一类q-变形代数与分形度量介质中Hausdorff导数的联系
On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric
摘要：
在最近的几十年里，出现了各种各样的形式主义，被用来处理复杂系统。其中，我们可以引用Tsallis版本的《非广延统计学》中的q演算，它在应用于广泛的不同系统时取得了不可否认的成功; Kaniadakis的方法，基于相对论和热力学之间的兼容性;分数阶微积分（FC），处理异常传输和其他自然现象的动力学，还有FC的一些局部版本，声称能够研究分形和多重分形空间，并通过分数阶微分方程描述这些空间中的动力学。我们可能会问的问题是，这些替代方法之间是否存在共同点。在这篇简短的文章中，我们讨论了在统计力学的两个不同背景下，即Tsallis框架和Kaniadakis场景中，q-变形代数之间可能的关系，定义在分形介质中的分数导数算子的局部形式，即所谓的Hausdorff导数，映射到具有分形测度的连续介质中。这种联系开辟了新的理论，令人满意地描述动态的分形度量，如多孔或颗粒介质中的运输。与其他替代定义的FC的可能的连接也被考虑。对复杂性的见解连接到粗粒度的时空和物理一般的概念指出。
英文摘要：
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.
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