Fisher-based thermodynamics for scale-invariant systems: Zipf's Law as an equilibrium state of a scale-free ideal gas

TL;DR

This paper develops a thermodynamic framework for scale-invariant systems, deriving Zipf's law as an equilibrium state of a scale-free ideal gas, supported by empirical data from various domains.

Contribution

It introduces a thermodynamic approach to scale-invariant systems, deriving Zipf's law from first principles and linking it to equilibrium states of a novel scale-free ideal gas model.

Findings

Zipf's law is derived as an equilibrium distribution of the scale-free ideal gas.

Empirical data from elections, city populations, and journal citations support the model.

Log-normal distribution predicted for constrained systems.

Abstract

We present a thermodynamic formulation for scale-invariant systems based on the principle of extreme information. We create an analogy between these systems and the well-known thermodynamics of gases and fluids, and study as a compelling case the non-interacting system -the scale-free ideal gas- presenting some empirical evidences of electoral results, city population and total cites of Physics journals that confirm its existence. The empirical class of universality known as Zipf's law is derived from first principles: we show that this special class of power law can be understood as the density distribution of an equilibrium state of the scale-free ideal gas, whereas power laws of different exponent arise from equilibrium and non-equilibrium states. We also predict the appearance of the log-normal distribution as the equilibrium density of a harmonically constrained system, and finally derive an equivalent microscopic description of these systems.