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