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

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