On Kinetic Equations for Particles with Internal Degress of Freedom

  • L. Waldmann
Part of the Acta Physica Austriaca book series (FEWBODY, volume 10/1973)


Already in his famous 1872-paper Boltzmann even included polyatomic dilute gases. More recently this question has been reconsidered from the quantum theoretical point of view. The simplest model in this context is a gas of particles with spin, which has to be described by a one particle distribution matrix with 2 S + 1 rows and columns (S = value of spin). The H-theorem can be traced back to the unitarity of the scattering matrix for the binary collision, suitably occurringin the kinetic equation. A special spin-dependent scattering amplitude is considered for which the classical limit can easily be effectuated. In conclusion some remarks on the classical Boltzmann equation for rigid dumb-bells are made.


Kinetic Equation Unitary Transformation Relative Momentum Optical Theorem Magnetic Quantum Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. Boltzmann, Wien. Der. 66, 275–370 (1872). “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”.Google Scholar
  2. [2]
    H.A. Lorentz, Wien, Ber. 95, 115–152 (1887).Google Scholar
  3. [3]
    L. Boltzmann, Wien, Ber. 95, 153–164 (1887).Google Scholar
  4. [4]
    L. Waldmann,Z. Naturforsch. 12a,660 (1957)ADSGoogle Scholar
  5. L. Waldmann,Z. Naturforsch. 13a,609 (1958).ADSMathSciNetGoogle Scholar
  6. [5]
    R.F. Snider,J. Chem. Phys. 32,1051 (1960).CrossRefADSMathSciNetGoogle Scholar
  7. [6]
    L. Waldmann,Nuovo Cimento 14,898 (1959)CrossRefMathSciNetGoogle Scholar
  8. L. Waldmann,Z. Naturforsch. 15a, 19 (1960).ADSMathSciNetGoogle Scholar
  9. [7]
    Y. Kagan and A.M. Afanasew, Zh. Eksp. Teor. Fiz. 41,1536 (1961),transi. Sov. Phys. JETP 14,1096 (1962).Google Scholar
  10. [8]
    L. Waldmann and S. Hess, Z. Naturforsch. 24a, 2010 (1969).ADSGoogle Scholar
  11. [9]
    H. Senftleben, Phys. Z. 31, 822, 961 (1930)Google Scholar
  12. J.J.M. Beenakker, G. Scoles, H.F.P. Knaap, and R.M. Jonkman, Phys. Letters 2, 5 (1962).CrossRefADSGoogle Scholar
  13. [10]
    Y. Kagan and L.A. Maksimov, Zh. Eksp. Teor. Fiz. 41, 842 (1961), transl. Sov. Phys. JETP 14, 604 (1962)Google Scholar
  14. J.J.M. Beenakker and F.R. McCourt, Ann. Rev. Phys. Chem. 21, 47 (1970).CrossRefADSGoogle Scholar
  15. J.S. Dahler and D.K. Hoffmann, in Transfer and Storage of Energy by Molecules, Vol. 3, ed. G.M. Burnett, Wiley, New York 1970.Google Scholar
  16. [11]
    S. Hess, Z. Naturforsch. 22a, 1871 (1967); 23a, 898 (1968); Physica 42, 633 (1969).Google Scholar
  17. F.M. Chen and R.F. Snider,J. Chem. Phys.46,3937(1967); 48,3185(1968);50,4082(1969).Google Scholar
  18. S. Hess and F.R. McCourt, Chem. Phys. Letters 5, 53 (1970).CrossRefADSGoogle Scholar
  19. F.R. McCourt and S. Hess,Z. Naturforsch. 25a,1169 (1970);26a,1234(1971).Google Scholar
  20. A. Tip and F.R. McCourt, Physica 52, 109 (1971).CrossRefADSGoogle Scholar
  21. [12]
    S. Hess, Z. Naturforsch.23a,597 (1968).Google Scholar
  22. [13]
    S. Hess, Phys. Letters 30A, 239 (1969).ADSGoogle Scholar
  23. F. Baas, Phys. Letters 36A, 107 (1971).ADSGoogle Scholar
  24. A.G.St. Pierre, W.E. Köhler and S. Hess, Z. Naturforsch. 27a, 721 (1972).ADSGoogle Scholar
  25. [14]
    V.G. Cooper, A.D. May, E.H. Hara, and H.F.P. Knaap, Can. J. Phys. 46,2019(1968).CrossRefADSGoogle Scholar
  26. R.A.J. Kèijser, M. Jansen, V.G. Cooper and H.F.P. Knaap, Physica 51,593(1971).CrossRefADSGoogle Scholar
  27. [15]
    S. Hess, Phys. Letters 29A, 108 (1969)ADSGoogle Scholar
  28. Z. Naturforsch. 24a, 1852 (1969); 25a, 350 (1970)Google Scholar
  29. Springer Tracts in Mod. Phys. 54, 136 (1970).Google Scholar
  30. S. Hess and H.F.P. Knaap, Z. Naturforsch. 26a, 1639 (1971).ADSGoogle Scholar
  31. [16]
    G.G. Scott, H.W. Sturner and R.M. Williamson, Phys. Rev. 158, 117 (1967).CrossRefADSGoogle Scholar
  32. [17]
    L. Waldmann, Z. Naturforsch. 22a, 1678 (1967).ADSGoogle Scholar
  33. [18]
    A.C. Levi and J.J.M. Beenakker, Phys. Letters 25A, 350 (1967).CrossRefADSGoogle Scholar
  34. [19]
    M.E. Larchez and T.W. Adair, Phys. Rev. A3, 2052 (1971).ADSGoogle Scholar
  35. S. Hess, Z. Naturforsch. 27a, 366 (1972).ADSGoogle Scholar
  36. [20]
    H. Hulsman, F.G. van Kuick, H.F.P. Knaap and J.J.M. Beenakker, Physica 57, 522 (1972).CrossRefADSGoogle Scholar
  37. [21]
    C.S. Wang Chang and G.E. Uhlenbeck, Eng. Res. Inst. Univ. Michigan Report CM 681 (1951).Google Scholar
  38. C.S. Wang Chang. Uhlenbeck and J. de Boer in Studies in Statistical Mechanics, ed. J. de Boer and G.E. Uhlenbeck, Vol.II, North Holland Publ. Comp., Amsterdam (1964).Google Scholar
  39. [22]
    S. Hess,Z. Naturforsch. 22a,1871 (1967).ADSGoogle Scholar
  40. [23]
    A. Tip, Physica 52,(1971).Google Scholar
  41. [24]
    R.F. Snider and B.C. Sanctuary, J. Chem. Phys. 55, 1555 (1971).CrossRefADSGoogle Scholar
  42. [25]
    L. Waldmann, Kinetic Theory of Dilute Gases with Internal Molecular Degrees of Freedom. In Fundamental Problems in Statistical Mechanics II, ed. E.G.D. Cohen, p. 276–305. Amsterdam, North Holland 1968.Google Scholar
  43. [26]
    L. Boltzmann, Vorlesungen über Gastheorie II. Teil, VII. Abschnitt; Leipzig 1898.Google Scholar
  44. [27]
    E.C.G. Stueckelberg, Helvet. Phys. Acta 25, 577 (1952) with a note due to W. Pauli.Google Scholar
  45. [28]
    L. Waldmann, Transporterscheinungen in Gasen von mittlerem Druck; In Handbuch d. Physik, ed. S. Flügge, 12, p. 484 f., Berlin-Göttingen-Heidelberg 1958.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • L. Waldmann
    • 1
  1. 1.Institut für Theoretische PhysikUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations