Boltzmann Equation Without the Molecular Chaos Hypothesisстатья
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Дата последнего поиска статьи во внешних источниках: 15 декабря 2021 г.
Аннотация:Abstract—A physically clear probabilistic model of a gas from hard spheres is considered using the theoryof random processes and in terms of the classical kinetic theory for the densities of distributionfunctions in the phase space: from the system of nonlinear stochastic differential equations (SDEs),first the generalized and then the random and nonrandom Boltzmann integro-differential equationsare derived, taking into account the correlations and fluctuations. The main feature of the originalmodel is the random nature of the intensity of the jump measure and its dependence on the processitself. For the sake of completeness, we briefly recall the transition to increasingly rough approximationsin accordance with a decrease in the dimensionless parameter, the Knudsen number. As a result,stochastic and nonrandom macroscopic equations are obtained that differ from the system of Navier–Stokes equations or systems of quasi-gas dynamics. The key difference of this derivation is a moreaccurate averaging over the velocity due to the analytical solution of the SDE with respect to the Wienermeasure, in the form of which the intermediate meso-model in the phase space is presented. Thisapproach differs significantly from the traditional one, which uses not the random process itself, butits distribution function. The emphasis is on the transparency of the assumptions when moving fromone level of detail to another, rather than on numerical experiments, which contain additional approximationerrors.Keywords: Boltzmann equation, Kolmogorov–Fokker–Planck equation, Navier–Stokes equation,equations of stochastic gas dynamics and quasi gas dynamics, random processes, SDE for Bernoulliand Wiener measures, particle method