Generalized thermodynamics underlying the laws of Zipf and Benford

  • Altamirano C
  • Robledo A
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Abstract

We show that the laws of Zipf and Benford, obeyed by scores of numerical data generated by many and diverse kinds of natural phe- nomena and human activity, are the focal expressions 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. We consider a generalized version of Benford’s law for data expressed in full and not by the first digit. The inverse functional of this expression is identified with Zipf’s law; but it naturally includes the bends or tails observed in real data for small and large rank. Thermodynamically, our version of Benford’s law expresses an (incomplete) Legendre transform between two entropy (or Massieu) potentials, while Zipf’s law is merely the ex- pression that relates the corresponding partition functions. Remarkably, we find that the entire problem is analogous to the transition to chaos via intermittency exhibited by low-dimensional nonlinear maps. Our re- sults also explain the generic form of the degree distribution of scale-free networks.

Author-supplied keywords

  • Benford's law
  • Fractal phase space
  • Generalized thermodynamics
  • Zipf's law

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Authors

  • Carlo Altamirano

  • Alberto Robledo

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