On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

TL;DR

This paper explores the relationship between q-deformed algebras, fractional calculus, and Hausdorff derivatives in fractal media, aiming to unify diverse approaches to modeling complex systems with fractal structures.

Contribution

It establishes a connection between q-deformed algebras in statistical mechanics and Hausdorff derivatives in fractal media, offering new insights into transport dynamics in complex systems.

Findings

Link between q-deformed algebras and Hausdorff derivatives

Application to transport in fractal media

Potential unification of different fractional calculus approaches

Abstract

Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the q-calculus in Tsallis' version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems; Kaniadakis' approach, based on the compatibility between relativity and thermodynamics; Fractional Calculus (FC), that deals with the dynamics of anomalous transport and other natural phenomena, and also some local versions of FC that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations. The question we might ask is whether or not there are common aspects that connect these alternative approaches. In this short communication, we discuss a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators defined in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. This connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics, such as porous or granular media. Possible connections with other alternative definitions of FC are also contempled. Insights on complexity connected to concepts like coarse-grained space-time and physics in general are pointed out.