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 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.
Highlights
As the formulation of new mathematical tools opens opportunities to describe systems of increasing complexity, entropy emerges as an important quantity in different areas
This work is organized as follows: in Section 2, the main aspects of thermodynamical fractals are reviewed; in Section 3, the fractal structure is analyzed in detail; in Section 4, a diagrammatic scheme to facilitate calculations is introduced, and some examples are given; in Section 5, it is shown that the temperature of the fractal system addressed here fluctuates according to the Euler Gamma Function, a kind of temperature fluctuation already associated with Tsallis statistics; in Section 6, we analyse the scale invariance of thermofractals in terms of the Callan–Symanzik equation, a result that may be of importance for applications in Hadron physics; in Section 7, our conclusions are presented
We analyze in detail a thermodynamical system recently proposed that presents a fractal structure in its thermodynamical functions, which leads to a natural description of its properties in terms of Tsallis statistics [12], and we show that such a system presents a fractal structure in its momentum space
Summary
As the formulation of new mathematical tools opens opportunities to describe systems of increasing complexity, entropy emerges as an important quantity in different areas. Being a way to relate formally Tsallis and Boltzmann statistics, the analysis of these fractals may shed some light on the open questions about the meaning of the entropic parameter and on the fundamental basis of the non-extensive statistics, and, in this way, it can contribute to a better understanding of entropy. In this regard, it is worth mentioning that fractals was one of the starting points for the formulation of the generalized statistics [1]. This work is organized as follows: in Section 2, the main aspects of thermodynamical fractals are reviewed; in Section 3, the fractal structure is analyzed in detail; in Section 4, a diagrammatic scheme to facilitate calculations is introduced, and some examples are given; in Section 5, it is shown that the temperature of the fractal system addressed here fluctuates according to the Euler Gamma Function, a kind of temperature fluctuation already associated with Tsallis statistics; in Section 6, we analyse the scale invariance of thermofractals in terms of the Callan–Symanzik equation, a result that may be of importance for applications in Hadron physics; in Section 7, our conclusions are presented
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