Derivation of the particle dynamics from kinetic equations
- 52 Downloads
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.
Key wordsBoltzmann-Enskog equation Boltzmann equation kinetic equations microscopic solutions reversibility paradox
Unable to display preview. Download preview PDF.
- 3.A. A. Vlasov, Many-Particle Theory and its Application to Plasma (Gordon and Breach, New York, 1961).Google Scholar
- 5.N. N. Bogoliubov, Problems of Dynamic Theory in Statistical Physics (Gostekhizdat, Moscow-Leningrad, 1946; North-Holland, Amsterdam, 1962; Interscience, New York, 1962).Google Scholar
- 6.N. N. Bogolyubov, Kinetic Equations and Green Functions in Statistical Mechanics (Institute of Physics of the Azerbaijan SSR Academy of Sciences, Baku, 1977), Preprint N. 57 [in Russian].Google Scholar
- 9.I. V. Volovich, “Time irreversibility problem and functional formulation of classical mechanics,” Vestnik Samara State Univ. 8/1, 35–54 (2008); arXiv:0907.2445v1 [cond-mat.stat-mech].Google Scholar
- 12.E. V. Piskovskiy and I. V. Volovich, “On the correspondence between Newtonian and functional mechanics,” in Quantum Bio-Informatics IV, pp. 363–372 (World Scientific, Singapore, 2011).Google Scholar
- 20.L. Arkeryd and C. Cercignani, “On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation,” Comm. PartialDiff. Eqns. 14(8-9), 1071–1090 (1989). L. Arkeryd and C. Cercignani, “Global existence in L 1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation,” J. Stat. Phys. 59(3/4), 845–867 (1990).MathSciNetzbMATHCrossRefGoogle Scholar