## 4 Discussion  
All shallow earthquakes are followed by an aftershock sequence. The statistical properties of aftershock sequences are associated with scaling relations such as that extracted in view of non-extensive statistical physics (NESP). In this study, we used a detailed temporal assessment of the aftershock sequences over the last five years of significant earthquakes in Greece with magnitudes that reach up to \(M_{w}=7.0\). We studied the strong events, such as the \(M_{w}\)6.0 Arkalochori (27 September 2021), the \(M_{w}\)6.3 Elassona (3 March 2021), the \(M_{w}\)7.0 Samos (30 October 2020), the \(M_{w}\)5.1 Parnitha (19 July 2019), the \(M_{w}\)6.6 Zakynthos (25 October 2018), the \(M_{w}\)6.5 Kos (20 July 2017) and the \(M_{w}\)6.1 Mytiene (12 June 2017) earthquakes. Based on non-expansive statistical physics, we analyzed the distribution of interevent times for each aftershock sequence for each main shock. In all cases, the cumulative distribution function \(P(>\)\(T)\) is defined by a \(Q\)-exponential in the early stage of the aftershock sequence where interevent times less than \(T_{c}\) are observed, where \(T_{c}\) is the crossover point between the non-additive and additive behavior. By fitting a \(Q\)-exponential function to the data up to a value close to \(T_{c}\), the parameter \(q\) is estimated for each aftershock sequence. In all the cases analyzed, the applicability of non-extensive statistical physics to the interevent times CDF is demonstrated, as well as, the existence of transition behavior from the power-law to exponential scaling for larger interevent times. Since the q entropic parameter is greater than one (\(q>\) 1) a sub-additive process is implied, supporting the conclusion that long-range memory exists in the early state of temporal evolution of aftershocks where mainly \(T<T_{c}\). Additionally, for aftershock sequences analyzed, the estimated Tsallis entropic \(q\)-values that describe the observed CDF are within the range of 1.62-1.71. In addition, the superposition of two aftershock mechanisms can be used to explain the observed scaling behavior and the deviation from the \(Q\)-exponential function for greater interevent times. For \(T>T_{c}\), a second mechanism--characterized by an exponential function--becomes apparent. The first mechanism, as presented by NESP, is dominant for \(T<T_{c}\). We, thus, introduce the generalization described in [20; 39; 40], to account for a transition from NESP (\(q\
eq 1\)) to BG (\(q=1\)) statistical mechanics, where:  
\[\frac{dp(T)}{dT}=-\beta_{1}p-\left(\beta_{q}-\beta_{1}\right)p^{q}, \tag{12}\]  
whose solution is  
\[p(T)=C\left[1-\frac{\beta_{q}}{\beta_{1}}+\frac{\beta_{q}}{\beta_{1}}e^{(q-1) \beta_{1}T}\right]^{1/1-q}, \tag{13}\]  
In Equation (13) the probability function \(p(T)\) decreases monotonically with increasing \(T\) for positive \(\beta_{q}\) and \(\beta_{1}\), where \(C\) is a normalization factor. As a result, when \((q-1)\beta_{1}<<1\), a \(q\)-exponential, \(p(T)\approx Cexp_{q}\left(-T/T_{q}\right)\), where \(T_{q}=1/\beta_{q}\), is an approximation of Equation (13), while for \((q-1)\beta_{1}>>1\), the asymptotic behavior of the probability distribution function \(p(T)\propto\left(\frac{\beta_{1}}{\beta_{1}}\right)^{1/(q-1)}e^{-\beta_{1}T}\), is an exponential one, where \(T_{c}=1/(q-1)\beta_{1}\) defines the crossover point from the non-additive to additive behavior [24; 41]. The \(T_{c}\) value suggests that the Tsallis entropy is prevalent in the early stages of aftershock evolution, while the traits of aftershock sequences which are associated with a NESP description, become less apparent as time passes and BG statistics are recovered [6; 9; 17; 22; 42; 43]. The super-statistical theory, which is complementary to NESP, is based on a superposition of ordinary local equilibrium statistical mechanics, with a Gamma distributed intensive parameter that varies over a fairly wide time scale. This approach can be used to explain the \(q\)-exponential behavior of the interevent times in aftershock sequences [14].