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Journal of Statistical Physics

, Volume 149, Issue 4, pp 738–753 | Cite as

Critical Exponents in Zero Dimensions

  • A. Alexakis
  • F. PétrélisEmail author
Article

Abstract

In the vicinity of the onset of an instability, we investigate the effect of colored multiplicative noise on the scaling of the moments of the unstable mode amplitude. We introduce a family of zero dimensional models for which we can calculate the exact value of the critical exponents β m for all the moments. The results are obtained through asymptotic expansions that use the distance to onset as a small parameter. The examined family displays a variety of behaviors of the critical exponents that includes anomalous exponents: exponents that differ from the deterministic (mean-field) prediction, and multiscaling: non-linear dependence of the exponents on the order of the moment.

Keywords

Stochastic processes Nonlinear physics Instabilities Multiplicative noise Critical exponents 

Notes

Acknowledgements

The authors would like to thank Stephan Fauve for his support and suggestions. Numerical computations were performed using the MESOPSL parallel cluster, the new supercomputing center of Paris Science and Letters, and their computational support is acknowledged.

References

  1. 1.
    Kadanoff, L.P., et al.: Rev. Mod. Phys. 39, 395 (1967) ADSCrossRefGoogle Scholar
  2. 2.
    Wilson, K.G., Fisher, M.E.: Phys. Rev. Lett. 28, 240–243 (1972) ADSCrossRefGoogle Scholar
  3. 3.
    Pétrélis, F., Alexakis, A.: Phys. Rev. Lett. 108, 014501 (2012) ADSCrossRefGoogle Scholar
  4. 4.
    Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (1992) Google Scholar
  5. 5.
    Arnold, L.: Random Dynamical System. Springer, Berlin (1998) Google Scholar
  6. 6.
    Fujisaka, H., Yamada, T.: Prog. Theor. Phys. 74, 918 (1985) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Fujisaka, H., Ishii, H., Inoue, M., Yamada, T.: Prog. Theor. Phys. 76, 1198 (1986) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Platt, N., Spiegel, E.A., Tresser, C.: Phys. Rev. Lett. 70, 279 (1993) ADSCrossRefGoogle Scholar
  9. 9.
    Aumaître, S., Mallick, K., Pétrélis, F.: J. Stat. Phys. 123(4), 909–927 (2006) MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Aumaître, S., Pétrélis, F., Mallick, K.: Phys. Rev. Lett. 95, 064101 (2005) ADSCrossRefGoogle Scholar
  11. 11.
    Monchaux, R., et al.: Phys. Rev. Lett. 98(4), 044502 (2007). ADSCrossRefGoogle Scholar
  12. 12.
    Monchaux, R., et al.: Phys. Fluids 21, 035108 (2009) ADSCrossRefGoogle Scholar
  13. 13.
    Pétrélis, F., Mordant, N., Fauve, S.: Geophys. Astrophys. Fluid Dyn. 101(3), 289–323 (2007) MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Young, W.R.: Woods Hole Oceanographic Institute Technical report No. WHOI-2000-07 (2000) Google Scholar
  15. 15.
    Touchette, H., Van der Straeten, E., Just, W.: J. Phys. A, Math. Theor. 43, 445002 (2010) ADSCrossRefGoogle Scholar
  16. 16.
    de Gennes, P.-G.: J. Stat. Phys. 119, 962 (2005) ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Laboratoire de Physique Statistique, Ecole Normale SupérieureCNRSParisFrance

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