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P-Adic Numbers Ultrametric Anal. Appl., 2012, Volume 4, Number 2, Pages 130–142 (Mi padic9)  

Derivation of the particle dynamics from kinetic equations

A. S. Trushechkinab

a National Research Nuclear University “MEPhI”, Kashirskoe Highway 31, 115409 Moscow, Russia
b Steklov Mathematical Institute, Russian Academy of Sciences , Gubkina St. 8, 119991 Moscow, Russia

Abstract: The microscopic solutions of the Boltzmann–Enskog equation discovered by Bogolyubov are considered. 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, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals.
This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00828-a
11-01-12114-ofi-m
Ministry of Education and Science of the Russian Federation NSh-2928.2012.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations
This work was partially supported by the Russian Foundation for Basic Research (projects 11-01-00828-a and 11-01-12114-ofi-m-2011), the grant of the President of the Russian Federation (project NSh-2928.2012.1), and the Division of Mathematics of the Russian Academy of Sciences.


DOI: https://doi.org/10.1134/S2070046612020057


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Document Type: Article
Language: English

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