Skip to main content
SpringerLink
Log in

Nonconcave Entropies in Multifractals and the Thermodynamic Formalism

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

Abstract

We discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as Legendre-Fenchel transforms of functions analogous to free energy functions. We show that the Legendre-Fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave. If the spectrum has no definite concavity, then the transform yields the concave envelope of the spectrum rather than the spectrum itself. Some mathematical and physical examples are given to illustrate this result, which lies at the root of the nonequivalence of the microcanonical and canonical ensembles. On a more positive note, we also show that the impossibility of expressing nonconcave multifractal spectra through Legendre-Fenchel transforms of free energies can be circumvented with the help of a generalized free energy function, which relates to a recently introduced generalized canonical ensemble. Analogies with the calculation of rate functions in large deviation theory are finally discussed.

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. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 1990).

    MATH  Google Scholar 

  2. C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993).

    MATH  Google Scholar 

  3. B. B. Mandelbrot, Multifractals and 1/f Noise (Springer, New York 1999).

    MATH  Google Scholar 

  4. U. Frisch and G. Parisi, in Turbulence and Predictibility of Geophysical Flows and Climate Dynamics, in M. Ghil, R. Benzi, and G. Parisi (eds.), North-Holland, Amsterdam (1985).

  5. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, Phys. Rev. A 33:1141 (1986).

    Google Scholar 

  6. R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A 17:3521 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  7. R. Benzi, L. Biferale, G. Paladin, A. Vulpiani and M. Vergassola, Phys. Rev. Lett. 67:2299 (1991).

    Article  ADS  Google Scholar 

  8. R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli and S. Succi. Phys. Rev. E 48, 29 (1993).

    Article  ADS  Google Scholar 

  9. G. Stolovitzky, K. R. Sreenivasan, and A. Juneja, Phys. Rev. E 48:3212 (1993).

    ADS  Google Scholar 

  10. D. Schertzer and S. Lovejoy, Phys. Chem. Hydrodyn. J. 6:623 (1985).

    Google Scholar 

  11. D. Schertzer, S. Lovejoy, F. Schmitt, Y. Chigirinskaya and D. Marsan, Fractals 5:427 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Schertzer and S. Lovejoy, J. Geophys. Res. 92:9693 (1987).

    Article  ADS  Google Scholar 

  13. D. Schertzer and S. Lovejoy, Fractals: Physical Origin and Consequences, in L. Pietronero (eds.), p. 49, Plenum, New York (1989).

  14. D. Schertzer and S. Lovejoy, (eds.), Scaling, Fractals and Non-Linear Variability in Geophysics (Kluwer, Boston, 1991).

    Google Scholar 

  15. F. Schmitt, D. Schertzer and S. Lovejoy, Appl. Stochastic Models Data Anal. 15:29 (1999).

    Article  MATH  Google Scholar 

  16. G. Paladin and A. Vulpiani, Phys. Rep. 156:147 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  17. E. Ott, W. D. Withers and J. A. Yorke, J. Phys. Stat. 36:687 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  18. R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970).

    MATH  Google Scholar 

  19. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer-Verlag, New York, 1985).

    MATH  Google Scholar 

  20. M. Costeniuc, R. S. Ellis, H. Touchette and B. Turkington, J. Stat. Phys. 119:1283 (2005).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. K. Huang, Statistical Mechanics (Wiley, New York, 1987).

    MATH  Google Scholar 

  22. R. S. Ellis, H. Touchette and B. Turkington, Physica A 335:518 (2004).

    Article  MathSciNet  ADS  Google Scholar 

  23. A.-M. S. Tremblay, Phys. Lett. A 116:329 (1986).

    Article  ADS  Google Scholar 

  24. B. Fourcade and A.-M. S. Tremblay, Phys. Rev. A 36:2352 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  25. P. Szépfalusy, T. Tél, A. Csordés and Z. Kovas, Phys. Rev. A 36:3525 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  26. A. Csordás and P. Szépfalusy, Phys. Rev. A 39:4767 (1989).

    Article  MathSciNet  ADS  Google Scholar 

  27. M. H. Jensen, in Universalities in Condensed Matter, in R. Jullien, L. Peliti, R. Rammal, and N. Boccara (eds.), pp. 233–235, Springer, Heidelberg (1988).

  28. H. Xu, N. Ouellette and E. Bodenschatz, Phys. Rev. Lett. 96:114503 (2006).

    Article  ADS  Google Scholar 

  29. T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47:1400 (1981).

    Article  ADS  Google Scholar 

  30. T. C. Halsey, P. Meakin and I. Procaccia, Phys. Rev. Lett. 56:854 (1986).

    Google Scholar 

  31. M. H. Jensen, A. Levermann, J. Mathiesen and I. Procaccia, Phys. Rev. E 65:046109 (2002).

    Google Scholar 

  32. H. Hata, T. Horita, H. Mori, T. Morita and K. Tomita, Prog. Theoret. Phys. 81:11 (1989).

    Article  MathSciNet  ADS  Google Scholar 

  33. H. Mori, H. Hata, T. Horita and T. Kobayashi, Prog. Theoret. Phys. Suppl. 99:1 (1989).

    MathSciNet  ADS  Google Scholar 

  34. H. Tominaga, H. Hata, T. Horita, H. Mori and K. Tomita, Prog. Theoret. Phys. 84:18 (1990).

    Article  ADS  Google Scholar 

  35. M. Kastner, Ph.D. thesis, Universität Erlangen-Nürnberg (2000).

  36. M. Pleimling, H. Behringer and A. Hüller, Phys. Lett. A 328:432 (2004).

    Article  ADS  MATH  Google Scholar 

  37. H. Behringer, M. Pleimling and A. Hüller, J. Phys. A 38:973 (2005).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. M. Sano, S. Sato and Y. Sawada, Prog. Theoret. Phys. 76:945 (1986).

    Article  MathSciNet  ADS  Google Scholar 

  39. H. Hata, T. Horita, H. Mori, T. Morita and K. Tomita, Prog. Theoret. Phys. 80:809 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  40. T. Horita, H. Hata, H. Mori, T. Morita, K. Tomita, S. Kuroki and H. Okamoto, Prog. Theoret. Phys. 80:793 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  41. K. Tomita, H. Hata, T. Horita, H. Mori, and T. Morita, Prog. Theoret. Phys. 80:953 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  42. K. Tomita, H. Hata, T. Horita, H. Mori, T. Morita, H. Okamoto and H. Tominaga, Prog. Theoret. Phys. 81:1124 (1989).

    Article  MathSciNet  ADS  Google Scholar 

  43. T. Yoshida and S. Miyazaki, Prog. Theoret. Phys. Suppl. 99:64 (1989).

    MathSciNet  ADS  Google Scholar 

  44. M. Costeniuc, R. S. Ellis, H. Touchette and B. Turkington, Phys. Rev. E 73:026105 (2006).

    Article  MathSciNet  ADS  Google Scholar 

  45. R. S. Ellis, K. Haven and B. Turkington, J. Stat. Phys. 101:999 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  46. H. Touchette, R. S. Ellis and B. Turkington, Physica A 340:138 (2004).

    Article  MathSciNet  ADS  Google Scholar 

  47. I. H. Dinwoodie and S. L. Zabell, Ann. Prob. 20:1147 (1992).

    MATH  MathSciNet  Google Scholar 

  48. D. Ioffe, Stat. Prob. Lett. 18:297 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  49. R. S. Ellis, Scand. Actuarial J. 1:97 (1995).

    MATH  Google Scholar 

  50. H. Touchette, Ph.D. thesis, McGill University (2003).

  51. D. Plachky and J. Steinebach, Per. Math. Hung. 6:343 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  52. O. E. Lanford, in Statistical Mechanics and Mathematical Problems, in A. Lenard (eds.), vol. 20 of Lecture Notes in Physics, pp. 1–113, Springer, Berlin (1973).

  53. G. Zohar, Stoc. Proc. Appl. 79:229 (1999).

    Google Scholar 

  54. D. Veneziano, Fractals 10:117 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  55. Y. Oono, Prog. Theoret. Phys. Suppl. 99:165 (1989).

    MathSciNet  ADS  Google Scholar 

  56. D. Ruelle, Thermodynamic Formalism, 2nd ed. (Cambridge University Press, Cambridge, 2004).

    MATH  Google Scholar 

  57. A. Imparato and L. Peliti, Phys. Rev. E 72:046114 (2005).

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary, University of London, London, E1 4NS, UK

    Hugo Touchette & Christian Beck

Authors
  1. Hugo Touchette
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Beck
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hugo Touchette.

Additional information

PACS numbers: 05.45.Df, 64.60.Ak, 65.40.Gr

Rights and permissions

Reprints and permissions

About this article

Cite this article

Touchette, H., Beck, C. Nonconcave Entropies in Multifractals and the Thermodynamic Formalism. J Stat Phys 125, 455–471 (2006). https://doi.org/10.1007/s10955-006-9174-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-006-9174-z

Key Words

Search

Navigation