Natural time analysis of critical phenomena

Abstract

A quantity exists by which one can identify the approach of a dynamical system to the state of criticality, which is hard to identify otherwise. This quantity is the variance κ 1 ð≡hχ 2 i − hχ i 2 Þ of natural time χ , where hfðχ Þi ¼ ∑ p k f ðχ k Þ and p k is the normalized energy released during the kth event of which the natural time is defined as χ k ¼ k∕N and N stands for the total number of events. Then we show that κ 1 becomes equal to 0.070 at the critical state for a variety of dynamical systems. This holds for criticality models such as 2D Ising and the Bak-Tang-Wiesenfeld sandpile, which is the standard example of self-organized criticality. This condition of κ 1 ¼ 0.070 holds for experimental results of critical phenomena such as growth of rice piles, seismic electric signals, and the subsequent seismicity before the associated main shock.short-term earthquake prediction | dynamic exponent | fractional Gaussian noise | fractional Brownian motion | Burridge-Knopoff "train" model I t has been shown that some unique dynamic features hidden behind can be derived from the time series of complex systems, if we analyze them in terms of natural time χ (1-3). For a time series comprising N events, we define an index for the occurrence of the kth event by χ k ¼ k∕N, which we term natural time. In doing so, we ignore the time intervals between consecutive events, but preserve their order and energy Q k . We, then, study the evolution of the pair (χ k , Q k ) by using the normalized power spectrumdefined by ΦðωÞ ¼ ∑ N k¼1 p k expðiωχ k Þ, where ω stands for the angular natural frequency andis the normalized energy for the kth event. In the time-series analysis using natural time, the behavior of ΠðωÞ at ω close to zero is studied for capturing the dynamic evolution, because all the moments of the distribution of the p k can be estimated from ΦðωÞ at ω → 0 (see ref. 4, p. 499). For this purpose, a quantity κ 1 is defined from the Taylor expansionWe found that this quantity, the variance of natural time χ k , is a key parameter for the distribution of energy within the natural time interval (0,1]. Note that χ k is "rescaled" as natural time changes to χ k ¼ k∕ðN þ 1Þ together with rescaling p k ¼ Q k ∕ ∑ Nþ1 n¼1 Q n upon the occurrence of any additional event. It has been demonstrated that this analysis enables recognition of the complex dynamic system under study entering the critical stage (1-3). This occurs when the variance κ 1 converges to 0.070. Originally the condition κ 1 ¼ 0.070 for the approach to criticality was theoretically derived for the seismic electric signals (SES) (1, 2), which are transient low frequency (≤1 Hz) electric signals that have been repeatedly observed before earthquakes (3,5,6). The experimental data showed that κ 1 obtained from SES activities in Greece and Japan attain the value 0.070 (1-3, 7-10). The emission of SES was attributed to a phase transition of second order. It was also shown empirically that the same condition κ 1 ¼ 0.070 holds for other time series, including turbule...

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