Boltzmann Equation without the Molecular Chaos Hypothesis.

A physically clear probabilistic model of a gas from hard spheres is considered using 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 (SDEs), first the generalized and then the random and nonrandom Boltzmann integro-differential equations are derived, taking into account the correlations and fluctuations. 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 increasingly rough approximations in 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 more accurate averaging over the velocity due to the analytical solution of the SDE with respect to the 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. [ABSTRACT FROM AUTHOR]