Generalized Thermodynamics Underlying the Laws of Zipf and Benford

Abstract

We demonstrate that the laws of Zipf and Benford, that govern scores of data generated by many and diverse kinds of human activity (as well as other data from natural phenomena), are the centerpiece 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 an especially severe fashion. Specifically, the restriction is that the accessible fraction of this space has fractal properties. We obtain 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 the Zipf’s law; but it naturally includes the tails observed in real data for small rank. Thermodynamically, our version of Benford’s law expresses a Legendre transform between two entropy (or Massieu) potentials, while Zipf’s law is merely the expression that relates the corresponding partition functions.