On a rigorous definition of microscopic solutions of the Boltzmann--Enskog equation

A. S. Trushechkin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2013), pp. 270-278.

Résumé

N. N. Bogolyubov discovered microscopic solutions of the Boltzmann–Enskog equation in kinetic theory of hard spheres. These solutions have the form of sums of the delta-functions and correspond to the exact microscopic dynamics. However, this was done at the “physical level” of rigour. In particular, Bogolyubov did not discuss the products of generalized functions in the collision integral. Here we give a rigorous sense to microscopic solutions by use of regularization. Also, starting from the Vlasov equaton, we obtain new kinetic equations for a hard sphere gas.

Détail

Keywords: kinetic equations, Boltzmann–Enskog equation, microscopic solutions, generalized functions.
Mots-clés : Vlasov equation

Bibliographie

[1] N. N. Bogolyubov, “Microscopic solutions of the Boltzmann–Enskog equation in kinetic theory for elastic balls”, Theoret. and Math. Phys., 24:2 (1975), 804–807 | DOI | MR

[2] N. N. Bogolubov, N. N. Bogolubov (Jr.), Introduction to Quantum Statistical Mechanics, 2nd, World Scientific, Singapore, 2009, 440 pp. | MR | Zbl

[3] A. S. Trushechkin, “Derivation of the particle dynamics from kinetic equations”, p-Adic Numb. Ultr. Anal. Appl., 4:2 (2012), 130–142 | DOI | MR | Zbl

[4] A. A. Vlasov, Theory of many particles, Gostekhizdat, Moscow, 1950, 348 pp.

[5] Arkeryd L., Cercignani C., “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 (1990), 845–867 | DOI | MR | Zbl

[6] D. Ya. Petrina, V. I. Gerasimenko, P. V. Malyshev, Mathematical foundations of classical statistical mechanics, Naukova Dumka, Kiev, 1985, 263 pp. | MR | Zbl

[7] C. Cercignani, Theory and application of the Boltzmann equation, Elsevier, New York, 1975, xii+415 pp. ; K. Cherchinyani, Teoriya i prilozheniya uravneniya Boltsmana, Mir, M., 1978, 495 pp. | MR | MR

[8] I. V. Volovich, “Time Irreversibility Problem and Functional Formulation of Classical Mechanics”, Vestnik SamGU. Estestvennonauchn. Ser., 2008, no. 8/1(67), 35–55 | MR

[9] I. V. Volovich, “Randomness in classical and quantum mechanics”, Found. Phys., 41:3 (2011), 516–528 | DOI | MR | Zbl

[10] A. I. Mikhaylov, “Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem”, p-Adic Numb. Ultr. Anal. Appl., 3:3 (2011), 205–211 | DOI | DOI | MR | Zbl

[11] E. V. Piscovskiy, I. V. Volovich,, “On the Correspondence Between Newtonian and Functional Mechanics”, Quantum Bio-Informatic IV, Quantum Probability and White Noise Analisis, 28, eds. L. Accardy, W. Freudenberg, M. Ohya, World Sci, Singapure, 2011, 363–372 | DOI | MR

[12] E. V. Piskovskiy, “On classical and functional approachs to mechanics”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 1(22), 134–139 | DOI

[13] A. S. Trushechkin, I. V. Volovich, “Functional classical mechanics and rational numbers”, p-Adic Numb. Ultr. Anal. Appl., 1:4 (2009), 361–367 | DOI | MR | Zbl

[14] I. V. Volovich, “Bogoliubov equations and functional mechanics”, Theoret. and Math. Phys., 164:3 (2010), 1128–1135 | DOI | DOI | Zbl

[15] V. V. Vedenyapin, Boltzmann and Vlasov Kinetic equations, Fizmatlit, Moscow, 2001, 112 pp.

[16] V. V. Kozlov, Thermal equilibrium in the sense of Gibbs and Poincaré, Institut Komp'yuternykh Issledovanij, Moscow, Izhevsk, 2002, 320 pp. | MR | Zbl

[17] V. V. Kozlov, Gibbs Ensembles and Non-equilibrium Statistical Mechanics, Institut Komp'yuternykh Issledovanij, Moscow, Izhevsk, 2008, 208 pp. | Zbl