## 1 Introduction
For a time series comprising \(N\) events, the \(k\)-th event's natural time \(\chi_{k}\) is defined as that event's normalization with respect to the total number of considered events, i.e., \(\chi_{k}=k/N\), serving as an index for its occurrence as shown in Figure 2. From its definition, it is clear that \(\chi_{k}\in[0,1]\). In addition to \(\chi_{k}\), we also define \(Q_{k}\) as a desired measure to characterize the signals. Its specific physical meaning is arbitrary, but energy and amplitude are common choices. Its corresponding normalized value is given by \(p_{k}=Q_{k}/\sum_{n=1}^{N}Q_{n}\). The pair \((\chi_{k},p_{k})\) serves as a basis for NT analysis. A given natural time \(\chi_{k}\)'s variance \(\kappa_{1}\) is calculated as:  
\[\kappa_{1}=\left\langle\chi^{2}\right\rangle-\left\langle\chi\right\rangle^{2 }\equiv\sum_{k=1}^{N}p_{k}\left(\frac{k}{N}\right)^{2}-\left(\sum_{k=1}^{N} \frac{k}{N}p_{k}\right)^{2} \tag{1}\]  
\(\kappa_{1}\) varies when a new \(k\)-th event occurs, once natural time \(\chi_{k}\) changes from \(k/N\) to \(k/(N+1)\) and \(p_{k}\) changes to \(Q_{k}/\sum_{i=1}^{N+1}Q_{i}\). As shown in [25], such changes may identify a dynamical system approaching a critical point. \(\kappa_{1}\) is often called the system's order parameter. Figure 1: \(b\)-value parameter extraction from an event set. Additionally, the entropy in natural time \(S\) can be defined as:  
\[S=\langle\chi\ln\chi\rangle-\langle\chi\rangle\ln\langle\chi\rangle\equiv\sum_{k=1 }^{N}p_{k}\chi_{k}\ln\chi_{k}-\left(\sum_{k=1}^{N}p_{k}\chi_{k}\right)\ln\left( \sum_{k=1}^{N}p_{k}\chi_{k}\right) \tag{2}\]  
Notice that, as observed by [30], this parameter is only based on numerical signal data and does not indicate a thermodynamic relation. Just like the variance \(\kappa_{1}\), the entropy \(S\) is also a dynamic parameter depending on the sequential order of events [31]. The third parameter in NT analysis is the entropy under time reversal \(S_{rev}\). It is computed by reversing the order of the energy data (\(Q_{k}\)), recalculating \(S\) in Equation (2) with this reversed series, and subtracting \(S\)'s original value. Finally, [PERSON] et al. [6], propose yet another order parameter, describing the \"average\" distance \(\langle D\rangle\) between the normalized power spectrum \(\Pi(\omega)\) of a sequence of events and its respective theoretical estimation \(\Pi(\omega)_{ideal}\), which represents the power spectrum in the critical state. Thus, \(\langle D\rangle=\langle|\Pi(\omega)-\Pi(\omega)_{ideal}|\rangle\to 0\) indicates that the system is approaching a critical region. \(\Pi(\omega)\) is defined as:  
\[\Pi(\omega)=\left|\sum_{k=1}^{N}p_{k}e^{i\omega\frac{\omega}{N}}\right|^{2} \tag{3}\]  
where \(\omega\) is the natural angular frequency, \(\omega=2\pi\phi\) with \(\phi\in(0,0.5)\) standing as the frequency in natural time. Its corresponding ideal value \(\Pi(\omega)_{ideal}\) can be shown to satisfy Equation (4):  
\[\Pi(\omega)_{ideal}=\frac{18}{5\omega^{2}}-\frac{6\cos\omega}{5\omega^{2}}- \frac{12\sin\omega}{5\omega^{3}} \tag{4}\]  
which, for \(\omega\to 0\), simplifies to \(\Pi(\omega)_{ideal}\approx 1-0.07\omega^{2}\). More detailed discussions of Equations (2)-(4) were presented in [25,32]. According to [PERSON] et al. [33,34,35,36], a dynamical system is considered in a labeled true critical state if three criteria are accomplished simultaneously:  
1. \(\langle D\rangle\) becomes smaller than \(10^{-2}\);
2. \(\kappa_{1}\) approaches 0.070 while descending from above;
3. Both \(S\) and \(S_{rev}\) become smaller than the entropy of uniform noise \(S_{u}=0.0966\). Lattice discrete element method (LDEM): As LDEM can simulate crack propagation and evolution, it captures internal changes in the model that resemble the emission of AE  
Figure 2: The NT analysis method: (**a**) a typical AE time series; (**b**) corresponding NT representation. signals during the damage process.