Journal of Statistical Physics

, Volume 22, Issue 1, pp 1–26 | Cite as

The stochastic Boltzmann equation and hydrodynamic fluctuations

  • Hiroshi Ueyama
Articles

Abstract

Based on the assumption of a kinetic equation in Г space, a stochastic differential equation of the one-particle distribution is derived without the use of the linear approximation. It is just the Boltzmann equation with a Langevin-fluctuating force term. The result is the general form of the linearized Boltzmann equation with fluctuations found by Bixon and Zwanzig and by Fox and Uhlenbeck. It reduces to the general Landau-Lifshitz equations of fluid dynamics in the presence of fluctuations in a similar hydrodynamic approximation to that used by Chapman and Enskog with respect to the Boltzmann equation.

Key words

Stochastic differential equation Langevin equation Boltzmann equation hydrodynamic fluctuations master equation kinetic equation hard-sphere system 

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References

  1. 1.
    L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, 1959), Chapter 17.Google Scholar
  2. 2.
    L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon Press, 1958), Chapter 12.Google Scholar
  3. 3.
    L. Onsager and S. Machlup,Phys. Rev. 91:1505 (1953).Google Scholar
  4. 4.
    R. F. Fox and G. E. Uhlenbeck,Phys. Fluids 13:1893, 2881 (1970).Google Scholar
  5. 5.
    M. Bixon and R. Zwanzig,Phys. Rev. 187:267 (1969).Google Scholar
  6. 6.
    M. Kac, inThe Boltzmann Equation, E. G. D. Cohen and W. Thirring, eds. (Acta Physica Austriaca, Suppl. X; Springer Verlag, 1973).Google Scholar
  7. 7.
    G. E. Uhlenbeck, inThe Boltzmann Equation, E. G. D. Cohan and W. Thirring, eds. (Acta Physica Austriaca, Suppl. X; Springer Verlag, 1973).Google Scholar
  8. 8.
    R. F. Fox,J. Math. Phys. 19:1993 (1978).Google Scholar
  9. 9.
    N. G. van Kampen,Phys. Lett. 50A:237 (1974).Google Scholar
  10. 10.
    N. G. van Kampen, inFluctuation Phenomena in Solids, R. E. Burgess, ed. (Academic Press, 1965).Google Scholar
  11. 11.
    J. Logan and M. Kac,Phys. Rev. A 13:458 (1976).Google Scholar
  12. 12.
    R. Graham,Phys. Rev. A 10:1762 (1964).Google Scholar
  13. 13.
    J. E. Keizer,Phys. Fluids 21:198 (1978).Google Scholar
  14. 14.
    F. L. Hinton,Phys. Fluids 13:857 (1970).Google Scholar
  15. 15.
    S. Watanabe,Stochastic Differential Equations (Sangyotosho Press, 1978) (in Japanese).Google Scholar
  16. 16.
    L. Arnold,Stochastic Differential Equations (Wiley, 1974).Google Scholar
  17. 17.
    G. E. Uhlenbeck, Appendix to M. Kac,Probability and Related Topics in Physical Sciences (Interscience, 1961).Google Scholar
  18. 18.
    M. Kac,Probability and Related Topics in Physical Sciences (Interscience, 1959).Google Scholar
  19. 19.
    R. Brout,Physica 22:509 (1956).Google Scholar
  20. 20.
    A. Onuki,J. Stat. Phys. 19:325 (1978).Google Scholar
  21. 21.
    A. V. Skorokhod,Studies in the Theory of Random Processes (Addison-Wesley, 1965), p. 34.Google Scholar
  22. 22.
    H. Ueyama,Physica 80A:98 (1975).Google Scholar
  23. 23.
    A. Onuki,J. Stat. Phys. 18:475 (1978).Google Scholar
  24. 24.
    R. Zwanzig, inLectures in Theoretical Physics, W. E. Brittinet al., eds. (Interscience, 1961), Vol. 3.Google Scholar
  25. 25.
    S. Chapman and T. G. Cowling,The Mathematical Theory of Nonuniform Gases (Cambridge Press, 1970).Google Scholar
  26. 26.
    L. Waldmann, inHandbuch der Physik, Vol. 12 (Springer, Berlin, 1958).Google Scholar
  27. 27.
    J. J. Hopfield and A. J. F. Bastin,Phys. Rev. 168:193 (1968).Google Scholar
  28. 28.
    M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen,Physica 45:127 (1969).Google Scholar
  29. 29.
    P. Langevin,Compt. Rend. 1908:530.Google Scholar
  30. 30.
    K. Itô,Mem. Am. Math. Soc. 4:1 (1951).Google Scholar
  31. 31.
    J. E. Moyal,J. Roy. Stat. Soc. B 11:150 (1949).Google Scholar
  32. 32.
    H. van Beijeren and M. H. Ernst,Physica 68:437 (1973).Google Scholar
  33. 33.
    H. H. U. Konijnendijk and J. M. J. van Leeuwen,Physica 64:342 (1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • Hiroshi Ueyama
    • 1
  1. 1.Max-Planck Institut für FestkörperforschungStuttgartWest Germany

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