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

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.

Axiomatic Local Metric Derivatives for Low-Level Fractionality with Mittag-Leffler Eigenfunctions

“…That is, the Leibniz rule or the chain rule could apply to non-local fractional operators, what actually happens not to be valid, unless we take the low-limit fracionality [14,13,21]. However, in a recent article [1], we have shown that the Jumarie's formalism seems to be actually related to a local deformed derivative operator and it is not really a fractional calculus formalism. Thus, we can consider it as an approximation, valid for low-fracionality limit, as shown in this contribution by means of the axiomatic form.…”

“…[1,2]. The local differential equation The differential equation, with the local Hausdorff derivative proposed in Ref.…”

“…In recent works, we have developed connections and a variational formalism for deformed or metric derivatives, considering the relevant space-time/ phase space as fractal or multifractal [1,2].…”

“…With this perspective, we argued that deformed or we should say, metric derivatives, similarly to the Fractional Calculus (FC), could allows us to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms in the dynamical governing equations. Also, we emphasized that the paradigm we adopt were different from the standard approach in the generalized statistical mechanics context [3,4,5], where the modification of entropy definition leads to the modification of algebra and consequently the derivative concept [1,2]. This was structured by mapping to a continuous fractal space [6,7,8] which leads naturally to the necessity of modifications in the derivatives, that we called deformed or better than, metric derivatives [1,2].…”

“…Also, we emphasized that the paradigm we adopt were different from the standard approach in the generalized statistical mechanics context [3,4,5], where the modification of entropy definition leads to the modification of algebra and consequently the derivative concept [1,2]. This was structured by mapping to a continuous fractal space [6,7,8] which leads naturally to the necessity of modifications in the derivatives, that we called deformed or better than, metric derivatives [1,2]. The modifications of derivatives, accordingly with the metric, bring to a change in the algebra involved, which in turn may conduct to a generalized statistical mechanics with some adequate definition of entropy.…”