Natural time analysis of critical phenomena.

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 = 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.