## 2 Principles of Non-Extensive Statistical Physics  
In this work, we use a generalized formulation of Boltzmann-Gibbs (BG) statistical physics, termed non-extensive statistical physics (NESP) [8; 9; 10; 11; 12], to investigate the distribution of the interevent times between the successive aftershocks. The fundamental benefit of NESP, is that it takes into account correlations on all length scales between system elements, resulting in asymptotic power-law behavior. NESP has been used in a wide variety of fields such as non-linear dynamical systems, including aftershock sequences [13], seismicity [5; 6; 7; 14; 15; 16], natural hazards [17], and complexity in volcanic areas [12], among others [8]. Such characteristics can be described in fracture-related phenomena. Non-extensive statistical physics is concerned with precisely such phenomena. Initially, NESP begins by defining entropy by [PERSON] [18]. This entropic functional is appropriate for characterizing complex systems with finite degrees of freedom, self-organized critically, and non-Markovian characteristics with long-range memory, properties as that commonly occur in geosciences [5; 19; 20; 21]. The present application of Tsallis entropy introduces the variable of \(T\) (i.e., the interevent times) between two successive aftershocks, where \(p(T)\)\(dT\) indicates the number of the parameter between \(T\) and \(T{+}dT\). An earthquake complex system, in a non-equilibrium state, can be described by an entropic functional \(S_{q}\) introduced by [PERSON] [18]  
\[S_{q}=k_{B}\frac{1-\sum_{i}p^{q}(T_{i})}{q-1} \tag{1}\]  
where \(k_{B}\) is [PERSON]'s constant, \(p(T)\) is the probability distribution of interevent times \(T\) and the index \(q\) expresses the degree of non-additivity of the system. The index \(q\) may violate the additivity principle of classical BG entropy [8; 18]. In [18] it was demonstrated that in the limit of \(q{\rightarrow}1\), the non-extensive entropy \(S_{q}\) recovers the Boltzmann-Gibbs (BG) one. In earth sciences, the cumulative distribution function is traditionally used in the framework of NESP [20; 21]. This expression is derived by maximizing \(S_{q}\) while imposing appropriate constraints and employing the Lagrange multipliers method, yielding to [8]:  
\[p(T)=\frac{\left[1-(1-q)\Big{(}\frac{T}{T_{q}}\Big{)}\right]^{\frac{1}{1-q}}}{Z_ {q}}=\frac{1}{Z_{q}}exp_{q}\left(-\frac{T}{T_{q}}\right) \tag{2}\]  
whith \(Z_{q}\) the so-called \(q\)-partition function  
\[Z_{q}=\int_{0}^{\infty}exp_{q}\big{(}-T/T_{q}\big{)}dT, \tag{3}\]  
and \(T_{q}\) denotes the generalized scaled interevent time. With respect to Equation (2) \(q\)-exponential function appears, defined as [8]  
\[exp_{q}(X)=[1+(1-q)X]^{\frac{1}{1-q}}, \tag{4}\]  
for \(1+(1-q)X\geq 0\), while in other cases \(exp_{q}(X)=0\). Equation (2) is further used to estimate the cumulative distribution function (CDF) of the interevent times:  
\[P(>T)=\frac{N(>T)}{N_{0}}, \tag{5}\]  
with \(N(>T)\), is the number of the interevent times with value greater than \(T\) and \(N_{0}\) their total number [22; 23]. By using Equation (2), \(P(>T)\) equals to Equation (6) which has the form of a \(q\)-exponential function, hereafter called Q-exponential one:  
\[P(>T)=exp_{Q}(-T/T^{*}), \tag{6}\]  
with:  
\[T^{*}=T_{q}Q \tag{7}\]  
\[q=2-\left(\frac{1}{Q}\right) \tag{8}\]  
The \(Q\)-logarithmic function is the inverse function of the \(Q\)-exponential and it is defined as:  
\[ln_{Q}P(>T)=\frac{P(>T)^{1-Q}-1}{1-Q}, \tag{9}\]  
Equation (9) demonstrates that the \(Q\)-logarithm [8; 24] of CDF of interevent times, is linearly scaled with \(T\) with an expression:  
\[ln_{Q}P(>T)=-\frac{T}{T^{*}}, \tag{10}\]  
with slope \(1/T^{*}\)[14]. According to the different values that the parameter \(q\) can take, three particular cases arise. More specifically, in the limit \(q\)\(\rightarrow\)\(1\), the \(q\)-exponential leads to the ordinary exponential function.