## 3 The Principles of Non-Extensive Statistical Physics as Applied in Drainage Systems  
A number of earth physics effects in different spatial and temporal scales, which includes rock and material properties, natural hazards, earthquake mechanics, plate tectonics, geomagnetic reversals and geological faults, have been interpreted in terms of the non-extensive statistical mechanics (NESP) view [36]. Here we recapitulate the main principles of NESP in which a cornerstone is the introduction of the non-extensive Tsallis entropy \(\mathrm{Sq}\)[28] in terms of the probability distribution \(\mathrm{p}(\mathrm{A})\) of a fundamental geometric parameter, \(\mathrm{A}\) that in our case could be the watershed area \(\mathrm{A}\):  
\[\mathrm{S_{q}}=\mathrm{k_{B}}\frac{1-\sum\mathrm{p^{q}(\mathrm{A})}}{\mathrm{ q}-1} \tag{1}\]  
where \(\mathrm{k_{B}}\) is Boltzmann's constant. The index \(\mathrm{q}\) is the degree of non-additivity. In the limit \(\mathrm{q}\)\(\rightarrow\)\(1\), \(\mathrm{Sq}\)\(\rightarrow\)\(\mathrm{S_{1}}\) and the approach reduced to the well-known Boltzmann-Gibbs (BG) entropy, with which the Tsallis entropy shares many common properties [28]. However, simple additivity is violated, because for a system composed of two statistically independent systems, \(\mathrm{U_{A}}\) and \(\mathrm{U_{B}}\), the Tsallis entropy satisfies:  
\[\mathrm{S_{q}(U_{A}+U_{B})}=\mathrm{S_{q}(U_{A})}+\mathrm{S_{q}(U_{B})}+\frac {1-\mathrm{q}}{\mathrm{k_{B}}}\mathrm{S_{q}(U_{A})}\mathrm{S_{q}(U_{B})}\]  
The last term on the right hand side of this equation describes the interaction between the two systems and is the origin of non-additivity. The index \(\mathrm{q}\) accounts for the memory, multifractality and long-range interaction between the elements (drainages) of the analyzed set, and for \(\mathrm{q}<1\), \(\mathrm{q}=1\) and \(\mathrm{q}>1\) respectively correspond to super-additivity, additivity and sub-additivity. This is the fundamental principle of non-extensive statistical mechanics. Figure 1: The ASTER GDEM of Greece with drainage basins polygonal data overlay. In order to estimate the expected probability distribution p(A) of the drainage area A, in terms of NESP, we maximized the non-extensive entropy under the appropriate constraints, using the Lagrange-multipliers method with the Lagrangian [28; 34; 37]:  
\[\mathrm{L_{q}}=-\int_{0}^{\infty}\mathrm{p^{q}(A)\ln_{q}p(A)dA-\lambda_{o}\;( \int_{0}^{\infty}p(A)dA-1)-\lambda_{1}\;(\int_{0}^{\infty}AP_{q}(A)dA-\langle A \rangle_{q})}\]  
The first constraint used refers to the normalization condition that reads as:  
\[\int_{0}^{\infty}p(A)dA=1\]  
Introducing the generalized expectation value (q-expectation value), \(<\)A\(>_{q}\) which is defined as:  
\[\mathrm{A_{q}}=\langle A\rangle_{q}=\int\limits_{0}^{\infty}AP_{q}(A)dA\]  
where the escort probability is given in [28] as:  
\[\mathrm{P_{q}(A)}=\frac{pq(A)}{\int_{0}^{\infty}pq\;(A)dA}\]  
the extremization of Sq with the above constraints yields to the probability distribution of p(A) as [34]:  
\[p(A)=C_{q}\left[1-\frac{1-q}{2-q}\bigg{(}\frac{A}{A_{q}}\bigg{)}\right]^{ \frac{1}{1-q}} \tag{2}\]  
where \(C_{q}\) is a normalization coefficient. We recall that the Q-exponential function introduced in NESP by [PERSON] (2009) is defined as [28]:  
\[\mathrm{exp_{Q}(X)}=\left\{ \begin{array}{ll}[1+(1-Q)X]1/(1-Q)&\text{if }(1+(1-Q)\;X\geq \;0)\\ 0&\text{if }(1+(1-Q)X\;<\;0)\end{array} \right.\]  
The normalized cumulative number of drainage with area greater than A can be obtained by integrating the probability density function p(A) as:  
\[P(>A)=\frac{N(>A)}{N_{0}}=\left[1+\bigg{(}\frac{q-1}{2-q}\bigg{)}\bigg{(} \frac{A}{A_{q}}\bigg{)}\right]^{\frac{q-2}{q-1}} \tag{3}\]  
where N(>A) is the number of drainages with area larger than A. In the latter expression, if we define \(q=2-\frac{1}{Q}\), this leads to:  
\[P(>A)=\mathrm{exp_{Q}}\bigg{(}-\bigg{(}\frac{A}{A_{q}}\bigg{)}\bigg{)}=\left[1 +(Q-1)\bigg{(}\frac{A}{A_{q}}\bigg{)}\right]^{-\frac{1}{Q-1}}, \tag{4}\]  
having a typical Q-exponential form.