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.
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References
N. N. Bogolyubov, “Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls,” Theor. Math. Phys. 24(2), 804–807 (1975).
N. N. Bogolubov and N. N. Bogolubov Jr., Introduction to Quantum Statistical Mechanics (World Scientific, Singapore, 2010).
A. A. Vlasov, Many-Particle Theory and its Application to Plasma (Gordon and Breach, New York, 1961).
V. V. Kozlov, Thermal Equilibrium in the Sense of Gibbs and Poincaré (Institute of Computer Research, Moscow-Izhevsk, 2002) [in Russian]. V.V. Kozlov, “Thermal equilibriumin the sense ofGibbs and Poincaré” Dokl.Math. 65(1), 125–128 (2002).
N. N. Bogoliubov, Problems of Dynamic Theory in Statistical Physics (Gostekhizdat, Moscow-Leningrad, 1946; North-Holland, Amsterdam, 1962; Interscience, New York, 1962).
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].
O. E. Lanford, “Time evolution of large classical systems,” Lect. Notes Phys. 38, 1–111 (1975).
C. Villani, “A review of mathematical topics in collisional kinetic theory,” Handbook Math. Fluid Dyn. 1, 71–305 (2002).
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].
I. V. Volovich, “Randomness in classical and quantum mechanics,” Found. Phys. 41(3), 516–528 (2011); arXiv: 0910.5391v1 [quant-ph].
A. I. Mikhailov, “Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem,” p-Adic Numb. Ultr. Anal. Appl. 3(3), 205–211 (2011).
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).
E. V. Piskovskiy, “On functional approach to classical mechanics,” p-Adic Numb. Ultr. Anal. Appl. 3(3), 243–247 (2011).
A. S. Trushechkin and I. V. Volovich, “Functional classicalmechanics and rational numbers,” p-Adic Numb. Ultr. Anal. Appl. 1(4), 361–367 (2009); arXiv:0910.1502 [math-ph].
I. V. Volovich, “Bogoliubov equations and functional mechanics,” Theor. Math. Phys. 164(3), 1128–1135 (2010).
N. Bellomo and M. Lachowicz, “On the asymptotic theory of the Boltzmann and Enskog equations: A rigorousH-theorem for the Enskog equation,” Springer Lecture Notes in Mathematics: Math. Aspects Fluid Plasma Dyn. 1191, 15–30 (1991).
C. Cercignani, Theory and Application of the Boltzmann Equation (Scottish Acad. Press, Edinburgh, 1975; Elsevier, New York, 1975). C. Cercignani, “On the Boltzmann equation for rigid spheres,” Transport Theory and Stat. Phys. 2(3), 211–225 (1972).
S.-Y. Ha and S.-E. Noh, “New a priori estimate for the Boltzmann-Enskog equation,” Nonlinearity 19(6), 1219–1232 (2006).
N. Bellomo and M. Lachowicz, “On the asymptotic equivalence between the Enskog and the Boltzmann equations,” J. Stat. Phys. 51(1/2), 233–247 (1988).
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).
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Trushechkin, A.S. Derivation of the particle dynamics from kinetic equations. P-Adic Num Ultrametr Anal Appl 4, 130–142 (2012). https://doi.org/10.1134/S2070046612020057
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DOI: https://doi.org/10.1134/S2070046612020057