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We report a similarity of fluctuations in equilibrium critical phenomena and non-equilibrium systems, which is based on the concept of natural time. The world-wide seismicity as well as that of San Andreas fault system and Japan are analyzed. An order parameter is chosen and its fluctuations relative to the standard deviation of the distribution are studied. We find that the scaled distributions fall on the same curve, which interestingly exhibits, over four orders of magnitude, features similar to those in several equilibrium critical phenomena ( e.g., 2D Ising model) as well as in non-equilibrium systems (e.g., 3D turbulent flow).

Three types of electric signals were analyzed: Ion current fluctuations in membrane channels (ICFMC), Seismic electric signals activities (SES), and "artificial" noises (AN). The wavelet transform, when applied to the conventional time domain, does not allow a classification of these signals, but does so in the "natural" time domain. A classification also becomes possible, if we study -(q) versus q, where chi stands for the "natural" time. For q values approximately between 1 and 2 the signals are classified and ICFMC lies between the other two types. For q=1, the "entropy" S identical with -ln of ICFMC almost equals that of a "uniform" distribution, while the AN and SES have larger and smaller S values, respectively. The recent [P. Varotsos, N. Sarlis, and E. Skordas, Phys. Rev. E 67, 021109 (2003)] finding that, in short time scales, both SES and AN (which are shown to be non-Markovian) result in comparable detrended fluctuation analysis exponents alpha in (1.0,1.5) is revisited. Even a Markovian dichotomous time series, in short time scales, leads to similar alpha exponents.

The Smoluchowski-Chapman-Kolmogorov functional equation is applied to the electric signals that precede rupture. The results suggest a non-Markovian character of the analyzed data. The rescaled range Hurst and detrended fluctuation analyses, as well as that related with the "mean distance a walker spanned," lead to power-law exponents, which are consistent with the existence of long-range correlations. A "universality" in the power spectrum characteristics of these signals emerges, if an analysis is made (not in the conventional time frame, but) in the "natural" time domain. Within this frame, it seems that certain power spectrum characteristics of ion current fluctuations in membrane channels distinguish them from the electric signals preceding rupture. The latter exhibit a behavior compatible with that expected from a model based on the random field Ising Hamiltonian at the critical point.

It has been shown that some dynamic features hidden in the time series of complex systems can be uncovered if we analyze them in a time domain called natural time χ . The order parameter of seismicity introduced in this time domain is the variance of χ weighted for normalized energy of each earthquake. Here, we analyze the Japan seismic catalog in natural time from January 1, 1984 to March 11, 2011, the day of the M9 Tohoku earthquake, by considering a sliding natural time window of fixed length comprised of the number of events that would occur in a few months. We find that the fluctuations of the order parameter of seismicity exhibit distinct minima a few months before all of the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-y period in the Japanese area. Among the minima, the minimum before the M9 Tohoku earthquake was the deepest. It appears that there are two kinds of minima, namely precursory and nonprecursory, to large earthquakes.criticality | seismic electric signals F or a time series comprised of N events, we define the natural time for the occurrence of the kth event by χ k = k=N (1), which means that we ignore the time intervals between consecutive events, but preserve their order. We also preserve their energy Q k . We then study the evolution of the pairðχ k ; p k Þ, whereQ n is the normalized energy. We postulated that the approach of a dynamical system to criticality can be identified by the variance κ 1 of natural time χ weighted for p k , namely,Earthquakes (EQs hereafter) exhibit complex correlations in time, space, and magnitude, and the opinion prevails (e.g., ref. 2 and references therein) that the EQs are critical phenomena. In natural time analysis of seismicity, the quantity κ 1 calculated from seismic catalogs serves as an order parameter (3, 4). Experiences have shown that the mainshock occurs in a few days to 1 wk after the κ 1 value in the candidate epicentral area approaches 0.070 (5). This was found useful in narrowing the lead time of EQ prediction. However, to trace the time evolution of κ 1 value, one needs to start the analysis of the seismic catalog at some time before the yet-to-occur mainshock. We chose, for the starting time for analysis, the initiation time of seismic electric signal (SES) activity. SESs are low-frequency (≤1 Hz) electric signals that precede EQs (6). The reason for this choice was based on the consideration that SESs are emitted when the focal zone enters the critical stage (7). In the case of the lack of SES data, as in the Tohoku EQ, we cannot adopt this approach. In this study, therefore, we instead examine the fluctuations of κ 1 near criticality, i.e., near the EQ occurrence. To compute the fluctuations, we apply the following procedure.First, take an excerpt comprised of W (≥100) successive EQs from the seismic catalog. We then form its subexcerpts consisting of the nth to (n + 5)th EQs, (n = 1, 2,. . .,W-5) and compute κ 1 for each of them. In so doing, we assign χ k = k=6 and the normalized energy p k = Q k = X 6 n = 1

Electric signals have been recently recorded at the Earth's surface with amplitudes appreciably larger than those hitherto reported. Their entropy in natural time is smaller than that, S u , of a "uniform" distribution. The same holds for their entropy upon time-reversal. This behavior, as supported by numerical simulations in fBm time series and in an on-off intermittency model, stems from infinitely ranged long range temporal correlations and hence these signals are probably Seismic Electric Signals (critical dynamics). The entropy fluctuations are found to increase upon approaching bursting, which reminds the behavior identifying sudden cardiac death individuals when analysing their electrocardiograms.

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