Possible thermodynamic structure underlying the laws of Zipf and Benford

Interdisciplinary Physics

Abstract

We show that the laws of Zipf and Benford, obeyed by scores of numerical data generated by many and diverse kinds of natural phenomena and human activity are related to the focal expression of a generalized thermodynamic structure. This structure is obtained from a deformed type of statistical mechanics that arises when configurational phase space is incompletely visited in a strict way. Specifically, the restriction is that the accessible fraction of this space has fractal properties. The focal expression is an (incomplete) Legendre transform between two entropy (or Massieu) potentials that when particularized to first digits leads to a previously existing generalization of Benford’s law. The inverse functional of this expression leads to Zipf’s law; but it naturally includes the bends or tails observed in real data for small and large rank. Remarkably, we find that the entire problem is analogous to the transition to chaos via intermittency exhibited by low-dimensional nonlinear maps. Our results also explain the generic form of the degree distribution of scale-free networks.

References

  1. 1.
    G.K. Zipf, Human Behavior and the Principle of Least-Effort (Addison-Wesley, 1949) Google Scholar
  2. 2.
    F. Benford, Proceedings of the American Philosophical Society 78, 551 (1938) Google Scholar
  3. 3.
    See J.G. van der Galien (2003) in http://en.wikipedia.org/wiki/Zipfs_law
  4. 4.
    L. Pietronero, E. Tosatti, V. Tosatti, A. Vespignani, Physica A 293, 297 (2001) ADSMATHCrossRefGoogle Scholar
  5. 5.
    G. Leech, P. Rayson, A. Wilson, Word Frequencies in Written and Spoken English: based on the British National Corpus (Longman, London, 2001) Google Scholar
  6. 6.
    H.G. Schuster, Deterministic Chaos. An Introduction, 2nd revised edn. (VCH, Weinheim, 1988) Google Scholar
  7. 7.
    F. Baldovin, A. Robledo, Europhys. Lett. 60, 518 (2002) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    B. Hu, J. Rudnick, Phys. Rev. Lett. 48, 1645 (1982) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    C. Altamirano, A. Robledo, in Complex Sciences, LNICST (Springer-Verlag, 2009), Vol. 5, p. 2232 Google Scholar
  10. 10.
    H. Touchette, Phys. Rep. 478, 1 (2009) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    H.B. Callen, Thermodynamics and an Introduction to Themostatistics, 2nd edn. (John Wiley & Sons, New York, 1985) Google Scholar
  12. 12.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998) CrossRefGoogle Scholar
  14. 14.
    R. Albert, A. Barabási, Rev. Mod. Phys. 74, 47 (2002) ADSMATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Instituto de FísicaUniversidad Nacional Autónoma de MéxicoD.F.Mexico
  2. 2.Grupo Interdisciplinar de Sistemas Complejos, Departamento de MatemáticasUniversidad Carlos III de MadridMadridSpain

Personalised recommendations