Advertisement

Foundations of Physics

, Volume 48, Issue 3, pp 271–294 | Cite as

On the Boltzmann–Grad Limit for Smooth Hard-Sphere Systems

  • Massimo Tessarotto
  • Claudio Cremaschini
  • Michael Mond
  • Claudio Asci
  • Alessandro Soranzo
  • Gino Tironi
Article
  • 67 Downloads

Abstract

The problem is posed of the prescription of the so-called Boltzmann–Grad limit operator (\(\mathcal {L}_{BG}\)) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator \(\mathcal {L}_{BG}\), the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is “no time-asymmetric ingredient” in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the “ab initio” axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.

Keywords

Boltzmann equation Hard-sphere classical dynamical system Boltzmann H-theorem Master kinetic equation 

Notes

Acknowledgements

The authors are grateful to the International Center for Theoretical Physics (Miramare, Trieste, Italy) for the hospitality during the preparation of the manuscript.

References

  1. 1.
    Boltzmann, L.: Weitere studien über das w ärmegleichgewicht unter gasmolekülen. Sitz. Akad. Wiss. 66, 275 (1872)MATHGoogle Scholar
  2. 2.
    Grad, H.: Principles of the Kinetic theory of Gases. Handbook der Physik, vol. XII, p. 205. Springer, Berlin (1958)Google Scholar
  3. 3.
    Loschmidt, J.: Uber den Zustand des Warmegleichgewichtes eines Systems von Korpern mit Rucksicht auf die Schwerkraft. Akad. Wiss. Wien 73, 128 (1876)Google Scholar
  4. 4.
    Boltzmann, L.: Weitere Studien iiber das Warmegleichgewicht unter Gasmolekule. Wien. Ber.66, 275 (1877) (in Boltzmann: Uber die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung, vol. II, pp. 112–148 (1909))Google Scholar
  5. 5.
    Ehrenfest, P., Ehrenfest-Afanassjewa, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell University Press, New York (1912)MATHGoogle Scholar
  6. 6.
    Cercignani, C.: H-Theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech. 34, 231 (1982)MathSciNetMATHGoogle Scholar
  7. 7.
    Lebowitz, J.L.: Boltzmann’s entropy and time’s arrow. Phys. Today 46, 32 (1994)CrossRefGoogle Scholar
  8. 8.
    Drory, A.: Is there a reversibility paradox? Recentering the debate on the thermodynamic time arrow. Stud. Hist. Phil. Sci. Part B 39, 889–913 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tessarotto, M., Cremaschini, C.: Axiomatic foundations of entropic theorems for hard-sphere systems. Eur. Phys. J. Plus 130, 91 (2015)CrossRefMATHGoogle Scholar
  10. 10.
    Tessarotto, M., Cremaschini, C., Tessarotto, M.: On the conditions of validity of the Boltzmann equation and Boltzmann H-theorem. Eur. Phys. J. Plus 128, 32 (2013)CrossRefMATHGoogle Scholar
  11. 11.
    Uffink, J., Valente, G.: Lanford’s theorem and the emergence of irreversibility. Found Phys. 45, 404 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lanford, O.E.: Time Evolution of Large Classical Systems. Lecture Notes in Physics, vol. 38. Springer, Berlin (1975)MATHGoogle Scholar
  13. 13.
    Lanford, O.E.: On the derivation of the Boltzmann equation. Soc. Math. France Asterisque 40, 117 (1976)MathSciNetGoogle Scholar
  14. 14.
    Lanford, O.E.: Hard-sphere gas in the Boltzmann-Grad limit. Physica 106A, 70 (1981)ADSCrossRefGoogle Scholar
  15. 15.
    Cercignani, C.: Theory and Applications of the Boltzmann Equation. Scottish Academic Press, Edinburgh (1975)MATHGoogle Scholar
  16. 16.
    Tessarotto, M., Asci, C., Cremaschini, C., Soranzo, A., Tironi, G., Tessarotto, M.: The Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids. Eur. Phys. J. Plus 127, 36 (2012)CrossRefGoogle Scholar
  17. 17.
    Ardourel, V.: Irreversibility in the derivation of the Boltzmann equation. Found Phys. 47, 471 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Enskog, D.: Kinetiche Theorie dr Wärmeleiting, Reibung und Selbs-diffusion in gewissenverdichteten Gasen und Flü ssigkeiten. Svensk Vetenskps Akad. 63, 4 (1921) (English trans: Brush, S.G., Kinetic Theory, vol. 3, Pergamon, New York, 1972)Google Scholar
  19. 19.
    Tessarotto, M., Cremaschini, C.: Theory of collisional invariants for the Master kinetic equation. Phys. Lett. A 378, 1760 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 1, Sect. 12; vol. 2, Sect. 22. University of California Press, California)Google Scholar
  22. 22.
    Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 1, Sect. 8. University of California Press, California)Google Scholar
  23. 23.
    Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 2, Sect. 38. University of California Press, California)Google Scholar
  24. 24.
    Cercignani, C.: 134 years of Boltzmann equation. In: Gallavotti, G., Reiter, W., Yngvason, J. (eds.) Boltzmann’s Legacy. ESI Lecture in Mathematics and Physics. European Mathematical Society, Prague (2008)Google Scholar
  25. 25.
    Munster, A.: Statistical Thormodynamics. Springer, New York (1969)Google Scholar
  26. 26.
    Villani, C.: Entropy Production and Convergence to Equilibrium for the Boltzmann Equation. In: Zambrini, J.C. (ed.) 14th Int. Congress on Math. Physics, 28 July–2 August 2003, University of Lisbon, Portugal. World Scientific Publishing Co., Singapore (2006)Google Scholar
  27. 27.
    Tessarotto, M., Cremaschini, C.: Modified BBGKY hierarchy for the hard-sphere system. Eur. Phys. J. Plus 129, 243 (2014)CrossRefMATHGoogle Scholar
  28. 28.
    Tessarotto, M., Cremaschini, C.: Theory of collisional invariants for the Master kinetic equation. Phys. Lett. A 379, 1206 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tessarotto, M., Asci, C., Cremaschini, C., Soranzo, A., Tironi, G.: Global validity of the Master kinetic equation for hard-sphere systems. Eur. Phys. J. Plus 130, 160 (2015)CrossRefGoogle Scholar
  30. 30.
    Tessarotto, M., Asci, C.: Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems. Phys. Lett. A 381, 1484 (2017)ADSCrossRefMATHGoogle Scholar
  31. 31.
    Tessarotto, M., Mond, M., Asci, C.: Microscopic statistical description of incompressible Navier-Stokes granular fluids. Eur. Phys. J. Plus 132, 213 (2017)CrossRefGoogle Scholar
  32. 32.
    Cercignani, C.: Mathematical Methods in Kinetic Theory. Plenum Press, New York (1969)CrossRefMATHGoogle Scholar
  33. 33.
    Asci, C.: Integration over an infinite-dimensional banach space and probabilistic applications. Int. J. Anal. (2014).  https://doi.org/10.1155/2014/404186
  34. 34.
    Asci, C.: Differentiation theory over infinite-dimensional banach spaces. J. Math. (2016).  https://doi.org/10.1155/2016/2619087

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Massimo Tessarotto
    • 1
    • 2
  • Claudio Cremaschini
    • 2
  • Michael Mond
    • 3
  • Claudio Asci
    • 1
  • Alessandro Soranzo
    • 1
  • Gino Tironi
    • 1
  1. 1.Department of Mathematics and GeosciencesUniversity of TriesteTriesteItaly
  2. 2.Institute of Physics, Faculty of Philosophy and ScienceSilesian University in OpavaOpavaCzech Republic
  3. 3.Department of Mechanical EngineeringBen Gurion University of the NegevBeershebaIsrael

Personalised recommendations