Journal of Statistical Physics

, Volume 152, Issue 2, pp 336–352 | Cite as

Anomalous System Size Dependence of Large Deviation Functions for Local Empirical Measure

Article

Abstract

We study the large deviation function for the empirical measure (the time-averaged density) of diffusing particles at one fixed position. We find that the large deviation function exhibits anomalous system size dependence in systems with translational symmetry if and only if they satisfy the following conditions: (i) there exists no macroscopic flow, and (ii) their space dimension is one or two. We investigate this anomaly by using a contraction principle. We also analyze the relation between this anomaly and the so-called long-time tail behavior on the basis of phenomenological arguments.

Keywords

Empirical measure Large deviation function Contraction 

References

  1. 1.
    Evans, D.J., Searles, D.J.: Phys. Rev. E 50, 1645 (1994) ADSCrossRefGoogle Scholar
  2. 2.
    Galavotti, G., Cohen, E.G.D.: J. Stat. Phys. 80, 931 (1995) ADSCrossRefGoogle Scholar
  3. 3.
    Bodineau, T., Derrida, B.: Phys. Rev. Lett. 92, 180601 (2004) ADSCrossRefGoogle Scholar
  4. 4.
    Derrida, B.: J. Stat. Mech. P07023 (2007) Google Scholar
  5. 5.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: J. Stat. Phys. 107, 635 (2002) ADSMATHCrossRefGoogle Scholar
  6. 6.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Phys. Rev. Lett. 94, 030601 (2005) ADSCrossRefGoogle Scholar
  7. 7.
    Maes, C., Netocny, K., Wynants, B.: Physica A 387, 2675 (2008) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Maes, C., Netocny, K.: arXiv:1206.3423
  9. 9.
    Garrahan, J.P., Jack, R.L., Lecomte, V., Pitard, E., van Duijvendijk, K., van Wijland, F.: Phys. Rev. Lett. 98, 195702 (2007) ADSCrossRefGoogle Scholar
  10. 10.
    Garrahan, J.P., Lesanovsky, I.: Phys. Rev. Lett. 104, 160601 (2010) ADSCrossRefGoogle Scholar
  11. 11.
    Mitsudo, T., Kato, N.: arXiv:1209.0879
  12. 12.
    Nemoto, T., Sasa, S.-I.: Phys. Rev. E 83, 030105 (2011) ADSCrossRefGoogle Scholar
  13. 13.
    Nemoto, T., Sasa, S.-I.: Phys. Rev. E 84, 061113 (2011) ADSCrossRefGoogle Scholar
  14. 14.
    Lacoste, D., Lau, A.W.C., Mallick, K.: Phys. Rev. E 78, 011915 (2008) ADSCrossRefGoogle Scholar
  15. 15.
    Li, J., Liu, Y., Ping, J., Li, S.-S., Li, X.-Q., Yan, Y.-J.: Phys. Rev. B 84, 115319 (2011) ADSCrossRefGoogle Scholar
  16. 16.
    Budini, A.A.: Phys. Rev. E 84, 011141 (2011) ADSCrossRefGoogle Scholar
  17. 17.
    Cox, J.T., Griffeath, D.: Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 543 (1984) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Benois, O.: Ann. Appl. Probab. 6, 269 (1996) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Landim, C.: Ann. Probab. 20, 206 (1992) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Chang, C.-C., Landim, C., Lee, T.-Y.: Ann. Probab. 32, 661 (2004) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Alder, B.J., Wainwright, T.E.: Phys. Rev. A 1, 18 (1970) ADSCrossRefGoogle Scholar
  22. 22.
    Pomeau, Y., Resibois, P.: Phys. Rep. 19, 63 (1975) ADSCrossRefGoogle Scholar
  23. 23.
    Chen, S.H., Huang, J.S.: Phys. Rev. Lett. 55, 1888 (1985) ADSCrossRefGoogle Scholar
  24. 24.
    Ramaswamy, S., Simha, R.A., Toner, J.: Europhys. Lett. 62, 196 (2003) ADSCrossRefGoogle Scholar
  25. 25.
    Otsuki, M., Hayakawa, H.: J. Stat. Mech. L 08003 (2009) Google Scholar
  26. 26.
    Wainwright, T.E., Alder, B.J., Gass, D.M.: Phys. Rev. A 4, 233 (1971) ADSCrossRefGoogle Scholar
  27. 27.
    Brillouin, L.: Wave Propagation in Periodic Structures. Dover, New York (1953) MATHGoogle Scholar
  28. 28.
    Gabrielli, D., Jona-Lasinio, G., Landim, C.: Phys. Rev. Lett. 77, 1202 (1996) MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Kanazawa, K., Sagawa, T., Hayakawa, H.: Phys. Rev. Lett. 108, 210601 (2012) ADSCrossRefGoogle Scholar
  30. 30.
    Dembo, A., Zeitouni, O.: Large Deviation Techniques and Applications, 2nd edn. Springer, New York (1998) CrossRefGoogle Scholar
  31. 31.
    Kipnis, C., Olla, S.: Stoch. Stoch. Rep. 33, 17 (1990) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Kipnis, C., Olla, S., Varadhan, S.R.S.: Commun. Pure Appl. Math. 42, 115 (1989) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Spohn, H.: J. Phys. A 16, 4275 (1983) MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Hatano, T., Sasa, S.-I.: Phys. Rev. Lett. 86, 3463 (2001) ADSCrossRefGoogle Scholar
  35. 35.
    Spinney, R.E., Ford, I.J.: Phys. Rev. Lett. 108, 170603 (2012) ADSCrossRefGoogle Scholar
  36. 36.
    Komatsu, T.S., Nakagawa, N., Sasa, S.-I., Tasaki, H.: Phys. Rev. Lett. 100, 230602 (2008) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Basic ScienceThe University of TokyoTokyoJapan

Personalised recommendations