Foundations of Physics

, Volume 1, Issue 2, pp 145–171 | Cite as

On the classical approximation in the quantum statistics of equivalent particles

  • Armand Siegel


It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This “IQC” (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the “N! paradox” is thereby resolved. The method of “correct Boltzmann counting” and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation.


Entropy Quantum Statistic Classical Statistic Microcanonical Ensemble Molecular Collision 
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  1. 1.
    S. N. Bose,Z. Physik 26, 178 (1924).Google Scholar
  2. 2.
    A. Einstein,Berl. Ber. 261 (1924); 3, 18 (1925).Google Scholar
  3. 3.
    E. Fermi,Z. Physik 36, 902 (1926).Google Scholar
  4. 4.
    P. A. M. Dirac,Proc. Roy. Soc. (London) A112, 661 (1926).Google Scholar
  5. 5.
    M. Planck,Verhand. Deut. Phys. Ges. 2, 202, 237 (1900).Google Scholar
  6. 6.
    D. ter Haar,Elements of Thermostatistics (Holt, Rinehart, and Winston, New York, 1966).Google Scholar
  7. 7.
    A. Sommerfeld,Thermodynamics and Statistical Mechanics (Academic Press, New York, 1964).Google Scholar
  8. 8.
    P. and T. Ehrenfest,The Conceptual Foundations of the Statistical Approach in Mechanics (English translation published by Cornell University Press, Ithaca, 1959).Google Scholar
  9. 9.
    F. Reif,Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 243–246.Google Scholar
  10. 10.
    E. P. Wigner,Phys. Rev. 40, 749 (1932).Google Scholar
  11. 11.
    J. G. Kirkwood,Phys. Rev. 45, 116 (1934).Google Scholar
  12. 12.
    J. G. Kirkwood,Phys. Rev. 44, 31 (1933).Google Scholar
  13. 13.
    O. Stern,Rev. Mod. Phys. 21, 534 (1949).Google Scholar
  14. 14.
    I. Prigogine,Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962), pp. 138, 139.Google Scholar
  15. 15.
    K. Huang,Statistical Mechanics (John Wiley and Sons, New York, 1963).Google Scholar
  16. 16.
    A. H. Wilson,Thermodynamics and Statistical Mechanics (Cambridge University Press, New York, 1957), p. 114.Google Scholar
  17. 17.
    G. S. Rushbrooke,Introduction to Statistical Mechanics (Oxford University Press, New York, 1949), p. 39.Google Scholar
  18. 18.
    Armand Siegel,Foundations of Physics 1, 57 (1970).Google Scholar
  19. 19.
    J. von Neumann,Mathematische Grundlagen der Quantenmechanik (Julius Springer, Berlin, 1932) (English translation:Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, N. J., 1955), Chapter 5, Section 4.Google Scholar
  20. 20.
    D. ter Haar,Elements of Statistical Mechanics (Rinehart and Company, New York, 1954).Google Scholar
  21. 21.
    J. R. Dorfman and E. G. D. Cohen, Difficulties in the kinetic theory of dense gases,J. Math. Phys. 8, 282 (1967).Google Scholar
  22. 22.
    J. de Boer and R. Byron Bird, Quantum theory and transport phenomena, inMolecular Theory of Gases and Liquids, J. O. Hirschfelder, C.F. Curtiss, and R.B. Bird, eds. (Wiley, New York, and Chapman and Hall, London, 1954).Google Scholar
  23. 23.
    P. Caldirola (ed.),Ergodic Theories (Academic Press, New York, 1961).Google Scholar
  24. 24.
    I. E. Farquhar,Ergodic Theory in Statistical Mechanics (Wiley-Interscience, New York, 1964).Google Scholar
  25. 25.
    J. H. Irving and Robert W. Zwanzig,J. Chem. Phys. 19, 1173 (1951).Google Scholar

Copyright information

© Plenum Publishing Corporation 1970

Authors and Affiliations

  • Armand Siegel
    • 1
    • 2
  1. 1.Boston UniversityBoston
  2. 2.Centre d'Études Nucléaires de Saclay91 Gif-sur-YvetteFrance

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