Generation of power law: Maximum entropy framework and superstatistics

Abstract

The ubiquitous nature of power law in a variety of systems is a wellknown phenomenon. We describe various mechanisms responsible for emergence of power law behavior. Maximum entropy based framework with appropriate moment constraints provides a useful approach for generating power law. It is found that maximization of Shannon entropy with either geometric or shifted geometricmean yields power tail behavior. Tsallis entropy maximization with arithmetic mean constraint also results in long tail distributions. A new framework based on superstatistics is discussed which also has the capability to generate heavy tail distributions. Illustrative examples from communication systems, computational neuroscience, Brownian motion in state dependent random force and social systems are briefly discussed.