Poincar\'{e}'s Observation and the Origin of Tsallis Generalized Canonical Distributions

TL;DR

This paper explores the geometric origins of Tsallis distributions, showing their relation to uniform distributions on spheres and Cauchy laws, and highlights their relevance in microcanonical ensembles like the ideal gas.

Contribution

It provides a geometric interpretation of Tsallis distributions for different q values, linking them to uniform sphere distributions and Cauchy laws, and demonstrates their application in microcanonical setups.

Findings

For q > 1, Tsallis distributions are marginals of uniform sphere distributions.

For q < 1, they are conditional distributions derived from Cauchy laws.

Illustrates relevance of Tsallis distributions in microcanonical ensembles like the ideal gas.

Abstract

In this paper, we present some geometric properties of the maximum entropy (MaxEnt) Tsallis- distributions under energy constraint. In the case q > 1, these distributions are proved to be marginals of uniform distributions on the sphere; in the case q < 1, they can be constructed as conditional distribu- tions of a Cauchy law built from the same uniform distribution on the sphere using a gnomonic projection. As such, these distributions reveal the relevance of using Tsallis distributions in the microcanonical setup: an example of such application is given in the case of the ideal gas.