The Distributions in Nature and Entropy Principle

TL;DR

This paper explores how maximum entropy principles lead to different distributions in nature, including bell-shaped and long-tail distributions, explaining phenomena like Zipf's law and Pareto's rule.

Contribution

It demonstrates that both bell-like and long-tail distributions arise from entropy maximization, linking statistical mechanics to empirical laws in nature.

Findings

Long-tail distribution explains Zipf's law and Pareto's rule.

Bell-like distribution occurs at low particle-to-box ratios.

Entropy maximization underpins diverse natural distributions.

Abstract

The derivation of the maximum entropy distribution of particles in boxes yields two kinds of distributions: a "bell-like" distribution and a long-tail distribution. The first one is obtained when the ratio between particles and boxes is low, and the second one - when the ratio is high. The obtained long tail distribution yields correctly the empirical Zipf law, Pareto's 20:80 rule and Benford's law. Therefore, it is concluded that the long tail and the "bell-like" distributions are outcomes of the tendency of statistical systems to maximize entropy.