Attempt to distinguish the origins of self-similarity by natural time analysis

TL;DR

This paper investigates the origins of self-similarity in complex systems using natural time analysis, distinguishing between process memory and infinite variance of increments, and relates findings to phenomena like earthquakes and solar flares.

Contribution

It introduces a method to differentiate the sources of self-similarity using natural time and analyzes empirical data to establish correlations with power-law exponents.

Findings

Seismic electric signals exhibit long-range temporal correlations.

A relationship between the power-law exponent b3 and 1 is identified.

The entropy maximizes around b3  1.6 to 1.7, consistent with various natural phenomena.

Abstract

Self-similarity may originate from two origins, i.e., the process memory and the process' increments ``infinite'' variance. A distinction is attempted by employing the natural time \chi. Concerning the first origin, we analyze recent data on Seismic Electric Signals, which support the view that they exhibit infinitely ranged temporal correlations. Concerning the second, slowly driven systems that emit bursts of various energies E obeying power-law distribution, i.e., P(E) ~ E^-\gamma, are studied. An interrelation between the exponent \gamma and the variance \kappa_1(= <\chi^2> - <\chi>^2) is obtained for the shuffled (randomized) data. In the latter, the most probable value of \kappa_1 is approximately equal to that of the original data. Finally, it is found that the differential entropy associated with the probability P(\kappa_1) maximizes for \gamma around 1.6 to 1.7, which is comparable to the value determined experimentally in diverse phenomena, e.g., solar flares, icequakes, dislocation glide in stressed single crystals of ice e.t.c. It also agrees with the b-value in the Gutenberg-Richter law of earthquakes.