Advertisement

Microscopic solutions of kinetic equations and the irreversibility problem

  • A. S. TrushechkinEmail author
Article

Abstract

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.

Keywords

Kinetic Equation Boltzmann Equation STEKLOV Institute Delta Function Mild Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. N. Bogolyubov, “Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls,” Teor. Mat. Fiz. 24(2), 242–247 (1975) [Theor. Math. Phys. 24, 804–807 (1975)].CrossRefGoogle Scholar
  2. 2.
    N. N. Bogolyubov and N. N. Bogolyubov, Jr., Introduction to Quantum Statistical Mechanics (Nauka, Moscow, 1984; World Sci., Hackensack, NJ, 2010).zbMATHGoogle Scholar
  3. 3.
    A. A. Vlasov, Many-Particle Theory (Gostekhizdat, Moscow, 1950); Engl. transl: Many-Particle Theory and Its Application to Plasma (New York, Gordon and Breach, 1961).Google Scholar
  4. 4.
    A. S. Trushechkin, “Derivation of the particle dynamics from kinetic equations,” p-Adic Numbers Ultrametric Anal. Appl. 4(2), 130–142 (2012); arXiv: 1201.3607 [math-ph].CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. S. Trushechkin, “Functional mechanics and kinetic equations,” in Quantum Bio-Informatics V (World Sci., Hackensack, NJ, 2013), QP-PQ: Quantum Probability and White Noise Analysis 30, pp. 339–350.CrossRefGoogle Scholar
  6. 6.
    L. Arkeryd and C. Cercignani, “On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation,” Commun. Partial Diff. Eqns. 14(8–9), 1071–1089 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    L. Arkeryd and C. Cercignani, “Global existence in L1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation,” J. Stat. Phys. 59(3–4), 845–867 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Y. Shizuta, “On the classical solutions of the Boltzmann equation,” Commun. Pure Appl. Math. 36(6), 705–754 (1983).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J. Polewczak, “Classical solution of the nonlinear Boltzmann equation in all R 3: Asymptotic behavior of solutions,” J. Stat. Phys. 50(3–4), 611–632 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    J. Polewczak, “Global existence and asymptotic behavior for the nonlinear Enskog equation,” SIAM J. Appl. Math. 49(3), 952–959 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R. J. DiPerna and P. L. Lions, “On the Cauchy problem for Boltzmann equations: Global existence and weak stability,” Ann. Math., Ser. 2, 130(2), 321–366 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    S.-Y. Ha and S. E. Noh, “Global weak solutions and uniform L p-stability of the Boltzmann-Enskog equation,” J. Diff. Eqns. 251(1), 1–25 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    N. V. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases (Oxford Univ. Press, Oxford, 2004).CrossRefzbMATHGoogle Scholar
  14. 14.
    D. N. Zubarev and V. G. Morozov, “Formulation of boundary conditions for the BBGKY hierarchy with allowance for local conservation laws,” Teor. Mat. Fiz. 60(2), 270–279 (1984) [Theor. Math. Phys. 60, 814–820 (1984)].CrossRefMathSciNetGoogle Scholar
  15. 15.
    D. N. Zubarev, V. G. Morozov, I. P. Omelyan, and M. V. Tokarchuk, “Kinetic equations for dense gases and liquids,” Teor. Mat. Fiz. 87(1), 113–129 (1991) [Theor. Math. Phys. 87, 412–424 (1991)].CrossRefMathSciNetGoogle Scholar
  16. 16.
    V. I. Gerasimenko and I. V. Gapyak, “Hard sphere dynamics and the Enskog equation,” Kinet. Relat. Models 5(3), 459–484 (2012); I. V. Gapyak and V. I. Gerasimenko, “On rigorous derivation of the Enskog kinetic equation,” arXiv: 1107.5572 [math-ph].CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics (Naukova Dumka, Kiev, 1985); Engl. transl.: Mathematical Foundations of Classical Statistical Mechanics: Continuous Systems (Taylor and Francis, London, 2002), Adv. Stud. Contemp. Math. 8.zbMATHGoogle Scholar
  18. 18.
    C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases (Springer, New York, 1994).CrossRefzbMATHGoogle Scholar
  19. 19.
    N. Bellomo and M. Lachowicz, “On the asymptotic theory of the Boltzmann and Enskog equations: A rigorous H-theorem for the Enskog equation,” in Mathematical Aspects of Fluid and Plasma Dynamics (Springer, Berlin, 1991), Lect. Notes Math. 1460, pp. 15–30.CrossRefGoogle Scholar
  20. 20.
    P. Resibois, “H-theorem for the (modified) nonlinear Enskog equation,” J. Stat. Phys. 19(6), 593–609 (1978).CrossRefMathSciNetGoogle Scholar
  21. 21.
    N. N. Bogolyubov, “Kinetic equations and Green’s functions in statistical mechanics,” Preprint No. 57 (Inst. Fiz. Akad. Nauk Azerb. SSR, Baku, 1977).Google Scholar
  22. 22.
    P. Résibois and M. De Leener, Classical Kinetic Theory of Fluids (J. Wiley & Sons, New York, 1977; Mir, Moscow, 1980).Google Scholar
  23. 23.
    H. Van Beijeren and M. H. Ernst, “The modified Enskog equation,” Physica 68(3), 437–456 (1973).CrossRefGoogle Scholar
  24. 24.
    O. E. Lanford III, “Time evolution of large classical systems,” in Dynamical Systems, Theory and Applications (Springer, Berlin, 1975), Lect. Notes Phys. 38, pp. 1–111.CrossRefGoogle Scholar
  25. 25.
    I. Gallagher, L. Saint-Raymond, and B. Texier, “From Newton to Boltzmann: The case of short-range potentials,” arXiv: 1208.5753v1 [math.AP].Google Scholar
  26. 26.
    M. Pulvirenti, C. Saffirio, and S. Simonella, “On the validity of the Boltzmann equation for short range potentials,” arXiv: 1301.2514 [math-ph].Google Scholar
  27. 27.
    H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).CrossRefzbMATHGoogle Scholar
  28. 28.
    N. N. Bogolyubov, Problems of a Dynamical Theory in Statistical Physics (Gostekhizdat, Moscow, 1946; Interscience, New York, 1962).Google Scholar
  29. 29.
    F. King, “BBGKY hierarchy for positive potentials,” PhD Thesis (Univ. California, Berkeley, CA, 1975).Google Scholar
  30. 30.
    C. Villani, “A review of mathematical topics in collisional kinetic theory,” in Handbook of Mathematical Fluid Dynamics (Elsevier, Amsterdam, 2002), Vol. 1, pp. 71–305.CrossRefGoogle Scholar
  31. 31.
    I. V. Volovich, “Time irreversibility problem and functional formulation of classical mechanics,” Vestn. Samar. Gos. Tekh. Univ., Estestvennonauchn. Ser., No. 8/1, 35–55 (2008); arXiv: 0907.2445 [cond-mat.stat-mech].Google Scholar
  32. 32.
    I. V. Volovich, “Randomness in classical mechanics and quantum mechanics,” Found. Phys. 41(3), 516–528 (2011); arXiv: 0910.5391v1 [quant-ph].CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    A. I. Mikhaylov, “The functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem,” Vestn. Samar. Gos. Tekh. Univ., Fiz.-Mat. Nauki, No. 1, 124–133 (2011) [p-Adic Numbers Ultrametric Anal. Appl. 3, 205–211 (2011)].Google Scholar
  34. 34.
    E. V. Piskovskiy and I. V. Volovich, “On the correspondence between Newtonian and functional mechanics,” in Quantum Bio-Informatics IV (World Sci., Hackensack, NJ, 2011), QP-PQ: Quantum Probability and White Noise Analysis 28, pp. 363–372.CrossRefGoogle Scholar
  35. 35.
    E. V. Piskovskiy, “On functional approach to classical mechanics,” Vestn. Samar. Gos. Tekh. Univ., Fiz.-Mat. Nauki, No. 1, 134–139 (2011) [p-Adic Numbers Ultrametric Anal. Appl. 3, 243–247 (2011)].Google Scholar
  36. 36.
    A. S. Trushechkin and I. V. Volovich, “Functional classical mechanics and rational numbers,” p-Adic Numbers Ultrametric Anal. Appl. 1(4), 361–367 (2009); arXiv: 0910.1502 [math-ph].CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    I. V. Volovich, “Bogoliubov equations and functional mechanics,” Teor. Mat. Fiz. 164(3), 354–362 (2010) [Theor. Math. Phys. 164, 1128–1135 (2010)].CrossRefGoogle Scholar
  38. 38.
    A. A. Vlasov, Statistical Distribution Functions (Nauka, Moscow, 1966) [in Russian].Google Scholar
  39. 39.
    V. V. Vedenyapin, The Boltzmann and Vlasov Kinetic Equations (Fizmatlit, Moscow, 2001) [in Russian]; see also: V. Vedenyapin, A. Sinitsyn, and E. Dulov, Kinetic Boltzmann, Vlasov and Related Equations (Elsevier, Amsterdam, 2011).Google Scholar
  40. 40.
    V. V. Kozlov, Thermal Equilibrium in the Sense of Gibbs and Poincaré (Inst. Komp’yut. Issled., Izhevsk, 2002) [in Russian].zbMATHGoogle Scholar
  41. 41.
    V. V. Kozlov, Gibbs Ensembles and Nonequilibrium Statistical Mechanics (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2008) [in Russian].Google Scholar
  42. 42.
    V. V. Kozlov, “The generalized Vlasov kinetic equation,” Usp. Mat. Nauk 63(4), 93–130 (2008) [Russ. Math. Surv. 63, 691–726 (2008)].CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia

Personalised recommendations