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

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.