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Fractal Structure and Non-Extensive Statistics

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Instituto de Física, Universidade de São Paulo, Rua do Matão Travessa R Nr.187, Cidade Universitária, CEP 05508-090 São Paulo, Brazil
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Instituto Tecnológico da Aeronáutica, 12228-900 São José dos Campos, Brazil
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Departamento de Física Teórica, Universidad del País Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain
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Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain
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Departamento de Física, CFM, Universidade Federal de Santa Catarina, CP 476, CEP 88040-900 Florianópolis, Brazil
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(9), 633; https://doi.org/10.3390/e20090633
Received: 27 June 2018 / Revised: 13 August 2018 / Accepted: 19 August 2018 / Published: 24 August 2018
(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)
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. View Full-Text
Keywords: fractal structure; non-extensive statistics; Tsallis statistics; self-similarity; scale invariance fractal structure; non-extensive statistics; Tsallis statistics; self-similarity; scale invariance
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Deppman, A.; Frederico, T.; Megías, E.; Menezes, D.P. Fractal Structure and Non-Extensive Statistics. Entropy 2018, 20, 633.

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