Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions

Abstract

In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the symbolic dynamics of two symbols produced by the 3D Ising model in its critical state. After this numerical verification of the reliability of PNA-STSM, three examples of how PNA-STSM can be applied to the category of systems that obey the dynamics of the on–off intermittency are presented. Usually, such time series, with on–off intermittency, present bimodal amplitude distributions (i.e., with two lobes). As has recently been found, the phenomenon of on–off intermittency is associated with the spontaneous symmetry breaking (SSB) of the second-order phase transition. Thus, the revelation that a system is close to SSB supports a deeper understanding of its dynamics in terms of criticality, which is quite useful in applications such as the analysis of pre-earthquake fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) signals. Beyond the case of on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic dynamics, even in cases in which there is an imbalance in the probability of the appearance of the two respective symbols since the two symbols are not considered separately but, instead, simultaneously, considering the information from both branches of the symbolic dynamics.

1. Introduction

In a series of works [1,2], it has been shown that the use of a device with circular rings carrying electric currents of the same intensity but in a random flow direction results in a symmetrical stratified magnetic field, i.e., a magnetic field that presents a zonal structure around an axis of symmetry. Furthermore, using the most recently introduced prime-numbers-based algorithm (PNA) to simulate the magnetic field of the specific device [3], the quantization of the above-mentioned stratified magnetic field is achieved, i.e., the magnetic field values’ zones can be converted to quantized magnetic field values. Thus, if a two-symbol symbolic time series (using, for example, the symbols “+1”, “−1”) is employed to determine the flow directions of the rings’ currents, a time-to-space mapping of the dynamics of the system producing the time series, is achieved [3]. This mapping enables the calculation of the exponents of the “waiting times” distributions in the space of the magnetic field positions, “$k,$” leading to the development of the method of symbolic time-series analysis referred to as the PNA-based symbolic-time-series-analysis method (PNA-STSM) [3]). The PNA-STSM can be applied to any two-symbol symbolic time series that is produced directly or indirectly (after coarse-graining) by any real or numerical dynamic system and presents the advantage that it can reliably expose the real dynamics of a complex system, even if a relatively short time series is available and the probabilities of the occurrence of each of the two symbols are not equal [3].

Waiting times are essential because their distribution can reveal the dynamics of the system generating the time series under analysis [3]. For example, if the distribution of the waiting times follows a power law with the exponent p$\in \left[\mathrm{1,2}\right),$ this is a strong indication that the system is in its critical state [4]. On the other end of the dynamical spectrum, the presence of exponential laws in these distributions suggests the complete absence of dynamics that create long scales; in such cases, long scales are truncated [5]. Between these two extremes of the dynamics spectrum, any intermediate state can be quantitatively determined from the competition of these two extreme cases (power law and exponential distributions).

Furthermore, waiting times can be defined in different ways. For example, in time series resulting from intermittent dynamics, the waiting times, also called laminar lengths [6], are defined as the time intervals between bursts, namely, the number of consecutive values of the time series that fall within specific thresholds that define the boundaries of a non-burst (laminar) values’ region. In a sequence of symbols, such as the symbolic dynamics of the symbols “+1” and “−1”, the waiting times are defined as the number of the same consecutive symbols (e.g., “+1”), which are interrupted by a number of consecutive appearances of the other symbol (e.g., “−1”), or vice versa. A key difference from the above-mentioned time domain (waiting-time-based) analysis approaches is that PNA-STSM introduces a space-domain analog of waiting times, called “$k$ waiting lengths,” which takes into account both symbols’ dynamics.

The first part of this work shows that PNA-STSM is an analysis method capable of correctly estimating the exponents characterizing the dynamics embedded in a time series. This is proven by comparing the results obtained by PNA-STSM with the results obtained by other well-known and reliable methods, as well as with the theoretical results for the 3D Ising model in the critical state. Subsequently, the main advantages of PNA-STSM over other analysis methods are highlighted. Three examples of dynamical systems are employed to demonstrate the efficacy of PNA-STSM. The first is a numerical example and refers to a special behavior of the 3D Ising model, and the second is from the area of nano-electronics, and the third concerns earthquake (EQ)-preparation processes. The common behaviors of these three systems are that they all obey the dynamics of on–off intermittency (of various forms) and that they all present bimodal (two-lobe) amplitude distributions, which, as has recently been shown [7], appears to be connected with the spontaneous symmetry breaking (SSB) phenomenon observed in the preparation of the second-order phase transition in finite systems. Through these examples, it is shown how to apply the PNA-STSM to systems that obey the dynamics of on–off intermittency and present bimodal amplitude distributions in order to extract the information on their critical state and the SSB. Moreover, it is demonstrated that beyond on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic-dynamics time series or for general two-symbol sequences because it is not affected by any imbalances in the probability of the appearance of the two symbols, since it simultaneously considers the information from both branches of the symbolic dynamics.

2. The 3D Ising Model and Its Waiting Times

The 3D Ising model is a well-known model in statistical mechanics describing specific ferromagnetic behavior [8,9]. It successfully describes the continuous phase transition in equilibrium, as well as more specialized topics, such as the SSB of the ${\phi }^4$ Landau free energy [10]. According to the theory of critical phenomena, all natural systems are classified into universality classes that are characterized by the values of the so-called critical exponents. The 1D Ising model has been analytically solved by Ernst Ising [11] and does not present phase transition for any finite temperature. Likewise, the 2D Ising model has been analytically solved by Onsager [12], and the six critical exponents of this model have been accurately calculated. Finally, the 3D Ising model has not been solved analytically yet, but only numerical solutions have been provided. However, it must be mentioned that in [13], and under suitable boundary conditions, the 3D Ising model can be described by the operator algebra and thus can be solved exactly. This theory has been studied using renormalization group methods, Monte-Carlo simulations, conformal bootstrap and other techniques [8,9,14,15].

Generally speaking, for a Z(N) spin system, spin variables are defined as: $s\left(a_i\right)=e^{i2\pi a_i/N}$ (lattice vertices $i=1,\dots ,i_{max}$), with $a_i=\mathrm{0,1},\mathrm{2,3},\dots ,N-1$. For $N=2$, one may derive the above-mentioned Ising models. Utilizing the Metropolis algorithm configurations at constant temperatures may be selected with Boltzmann statistical weights $e^{-\beta \mathcal{H}}$, where $\mathcal{H}$ is the Hamiltonian of the spin system for which the nearest neighbors’ interactions without external field can be written as:

$$\mathcal{H}=-\sum _{<i,j>}{J}_{ij}{s}_i{s}_j, s_j, s_j=\pm 1.$$

(1)

It is known that this model undergoes a second-order phase transition when the temperature drops below a critical value [10,16]. Thus, for a lattice of ${20}^3$ nodes in three dimensions (3D Ising model), the critical (or pseudocritical for finite-size lattices) temperature has been found to be $T_c=4.545$($J_{ij}=$1). Considering the sweep of the whole lattice as an algorithmic time unit, the possible values that spin can have in the model are ±1. The quantity recorded in a numerical experiment (conducted using the Metropolis algorithm) is the mean magnetization (space averaged magnetization density) $M$, which plays the role of the order parameter. The trajectory generated by the numerical experiment is a “time series” of the fluctuations of the order parameter. This means that for a simulation performed for $N_{iter}$ lattice sweeps, the generated time series of $M$ will consist of $N_{iter}$-points, with a time step equal to the algorithmic time of the whole lattice sweep. Figure 1 presents the distribution of the resulting mean magnetization $M$ values at the pseudocritical point $T_c=4.545$ for $N_{iter}=\mathrm{200,000}$.

Calculating the distribution of waiting times (see Section 1), one may extract useful and important quantitative information about the dynamics of the studied system. In the cases that this distribution turns out to be a scale-free function (i.e., a power law), the signature of the critical state of the system could be revealed. Therefore, the correct calculation of waiting times is of great importance. As it has been shown in [4,17], the fluctuations of the order parameter at the critical point follow the dynamics of Type I intermittency, for which it is known that the distribution $P\left(L\right)$ of the appropriately defined waiting times $L$ (laminar lengths, see Section 1) follows a power law of the form [6]:

$$P\left(L\right)~L^{-p} .$$

(2)

As shown in [4], the exponent $p$ is related to the isothermal critical exponent $\delta$, as:

$$p=1^{\prime }+\frac{1}{\delta } ,$$

(3)

while it holds that $\Gamma \left(M\right)~{{\rm M}}^{\delta +1}$ [4], where $\Gamma \left(M\right)$ is the effective action for the mean magnetization. It is also mentioned that the exponent $p$ is related to other well-known exponents through the isothermal critical exponent $\delta$ (one of the six critical exponents, related one another through four relations, known as “scaling laws” [16]). Moreover, it has been shown that $p$ is related to the spectral exponent, the Hurst exponent, and the fractal dimension [18], as well as to the exponent describing Lévy flights and the Tsallis non-extensive parameter $q$ [19].

Since $\delta >1$ [16], one may deduce from Equation (3) that the critical state exists when $p\in \left[\mathrm{1,2}\right)$. Beyond the critical point, another type of intermittency dynamics [20] that describe the dynamics of the order parameter fluctuations at the beginning of a tricritical crossover, has been found. This happens around the point where the second-order phase transition and the first-order phase transition lines meet in the parametric space of Ginzburg–Landau free energy (tricritical point according to Griffiths [16]). In this case, the exponent $p$ of the power law distribution of laminar lengths is given as [4]:

$$p=\frac{\delta +1}{\delta +2} ,$$

(4)

from which it follows that $p\in \left(\mathrm{0.66,1}\right)$ [21]. At this point, it should be mentioned that when the distribution of the waiting times follows a power law with an exponent $p\in \left(0, 0.66\right)$ it probably refers to some subcritical region without clear boundaries that can be predicted by the theory, and it does not indicate criticality (critical or tricritical point). In our analyses of dynamical systems, real and numerical, coming from different disciplines, we have not yet come across such a case.

In [4], the value of the exponent $p$ for the critical 3D Ising model has been estimated with very good agreement with the one predicted from the theory of critical phenomena, using the notion of waiting times in a time series (see Section 1). Specifically, using the method of critical fluctuations (MCF) [4,22], which has been introduced for this purpose, it has been found that the exponent $p$ obtains the value $p=1.21\pm 0.02$ for the critical 3D Ising model [4]. Exactly at the pseudocritical point, the characteristic exponent is the isothermal critical exponent $\delta$. Notice that for the 3D Ising model this exponent has the value $\delta =4.8$ [23]. Consequently, from Equation (4), the theoretical value of the exponent $p$ is $p=1.208$, and therefore, the value estimated using the MCF for the numerical experiment ($p=1.21$) is in perfect agreement with the theoretical one.

Another way to calculate the exponent $p$ of the distribution of the waiting times in the critical 3D Ising model is to resort to symbolic dynamics based on two symbols in the time domain. This can be achieved by appropriately converting the mean magnetization time series to a symbolic time series and then by defining the waiting times as the number of the same consecutive symbols (see Section 1). Given that the distribution of Figure 1 is almost symmetrical around $M=0$, a straightforward way to obtain a two-state coarse-graining description of the mean magnetization value for the specific numerical experiment (as mentioned, 200,000 configurations of a ${20}^3$ 3D Ising lattice at the pseudocritical temperature $T_c=4.545$) is to assign positive and negative time series values to the symbols “+1” and “−1”, respectively. Thus, the simulated mean magnetization time series is converted into a two-symbol symbolic time series. By calculating the waiting times of the obtained mean magnetization symbolic time series, one may create the distributions for each of the two symbols, which are shown in Figure 2. As in the MCF, one may employ the truncated power law function $g\left(L\right)$ for each symbol in order to model the distribution of waiting times $P\left(L\right)$:

$$g\left(L\right)=\mathrm{ }{p}_1·L^{-p_2}·e^{-L{p}_3} ,$$

(5)

In the case that $p_3=0$, then $p_2$ is equal to the power law exponent $p$.

For the waiting times corresponding to the symbol “+1”, their distribution indeed follows a power law with an exponent value $p=1.159\pm 0.017$ (Figure 2a); while for the waiting times corresponding to the symbol “−1”, again a power law distribution holds, but this time with a different exponent value of $p=1.182\pm 0.026$ (Figure 2b). Apparently, these exponent values are not the same due to the small asymmetry between the two symbols, 49% (“+1”’s) vs. 51% (“−1”’s) probability of occurrence, which is reflecting the small asymmetry between the positive and negative values of the original mean magnetization time series (also 49% vs. 51%, see the distribution in Figure 1). It is also noted that considering the error limits, the “+1” branch leads to $p$ values in the range between 1.142 and 1.176, while the “−1” branch leads to $p$ values in the range between 1.156 and 1.208, which intersect one another. However, the resulting higher value for the “+1” branch (1.176) is close but lower than the nominal value of the “−1” branch (1.182). Thus, one cannot claim that the two nominal values (1.159 and 1.182) coincide within their error limits.

Summing up, based on the same numerical data of 200,000 configurations for a ${20}^3$ lattice at the pseudocritical temperature $T_c=4.545$, the MCF estimates for the exponent $p$ a value of 1.21, whereas the waiting times analysis of the corresponding two-symbol symbolic time series in the time domain leads to two different exponent $p$ values, 1.159 and 1.182, for each symbolic branch; while we know that for the 3D Ising model, the theoretically calculated value of the waiting times distribution’s power law exponent is $p=1.208$. From the above, it is clear that although the existence of dynamics in a 3D Ising two-symbol symbolic time series can be revealed in the time domain via the scaling behavior of the waiting times, the quantitative result is ambiguous, even for time series with only a slight asymmetry in the distribution of their values, as has also been shown in [3] for the case of the 2D Ising model. Furthermore, the 3D Ising model case proves that this result is not only ambiguous but also far from the one theoretically calculated. However, it has been proven in [3] that PNA-STSM can overcome this ambiguity issue, even for time series of relatively short length, in which one of the symbols appears more often than the other. So, in Section 3, after a brief presentation of the key notions of the PNA-STSM, the case of the 3D Ising symbolic time series is analyzed using the PNA-STSM, leading not only to unambiguous but also accurate exponent $p$ estimation.

3. Application of PNA-STSM to the Critical 3D Ising Model

3.1. The PNA-STSM Time Series Analysis Method

By briefly presenting the application of PNA-STSM in the case of a two-symbol symbolic time series, we use as an example the symbolic time series of the mean magnetization fluctuations of the critical 3D Ising model that has been presented in Section 2. For a detailed description of the PNA-STSM, one may refer to [3], while in the Appendix of [3], one may find a Fortran implementation of the main part of the method.

The block diagram depicted in Figure 3 schematically presents all stages of the PNA-STSM, which accepts as input any two-symbol symbolic time series directly or indirectly (after coarse-graining) produced by any real or numerical dynamical system.

In order to apply the PNA-STSM to the 3D Ising model at the pseudocritical temperature $T_c=4.545$, one begins from the two-symbol symbolic time series obtained, as described in Section 2. The symbolic time series is fed to the “transformation” from the time domain (t) to the space domain ($k$), where $k$ (see Figure 3) denotes the ring positions of the employed current carrying circular rings device of Figure 4. The electric currents are of the same intensity $I$, but their flow direction ($+I,-I$) is determined by the corresponding symbol of the symbolic time series. Specifically, if the two-symbol symbolic time series is $\left\{x\left(t_i\right)\right\}$, $t_i=i·\tau$, $\tau$ being the sampling period and $i=\mathrm{1,2},\dots , N$, then for $i=k=1$, if $x\left(t_1\right)=-1$, then $I_1=-1$, while if $x\left(t_1\right)=+1$, then $I_1=+1$, etc. This way, a mapping from the time domain (t) to the space domain (ring positions $k$) is achieved. Each resulting spatial pattern of the magnetic field is unique and determined by the respective symbolic dynamics of the driving time series.

Each ring is circular, with radius $\alpha$, $d$ is the distance between two consecutive rings (Figure 4), while $c=a/d$ is a parameter determining the behavior of the device. Using the PNA algorithm, results in a quantized magnetic field of six values (three positive and three symmetric negative) in the case $c^2\ll 1$, while if $c\ll 1$, only the central positive and negative ones $B_k=\pm 0.5$ remain in the pattern of the magnetic field [3]. Thus, these central values possess a special property since their existence does not depend on the value of the geometric characteristics of the device. On the other hand, as has been shown in [3], in each half-space of the quantized magnetic field (positive or negative), the probability of the appearance of a quantized magnetic field value at the central level, $B_k=+0.5$ or $B_k=-0.5$, is equal to the sum of the probabilities of the appearance of the quantized magnetic field values at the two other value levels of the corresponding half-space.

As already mentioned in Section 1, the PNA-STSM introduced a space domain analog of the waiting times called the “$k$ waiting lengths”. The latter are calculated by counting the number of consecutive ring positions for which the quantized magnetic field value remains equal to “the value of interest” [3]. The simultaneous choice of both central levels, $B_k=\pm 0.5$ as “the value of interest”, respects the above-mentioned symmetry. This way, an equivalent description is obtained when ones considers either the distribution at both central levels or at the other four possible quantized magnetic field levels. This justifies the definition of the “$k$ waiting lengths” as a representation of the dynamics of the alteration of the magnetic field between the central values $B_k=\pm 0.5$ and any of the other four (two positive and two negative) possible values (see Figure 3). Thus, when one calculates the “$k$ waiting lengths” in the PNA-STSM, it practically counts the number of consecutive ring positions for which the quantized magnetic field value is either $B_k=+0.5$ or $B_k=-0.5$, thus taking into account the positive and the negative subspace of the symbolic dynamics in the space-domain, simultaneously [3].

The calculation of the “$k$ waiting lengths” that considers both subspaces simultaneously is the most important choice in the PNA-STSM since it provides a single statistical representation of the inputted symbolic time series in the space domain. This way, any (even slight) asymmetry in the driving symbolic time series does not affect the obtained result. Namely, if the distribution of the “$k$ waiting lengths” follows a power law, then the value of the power law exponent $p$ estimated using PNA-STSM is unambiguous [3], in contrast to the time domain approach demonstrated for the 3D Ising case in Section 2.

Finally, it is mentioned that in all cases of the PNA-STSM application presented in the following sections, the following parameters have been used for the current carrying circular rings device: $a=1$, $d=10$, and $I=1$. Therefore, $c=a/d=0.1$ and $c^2\ll 1$. Moreover, it should be mentioned that the time needed for carrying out the involved calculations appears to present a power law dependence on the length of the time series. However, the calculations are not particularly time-consuming, e.g., for a rather weak personal computer (Intel^® Core™2 Duo processor T5670, with clock speed 1.8 GHz, bus speed 800 MHz, and 4 GB DDR2 RAM @667 MHz) a 100,000-points-long time series takes only 1.25 min to be analyzed, whereas a 200,000-points-long time series takes less than 5 min to be analyzed.

3.2. Application of PNA-STSM to the Critical 3D Ising Model

In this section, we present the results obtained by the application of PNA-STSM to the two-symbol symbolic time series that was analyzed in the time domain in Section 2 ($N_{iter}=\mathrm{200,000}$ configurations in a ${20}^3$ 3D Ising lattice at the pseudocritical temperature $T_c=4.545$). Figure 5a portrays a 1000-point segment of the 3D Ising mean magnetization time series $M\left(t\right)$ of the specific numerical experiment, while Figure 5b shows the corresponding quantized magnetic field pattern produced after the “transformation” of the corresponding two-symbol symbolic time series from the time domain (t) to the space-domain ($k$). Three positive and three symmetric negative magnetic field values were produced. The positive quantized values of the resulting magnetic field, calculated with an accuracy of four decimal points, are: $B_k=0.500$, $B_k=0.4988$, and $B_k=0.5012$. Similarly, in the negative half-space, the corresponding negative ones appear. From Figure 5b, one can easily identify the existence of the same magnetic field value at consecutive positions ($k$), interrupted by gaps, during which the magnetic field gets one of the other possible quantized values.

For the calculation of the “$k$ waiting lengths” distributions, the whole quantized magnetic field pattern in the space domain, i.e., for $k=1, 2,\dots , \mathrm{200,000}$, was used. Figure 6 presents the “$k$ waiting lengths” distribution according to the PNA-STSM, i.e., by counting the number of consecutive ring positions for which the quantized magnetic field value is either $B_k=+0.5$ or $B_k=-0.5$, interrupted by magnetic field values at any of the other four possible levels of the produced spatial pattern. Since $p_3\approx 0 \left(p_3=0.01\right)$, the distribution of the “$k$ waiting lengths” in Figure 6 actually follows a power law with exponent $p\left(=p_2\right)=1.202\pm 0.030$. Thus, by taking into account both positive and negative branches, the PNA-STSM not only provides an unambiguous (a single power law exponent) but also an accurate result. Indeed, the estimated exponent’s nominal value (1.202) is much closer to the theoretically calculated value ($p=1.208$) of the 3D Ising model at the pseudocritical temperature than the two different power law exponent’s nominal values that were calculated by analyzing the waiting times (time domain analysis) at each of the two symbols of the symbolic time series, i.e., 1.159 and 1.182 at “+1” and “−1”, respectively (see Section 2). We also note that concerning the exponential factor of the fitting function of Equation (5), the time domain analysis resulted in $p_3=0.02$ for both symbols (see Figure 2), while the PNA-STSM led to a value for $p_3$ closer to zero $\left(p_3=0.01\right)$. It is apparent that the PNA-STSM-obtained distribution is closer to the power law.

Finally, it should be mentioned that the time domain analysis could lead to comparable results with the PNA-STSM only to the asymptotic limit $N_{iter}\to \infty$, for which the distribution of Figure 1 becomes perfectly symmetric. However, for real signals of finite length, this is impossible; there will always be, even slight, asymmetry. Thus, the time domain analysis of the waiting times at each of the symbols of such a two-symbol time series will always lead to different exponent values. On the other hand, the estimation of a single exponent as the average value of the two calculated exponents usually does not apply since the phenomena producing the symbolic dynamics may be strongly nonlinear. In such cases, the waiting times time domain analysis is generally leading to ambiguous and inaccurate results, rendering it inferior to the PNA-STSM.

4. Analysis of Time Series Presenting Two-Lobes’ Amplitude Distributions Using PNA-STSM

As has recently been shown [7,24] and according to the ${\phi }^4$ theory, systems of finite size that undergo a second-order phase transition present a hysteresis zone between the pseudocritical point and the completion of the SSB, where the critical state continues to survive, also referred to as the “SSB-zone”. Within this hysteresis zone, the single minimum of the Landau free energy, appearing at the value of the control parameter that corresponds to the symmetrical phase (pseudocritical point), is replaced by a degenerate set of minima. These minima “communicate” until the SSB is completed by appropriately changing the value of the control parameter. In terms of the distribution of the order parameter’s values, this hysteresis zone is manifested by the change of the distribution form from unimodal (one lobe) to bimodal (two lobes). This happens because a fixed point (the Landau free energy minimum) attracts a high number of values of the order parameter close to it. Thus, the appearance of the degenerate set of minima changes the form of the distribution to bimodal.

As the control parameter, which in thermal systems is the temperature, falls below the pseudocritical temperature, these lobes reduce their communication, and as soon as their complete separation is achieved, the SSB is completed. Essentially, it is a degeneration of the critical state, which continues as long as the temperature drops until the complete separation of the two lobes is completed [7].

Beyond thermal systems, the appearance of two lobes in the distribution of an observable is found in many natural systems and beyond. Then, the question raised for all these systems is whether they could be found in the conditions of the above-mentioned hysteresis zone, like the thermal systems. The answer to this question is a very important piece of information in understanding how the critical state is organized in nature, but also what the consequences of the completion of SSB are (such as the imminent occurrence of an extreme event, e.g., earthquake, geomagnetic storm, stock market crash, etc.).

The answer to the above-raised question would be positive if the examined observable is a quantity that possesses the characteristics of the order parameter and if the distribution of the waiting times or, in the case of the PNA-STSM analysis, of the “$k$ waiting lengths” is a power law with an exponent $p\in \left[\mathrm{1,2}\right)$, as already explained in Section 2. Since the MCF requires a form of stationarity in the analyzed time series, it is not possible to apply it to time series that present bimodal amplitude distributions. On the other hand, in real systems, ideal conditions that allow the appearance of the two lobes with equal probabilities do not occur. This means that the solution must consider the information embedded in both lobes simultaneously, which points to the use of PNA-STSM.

As already mentioned in Section 1, in the following we present the application of PNA-STSM to three different time series, obtained from three different systems. In specific, the cases of:

(a)

The 3D Ising mean magnetization time series at some temperature (below the pseudocritical one) within the hysteresis zone which confirms that the critical state survives in the SSB-zone;

(b)

The time series of a nano-MOSFET noise current, and

(c)

A MHz fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) time series recorded prior to a strong earthquake event, are studied in Section 4.1, Section 4.2, and Section 4.3, respectively.

4.1. The Case of the Hysteresis Zone in 3D Ising Model

The first case studied regards the analysis of the mean magnetization time series at the temperature $T=4.51 (<T_c=4.545)$, which is within the hysteresis zone for the 3D Ising model. The numerical experiment was conducted for $N_{iter}=\mathrm{300,000}$. As can be seen in Figure 7a, the obtained time series indicates dynamics similar to the on–off intermittency dynamics, while the corresponding distribution presents two (asymmetric) lobes. The separation point between the lobes is located at the position $M=0$ (Figure 7b), which is the critical point according to the ${\phi }^4$ critical theory. It is also noted that the two lobes are not completely separated, instead they “communicate” with one another (SSB has not been completed for $T=4.51$).

After converting the mean magnetization time series (as in the case of the critical state in Section 2) into a two-symbol (“+1”, “−1”) symbolic time series, one can apply the PNA-STSM, as presented in Section 3. It is mentioned that the probabilities of appearance of the two symbols are not the same (46% “+1” vs., 54% “−1”) in the analyzed symbolic time series, thus the specific symbolic time series has statistics closer to the ones of real time series. The result of the PNA-STSM analysis, is that the distribution of the “$k$ waiting lengths” at the central values $\pm 0.5$ of the quantized magnetic field is very close to a power law since $p_3=0.02$, with power law exponent $p=\left(p_2=\right)1.21$ (see Figure 7c). The obtained exponents’ values, i.e., $p_2\in \left[\mathrm{1,2}\right)$ and $p_3\approx 0$, indicate that the critical state survives within the hysteresis zone, which is something expected since the studied temperature $T=4.51$ is very close to the pseudocritical temperature $T_c=4.545$.

Analyzing the specific symbolic time series in the time domain, i.e., by analyzing the waiting times at each of the two symbols separately (see Section 2), one arrives at two different results: for waiting times for the “+1” symbol distribution, the estimated exponents’ values are $p_2=1.15\in \left[\mathrm{1,2}\right), { p}_3=0.02$, whereas for the waiting times for the symbol “−1” distribution, the estimated values are $p_2=1.10\in \left[\mathrm{1,2}\right), p_3=0.04$. Both sets of exponents’ values indicate that the critical state survives at $T=4.51$, but the estimated power law exponent $p(=p_2)$ is different for each symbol.

Thus, one cannot reach a definite quantitative result by employing the time domain analysis. On the contrary, the use of the PNA-STSM, which considers both symbolic dynamics’ branches simultaneously, leads to a single power law exponent ($p=1.21$), providing a definite quantitative result. It should be finally mentioned that the value of the exponent $p$ estimated using the PNA-STSM cannot be obtained by any linear combination of the two different power law exponent values estimated using the time domain analysis, e.g., as their mean value. This also indicates the strong nonlinearity and complexity of the examined system.

4.2. The Case of the Nano-MOSFET Electronic Device

The second case studied regards an ultrathin body and buried box (UTBB) fully depleted silicon-on-insulator (FD-SOI) nano-MOSFET. This device was fabricated with channel dimensions W = 0.5 μm and L = 30 nm, while the silicon film thickness was t_Si = 7 nm [25]. Finally, the box thickness was 25 nm, and the equivalent front gate oxide thickness was (EOT) t_ox = 1.55 nm (TiN/HfSiON stack) [25]. Fully depleted silicon-on-insulator MOSFETs [26,27] exhibit low threshold voltage and threshold voltage controllability (by the second gate bias) at the nanoscale. Because of its compatibility to the standard planar CMOS technology, it is capable of further downscaling; therefore, the characteristics of noise currents are crucial to study. Next to typical noise analysis [25], UTBB-FD-SOI nano-MOSFET has also been studied from the point of view of nonlinear dynamics [28], critical phenomena [29], as well as Tsallis non-extensive statistics [30]. All three approaches confirmed the deterministic origin of demonstrated random telegraph noise (RTN).

In the following, we present the analysis of the above-mentioned nano-MOSFET’s drain current $I_d$ time series of a duration of 5 s, sampled at a sampling period of $5\times {10}^{-5} \mathrm{s}$, thus consisting of 100,000 points. The drain current has been acquired for bias conditions: $V_d=30 \mathrm{V}$, $V_{bg}=0 \mathrm{V}$, and $V_g=300 \mathrm{m}\mathrm{V}$, while $V_g$ acted as a control parameter. Figure 8a portrays a small segment of the analyzed time series to illustrate that $I_d$ values fluctuate around two current levels, hereafter referred to as “HIGH values” and “LOW values”, respectively. In the case of the specific nano-MOSFET on–off intermittency occurs between HIGH and LOW values. One can easily identify the bimodal form of the corresponding time series values’ histogram (Figure 8b), while the two lobes are not completely separated from each other; thus, they “communicate”. If one defines as “separation point” between the two lobes of the distribution of the histogram bin of the smallest statistic ($I_d=1.35\times {10}^{-7} \mathrm{A}$), a straightforward way to obtain a two-state coarse-graining description of the $I_d$ value for the specific experiment is to assign time series values $I_d\ge 1.35\times {10}^{-7} \mathrm{A}$ and $I_d<1.35\times {10}^{-7} \mathrm{A}$ to the symbols “+1” (HIGH value) and “−1” (LOW value), respectively.

As soon as the original real-valued time series was converted into a two-symbol symbolic time series, PNA-STSM analysis was applied (see Section 3.1). The result is presented in Figure 8c, showing that the distribution of the “$k$ waiting lengths” at the central values $\pm 0.5$ of the quantized magnetic field is an exact power law ($p_3=0.00$) with power law exponent $p=\left(p_2=\right)1.02\pm 0.04$. Since the nominal value of the estimated power law exponent is $p=1.02\in \left[\mathrm{1,2}\right)$, we confirm that the dynamics embedded in the $I_d$ symbolic time series are indeed critical dynamics; but notice that at the lower error limit, the tricritical dynamics appears ($p_{-error}=1.02-0.04=0.98\in (\mathrm{0.66,1})$). This entanglement of the critical and the tricritical points is confirmed by the theory of critical phenomena. Indeed, according to the ${\phi }^6$ theory in the parametric space of the Landau free energy, where the coefficients of the ${\phi }^2$ and ${\phi }^4$ terms of the Landau polynomial serve as the parameters, meeting of the lines of the second-order phase transition and the first-order phase transition takes place [16]. Thus, the critical point, which is the endpoint of the second-order phase transition line, meets the tricritical point, which is the starting point of the first-order phase transition line. Therefore, the result obtained by the PNA-STSM indicates exactly the above-mentioned meeting that the theory predicts.

It is also interesting to see the results obtained if one analyzes the specific symbolic time series in the time domain, i.e., by analyzing the waiting times at each of the two symbols separately (see Section 2). The time domain analysis of the waiting times for the “+1” symbol distribution leads to the set of exponents $p_2=0.980\pm 0.027, p_3=0.00$; while for the distribution of the waiting times for the “−1” symbol, one obtains $p_2=1.100\pm 0.032, p_3=0.00$. In both cases, an exact power law fits the waiting times distribution. However, although the probabilities of the appearance of the two symbols are very close, 49% (“+1”) vs. 51% (“−1”), the two estimated power law exponents differ. Importantly, on the one hand, the nominal value of the power law exponent estimated for the “−1” symbol distribution is $p=1.10\in \left[\mathrm{1,2}\right)$, indicating a critical state; on the other hand, the nominal value of the power law exponent estimated for the “+1” symbol distribution is $p=0.98\in \left(\mathrm{0.66,1}\right)$, which denotes a tricritical state. Additionally, considering the error limits of the two estimated $p$ values, one finds that the $p$ value resulting from the “−1” branch is clearly in the critical state, contrary to the one resulting from the “+1” branch that is mainly in the tricritical state, but also marginally enters the critical state. Thus, by means of the time domain waiting times analysis, one cannot definitely answer the question of what are the dynamics to which the drain current fluctuations obey. On the contrary, as already shown above, the PNA-STSM analysis, which simultaneously considers the information from both symbols, provides a clear answer.

4.3. The Case of the Presismic MHz Fracto-Electromagnetic Emission

As has been shown in [31,32], earthquake (EQ) preparation processes present remarkable analogies to thermal systems, as studied through the analysis of FEME/FEMR in the MHz band. Specifically, the spin organization mechanism of the 3D Ising model presents striking similarities, both in qualitative and quantitative terms, with the pre-seismic organization of a strong EQ. As already mentioned in Section 4, in finite thermal systems, the transition from the critical state to the SSB occurs gradually as the temperature drops. The characteristic of this temperature transition zone is the communication of the two lobes of the mean magnetization distribution. As soon as this communication stops, the two lobes completely separate, signifying the completion of the SSB. In the FEME/FEMR time series recorded before the EQ occurrence in the MHz band, there can be found segments of the signal values distributions which are bimodal, while the two lobes communicate with one another, as it happens with the mean magnetization of the 3D Ising model within the zone between the critical state and the SSB.

In the following example, we refer to a shallow, very strong EQ ($M_w=6.9$) which occurred on 14 February 2008 offshore Methoni town (South-West Peloponnese, Greece). Two days before the EQ occurrence, the 41 MHz receiver of the ELSEM-Net (hELlenic Seismo-ElectroMagnetics Network, http://elsemnet.uniwa.gr, (accessed on 8 June 2023) station located in Zakynthos (Zante) Island recorded the 40,000-sec-long time series excerpt depicted in Figure 9a (sampling period = 1 s, the receiver output voltage, $V_E$, is directly related to the received electric field level at 41 MHz). As shown in Figure 9b, the specific time series excerpt presents a bimodal amplitude distribution, while the two lobes are again not completely separated.

Although Figure 9a does not resemble the familiar structure of on–off intermittency encountered in the two previous time series of the examples utilized (Section 4.1 and Section 4.2), one could consider that it presents a kind of on–off intermittency with only a single alternation between two value levels (HIGH and LOW), around which fluctuations occur. If one defines as “separation point” between the two lobes of the distribution of Figure 9b, the histogram bin of the smallest statistic ($V_E=837 \mathrm{m}\mathrm{V}$), a straightforward way to obtain a two-state coarse-graining description of the $V_E$ values could be established. For the specific recording, we assigned time series values $V_E\ge 837 \mathrm{m}\mathrm{V}$ and $V_E<837 \mathrm{m}\mathrm{V}$ to the symbols “+1” (HIGH value) and “−1” (LOW value), respectively. It must be noted that the exact position of the “separation point” does not substantially affect the results.

After converting the real-valued time series of the MHz FEME/FEMR recordings into a two-symbol symbolic time series, the PNA-STSM analysis was applied (see Section 3.1). The result is presented in Figure 9c, showing that the eight first points of the distribution of the “$k$ waiting lengths” at the central values $\pm 0.5$ of the quantized magnetic field can be fitted by a power law with exponent $p=1.36\in \left[\mathrm{1,2}\right)$, which declares the critical state. In studying MHz FEME/FEMR precursors, it is very important to reveal the information embedded in time series excerpts, such as the presented one, that present bimodal amplitude distributions because such time series segments herald the upcoming complete separation of the two lobes. As already mentioned, the complete separation of the two lobes signifies the completion of the SSB, which leads to preferred directions in the organization of the fractures in the focal area [31,32]. In other words, it leads to ruptures that may result in seismic events. In Figure 9d, we present the values’ distribution of such an SBB time series excerpt that was recorded after the critical segment of Figure 9a. The specific SSB excerpt was 18,000-sec-long and was completed just 3.5 h before the examined EQ.

We finally mention that if one tries to analyze the same symbolic time series in the time domain, i.e., by analyzing the waiting times at each of the two symbols separately (see Section 2), a very different, in fact false, picture is drawn. As shown in [33], a power law distribution of the waiting times with exponent $p\in \left(1, 1.5\right)$ indicates that the critical organization is achieved by the domination of the Lévy process, which may result in the occurrence of a strong EQ, while if $p\in \left(\mathrm{1.5,2}\right)$, then a slow transition from the Lévy to the Gaussian process appears, which means that a strong EQ should not be expected to follow. Our “a posteriori” analysis by means of the PNA-STSM yielded a single power law exponent $p=1.36\in \left(1, 1.5\right)$, which is a result in perfect agreement with the fact that a strong EQ indeed occurred after the analyzed MHz FEME/FEMR time series excerpt of Figure 9a. On the contrary, the time domain analysis of the waiting times at the “+1” symbol distribution estimates an exponent $p=1.64\in \left(\mathrm{1.5,2}\right)$, which leads to the false conclusion that a strong EQ should not be expected to follow, whereas time domain analysis of the waiting times at the “−1” symbol results to a waiting times distribution that does not follow any power law. Therefore, PNA-STSM is the only credible way to retrieve the information embedded in MHz FEME/FEMR precursors time series excerpts that present bimodal amplitude distributions, such as the presented one.

5. Conclusions

The results emerging from this work strongly lead to the conclusion that the PNA-STSM can unambiguously and accurately estimate the power law exponent $p$ that characterizes the waiting times of the 3D Ising model at the critical state. Having proved the accuracy of the method, three examples of the application of the PNA-STSM have been presented, demonstrating the way one may apply this method to time series, as long as these follow the dynamics of on–off intermittency and present a bimodal amplitude distribution. In all three cases, it was found that the systems producing the studied time series were in a critical state, which was further leading to an SSB of a second-order phase transition.

Finally, it was shown that, contrary to the standard time domain analysis of the waiting times, the PNA-STSM is capable of providing safe results for the dynamics of any symbolic dynamics of two symbols. This is due to the fact that the probability of the appearance of any of the two symbols is not affected by any imbalance because it simultaneously considers the information from both branches of the symbolic dynamics.