Variational principle for the Pareto power law

TL;DR

This paper introduces a variational principle-based mechanism explaining the emergence of power law distributions in complex systems, demonstrated through mechanical models, network diffusion, and wealth exchange scenarios.

Contribution

It presents a novel variational approach within canonical statistical mechanics to explain power law tails in diverse complex systems.

Findings

Power law tails can arise from superpositions of equilibrium energy densities.

The mechanism is demonstrated through exactly solvable mechanical models.

Applications include complex network diffusion and wealth exchange models.

Abstract

A mechanism is proposed for the appearance of power law distributions in various complex systems. It is shown that in a conservative mechanical system composed of subsystems with different numbers of degrees of freedom a robust power-law tail can appear in the equilibrium distribution of energy as a result of certain superpositions of the canonical equilibrium energy densities of the subsystems. The derivation only uses a variational principle based on the Boltzmann entropy, without assumptions outside the framework of canonical equilibrium statistical mechanics. Two examples are discussed, free diffusion on a complex network and a kinetic model of wealth exchange. The mechanism is illustrated in the general case through an exactly solvable mechanical model of a dimensionally heterogeneous system.