# Fractal Structure and Non-Extensive Statistics

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## Abstract

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## 1. Introduction

## 2. Fractals and Tsallis Statistics

- It presents a complex structure with a number ${N}^{\prime}$ of compound systems that present the same properties as the parent system.
- The internal energy, E, and the kinetic energy, F, of each compound system are such that the ratio $E/F$ follows a distribution $\tilde{P}(\epsilon )$.
- At some level of the internal struture, the fluctuations of internal level of the compund systems are small enough to be disconsidered, and then their internl energy can be regarded as constant.

## 3. Fractal Structure

## 4. Diagrammatic Representation

- A line corresponds to a term$${\int}_{-\infty}^{\infty}{d}^{3}\pi {\u03f5}^{-3/2}{e}^{-f}\phantom{\rule{0.166667em}{0ex}}$$
- A vertex corresponds to the term$${(2\pi )}^{-3/2}{\displaystyle \prod _{i=1}^{{N}^{\prime}}}\phantom{\rule{0.166667em}{0ex}}\delta \left(f-{\displaystyle \sum _{j=1}^{{N}^{\prime}}}{f}_{j}\right)\phantom{\rule{0.166667em}{0ex}}$$
- To each final line, i.e., those lines that do not finish in a vertex, the associated term reads$${\int}_{0}^{\infty}A\phantom{\rule{0.166667em}{0ex}}kT\phantom{\rule{0.166667em}{0ex}}{e}^{-\u03f5}{\left[\tilde{P}(\epsilon )\right]}^{\nu}d\u03f5\phantom{\rule{0.166667em}{0ex}}$$

## 5. Temperature Fluctuation in Thermofractals

## 6. Callan–Symanzik Equation for Thermofractals

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Useful Formulae

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**Figure 1.**Basic diagrams for the fractal structure: (

**a**) main fractal; (

**b**) vertex; (

**c**) final fractal.

**Figure 3.**The same diagram of Figure 2 represented as a linear graph. This is possible by rearranging terms in the summation of different contributions and using the merging property of thermofractals.

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Deppman, A.; Frederico, T.; Megías, E.; Menezes, D.P. Fractal Structure and Non-Extensive Statistics. *Entropy* **2018**, *20*, 633.
https://doi.org/10.3390/e20090633

**AMA Style**

Deppman A, Frederico T, Megías E, Menezes DP. Fractal Structure and Non-Extensive Statistics. *Entropy*. 2018; 20(9):633.
https://doi.org/10.3390/e20090633

**Chicago/Turabian Style**

Deppman, Airton, Tobias Frederico, Eugenio Megías, and Debora P. Menezes. 2018. "Fractal Structure and Non-Extensive Statistics" *Entropy* 20, no. 9: 633.
https://doi.org/10.3390/e20090633