Geodesic
Microscopic solutions of kinetic equations and the irreversibility problem
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and analysis, Tome 285 (2014), pp. 264-287

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As established by N. N. Bogolyubov, the Boltzmann–Enskog kinetic equation admits the so-called microscopic solutions. These solutions are generalized functions (have the form of sums of delta functions); they correspond to the trajectories of a system of a finite number of balls. However, the existence of these solutions has been established at the “physical” level of rigor. In the present paper, these solutions are assigned a rigorous meaning. It is shown that some other kinetic equations (the Enskog and Vlasov–Enskog equations) also have microscopic solutions. In this sense, one can speak of consistency of these solutions with microscopic dynamics. In addition, new kinetic equations for a gas of elastic balls are obtained through the analysis of a special limit case of the Vlasov equation. 
@article{TM_2014_285_a17,
     author = {A. S. Trushechkin},
     title = {Microscopic solutions of kinetic equations and the irreversibility problem},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {264--287},
     publisher = {mathdoc},
     volume = {285},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_285_a17/}
}
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A. S. Trushechkin. Microscopic solutions of kinetic equations and the irreversibility problem. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and analysis, Tome 285 (2014), pp. 264-287. http://geodesic.mathdoc.fr/item/TM_2014_285_a17/