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Foundations of Physics

, Volume 45, Issue 4, pp 404–438 | Cite as

Lanford’s Theorem and the Emergence of Irreversibility

  • Jos Uffink
  • Giovanni ValenteEmail author
Original Paper

Abstract

It has been a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A result by Lanford (Dynamical systems, theory and applications, 1975, Asterisque 40:117–137, 1976, Physica 106A:70–76, 1981) shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. Here, we examine how and in what sense Lanford’s theorem succeeds in deriving this remarkable result. Many authors have expressed different views on the question which of the ingredients in Lanford’s theorem is responsible for the emergence of irreversibility. We claim that these interpretations miss the target. In fact, we argue that there is no time-asymmetric ingredient at all.

Keywords

Statistical mechanics Irreversibility Time-reversal invariance Lanford 

Notes

Acknowledgments

The authors would like to thank Herbert Spohn for extensive comments on the technical and conceptual subtleties of Lanford’s theorem, as well as two anonymous referees for helpful remarks.

References

  1. 1.
    Alexander, R.K.: The infinite hard-sphere system, Ph.D. Thesis, University of California at Berkeley (1975)Google Scholar
  2. 2.
    Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wien. Ber. 66, 275–370 (1872); in Boltzmann (1909) vol.I, pp. 316–402Google Scholar
  3. 3.
    Boltzmann, L.: Über das Wärmegleichgewicht Von Gasen, auf welche äußere Kräfte wirken, Wien. Ber. 72, 427–457 (1872); in Boltzmann (1909) vol.II, pp. 1–30Google Scholar
  4. 4.
    Boltzmann, L.: Bemerkungen über einige probleme der mechanische Wärmetheorie, Wien. Ber. 66, 275–370 (1877); in Boltzmann (1909) vol.II, pp. 112–148Google Scholar
  5. 5.
    Boltzmann, L.: On certain questions of the theory of gases. Nature, 51, 413–415 (1895); in Boltzmann (1909) vol.III, pp. 535–544Google Scholar
  6. 6.
    Boltzmann L.: Zu Hrn Zermelos Abhandlung Über die mechanische Erklärung irreversibler Vorgänge, Wied. Ann. 60, 392–398 (1897); in Boltzmann (1909), Vol III, paper 119Google Scholar
  7. 7.
    Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen. Chelsea, Leipzig (1909)Google Scholar
  8. 8.
    Cercignani, C.: 134 years of Boltzmann equation. In: Gallavotti, G., Gallavotti, W.L., Yngvason, J. (eds.) Boltzmann’s Legacy, pp. 107–128. Zürich: European Mathematical Society, Zürich (2008)CrossRefGoogle Scholar
  9. 9.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Diluted Gases. Springer, New York (1994)CrossRefGoogle Scholar
  10. 10.
    Culverwell, E.P.: Dr. Watson’s proof of Boltzmann’s theorem on permanence of distributions. Nature 50, 617 (1894)CrossRefADSzbMATHGoogle Scholar
  11. 11.
    Drory, A.: Is there a reversibility paradox? Recentering the debate on the thermodynamic time arrow. Stud. Hist. Philos. Mod. Phys. 39, 889–913 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ehrenfest, P., Ehrenfest-Afanassjewa, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell University Press, New York (1912)Google Scholar
  13. 13.
    Feynman, R., Leighton, R., Sands, M.: The Feynman Lectures on Physics. Addison Wesley, Reading (1964)Google Scholar
  14. 14.
    Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two-dimensional rare gas in a vacuum. Commun. Math. Phys. 105, 189–203 (1986); erratum ibidem 121 143–146 (1989)Google Scholar
  15. 15.
    Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Theoretical Physics, vol. 38, pp. 1–111. Springer, Berlin (1975)CrossRefGoogle Scholar
  16. 16.
    Lanford, O.E.: On the derivation of the Boltzmann equation. Asterisque 40, 117–137 (1976)MathSciNetGoogle Scholar
  17. 17.
    Lanford, O.E.: The hard sphere gas in the Boltzmann-Grad limit. Physica 106A, 70–76 (1981)CrossRefADSGoogle Scholar
  18. 18.
    Lebowitz, J.L.: Microscopic dynamics and macroscopic laws. In: Horton Jr, C.W., Reichl, L.E., Szebehely, V.G. (eds.) Long-Time Prediction in Dynamics, pp. 220–233. Wiley, New York (1983)Google Scholar
  19. 19.
    Loschmidt, J.: Über die Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft. Wien. Ber. 73, 139 (1877)Google Scholar
  20. 20.
    Goldstein, S.: Boltzmann’s approach to statistical mechanics. In: Bricmon, J., et al. (eds.) Chance in Physics: Foundations and Perspectives. Lecture Notes in Physics, pp. 39–54. Springer, Berlin (2001)CrossRefGoogle Scholar
  21. 21.
    Schrödinger, E.: Irreversibility. Proc. R. Ir. Acad. 53, 189–195 (1950)Google Scholar
  22. 22.
    Spohn, H.: Kinetic equations from Hamiltonian dynamics: markovian limits. Rev. Mod. Phys. 53, 569–615 (1980)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  24. 24.
    Spohn, H.: Loschmidt’s reversibility argument and the H-theorem. In: Fleischhacker, W., Schlonfeld, T. (eds.) Pioneering Ideas for the Physical and Chemical Sciences, Loschmidt’s Contributions and Modern Developments in Structural Organic Chemistry, Atomistics and Statistical Mechanics, pp. 153–158. Plenum Press, New York (1997)Google Scholar
  25. 25.
    Spohn, H.: On the integrated form of the BBGKY hierarchy for hard spheres. http://arxiv.org/pdf/math-ph/0605068 (2006). Accessed 25 May 2006
  26. 26.
    Uffink, J.: Compendium of the foundations of classical statistical physics. In: Butterfield, J., Earman, J. (eds.) Handbook for the Philosophy of Physics, pp. 923–1074. Elsevier, Amsterdam (2008)Google Scholar
  27. 27.
    Valente, G.: The approach towards equilibrium in Lanford’s theorem. Eur. J. Philos. Sci. 4(3), 309–335 (2014)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Villani, C.: (Ir)reversibilité et entropie. Séminaire Poincaré XV, Le Temps, pp. 17–75. Institut Henri Poincaré, Paris (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of MinnesotaMinneapolisUSA
  2. 2.PittsburghUSA

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