Boltzmann equation without molecular chaos hypothesis
S. V. Bogomolova, T. V. Zakharovaba
a Lomonosov Moscow State University
b Federal Research Center "Computer Science and Control" of the RAS
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.
Boltzmann equation, Kolmogorov-Fokker-Planck equation, Navier-Stokes equations; random processes, SDE with respect to Bernulli and Wiener measures, particle methods.
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S. V. Bogomolov, T. V. Zakharova, “Boltzmann equation without molecular chaos hypothesis”, Matem. Mod., 33:1 (2021), 3–24
Citation in format AMSBIB
\by S.~V.~Bogomolov, T.~V.~Zakharova
\paper Boltzmann equation without molecular chaos hypothesis
\jour Matem. Mod.
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