A short account of a connection of Power Laws to the Information Entropy

TL;DR

This paper derives power law distributions from Shannon's entropy formalism, linking complex system behavior, entropy, and evolution, and explains city population distributions including Zipf's law.

Contribution

It introduces a novel derivation of power laws from entropy principles, connecting physical assumptions to observed city population distributions.

Findings

Power law distribution derived from Shannon's entropy formalism.

Zipf's law emerges as a special case within the model.

Complex systems maintain constant entropy to enhance survival.

Abstract

We use the formalism of 'Maximum Principle of Shannon's 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 Boltzmann's 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 Boltzmann's 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 Zipf's law comes out as a natural special point of the model. We also suggest further directions of theory.