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)(≡<χ(2)> - <χ>(2)) of natural time χ, where <f(χ)> = Σp(k)f(χ(k)) and p(k) is the normalized energy released during the kth event of which the natural time is defined as χ(k) = 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) = 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.

Fig. 1.

The values of κ₁ as a function of dynamic critical exponent z. Various dynamical universality classes are depicted according to their dynamic critical exponent values (see tables IV, VII, IX, and XI of ref. 19). Models A and B correspond to nonconserved or conserved order parameter dynamics as defined by Hohenberg and Halperin (33).

Fig. 2.

(A) Evolution of as a function of the number k of MCS, after an abrupt quench to close but below T_c, up to k = 10⁴. (B) Log–log plot of A. The broken line corresponding to z = 2.165 (see ref. 20) is drawn as a guide to the eye. (C) The evolution of as a function of κ₁ when |M_k| is analyzed in natural time. The average (μ) and the one standard deviation (μ ± σ) values of κ₁ are drawn with the thick and thin lines. The results are obtained by 10³ runs of the model for various L.

Fig. 3.

Centrally fed sandpile. The evolution of κ₁ versus the number of avalanches for D = 2 to D = 7. The initial condition is z_i = 0. The κ₁ values fluctuate around κ₁ = 0.056, 0.064, 0.069, 0.071, 0.073, and 0.075 for D = 2 to 7, respectively. The value of κ₁ = 0.070 is also drawn with the broken horizontal line for the sake of comparison.