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.

arXiv comment12 pages. Original Article

References in corpus (24)

• Quantum simulation of the Dirac equation
• Non Linear Kinetics underlying Generalized Statistics
• Time-space fabric underlying anomalous diffusion
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Fractional differentiability of nowhere differentiable functions and dimensions
• Fractal universe and quantum gravity
• Local fractional Fokker-Planck equation
• Fractional Hydrodynamic Equations for Fractal Media
• Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
• Geometry and field theory in multi-fractional spacetime
• Probing the quantum nature of spacetime by diffusion
• Fractional Calculus as a Macroscopic Manifestation of Randomness
• Subordination model of anomalous diffusion leading to the two-power-law relaxation responses
• Generalized space and linear momentum operators in quantum mechanics
• Functional characterization of generalized Langevin equations
• Dissipative Quantum Systems with Potential Barrier. General Theory and Parabolic Barrier
• Common aspects of q-deformed Lie algebras and fractional calculus
• q-deformed Lie algebras and fractional calculus
• Varying electric charge in multiscale spacetimes
• Fractional Quantum Field Theory, Path Integral, and Stochastic Differential Equation for Strongly Interacting Many-Particle Systems
• Fractional Dirac Bracket and Quantization for Constrained Systems
• The variant of post-Newtonian mechanics with generalized fractional derivatives
• Anomalous g-Factors for Charged Leptons in a Fractional Coarse-Grained Approach
• Non-Boltzmann Statistics as an Alternative to Holography

Cited by in corpus (6)

• Fractal derivatives, fractional derivatives and $q$-deformed calculus
• Variational Approach and Deformed Derivatives
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• Structural scale $q-$derivative and the LLG-Equation in a scenario with fractionality
• A deformed derivative model for turbulent diffusion of contaminants in the atmosphere
• Structural derivatives on time scales