimensionlessization parameter, the Knudsen number. As a result, stochastic and nonrandom macroscopic equations are obtained, which differ from the system of Navier-Stokes equations or systems of quasi-gas dynamics. The key difference of this derivation is a more accurate averaging over the velocity due to the analytical solution of the SDE with respect to Wiener measure, in the form of which the intermediate meso-model in the phase space is presented. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. The emphasis is on the transparency of the assumptions when moving from one level of detail to another, rather than on numerical experiments, which contain additional approximation errors.

Détail

Keywords: Boltzmann equation, Navier-Stokes equations; random processes, SDE with respect to Bernulli and Wiener measures, particle methods.
Mots-clés : Kolmogorov-Fokker-Planck equation