Double power laws, fractals and self-similarity

Abstract

a b s t r a c tPower law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems.In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

Persistence analysis of velocity and temperature fluctuations in convective surface layer turbulence

“…Later, Katul et al 27 also observed the same, when they investigated the persistence PDFs of the burst events in the sensible heat flux in a convective surface layer. Pinto, Lopes, and Tenreiro Machado 28 showed that the double power-law feature in a distribution function is related to the presence of two sets of fractals with two different fractal dimensions associated with two different scale-free processes. However, Narasimha et al 29 found that the persistence PDFs of the momentum flux events in a near-neutral surface layer followed an exponential distribution, suggestive of a Poisson type process.…”

Double Power-Law in Leucosome Width Distribution: Implications for Recognizing Melt Movement in Migmatites

“…In this work we refer to the general shapes of power-law distributions in Figure 1 as single untruncated ( Figure 1A) or truncated ( Figure 1B) power-law distributions, and as double power-law distributions with a positive kink ( Figure 1C) or a negative kink ( Figure 1D). In Earth sciences, double power-law distributions have been shown to describe phenomena such as the severity of tornadoes in the United States (Pinto et al, 2014), the size of FIGURE 1 | Simplified sketch of the different leucosome width distribution types. (A) In migmatites showing this leucosome distribution pattern, leucosome widths follow a single untruncated power-law distribution.…”

“…forest fires in Portugal (Pinto et al, 2014), the daily flow above the annual flow in some hydrological systems (Segura and Pitlick, 2010;Segura et al, 2013), and the seismic moment of global earthquakes (Corral and González, 2019).…”

Analysis of global terrorism dynamics by means of entropy and state space portrait

“…Self-similarity is related to the concept of scale invariance, which is mathematically described by power-law (PL) distributions [47,49,61]. Scale invariance has a statistical meaning, whereas self-similarity is a geometric concept [50]. The fractal dimension is a measure of how much the fractal fills the space as we zoom from larger to smaller scales, for which there are several definitions and, in general, do not coincide [10,21,57].…”

“…Geometric self-similarity is ubiquitous in nature. In fact, many objects can be regarded as natural fractals, namely clouds, coastlines, snowflakes, crystals, blood veins and trees [45,50]. Fractals are usually characterized by statistical indices that measure the geometric complexity of an object, i.e., how the detail in a pattern changes with the scale and reach collective goals, in order to bring change to status quo situations they hate [48].…”