Fractal structure and non extensive statistics

TL;DR

This paper explores how fractal structures in thermodynamics can lead to non-extensive statistics, specifically Tsallis statistics, by analyzing temperature fluctuations and scale invariance in such systems.

Contribution

It introduces a diagrammatic method to analyze fractal thermodynamic systems and links fractal scale invariance to non-extensive statistical behavior.

Findings

Fractal thermodynamic systems exhibit temperature fluctuations described by the Euler Gamma Function.

The fractal structure can generate non-extensive statistics consistent with Tsallis formalism.

Fractal scale invariance relates to the Callan-Symanzik Equation in thermodynamics.

Abstract

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 has been discussed in details 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 along the last several years, in particular a fractal structure in thermodynamics 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 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 evidenced the connections between those fluctuations and Tsallis statistics. Finally, the fractal scale invariance is discussed in terms of the Callan-Symanzik Equation.