A short account of a connection of power laws to the information entropy

We use the formalism of “maximum principle of Shannonʹs entropy” to derive the general power law distribution function, using what seems to be a reasonable physical assumption, namely, the demand of a constant mean “internal order” (Boltzmann entropy) of a complex, self-interacting, self-organized system. Since the Shannon entropy is equivalent to the Boltzmannʹs entropy under equilibrium, non-interacting conditions, we interpret this result as the complex system making use of its intra-interactions and its non-equilibrium in order to keep the equilibrium Boltzmannʹs entropy constant on the average, thus enabling it an advantage at surviving over less ordered systems, i.e., hinting towards an “Evolution of Structure”. We then demonstrate the formalism using a toy model to explain the power laws observed in Cities’ populations and show how Zipfʹs law comes out as a natural special point of the model. We also suggest further directions of theory.