Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus.

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo's derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.

Fractal Structure and Non-Extensive Statistics.

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan-Symanzik equation.

Gravitational anomaly and transport phenomena.

Quantum anomalies give rise to new transport phenomena. In particular, a magnetic field can induce an anomalous current via the chiral magnetic effect and a vortex in the relativistic fluid can also induce a current via the chiral vortical effect. The related transport coefficients can be calculated via Kubo formulas. We evaluate the Kubo formula for the anomalous vortical conductivity at weak coupling and show that it receives contributions proportional to the gravitational anomaly coefficient. The gravitational anomaly gives rise to an anomalous vortical effect even for an uncharged fluid.