Noninteracting Particles

  • Walter T. GrandyJr.
Part of the Fundamental Theories of Physics book series (FTPH, volume 19)


When we set  = exp(−βĤ N ) in Eq. (4-105) we obtain a completely general expression for the canonical partition function describing an N-particle system in thermal equilibrium:
$${\text{Tr}}{{e}^{ - }}^{{\beta {{{\widehat{{\text{H}}}}}_{N}}}} = \frac{1}{{N!}}{{\sum\limits_{{{{\lambda }_{1}} \cdots {{\lambda }_{N}}}} {\left[ {\sum\limits_{{P'}} {{{\varepsilon }^{{P'}}}\hat{P}'} \left\langle {{{\lambda }_{1}} \cdots {{\lambda }_{N}}\left| {{{e}^{ - }}^{{\beta {{{\widehat{{\text{H}}}}}_{N}}}}} \right|{{{\lambda '}}_{1}} \cdots {{{\lambda '}}_{N}}} \right\rangle } \right]} }_{{{{{\lambda '}}_{i}} = {{{\lambda '}}_{1}}_{i}}}}, $$
where Ĥ N is the N-particle Hamiltonian, including interactions.


Partition Function Pauli Principle Canonical Partition Function Boltzmann Statistic Noninteracting Particle 
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  1. Abramowitz, M., and I.A. Stegun (eds.): 1964, Handbook of Mathematical Functions, AMS 55, Natl. Bur. Standards, Washington.Google Scholar
  2. Beckmann, R., F. Karsch, and D.E. Miller: 1979, ‘Bose-Einstein Condensation of a Relativistic Gas in d Dimensions’, Phys. Rev. Letters 43, 1277.ADSCrossRefGoogle Scholar
  3. Boltzmann, L.: 1897, Vorlesungen über die Principe der Mechanik, Vol.1, Barth, Leipzig, p.9.zbMATHGoogle Scholar
  4. Bose, S.N.: 1924, ‘Plancks Gesetz und Lichtquantenhypothese’, Z. Phys. 26, 178.ADSCrossRefGoogle Scholar
  5. Chandrasekhar, S.: 1939, Introduction to the Study of Stellar Structure, Univ. Chicago PressGoogle Scholar
  6. Chicago [Reprinted: Dover, New York, 1957].Google Scholar
  7. Clayton, D.D.: 1968, Principles of Stellar Evolution and Nucleosynthesis, Mraw-Hill, New York.Google Scholar
  8. de Boer, J.: 1973, ‘Some Reflections on the Two-Fluid Model and Bose-Einstein Condensation’, Physica 69, 193.ADSCrossRefGoogle Scholar
  9. Dennison, D.M.: 1927, ‘A Note on the Specific Heat of the Hydrogen Molecule’, Proc. Roy. Soc. (London) A115, 483.ADSGoogle Scholar
  10. Dirac, P.A.M.: 1926, ‘On the Theory of Quantum Mechanics’, Proc. Roy. Soc. (London) A112, 661.ADSGoogle Scholar
  11. Einstein, A.: 1924, ‘Quantentheorie des einatomigen Gases’, Sitz. Preuss. Akad. Wiß. Phys.-Math. Kl., 261.Google Scholar
  12. Einstein, A.: 1925, ‘Quantentheorie des einatomigen idealen Gases.2. Abhandlung’, Sitz. Preuss. Akad. Wiß. Phys.-Math. Kl., 3.Google Scholar
  13. Erdelyi, A. (ed.): 1954, Higher Transcendental Functions, Vol.1, Mraw-Hill, New York.Google Scholar
  14. Fermi, E.: 1926, ‘Zur Quantelung des idealen einatomigen Gases’, Z. Phys. 36, 902.ADSCrossRefGoogle Scholar
  15. Fowler, R.H.: 1926, ‘Dense Matter’, Mon. Not. Roy. Astron. Soc. 87, 114.ADSGoogle Scholar
  16. Glaser, W.: 1935, ‘Korpuskel und Lichtquanten’,Z. Phys. 94, 677.ADSzbMATHCrossRefGoogle Scholar
  17. Grandy, W.T., Jr.: 1970, Introduction to Electrodynamics and Radiation, Academic Press, New York.Google Scholar
  18. Grandy, W.T., Jr., and S.G. Rosa: 1981, ‘Applications of Mellin Transforms to the Statistical Mechanics of Ideal Quantum Gases’, Am. J. Phys. 49, 570.ADSCrossRefGoogle Scholar
  19. Gunton, J.D., and M.J. Buckingham: 1968, ‘Condensation of the Ideal Bose Gas as a Cooperative Transition’, Phys. Rev. 166, 152.ADSCrossRefGoogle Scholar
  20. Haber, H.E., and H.A. Weldon: 1981, ‘Thermodynamics of an Ultrarelativistic Ideal Bose Gas’, Phys. Rev. Letters 46, 1497.ADSCrossRefGoogle Scholar
  21. Haber, H.E., and H.A. Weldon: 1982, ‘On the Relativistic Bose-Einstein Integrals’, J. Math. Phys. 23, 1852.MathSciNetADSCrossRefGoogle Scholar
  22. Hohenberg, P.C.: 1967, ‘Existence of Long-Range Order in One and Two Dimensions’, Phys. Rev. 158, 383.ADSCrossRefGoogle Scholar
  23. Hwang, I.K., and W.T. Grandy, Jr.: 1969, ‘Theory of Photons in a Fully Ionized Gas.I. Photon Momentum Distribution’, Phys. Rev. 177, 359.ADSCrossRefGoogle Scholar
  24. Imry, Y.: 1969, ‘Effective Long-Range Order and Phase Transitions in Finite Macroscopic One and Two Dimensional Systems’, Ann. Phys. (N.Y.) 51, 1.ADSCrossRefGoogle Scholar
  25. Iwata, G.: 1960, ‘Applications of Mellin Transforms to Some Problems of Statistical Mechanics’, Prog. Theor. Phys. (Tokyo) 24, 1118.MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. Johnston, J.R.: 1970, ‘Coherent States in Superfluids: The Ideal Einstein-Bose Gas’, Am. J. Phys. 38, 516.ADSCrossRefGoogle Scholar
  27. Jüttner, F.: 1911a, ‘Das Maxwell’sche Gesetz der Geschwindigkeitverteilung in der Relativitätstheorie’, Ann. d. Phys. 34, 856.CrossRefGoogle Scholar
  28. Jüttner, F.: 1911b, ‘Die Dynamik eines bewegten Gases in der Relativtheorie’,Ann. d. Phys. 35, 145.CrossRefGoogle Scholar
  29. Jüttner, F.: 1928, ‘Die relativistische Quantentheorie des idealen Gases’, Z. Phys. 47, 542.ADSCrossRefGoogle Scholar
  30. Landau, L. J., and I.F. Wilde: 1979, ‘On the Bose-Einstein Condensation of an Ideal Gas’, Commun. Math. Phys. 70, 43.MathSciNetADSCrossRefGoogle Scholar
  31. Landsberg, P.T., and J. Dunning-Davies, Jr.: 1965, ‘Ideal Relativistic Bose Condensation’, Phys. Rev. 138A, 1049.MathSciNetADSCrossRefGoogle Scholar
  32. Leonard, A.: 1968, ‘Exact Inversion of the Fugacity-Density Relation for Ideal Quantum Gases’, Phys. Rev. 175, 221.ADSCrossRefGoogle Scholar
  33. London, F.: 1954, Superßuids, Vol.11, Wiley, New York.Google Scholar
  34. Nieto, M.M.: 1970, ‘Exact State and Fugacity Equations for the Ideal Quantum Gases’, J. Math. Phys. 11, 1346.ADSzbMATHCrossRefGoogle Scholar
  35. Pais, A.: 1979, ‘Einstein and the Quantum Theory’, Rev. Mod. Phys. 51, 863.MathSciNetADSCrossRefGoogle Scholar
  36. Planck, M.: 1900, ‘Zur Theorie des Gesetzes der Energievertheilung in Normalspektrum’, Verh. Deut. Phys. Ges. 2, 237.Google Scholar
  37. Robinson, J.E.: 1951, ‘Note on the Bose-Einstein Integral Functions’, Phys. Rev. 83, 678.ADSzbMATHCrossRefGoogle Scholar
  38. Schiff, L.I.: 1955, Quantum Mechanics, 2nd ed., Mraw-Hill, New York, See.40.zbMATHGoogle Scholar
  39. Sondheimer, E.H., and A.H. Wilson: 1951, ‘The Diamagnetism of Free Electrons’,Proc. Roy. Soc. (London) A210, 173.ADSGoogle Scholar
  40. ’t Hooft, A.H., and J. de Boer: 1970, ‘Symmetry Breaking in the Ideal Bose-Einstein Gas’, Proc. Kon. Akad. Wet. B73, 433, 446.Google Scholar
  41. Ziff, R.M., G.E. Uhlenbeck, and M. Kac: 1977, ‘The Ideal Bose-Einstein Gas Revisited’,Phys. Repts. 32,169.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1987

Authors and Affiliations

  • Walter T. GrandyJr.
    • 1
  1. 1.Department of Physics and AstronomyUniversity of WyomingUSA

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