  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.
