## 4 Results and Discussion
### The Variability of Index \(q\)  
In the case of the UC experiment, a total of \(N=898\) AE hits were recorded, which were divided into 5 groups of \(n=180\) consecutive AE hits. Each group was derived from the previous one by sliding by \(n/2=90\) AE hits. Thus, \(k=9\) AE hit groups were created, and for each group, the values of the CDFs were calculated after they were rendered graphically \([P(>\delta\tau)\) vs \(\delta\tau]\). The fitting based on Equation (5) made it possible to determine the values of the entropic index \(q\) and the parameter \(\beta_{q}\). For each AE hit group, the mean value of the mechanical stress \(\overline{S}\) was also calculated. Table 1 includes the values of the entropic indices \(q\), the parameters \(\beta_{q}\) and \(\tau_{q}\), and the corresponding average normalized stress \(\overline{S}_{n}=\overline{S}/S_{f}\), with \(S_{f}\) being the ultimate stress strength for each group of AE hits. In the same table, the quantity \(F\) is included, which expresses the average occurrence frequency of the AE hits in each group and is calculated as the inverse of the average value of the time intervals of each group \(\bigl{(}F=1/\overline{\delta\tau}\bigr{)}\). An average value is related to both the entropic index \(q\) and the time parameter \(\beta_{q}\) through the relation [29]:  
\[\overline{\delta\tau}=\frac{1}{\beta_{q}}\cdot\frac{B\bigl{(}2,\frac{2-q}{q-1 }\bigr{)}}{\bigl{(}q-1\bigr{)}^{2}} \tag{6}\]  
where \(B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt\) is the so-called beta function [44], with \(x=2\) and \(y=\frac{2-q}{q-1}\). If \(Q(q)\) denotes the function: \((q-1)^{2}/B\Bigl{(}2,\frac{2-q}{q-1}\Bigr{)}\), it results that:  
\[\frac{1}{\beta_{q}}=\frac{Q(q)}{F} \tag{7}\]  
Figure 5 shows the evolution of the entropic index \(q\) and the acoustic activity, expressed in terms of the quantity \(F\), as functions of the normalized stress \(\overline{S}_{n}\). For the values of the \begin{table}
\begin{tabular}{c c c c c} \hline AE Hit Group & \(\frac{-}{S_{n}}\) & \(q\) & \(\beta_{q}\) (s\({}^{-1}\)) & \(F\) (s\({}^{-1}\)) \\ \hline
1 & 0.398 & 1.15 & 2.56 & 2.22 \\
2 & 0.581 & 1.27 & 5.81 & 4.59 \\
3 & 0.692 & 1.26 & 8.48 & 6.69 \\
4 & 0.779 & 1.28 & 9.66 & 7.84 \\
5 & 0.857 & 1.32 & 11.92 & 9.06 \\
6 & 0.918 & 1.39 & 19.33 & 12.81 \\
7 & 0.957 & 1.47 & 37.99 & 22.10 \\
8 & 0.978 & 1.36 & 73.84 & 41.54 \\
9 & 0.988 & 1.16 & 102.03 & 77.19 \\ \hline \end{tabular}
\end{table} 
Table 1: The entropic and mechanical parameters during the UC experiment for each consequent AE hit group. index \(q\), the stress limits (asterisks) of the normalized stress \(\overline{S}_{n}\) that correspond to each AE hit group are also shown. During the initial stages of the loading protocol for stress values \(\overline{S}_{n}<0.8\), the entropic index \(q\) attains low values (\(1.15<q<1.30\)), which is indicative of the low degree of organization regarding the system of microcracks created due to the closure and rubbing of the pre-existing microcracks and the discontinuities inside the marble specimen. At the same time, the acoustic activity showed a weakly increasing trend, which intensified progressively with the increase in stress (\(\overline{S}_{n}\)). When the normalized stress \(\overline{S}_{n}\) approached levels ranging from 0.8 to 0.96, the values of \(q\) showed an increasing trend, reaching a maximum value of 1.47. At this stage, the mechanisms of creation, development, and propagation of microcracks seemed to be characterized by a higher degree of organization, and the formation of a large number of new microcracks was observed as well as an increase in the dimensions of the already existing microcracks, which was also related to the increased acoustic activity. Finally, for \(\overline{S}_{n}>0.96\), a noticeable decrease in the index \(q\) values was observed, and the last group of AE hits corresponded to a \(q\) index value of 1.16.