# Fisher-information and the thermodynamics of scale-invariant systems

## TL;DR

This paper develops a thermodynamic framework for scale-invariant systems using Fisher's information, revealing analogies with gases, and provides empirical evidence for scale-free ideal gases in diverse real-world data.

## Contribution

It introduces a thermodynamic formulation for scale-invariant systems based on Fisher information minimization, linking them to ideal gas behavior and explaining Zipf's law thermodynamically.

## Key findings

- Empirical evidence of scale-free ideal gases in electoral results, city populations, and citations.
- Thermodynamic interpretation of Zipf's law as the surface of a finite system.
- Microscopic description consistent with previous numerical simulations.

## Abstract

We present a thermodynamic formulation for scale-invariant systems based on the minimization with constraints of Fisher's information measure. In such a way a clear analogy between these systems's thermal properties and those of gases and fluids is seen to emerge in natural fashion. We focus attention on the non-interacting scenario, speaking thus of scale-free ideal gases (SFIGs) and present some empirical evidences regarding such disparate systems as electoral results, city populations and total citations in Physics journals, that seem to indicate that SFIGs do exist. We also illustrate the way in which Zipf's law can be understood in a thermodynamical context as the surface of a finite system. Finally, we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature.