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Changes of Scale

  • Jan Naudts

Abstract

The Tsallis entropy function is directly related to Renyi’s alpha-entropies. It is shown that in the context of closed mechanical systems the use of Renyi’s entropy function leads to an acceptable definition of the thermodynamic temperature. Since Renyi’s entropies are well known in the physics community because of their appearance in the theory of fractal measures a digression in this direction is included. It is shown that the thermodynamic formalism of fractals fits into the standard non-deformed formalism of statistical mechanics. The Sharma-Mittal entropy functions are mentioned at the end of the chapter.

Keywords

Exponential Family Entropy Function Fractal Measure Multifractal Analysis Discrete Probability Distribution 
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References

  1. 1.
  2. 2.
    Selected Papers of Alfréd Rényi, Vol. 2. Akadémiai Kiadó, Budapest (1976) Google Scholar
  3. 3.
    Baeten, M., Naudts, J.: On the thermodynamics of classical microcanonical systems. arxiv:1009.1787 (2010)
  4. 4.
    Beck, C., Schlögl, F.: Thermodynamics of chaotic systems: an introduction. Cambridge University Press (1997) Google Scholar
  5. 5.
    Czachor, M., Naudts, J.: Thermostatistics based on Kolmogorov-Nagumo averages: Unifying framework for extensive and nonextensive generalizations. Phys. Lett. A 298, 369–374 (2002) MATHCrossRefGoogle Scholar
  6. 6.
    Frank, T., Daffertshofer, A.: Exact time-dependent solutions of the Rényi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharma and Mittal. Physica A 285, 351–366 (2000) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frank, T., Plastino, A.: Generalized thermostatistics based on the Sharma-Mittal entropy and escort mean values. Eur. Phys. J. B 30, 543–549 (2002) CrossRefGoogle Scholar
  8. 8.
    Halsey, T.C., Jensen, M., Kadanoff, L., Procaccia, I., Shraiman, B.: Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mandelbrot, B.: Information theory and psycholinguistics: a theory of word frequencies. In: P. Lazarsfield, N. Henry (eds.) Readings in mathematical social sciences, pp. 151–168. MIT Press (1966) Google Scholar
  10. 10.
    Mandelbrot, B.: The fractal geometry of nature. Freeman, New York (1982) MATHGoogle Scholar
  11. 11.
    Masi, M.: A step beyond Tsallis and Rényi entropies. Phys. Lett. A 338, 217–224 (2005) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Montemurro, M.: Beyond the Zipf-Mandelbrot law in quantitative linguistics. Physica A 300, 567–578 (2001) MATHCrossRefGoogle Scholar
  13. 13.
    Riedi, R.: Multifractal processes. In: P. Doukhan, G. Oppenheim, M. Taqqu (eds.) Theory of Long-Range Dependence. Birkhäuser (2002) Google Scholar
  14. 14.
    Sharma, B., Mittal, D.: New nonadditive measures of inaccuracy. J. Math. Sci. 10, 28 (1975) MathSciNetGoogle Scholar
  15. 15.
    Tsallis, C., Bemski, G., Mendes, R.: Is re-association in folded proteins a case of nonextensivity? Phys. Lett. A 257, 93–98 (1999) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of AntwerpAntwerpBelgium

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