Escort distributions and Tsallis entropy

TL;DR

This paper provides a geometric justification for escort distributions in Tsallis entropy using hyperbolic Riemannian metrics, linking the non-extensive parameter to geometric properties and extending results to continuous systems.

Contribution

It introduces a geometric framework based on Riemannian metrics to explain escort distributions and the behavior of systems described by Tsallis entropy.

Findings

Escort distributions derived from Riemannian measure mapping.

Geometric interpretation of the non-extensive parameter.

Polynomial growth of sample space volume for continuous systems.

Abstract

We present an argument justifying the origin of the escort distributions used in calculations involving the Tsallis entropy. We rely on an induced hyperbolic Riemannian metric reflecting the generalized composition property of the Tsallis entropy. The mapping of the corresponding Riemannian measure on the space of thermodynamic variables gives the specific form of the escort distributions and provides a geometric interpretation of the non-extensive parameter. In addition, we explain the polynomial rate of increase of the sample space volume for systems described by the Tsallis entropy, thus extending the previously reached conclusions for discrete systems to the case of systems whose evolution is described by flows on Riemannian manifolds.