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

• We connected qq-deformed algebras and the Hausdorff derivatives.
• The physical basis involved is the mapping into the fractal continuum.
• Local fractional derivative is comparable to the definition of the qq-derivative.
• The qq-derivative was considered in the context of nonextensive statistics.
• An attempt to connect with Kaniadakis’ formalism has also been contemplated.

Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the qq-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 qq-deformed algebras in two different contexts of Statistical Mechanics, namely, Tsallis’ framework and 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 contemplated. Insights on complexity connected to concepts like coarse-grained space–time and physics in general are pointed out.