# Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization

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## Abstract

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## 1. Introduction

- Earthquakes spatial distributions [24],
- Earthquake slip patterns [25],
- Structural geology [27],
- Galaxies clustering [28],
- Self-organized criticality (SOC) systems [29],
- High energy collisions data [32],
- Fractal electrodynamics [33],
- Fractal structures of spacetime and mass [34],
- Snowflakes dendrites distribution [35],
- Neuropsychiatric disorders [38],
- Ecology [39],
- Economics [40],
- Urbanism [41],
- Laws [42],

## 2. Results and Discussion

#### 2.1. Entropy of Fractals Cracks Distribution

#### Entropy Change in Terms of Spatial Parameters

#### 2.2. Entropy Change in Terms of External Stress Change

#### Seismic Moment and Entropy

#### 2.3. Entropy and Fractal Geometry Generalization for Linear Nonequilibrium Thermodynamics

#### Multifractal Entropy for Linear Nonequilibrium Thermodynamics

#### 2.4. Discussion

## 3. Conclusions

- As Equation (8) is always positive, it is implied that the generation of cracks are the manifestation of irreversible process.
- The pre-failure and failure process can be linked by means of the entropy changes.
- The seismic moment and magnitude exist if external stress, that increases of the entropy of the lithosphere, and increases in the number of cracks and electromagnetic signals also exist.
- It is possible to estimate an expected seismic magnitude in terms of the entropy change/stress change.
- Entropy rapidly increases before earthquakes.
- No entropy increase, no earthquake.
- The seismo-electromagnetic theory explains the non-seismic pre-earthquakes signals and gives physical foundations to the generation of earthquakes.

- The tendency in which nature creates fractals corresponds to a geometrical manifestation of that tendency in which the universe increases the entropy.
- Fractals rising in several fields and topics reveals the increase of ‘disorder’ of those systems.
- The phenomenological coefficients can describe geometrical properties of forces and fluxes.
- The Constructal law is one geometrical application of Onsager’s relations.
- The entropy density is defined as $d\widehat{s}$, which represents the quadratic time derivative of those forces (${\dot{F}}^{2}$) that generate the fractal geometry ${V}_{fr}$. No changing force $F$ implies no fractality.
- More work must be done in order to link metric tensor, fractal entropy and multiscale thermodynamics.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Different values of entropy change in terms of spatial parameters. Specifically, in terms of maximum fractal length l_max and fractal dimension D. Note that the entropy is larger when l

_{max}is larger and D is smaller. (

**b**) Representation of randomly located fractal volumes which are characterized by a large D value. (

**c**) The same volume distribution but considering lower values of D. It is possible to observe that the volume size distribution is different when a large of smaller values of D are considered. When D is small, the domain is filled by a large number of small volume and few large volumes. The opposite is found for large D values.

**Figure 2.**(

**a**) Relation between entropy change $dS$ and the maximum fractal crack of length ${l}_{max}$. The black curve shows how the entropy increases in terms of the volume growth and $d\sigma /dt$ is constant. The red curve shows the incorporation of the sigmoid function in $d\sigma /dt$. The purple point shows the $dS$ and ${l}_{max}$ where the earthquake occurs for the sigmoid function. (

**b**) Stress evolution $\sigma $ (black curve) and shear stress change $d\sigma /dt$ (red curve) of the lithosphere prior to and after the main failure. Here, the earthquake time is $t={t}_{C}=5$ a. u. Note that a. u. means arbitrary units. (

**c**) Entropy increases by using Equation (10). It is shown that there are two main behaviors: the initial slow increase that lasts up to $t\sim 3.5$ a. u. (this trend is represented as a black dotted line) and the fast increases between 3.5 a. u. and ${t}_{C}=5$ a.u. (

**d**) Magnitude expected in terms of the stress change.

**Figure 3.**Schematic representation of the of shear stress change $d\sigma /dt$, entropy change $dS/dt$ given by Equation (10) and the growth of microcracks prior the main failure. Initially (

**a**) the almost zero $d\sigma /dt$ generates no considerable stress change nor microcrack growth (blue circles). (

**b**) The $d\sigma /dt$ increase (red curve) is determined by a small linear increase of $dS/dt$ (blue circles). (

**c**) The fast entropy increases y related to the fast increase of the uniaxial stress. Finally, (

**d**) shows that the maximum entropy change is found right before the impending main failure (marked as green area).

**Figure 4.**Schematic representation of the seismo-electromagnetic domain. It is possible to observe that the cascade of physical phenomena start with stress changes that increase the entropy of the lithosphere. Then, the green arrows represent the seismo-electromagnetic branch that explain the observed seismic and non-seismic pre-earthquake measurements. The black arrows represent the classical seismic domain. Note that the physics that explain earthquake occurrences come from the seismo-electromagnetic domain. That is, the change of b-value, the main earthquake, and secondary effects and fault’s frictional changes are due the entropy increases. Those green-black arrows represent relation that can be stablished by classical seismology and seismo-electromagnetic phenomena.

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Venegas-Aravena, P.; Cordaro, E.G.; Laroze, D.
Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization. *Entropy* **2022**, *24*, 1337.
https://doi.org/10.3390/e24101337

**AMA Style**

Venegas-Aravena P, Cordaro EG, Laroze D.
Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization. *Entropy*. 2022; 24(10):1337.
https://doi.org/10.3390/e24101337

**Chicago/Turabian Style**

Venegas-Aravena, Patricio, Enrique G. Cordaro, and David Laroze.
2022. "Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization" *Entropy* 24, no. 10: 1337.
https://doi.org/10.3390/e24101337