Zeitschrift für Physik B Condensed Matter

, Volume 76, Issue 3, pp 403–411 | Cite as

Correlation functions near instabilities in systems driven by parametric noise

  • J. Casademunt
  • A. Hernández-Machado


The steady-state correlation functions of non-linear stochastic processes driven by parametric noise are studied. A systematic method proposed by Nadler and Schulten is applied beyond the lowest order for the first time in this context. It is reformulated in a way which admits generalization to noises other than Gaussian and white. The explicit results obtained close to the instability point for the Verhulst model with Gaussian white noise improve considerably those previously obtained by continued-fraction techniques.


Spectroscopy Neural Network State Physics Correlation Function Complex System 
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.
    Moss, F., McClintock, P.V.E. (eds.): Noise in nonlinear dynamical systems. Cambridge: Cambridge University Press 1989Google Scholar
  2. 2.
    Bermejo, F.J., Pesquera, L. (eds.). Dynamics of non-linear optical systems. Singapore: World Scientific 1989Google Scholar
  3. 3.
    Kaminishi, K., Roy, R., Short, R., Mandel, L.: Phys. Rev. A24, 370 (1981)Google Scholar
  4. 4.
    Short, R., Mandel, L., Roy, R.: Phys. Rev. Lett.49, 647 (1982)Google Scholar
  5. 5.
    Lett, P.D., Gage, E.C.: Phys. Rev. A39, 1193 (1989)Google Scholar
  6. 6.
    Graham, R., Hönerbach, M., Schenzle, A.: Phys. Rev. Lett.48, 1396 (1982)Google Scholar
  7. 7.
    Dixit, S.W., Sahni, P.S.: Phys. Rev. Lett.50, 1273 (1983)Google Scholar
  8. 8.
    Hernández-Machado, A., San Miguel, M., Katz, S.: Phys. Rev. A31, 2362 (1985)Google Scholar
  9. 9.
    Leiber, Th., Jung, P., Risken, H.: Z. Phys. B-Condensed Matter68, 123 (1987)Google Scholar
  10. 10.
    Aguado, M., Hernández-García, E., San Miguel, M.: Phys. Rev. A38, 5670 (1989)Google Scholar
  11. 11.
    Grossmann, S.: Phys. Rev. A17, 1123 (1978)Google Scholar
  12. 12.
    Fujisaka, H., Grossmann, S.: Z. Phys. B-Condensed Matter43, 69 (1981)Google Scholar
  13. 13.
    Hernández-Machado, A., San Miguel, M., Sancho, J.M.: Phys. Rev. A29, 3388 (1984)Google Scholar
  14. 14.
    Faetti, S., Festa, C., Fronzoni, L., Grigolini, P., Marchesoni, F., Palleschi, V.: Phys. Lett.99A, 25 (1984)Google Scholar
  15. 15.
    Mori, H.: Prog. Theor. Phys.33, 423 (1965);34, 399 (1965)Google Scholar
  16. 16.
    Zwanzig, R.: In: Lectures in theoretical physics. Brittin, W., Dunham, L. (eds.), Vol. 3, p. 135. New York: Wiley 1961Google Scholar
  17. 17.
    Hernández-Machado, A., Rodriguez, M.A., San Miguel, M.: Ann. Nucl. Energy12, 471 (1985)Google Scholar
  18. 18.
    Jung, P., Risken, H.: Z. Phys. B-Condensed Matter59, 469 (1985)Google Scholar
  19. 19.
    Nadler, W., Schulten, K.: J. Chem. Phys.82, 151 (1985)Google Scholar
  20. 20.
    Nadler, W., Schulten, K.: Z. Phys. B-Condensed Matter,59, 53 (1985)Google Scholar
  21. 21.
    Nadler, W., Schulten, K.: Z. Phys. B-Condensed Matter72, 535 (1988)Google Scholar
  22. 22.
    Nadler, W.: Z. Phys. B-Condensed Matter73, 271 (1988)Google Scholar
  23. 23.
    Nadler, W., Schulten, K.: Phys. Rev. Lett.51, 1715 (1983)Google Scholar
  24. 24.
    Horsthemke, W., Lefever, R.: Noise-induced transitions. In: Springer Series in Synergetics. Vol. 15. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  25. 25.
    Kai, S., Kai, T., Takata, M.: J. Phys. Soc. Jpn.47, 1379 (1979)Google Scholar
  26. 26.
    Kawakubo, T., Yanagita, A., Kabashima, S.: J. Phys. Soc. Jpn.50, 1451 (1981)Google Scholar
  27. 27.
    Dutré, W.L., Debosscher, A.F.: Nucl. Sci. Eng.62, 355 (1977)Google Scholar
  28. 28.
    Graham, R., Schenzle, A.: Phys. Rev. A25, 1731 (1982)Google Scholar
  29. 29.
    Brenig, L., Banai, N.: Physica5D, 208 (1982)Google Scholar
  30. 30.
    Suzuki, M., Kaneko, K., Sasagawa, F.: Prog. Theor. Phys.65, 828 (1981)Google Scholar
  31. 31.
    Casademunt, J., Sancho, J.M.: Phys. Lett. A123 271 (1987); Casademunt, J., Sancho, J.M.: J. Stat. Phys. (1989) (in press)Google Scholar
  32. 32.
    Casademunt, J., Mannella, R., McClintock, P.V.E., Moss, F., Sancho, J.M.: Phys. Rev. A35, 5183 (1987)Google Scholar
  33. 33.
    The relations (2.4) and (2.6) can be easily obtained by expandingC(t) or exp(−ωt) of (2.2) respectively in powers oft Google Scholar
  34. 34.
    The truncated continued fraction expansion is nothing but a rational function of ω which coincides with the exact Laplace transformC(ω) in a given number of the first coefficients of the 1/ω-expansionGoogle Scholar
  35. 35.
    In (2.11) it is implicitly assumed that the processx(t) is Markovian. For non-Markovian processes the method is still valid when reformulated in its equivalent multivariable Markovian form. For more details see[30]Google Scholar
  36. 36.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series and products. New York: Academic Press 1980Google Scholar
  37. 37.
    It has been argued that for additive noise models, a divergence of a relaxation time can only occur in the deterministic limit (D→0) as the so-called asymptotic critical slowing down of [30]Google Scholar
  38. 38.
    In general one could characterize long-time tails decaying with different exponents according to which relaxation moment diverges firstGoogle Scholar
  39. 39.
    Casademunt, J., Jiménez-Aquino, J.I., Sancho, J.M.: Physica A 156, 628 (1989)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. Casademunt
    • 1
  • A. Hernández-Machado
    • 1
  1. 1.Department d'Estructura i Constituents de la MatèriaUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations