Moduli of curve families and (quasi-)conformality of power-law entropies

TL;DR

This paper explores the mathematical properties of curve family moduli in metric spaces and their relation to power-law entropies like Tsallis entropy, focusing on conformality and invariance under transformations.

Contribution

It introduces a novel connection between the moduli of curve families and non-additive entropies, analyzing their behavior under quasi-conformal maps in diverse spaces.

Findings

Moduli calculations for cylinders and annuli in Euclidean space.

Insights into the invariance of Tsallis entropy under Möbius transformations.

Analysis of quasi-conformal maps' impact on power-law entropies.

Abstract

We present aspects of the moduli of curve families on a metric measure space which may prove useful in calculating, or in providing bounds to, non-additive entropies having a power-law functional form. We use as paradigmatic cases the calculations of the moduli of curve families for a cylinder and for an annulus in $\mathbb{R}^n$. The underlying motivation for these studies is that the definitions and some properties of the modulus of a curve family resembles those of the Tsallis entropy, when the latter is seen from a micro-canonical viewpoint. We comment on the origin of the conjectured invariance of the Tsallis entropy under M\"obius transformations of the non-extensive (entropic) parameter. Needing techniques applicable to both locallly Euclidean and fractal classes of spaces, we examine the behavior of the Tsallis functional, via the modulus, under quasi-conformal maps. We comment on properties of such maps and their possible significance for the dynamical foundations of power-law entropies.