Skip to main content
Log in

Fluctuating hydrodynamics and Brownian motion

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 6×6 friction tensorγ ij (t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Einstein,Investigations on the Theory of the Brownian Movement, Dover Publications, New York (1956).

    Google Scholar 

  2. N. Wax (ed.),Selected Papers on Noise and Stochastic Processes, Dover Publications, New York (1954).

    Google Scholar 

  3. L. Onsager and S. Machlup,Phys. Rev. 36:823 (1953).

    Google Scholar 

  4. L. D. Landau and E. M. Lifshitz,Fluid Mechanics, Pergamon Press, London (1959).

    Google Scholar 

  5. M. S. Green,J. Chem. Phys. 22:398 (1954).

    Google Scholar 

  6. J. L. Lebowitz and E. Rubin,Phys. Rev. 131:1381 (1963).

    Google Scholar 

  7. P. Mazur and I. Oppenheim,Physica 50:241 (1970).

    Google Scholar 

  8. H. A. Lorentz,Lessen over Theoretische Natuurkunde. Vol. V. Kinetische Problemen, E. J. Brill, Leiden (1921).

    Google Scholar 

  9. A. Widom,Phys. Rev. A3:1394 (1971).

    Google Scholar 

  10. B. J. Alder and T. E. Wainwright,Phys. Rev. A1:18 (1970).

    Google Scholar 

  11. R. Zwanzig and M. Bixon,Phys. Rev. A2:2005 (1970).

    Google Scholar 

  12. J. R. Dorfman and E. G. D. Cohen,Phys. Rev. Letters 25:1257 (1970); M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen,Phys. Rev. Letters 25:1254 (1970),Phys. Letters 34A:419 (1971); K. Kawasaki,Progr. Theor. Phys. (Kyoto) 45:1691 (1971).

    Google Scholar 

  13. R. Zwanzig,J. Res. Natl. Bur. Std. (U.S.) 68B:143 (1964).

    Google Scholar 

  14. R. F. Fox and G. E. Uhlenbeck,Phys. Fluids 13:1893 (1970).

    Google Scholar 

  15. T. S. Chow and J. J. Hermans,J. Chem. Phys. 56:3150 (1972).

    Google Scholar 

  16. R. Kubo,Rept. Progr. Phys. 29:255 (1966).

    Google Scholar 

  17. N. G. van Kampen, inStochastic Processes in Chemical Physics, K. E. Shuler, ed., Interscience, New York (1969).

    Google Scholar 

  18. L. Onsager,Phys. Rev. 37:405 (1931);38:2265 (1931); H. B. G. Casimir,Rev. Mod. Phys. 17:343 (1945).

    Google Scholar 

  19. H. Lamb,Hydrodynamics, Dover Publications, New York (1945), pp. 642–644.

    Google Scholar 

  20. M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen,Phys. Rev. A4:2055 (1971).

    Google Scholar 

  21. N. K. Ailawadi and B. J. Berne, unpublished report at the IUPAP Conference, 1971; see alsoJ. Chem. Phys. 54:3569 (1971).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hauge, E.H., Martin-Löf, A. Fluctuating hydrodynamics and Brownian motion. J Stat Phys 7, 259–281 (1973). https://doi.org/10.1007/BF01030307

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01030307

Key words

Navigation