Skip to main content
SpringerLink
Log in

The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this investigation, exact particular realizations are sought for the microscopic statistical description which is associated with the classical dynamical system (CDS) formed by N identical smooth hard spheres subject to elastic collisions (S N -CDS). The problem is posed in the framework of the ab initio statistical description of S N -CDS recently developed. It is shown that the Liouville equation associated with SN-CDS admits an exact particular solution for the N-body probability density function (PDF). This is factorized in terms of the i-th particle 1-body PDF (for all i = 1, N) via suitable weighting factors, which are denoted here as particle occupation coefficients. The latter are found to depend functionally only on the 1-body PDFs which are associated with each of the remaining particles belonging to S N -CDS. Furthermore, the 1-body PDF is proved to obey a well-defined statistical equation, referred to here as Master kinetic equation. This is an exact kinetic equation which takes into account the occurrence of configuration-space correlations due to the finite size of the extended particles, while depending functionally on the same 1-body PDF only. The asymptotic approximation of the Master equation, which holds in validity of the Boltzmann-Grad limit, is shown to recover in a suitable asymptotic sense the customary Boltzmann equation. Finally, a critical analysis is presented of the original and modified versions of the Enskog kinetic equation, as well as of some of the non-linear kinetic approaches formulated in the past for dense granular gases. Their conditions of validity and main differences with respect to the present theory are pointed out.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Massimo Tessarotto, Claudio Cremaschini, Marco Tessarotto, Eur. Phys. J. Plus 128, 32 (2013).

    Article  Google Scholar 

  2. M. Tessarotto, C. Cremaschini, Phys. Lett. A 378, 1760 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  3. Y.G. Sinai, Russ. Math. Surv. 25, 137 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  4. Y.G. Sinai, Dynamical Systems II: Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics (Springer-Verlag, Berlin, 1989).

  5. D.V. Anosov, Y.G. Sinai, Russ. Math. Surv. 22, 103 (1967).

    Article  MathSciNet  Google Scholar 

  6. C. Cercignani, Theory and applications of the Boltzmann equation (Scottish Academic Press, Edinburgh and London, 1975).

  7. C. Cercignani, Mathematical methods in kinetic theory (Plenum Press, New York, 1969).

  8. L. Boltzmann, Wien. Ber. 66, 275 (1872) in Wissenschaftliche Abhandlungen.

    MATH  Google Scholar 

  9. H. Grad, Principles of the kinetic theory of gases, in Handbook der Physik Vol. 12 (1958) p. 205.

  10. M. Born, H.S. Green, Proc. R. Soc. A 188, 10 (1946) DOI:10.1098/rspa.1946.0093.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. N.N. Bogoliubov, J. Exp. Theor. Phys. 16, 691 (1946) (in Russian).

    Google Scholar 

  12. N.N. Bogoliubov, J. Phys. (USSR) 10, 265 (1946).

    Google Scholar 

  13. J.G. Kirkwood, J. Chem. Phys. 14, 180 (1946).

    Article  ADS  Google Scholar 

  14. J. Yvon, in Actual. Sci. Indust., N. 203 (Paris, Hermann, 1935).

  15. D. Enskog, Kungl. Svensk. Vetenskps. Akad. 63, 4 (1921) (English translation by S.G. Brush, Kinetic Theory.

    Google Scholar 

  16. K. Uchiyama, Hiroshima Math. J. 18, 245 (1988).

    MATH  MathSciNet  Google Scholar 

  17. C. Cercignani, R. Illner, Arch. Rat. Mech. Anal. 134, 1 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  18. M.H. Ernst, H. van Beijeren, Physica 68, 437 (1973).

    Article  ADS  Google Scholar 

  19. M.H. Ernst, H. van Beijeren, Physica 70, 225 (1973).

    Article  ADS  Google Scholar 

  20. M.H. Ernst, H. van Beijeren, Phys. Lett. A 43, 367 (1973).

    Article  ADS  Google Scholar 

  21. E.G.D. Cohen, Physica A 118, 17 (1983).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. J.M.J. van Leeuwen, A. Weyland, Physica 36, 457 (1967).

    Article  ADS  Google Scholar 

  23. J.M.J. van Leeuwen, A. Weyland, Physica 39, 35 (1968).

    Google Scholar 

  24. M.H. Ernst, J.R. Dorfman, W.R. Hoegy, J.M.J. van Leeuwen, Physica 45, 127 (1969).

    Article  ADS  Google Scholar 

  25. J.R. Dorfman, H. van Beijeren, The Kinetic Theory of Gases, in Statistical Mechanics, Part B: Time-Dependent Processes, edited by B.J. Berne (Plenum Press, New York, 1977).

  26. J. Loschmidt, Akad. Wissenschaf. Wien 73, 128 (1876).

    Google Scholar 

  27. E. Zermelo, Ann. Phys. 57, 485 (1896) (English translation by S.G. Brush, The kinetic theory of gases.

    Article  Google Scholar 

  28. E. Zermelo, Ann. Phys. 59, 4793 (1896) (English translation by S.G. Brush, The kinetic theory of gases.

    Google Scholar 

  29. L. Boltzmann, Ann. Phys. 57, 773 (1896) (English translation by S.G. Brush, The kinetic theory of gases.

    Article  Google Scholar 

  30. L. Boltzmann, Ann. Phys. 60, 392 (1897) (English translation by S.G. Brush, The kinetic theory of gases.

    Article  Google Scholar 

  31. L. Boltzmann, Vorlesungen über Gasstheorie, Vol. 2 (J.A. Barth, Leipzig, 1896-1898) English translation by H. Brush, Lectures on gas theory (University of California Press, 1964).

  32. M. Tessarotto, M. Ellero, AIP Conf. Proc. 762, 108 (2005).

    Article  ADS  Google Scholar 

  33. M. Ellero, M. Tessarotto, Physica A 355, 233 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  34. M. Tessarotto, M. Ellero, Physica A 373, 142 (2007).

    Article  ADS  Google Scholar 

  35. M. Tessarotto, M. Ellero, Proceedings of the 25th RGD (International Symposium on Rarefied Gas Dynamics, St. Petersburg, Russia, July 21-28, 2006), edited by M.S. Ivanov, A.K. Rebrov (Novosibirsk Publ. House of the Siberian Branch of the Russian Academy of Sciences, 2007) p. 1001, arXiv:physics/0611113.

  36. C. Cremaschini, M. Tessarotto, AIP Conf. Proc. 1084, 188 (2008).

    Article  ADS  Google Scholar 

  37. Marco Tessarotto, Claudio Cremaschini, Piero Nicolini, Massimo Tessarotto, AIP Conf. Proc. 1084, 182 (2008).

    Article  Google Scholar 

  38. M. Tessarotto, C. Cremaschini, Physica A 392, 3962 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  39. Marco Tessarotto, Claudio Cremaschini, Massimo Tessarotto, Physica A 388, 3737 (2009).

    Article  MathSciNet  Google Scholar 

  40. Massimo Tessarotto, Claudio Asci, Claudio Cremaschini, Alessandro Soranzo, Gino Tironi, Eur. Phys. J. Plus 127, 36 (2012).

    Article  Google Scholar 

  41. M. Tessarotto, M. Ellero, P. Nicolini, Phys. Rev. A 75, 012105 (2007).

    Article  ADS  Google Scholar 

  42. M. Tessarotto, M. Ellero, P. Nicolini, AIP Conf. Proc. 1084, 33 (2008).

    Article  ADS  Google Scholar 

  43. M. Tessarotto, M. Ellero, N. Aslan, M. Mond, P. Nicolini, AIP Conf. Proc. 1084, 224 (2008).

    Article  ADS  Google Scholar 

  44. M. Tessarotto, M. Ellero, D. Sarmah, P. Nicolini, AIP Conf. Proc. 1084, 170 (2008).

    Article  ADS  Google Scholar 

  45. M. Tessarotto, Magnetohydrodyn. J. 45, 3 (2009).

    Google Scholar 

  46. E. Fonda, M. Tessarotto, P. Nicolini, M. Ellero, AIP Conf. Proc. 1084, 164 (2008).

    Article  ADS  Google Scholar 

  47. O.E. Lanford, Time evolution of large classical systems, in Dynamical Systems Theory and Applications, in Lecture Notes in Physics, edited by E.J. Moser, Vol. 38 (Springer-Verlag, Berlin, 1975).

  48. O.E. Lanford, Physica A 106, 70 (1981).

    Article  ADS  Google Scholar 

  49. C. Cercignani, The Boltzmann equation and its applications, in Applied Mathematical Sciences, Vol. 67 (Springer-Verlag, Berlin, 1988).

  50. C. Cercignani, 134 years of Boltzmann equation, in Boltzmann’s legacy, edited by G. Gallavotti, W. Reiter, J. Yngvason, ESI Lecture in Mathematics and Physics (European Mathematical Society, 2008) p. 107.

  51. S. Chapman, T.G. Cowling, The mathematical theory of non-uniform gases (Cambridge University Press, 1939).

  52. H. Van Beijeren, M.H. Ernst, Phys. Lett. A 43, 367 (1973).

    Article  ADS  Google Scholar 

  53. H. Van Beijeren, M.H. Ernst, Physica 68, 437 (1973).

    Article  ADS  Google Scholar 

  54. H. Van Beijeren, Physica 70, 225 (1973).

    Article  ADS  Google Scholar 

  55. C. Cercignani, M. Lampis, J. Stat. Phys. 53, 655 (1988).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  56. K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, R.E. Goldstein, Phys. Rev. Lett. 103, 198103 (2009).

    Article  ADS  Google Scholar 

  57. F.G. Woodhouse, R.E. Goldstein, Phys. Rev. Lett. 109, 168105 (2012).

    Article  ADS  Google Scholar 

  58. J.M. Montanero, V. Garzó, A. Santos, J.J. Brey, J. Fluid Mech. 389, 391 (1999).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  59. C.-H. Lee, C.-J. Huang, Phys. Fluids 24, 073303 (2012).

    Article  ADS  Google Scholar 

  60. G.P. Bewley, K.R. Sreenivasan, D.P. Lathrop, Exp. Fluids 44, 887 (2008).

    Article  Google Scholar 

  61. C.E. Shannon, Bell Syst. Tech. J. 27, 623 (1948).

    Article  MathSciNet  Google Scholar 

  62. P. Nicolini, M. Tessarotto, Phys. Plasmas 13, 052901 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  63. C. Cremaschini, M. Tessarotto, Phys. Rev. E 87, 032107 (2013).

    Article  ADS  Google Scholar 

  64. R. Illner, M. Pulvirenti, Transp. Th. Stat. Phys. 16, 997 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  65. Y.L. Klimontovich, Zh. Eksp. Teor. Fiz. 33, 928 (1958) (Sov. Phys. JETP 6.

    Google Scholar 

  66. Y.L. Klimontovich, Kinetic Theory of nonideal gases and nonideal plasmas (Nauka, Moscow, 1975) (Pergamon Press, 1982).

  67. Y.L. Klimontovich, Statistical theory of open systems, Vol. I (Janus, Moscow, 1995) (Kluwer, Dordrecht, 1998).

  68. E.T. Jaynes, Phys. Rev. 106, 620 (1957).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  69. E.T. Jaynes, Phys. Rev. 108, 171 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  70. T.P.C. van Noije, M.H. Ernst, Gran. Matt. 1, 57 (1998).

    Article  Google Scholar 

  71. J.J. Brey, D. Cubero, M.J. Ruiz-Montero, Phys. Rev. E 59, 1256 (1999).

    Article  ADS  Google Scholar 

  72. W. Losert, D.G. W. Cooper, J. Delour, A. Kudrolli, J.P. Gollub, Chaos 9, 682 (1999).

    Article  ADS  MATH  Google Scholar 

  73. A. Kudrolli, J. Henry, Phys. Rev. E 62, R1489 (2000).

    Article  ADS  Google Scholar 

  74. V. Garzó, J.W. Dufty, C.M. Hrenya, Phys. Rev. E 76, 031303 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  75. E.G.D. Cohen, in Transport Phenomena in Fluids, edited by H.J.M. Hartley, Marcel Dekker (New York, USA, 1969).

  76. H.J.M. Hanley, R.D. McCarty, E.G.D. Cohen, Physica 60, 322 (1972).

    Article  ADS  Google Scholar 

  77. H.J.M. Hartley, E.G.D. Cohen, Physica A 83, 215 (1976).

    Article  ADS  Google Scholar 

  78. M.H. Ernst, H. van Beijeren, Physica 68, 437 (1973).

    Article  ADS  Google Scholar 

  79. A.W. Silva, G.M. Alves, G.M. Kremer, Physica A 387, 1733 (2008).

    Article  ADS  Google Scholar 

  80. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54, 1523 (1971).

    Article  ADS  Google Scholar 

  81. H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68, 1259 (1996).

    Article  ADS  Google Scholar 

  82. T.P.C. van Noije, M.H. Ernst, in Lecture Notes in Physics, Vol. 564, edited by T. Poschel, S. Luding (Springer-Verlag, 2001).

  83. H. Grad, Singular limits of solutions of Boltzmann’s equation, in Proceedings of the 8th International Symposium on Rarefied Gas Dynamics, New York, July 1972 (NYU Courant Institute of Mathematical Sciences, 1972) Int. Rep. MF-77, AFOSR-TR-72-1843.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127, Trieste, Italy

    Massimo Tessarotto

  2. Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601, Opava, Czech Republic

    Massimo Tessarotto & Claudio Cremaschini

Authors
  1. Massimo Tessarotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudio Cremaschini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudio Cremaschini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tessarotto, M., Cremaschini, C. The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system. Eur. Phys. J. Plus 129, 157 (2014). https://doi.org/10.1140/epjp/i2014-14157-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2014-14157-4

Keywords

Search

Navigation