On the Derivation of power-law distributions within standard statistical mechanics

TL;DR

This paper demonstrates that power-law distributions similar to Tsallis distributions can be derived from classical statistical mechanics by assuming microcanonical distributions are separable from total energy, linking the q-parameter to physical system properties.

Contribution

It introduces a derivation of Tsallis-type power-law distributions within classical mechanics based on a separability assumption, connecting the q-parameter to system size and interactions.

Findings

Power-law distributions can be derived from classical mechanics with a separability assumption.

The q-parameter in Tsallis distributions relates to a separation constant Q.

In many cases, Q tends to 1, recovering standard thermodynamics.

Abstract

We show that within classical statistical mechanics it is possible to naturally derive power law distributions which are of Tsallis type. The only assumption is that microcanonical distributions have to be separable from of the total system energy, which is reasonable for any sensible measurement. We demonstrate that all separable distributions are parametrized by a separation constant Q which is one to one related to the q-parameter in Tsallis distributions. The power-laws obtained are formally equivalent to those obtained by maximizing Tsallis entropy under q constraints. We further ask why nature fixes the separation constant Q to 1 in so many cases leading to standard thermodynamics. We answer this with an explicit example where it is possible to relate Q to sytem size and interaction parameters, characterizing the physical system. We argue that these results might be helpful to explain the ubiquity of Tsallis distributions in nature.