Derivation of the particle dynamics from kinetic equations

Abstract

We consider the microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, we construct the so called approximate microscopic solutions. These solutions are continuously differentiable and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the time-reversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.Comment: 18 pages; some misprints have been corrected, some references have been adde

Microscopic solutions of the Boltzmann-Enskog equation in the series representation

“…Conclusions. The presented way of giving a rigorous sense to the microscopic solutions of the Boltzmann-Enskog equation appears to be the most natural one, in comparison to the previously proposed variants [23,24,25,26], since it does not involve regularizations of delta functions, but gives direct sense to weak solutions of the Boltzmann-Enskog equation by means of a notion of series solution. Here we have introduced a series expansion that involves only partial chronological ordering of collisions.…”

“…A difficulty with the microscopic solutions (3) is that their formal substitution into (1) yields products of delta functions and, hence, it is ill-defined. A rigorous sense to these solutions was given in [25] by means of regularizations for delta functions and the collision integral (see also [23,26] for other variants). On the other hand it was proven in [19] that the empirical distributions (3) (more precisely the family of the empirical marginals) solve the BBGKY hierarchy for hard spheres.…”

On the evolution of the empirical measure for the Hard-Sphere dynamics

“…As in the case of the H-S hierarchy, the Boltzmann-Enskog collision operator appearing in the right hand side of (1.2) is not well defined when evaluated in f = µ N , so that a discussion on the precise mathematical meaning of the microscopic solutions of [4] is required. In [18,19] a suitable regularization of µ N has been used to give a sense to (1.2) in terms of a limiting procedure. In this paper, we will approach the problem in a different way.…”

О Строгом Определении Микроскопических Решений Уравнения Больцмана - Энскога

“…Основной целью настоящей работы является придание строгого смысла этим решениям. Один из способов приведён нами в более ранней работе [3]. Здесь мы приведём другой, более естественный способ.…”