Possible thermodynamic structure underlying the laws of Zipf and Benford

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.