# A Non-Extensive Statistical Mechanics View on Easter Island Seamounts Volume Distribution

## Abstract

**:**

## 1. Introduction

## 2. Regional Tectonic Setting and Bathymetric Data Used

## 3. Principles of Non-Extensive Statistical Physics and Estimation of Seamount Frequency-Volume Distribution

_{q}[14,15], which is non-additive in the sense that it is not proportional to the number of the system’s elements, as in the Boltzmann-Gibbs entropy S

_{BG}. The Tsallis entropy S

_{q}reads as:

_{B}is Boltzmann’s constant; p

_{i}is a set of probabilities; W is the total number of microscopic configurations; and q the entropic index.

_{BG}is obtained; ${S}_{BG}=-{k}_{B}{{\displaystyle \sum}}_{i=1}^{W}{p}_{i}ln{p}_{i}$. We note that for q = 1, we obtain the well-known exponential distribution [14]. The cases q > 1 and q < 1 correspond to sub-additivity and super-additivity, respectively. Although Tsallis entropy shares a lot of common properties with the Boltzmann-Gibbs entropy, S

_{BG}is additive, whereas S

_{q}(q ≠ 1) is non-additive [15]. According to this property, S

_{BG}exhibits only short-range correlations, and the total entropy depends on the size of the systems’ elements. Alternatively, S

_{q}allows all-length scale correlations and seems more adequate for complex dynamical systems, especially when long-range correlations between the elements of the system are present.

_{q}which is defined as: ${V}_{q}={\langle V\rangle}_{q}={{\displaystyle \int}}_{0}^{\infty}V{P}_{q}\left(V\right)dV$, where the escort probability is given in [15] as: ${P}_{q}\left(V\right)=\frac{{p}^{q}\left(V\right)}{{{\displaystyle \int}}_{0}^{\infty}{p}^{q}\left(V\right)dV}$, the extremization of S

_{q}with the above constraints yields to the probability distribution of p(V) as [18,19,20,21]:

_{R}<< β

_{Q}defines three regions. The asymptotic behavior of the first region related to the very small values of the volume V and is

_{Q}and β

_{1}positive parameters that lead to a function p(V) that decreases monotonically with increasing V as presented in [24].

_{c}is the crossover point between the anomalous (Q ≠ 1) to normal (R = 1) statistical mechanics. Equation (7a) lead to the Q-exponential of Equation (2) for a cumulative distribution function $P(>V)$.

## 4. ESC Data Analysis and Discussion

_{c}≈ 1200 km

^{3}. The analysis of ESC volume-distribution (see Figure 2) leads to Q = 2.20 with an error of 0.01 which leads to q = 1.54. For V > V

_{c}an exponential function describes the data, in agreement with the asymptotic behavior presented in Equations (7a) and (7b).

_{o}we lead to a power law description of the distribution function and in such a case the cumulative distribution is $P\left(>V\right)\cong C{\left(\frac{V}{{V}_{q}}\right)}^{-\frac{2-q}{q-1}}~{V}^{-\beta}$ with an exponent $\beta =\frac{2-q}{q-1}$ and $C={(\frac{q-1}{2-q})}^{\raisebox{1ex}{$2-q$}\!\left/ \!\raisebox{-1ex}{$1-q$}\right.}$ in agreement with the power law empirically used to describe the seamount distribution [40] with β = 0.85 close to that presented in [40]. To have an estimation of V

_{ο}we select the volume where the power law approximation of P(>V) takes the value P(>V) = 1 leading to ${V}_{o}={V}_{q}{(\frac{2-q}{q-1})}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. We observe that β < 1 leading to $1.5<q$, in agreement with previous published results on earth physics processes in a broad range of scales from laboratory up to geodynamic one [16,17]. For V

_{q}= 24 km

^{3}as estimated for the fitting of Q-exponential (see Figure 2) we lead to V

_{o}= 20 km

^{3}which is similar with the cut off used in [40].

## 5. Conclusions

## Conflicts of Interest

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**Figure 1.**Map of the topography of the southeastern Pacific created using the Global Topography database. The study area is indicated by the dotted which includes Easter Island (EI) and Salas y Gómez Island (SGI). The Easter Island/Salas y Gomez Chain (ESC) is extracted in the map (modified from [10]).

**Figure 2.**The cumulative distribution function P(>V) of ESC volume data (diamond blue) as reported in [40], fitted with a Q-exponential (red line) with Q = 2.20 and a geometric factor V

_{q}≈ 24 km

^{3}. The Q = 2.20 leads to q = 1.54.

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Vallianatos, F.
A Non-Extensive Statistical Mechanics View on Easter Island Seamounts Volume Distribution. *Geosciences* **2018**, *8*, 52.
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Vallianatos F.
A Non-Extensive Statistical Mechanics View on Easter Island Seamounts Volume Distribution. *Geosciences*. 2018; 8(2):52.
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2018. "A Non-Extensive Statistical Mechanics View on Easter Island Seamounts Volume Distribution" *Geosciences* 8, no. 2: 52.
https://doi.org/10.3390/geosciences8020052