Similarity of fluctuations in correlated systems: The case of seismicity

We report a similarity of fluctuations in equilibrium critical phenomena and nonequilibrium systems, which is based on the concept of natural time. The worldwide seismicity as well as that of the 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., two-dimensional Ising model) as well as in nonequilibrium systems (e.g., three-dimensional turbulent flow).

Figure 1

How a series of seismic events in conventional time (a) is read in the natural time (b). This example refers to the first 11 small earthquakes [cf. the month/date is marked on the horizontal axis in (a)] that occurred after the SES activity recorded on April 18, 1995 and preceded the mainshock $(M=6.6)$ of May 13, 1995.

Figure 2

How the variance ${\kappa }_1$ evolves event by event during the following period: from the detection [31] of the SES activity on April 18, 1995 until the occurrence of the $M=6.6$ mainshock (labeled 18) on May 13, 1995. All the EQs used in the calculation are tabulated in Ref. 30 [cf. the first 11 EQs—out of 18—are those depicted in Fig. 1a].

Figure 3

(Color online) Validity of Eq. (2) for SCEC and Japan. We first determine (see also Appendix ) for SCEC (circles) and Japan (crosses), for each $\varphi$ value, the value ${\Pi }_p\left(\varphi \right)$ at which $P\left[\Pi \left(\varphi \right)\right]$ maximizes. Two such examples are shown in (a) and (b) for $\varphi =0.05$ and $\varphi =0.4$, respectively. In (c), we plot the resulting ${\Pi }_p\left(\varphi \right)$ values vs $\varphi$ for SCEC (circles) and Japan (crosses); the solid line corresponds to Eq. (2). The percentage difference between ${\Pi }_p\left(\varphi \right)$ and ${\Pi }_{th}\left(\varphi \right)$ [obtained from Eq. (2)] is shown in (d).

Figure 4

(Color online) Universality of the probability density function of $\Pi \left(\varphi \right)$ for EQs in the natural time domain. The log-linear plot of $\sigma P\left(X\right)$ vs $(X-⟨X⟩)∕\sigma$, where $X$ stands for $\Pi \left(\varphi \right)$ for $\varphi \approx 0$. Crosses and circles correspond to Japan $(M⩾3.5)$ and SCEC $(M⩾2.0)$, respectively. The inset depicts the corresponding results for the “shuffled” data (black curve) and the original data (red crosses) in Japan. The same graph is obtained for three different regions in Japan (see Fig. 5).

Figure 5

(Color online) The same as Fig. 4, but for the regions A (red), B (green), and C (blue) in Japan (b). A map of these regions is shown in (a).

Figure 6

(Color online) The common feature of fluctuations in different correlated systems. The log-linear plot of $\sigma P\left(X\right)$ vs $(X-⟨X⟩)∕\sigma$ for WWS (triangles), Japan (crosses), and SCEC (circles). The magnitude threshold $M>5.7$ for WWS and Japan (while $M⩾4.0$ for SCEC) was used, see the text. Furthermore, the dotted curve shows the results obtained for the 2D $XY$ model (with [33] inverse Kosterlitz-Thouless transition temperature $K_{KT}\approx 1.2$) $(X=\sqrt{M_x^2+M_y^2}),K=2.0$ for $L=10(N=100)$ which has been shown [1] to describe the experimental results for 3D turbulent flow. The results of the 2D Ising model $K=0.4707$ (while $K_c\approx 0.4407$), either for $L=128$ (dashed) or $L=256$ (solid line), are also plotted.

Figure 7

(Color online) The linear-linear (a) and log-linear (b) plots of $\sigma P\left(X\right)$ vs $(X-⟨X⟩)∕\sigma$, where $X$ stands for $\Pi \left(\varphi \right)$ for $\varphi \approx 0$. The results for Japan $(M⩾3.5)$ and SCEC $(M⩾2.0)$ are plotted along with those deduced from a series of independent and identically distributed ${\left(M_0\right)}_k$ samples from a Poisson distribution with mean values 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, and 200: from the bottom to the top in the maxima appearing in (a), respectively, and from the upper to the lower left branch in (b), respectively.

Figure 8

(Color online) The linear-linear (a) and log-linear (b) plots of $\sigma P\left(X\right)$ vs $(X-⟨X⟩)∕\sigma$, where $X$ stands for $\Pi \left(\varphi \right)$ for $\varphi \approx 0$. The results for Japan $(M⩾3.5)$ and SCEC $(M⩾2.0)$ are plotted along with those deduced from shuffled artificially generated EQ data obeying the Gutenberg-Richter law for various values of the exponent $b=0.5$, 0.7, 0.9, 1.0, 1.1, 1.2, 1.3, 1.5, and 2.0 from the lower to the upper curve at the value $(X-⟨X⟩)∕\sigma \approx -0.5$ in (a), and from the upper to the lower curve at the value $(X-⟨X⟩)∕\sigma \approx -4$ in (b).

Figure 9

(Color online) Two examples of the procedure used in the determination of ${\Pi }_p\left(\varphi \right)$: (a) for SCEC at $\varphi =0.03$, and (b) for Japan at $\varphi =0.4$. The (red) curves in each case show the cubic polynomial fit which was used in the range $[a,b]$ around the maximum. The (red) arrow indicates the position of ${\Pi }_p\left(\varphi \right)$.