Skip to main content
SpringerLink
Log in

A generalized Langevin equation for dealing with nonadditive fluctuations

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

Abstract

A suitable extension of the Mori memory-function formalism to the non-Hermitian case allows a “multiplicative” process to be described by a Langevin equation of non-Markoffian nature. This generalized Langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well-known theoretical approach developed by Kubo, the stochastic Liouville equation (SLE) theory. It is shown, furthermore, that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath. The stochastic process under study can also be described by a Fokker-Planck-like equation, which results in a Gaussian equilibrium distribution for the variable of interest. The main flaw of the SLE theory, that resulting in an uncorrect equilibrium distribution, is therefore completely eliminated.

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

Access this article

Log in via an institution

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. R. Kubo, inStochastic Processes in Chemical Physics, K. E. Shuler, ed. (Wiley, New York, 1969), p. 101.

    Google Scholar 

  2. R. F. Fox,j. Math. Phys. 13:1196 (1972).

    Google Scholar 

  3. R. F. Fox,Phys. Rep. 48:179 (1978).

    Google Scholar 

  4. M. Ferrario and P. Grigolini,J. Math. Phys. 20:2567 (1979).

    Google Scholar 

  5. M. Ferrario and P. Grigolini,Chem. Phys. Lett. 62:100 (1979).

    Google Scholar 

  6. M. W. Evans, M. Ferrario, and P. Grigolini,Chem. Phys. Lett. 71:139 (1980).

    Google Scholar 

  7. M. Tokuyama and H. Mori,Prog. Theor. Phys. 55:411 (1976).

    Google Scholar 

  8. N. Hashitsume, F. Shibata, and M. Shingū,J. Stat. Phys. 17:155 (1977).

    Google Scholar 

  9. F. Shibata, Y. Takahashi, and N. Hashitsume,J. Stat. Phys. 17:171 (1977).

    Google Scholar 

  10. H. Mori,Prog. Theor. Phys. 33:423 (1965).

    Google Scholar 

  11. H. Mori,Prog. Theor. Phys. 34:399 (1965).

    Google Scholar 

  12. P. Grigolini,Nuovo Cimento,63B:174 (1981).

    Google Scholar 

  13. M. W. Evans, M. Ferrario, P. Grigolini,Molec. Phys. 39:1369 (1980).

    Google Scholar 

  14. A. S. Adelman,J. Chem. Phys. 16:124 (1976).

    Google Scholar 

  15. R. F. Fox,J. Stat. Phys. 16:259 (1977).

    Google Scholar 

  16. R. F. Fox,J. Math. Phys. 18:2331 (1977).

    Google Scholar 

  17. P. Hänggi,Z. Phys. B31:407 (1978).

    Google Scholar 

  18. M. San Miguel and J. M. Sancho,J. Stat. Phys. 22:605 (1980).

    Google Scholar 

  19. P. Hänggi, H. Thomas, H. Grabert, and P. Talkner,J. Stat. Phys. 18:155 (1978).

    Google Scholar 

  20. T. Karasudani, K. Nagano, H. Okamoto, H. Mori,Prog. Theor. Phys. 61:850 (1979).

    Google Scholar 

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

    Google Scholar 

  22. H. Mori and H. Fujisaka,Prog. Theor. Phys. 49:764 (1973).

    Google Scholar 

  23. W. R. Schneider,Z. Phys. B24:135 (1976).

    Google Scholar 

  24. P. Grigolini, M. Ferrario, and M. W. Evans,Z. Phys. B,41:165 (1981).

    Google Scholar 

  25. M. Ferrario and P. Grigolini,J. Chem. Phys. 74:235 (1981).

    Google Scholar 

  26. H. Grabert, P. Talkner, and P. Hänggi,Z. Phys. B26:389 (1977);B29:273 (1978); H. Grabert, P. Hänggi, and P. Talkner,J. Stat. Phys. 22:537 (1980).

    Google Scholar 

  27. M. Tokuyama,Physica 102A:399 (1980).

    Google Scholar 

  28. S. A. Adelman,J. Chem. Phys. 71:4471 (1979).

    Google Scholar 

  29. A. E. Stillman and J. H. Freed,J. Chem. Phys. 72:550 (1980).

    Google Scholar 

  30. R. J. Abbott and D. W. Oxtoby,J. Chem. Phys. 72:3972 (1980).

    Google Scholar 

  31. G. Ciccotti and J. P. Ryckaert,Molec. Phys. 40:141 (1980).

    Google Scholar 

  32. F. de Pasquale and P. Tombesi,Phys. Lett. 72A:7 (1979); F. de Pasquale, P. Tartaglia, and P. Tombesi,Physica 99A:581 (1979); F. de Pasquale, P. Tartaglia, and P. Tombesi,Phys. Lett. 78A:129 (1980).

    Google Scholar 

  33. U. Balucani, V. Tognetti, and R. Vallauri,Phys. Lett. A64:387 (1978);Phys. Rev. A 19:177 (1978).

    Google Scholar 

  34. R. Kubo,J. Math. Phys. 4:174 (1963).

    Google Scholar 

  35. N. G. Van Kampen,Phys. Rep. 24C:171 (1976).

    Google Scholar 

  36. S. A. Rice, inExcited States, E. C. Lim, ed., Vol. 2 (Academic Press, New York, 1975), p. 111; K. E. Freed,Topics Current Chem. 31:105 (1972); J. Jortner and S. Mukamel,Proceedings of the First International Congress on Quantum Chemistry, Menton, France, R. Daudel and B. Pullman, eds. (D. Reidel Publishing Company, Dordrecht 1974), p. 145.

    Google Scholar 

  37. K. O. Friedrichs,Commun. Pure Appl. Math. 1:361 (1948).

    Google Scholar 

  38. J. L. Peitenpol,Phys. Rev. 162:1301 (1967).

    Google Scholar 

  39. J. W. Middleton and W. C. Schieve,Physica 63:139 (1973); A. P. Grecos and I. Prigogine,Physica 59:77 (1972); W. C. Schieve,Lecture Notes in Physics 28:1 (1974).

    Google Scholar 

  40. A. Laubereau and W. Kaiser,Rev. Mod. Phys. 50:607 (1978).

    Google Scholar 

  41. J. Jortner,SPIE 113:88 (1967).

    Google Scholar 

  42. M. Abramowitz and A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1972), p. 775.

    Google Scholar 

  43. B. Spain and M. G. Smith,Function of Mathematical Physics (Van Nostrand Reinhold Company, London, 1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto di Fisica dell'Università di Pisa e GNSM del CNR, Piazza Torricelli 2, 56100, Pisa, Italy

    Paolo Grigolini

Authors
  1. Paolo Grigolini
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grigolini, P. A generalized Langevin equation for dealing with nonadditive fluctuations. J Stat Phys 27, 283–316 (1982). https://doi.org/10.1007/BF01008940

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01008940

Key words

Search

Navigation