## 5 Discussion
According to NESP the generalized probability distribution  
\[\mathrm{p}=\mathrm{p}(\mathrm{A})\propto\exp_{\mathrm{q}}\left(-\left(\frac{ \mathrm{A}}{\mathrm{A}_{\mathrm{q}}}\right)\right)\]  
can be obtained by solving the nonlinear differential equation  
\[\frac{\mathrm{d}\mathrm{p}}{\mathrm{dA}}=-\beta_{\mathrm{q}}\mathrm{p}^{ \mathrm{q}},\text{ where }\beta_{\mathrm{q}}=\frac{1}{\mathrm{A}_{\mathrm{q}}} \text{ and }\mathrm{q}\
eq 1;\]  
while Boltzmann-Gibbs (BG) are approached, with the BG entropy optimized when q = 1. We now use the above differential equation path in order to further generalize the anomalous equilibrium distribution, in such a way as to have a crossover from anomalous (q \(\
eq\) 1) to normal (q = 1) statistical mechanics, while increasing the drainage's area. Following [63] we consider the differential equation  
\[\frac{\mathrm{d}\mathrm{p}}{\mathrm{dA}}=-\beta_{1}\mathrm{p}-\left(\beta_{ \mathrm{q}}-\beta_{1}\right)\mathrm{p}^{\mathrm{q}},\]  
whose solution is  
\[\mathrm{p}(\mathrm{A})=C\ [\ 1-\frac{\beta_{\mathrm{q}}}{\beta_{1}}+\ \frac{\beta_{ \mathrm{q}}}{\beta_{1}}\mathrm{e}^{(\mathrm{q}-1)\beta_{1}\mathrm{A}}]^{- \frac{1}{\mathrm{q}-1}}\, \tag{7}\]  
where C is a normalization factor. For positive \(\beta_{\mathrm{q}}\) and \(\beta_{1}\), \(\mathrm{p}(\mathrm{A})\) decreases monotonically with increasing A. It can be easily verified that in the case where \(\beta_{\mathrm{q}}>>\beta_{1}\), equation (7) defines three regions, according to the area A of the drainage. We will call these \"drainage regions\", small, intermediate and large drainage area regions, respectively. The asymptotic behavior of the probability distributions in these areas is  
\[\mathrm{p}(\mathrm{A})\propto 1-\beta_{\mathrm{q}}\mathrm{A}\mathrm{for}\ 0\leq\mathrm{A}\leq\ A_{\mathrm{c1}}\text{ where }\mathrm{A}_{\mathrm{c1}}=\ \frac{1}{\mathrm{q}-1}\frac{1}{\beta_{ \mathrm{q}}}\,\]  
\[\mathrm{p}(\mathrm{A})\propto\left[(\mathrm{q}-1)\beta_{\mathrm{q}}\right]^{ \frac{-1}{(\mathrm{q}-1)}}\mathrm{A}^{-\frac{1}{(\mathrm{q}-1)}}\mathrm{ for}\ \mathrm{A}_{\mathrm{c1}}\leq\mathrm{A}\leq\ A_{\mathrm{c2}}\text{ where }\mathrm{A}_{\mathrm{c2}}=\ \frac{1}{\mathrm{q}-1}\frac{1}{\beta_{1}}\,\]  
\[\text{q}=2-\frac{1}{\text{Q}}.\]0  
where \(\mathrm{A}_{\mathrm{c1}}\) and \(\mathrm{A}_{\mathrm{c2}}\) are the cross-over points between the three regions. The observed crossover from hierarchical (q \(\
eq\) 1) to exponential (q = 1) pattern, with increasing of the parameter A deserves special attention. In this case, in equation (7) expanding the exponential term in it can be easily concluded that in the case where \((\mathrm{q}-1)\beta_{1}\mathrm{A}\ll 1\) the asymptotic behavior of the probability distributions simplified as a q-exponential  
\[\text{q}=2-\frac{1}{\text{Q}}.\]1while  
\[\text{q}=2-\frac{1}{\text{Q}}.\]2  
where Ac is the crossover point between the hierarchical (q \(\
eq\) 1) to exponential (q = 1) statistical mechanics. Equation (8a) leads to the q-exponential of Equation (6) for a cumulative distribution function P(\(>\) A). The latter Equation (8a,b) explain the q-exponential description of the distribution of drainage areas along with the exponential tail observed in some of the cases, as a result of generalized non-extensive statistical mechanics. Here we state that the estimation of the entropic parameter q which is mainly in the range 1.45-1.55, indicates a sub-additive system, with significant long range interactions. The q non-extensive parameter increases as the mean elevation increases, implying that watersheds in higher elevations interact stronger than those of lower mean elevations. These strong interactions are probably related to the limited space in which they have to be developed. As a result, the area distribution between smaller and bigger basins follows a hierarchical pattern. On the other hand, watersheds of low elevation are more sensitive to tectonic (presence of faults), climatic (precipitation) and anthropogenic (land use changes) factors.