Solar flares are commonly accompanied by coronal mass ejections (CME), and thus CMEs display similar size distributions and waiting time distributions as solar flares do. However, some studies report relatively steep power law slopes with values of , which most likely are caused by a bias due to neglecting background subtraction in GOES data. The datasets from LASCO/SOHO are not affected by this background bias, because the white light background from CMEs appears to be sufficiently faint or nonexisting. Waiting time distributions are sampled from a variety of CME and flare catalogs, such as CDAW, LASCO/SOHO, ARTEMIS, CACTus SEEDS, and CORIMP. These waiting time distributions are found to be consistent with the theoretical prediction of the standard FD-SOC model.

Self-organized criticality (SOC) is a theoretical concept that describes the statistics of nonlinear processes. It is a fundamental principle common to many nonlinear dissipative systems in the universe. Due to its ubiquity, SOC is a law of nature, for which we derive a theoretical framework and specific macroscopic physical models. Introduced by Bak, Tang, and Wiesenfeld in 1987, the SOC concept has been applied to laboratory experiments of sandpiles, to human activities such as population growth, language, economy, traffic jams, or wars, to biophysics, geophysics, magnetospheric physics, solar physics, stellar physics, and to galactic physics and cosmology. From an observational point of view, the hallmark of SOC behavior is the power law shape of occurrence frequency distributions of spatial, temporal, and energy scales, implying scale-free nonlinear processes. Power laws are neither a necessary nor a sufficient condition for SOC behavior, because intermittent turbulence produces power law-like size distributions also. A novel trend that is ongoing in current SOC research is a paradigm shift from “microscopic” scales toward “macroscopic” modeling based on physical scaling laws.

Empirically we find that many phenomena in planetary science reveal power law-like size distributions with a power law slope of for differential size distributions, or for cumulative size distributions. These observational results fully agree with theoretical predictions of the FD-SOC model. These observations include lunar craters, Saturn ring particles, near-Earth objects, Jovian Trojans, asteroids, Neptune Trojans, the Kuijper belt, and extrasolar planets. Mars fluvial systems and dust storms reveal fractal structures. Terrestrial gamma-ray flashes indicate also scale-free power law slopes. It appears that the scale-free behavior of planetary phenomena could result from both accumulation and fragmentation processes.

Solar flare hard X-ray events are produced by the electron thick-target bremsstrahlung process at electron energies of ~20 keV. Large statistical samples of hard X-ray fluxes, fluences, energies, flare durations, and waiting times have been observed with instruments from three different spacecraft (HXRBS/SMM, BATSE/CGRO, and RHESSI) from three different solar cycles and analyzed with different automated event detection methods. Despite of this large variety of data, all datasets reveal self-consistent results, for instance, power law peak fluxes with a slope of , which match the theoretical prediction of the fractal-diffusive SOC model, that is, . Systematic errors and uncertainties of these datasets include insufficient fitting ranges, spacecraft orbital data gaps, finite-size effects, south Atlantic anomaly data gaps, instrumental sensitivity, incomplete samples, thresholded event selection, and background subtraction.

A key result of solar flare statistics is the continuity of size distributions over nine orders of magnitude, consisting of nanoflares, microflares, and large flares, covering a range of ~1024–1033 ergs in energy. The FD-SOC model predicts power law distribution functions with a slope of when the energy of flare events are derived from the flare event 2-D area , but a flatter slope of , if the flare energies are derived from the volume-integrated total flux of the 3-D flare volume. These predictions match the observations of EUV nanoflares and microflares. These scaling laws imply more energy is distributed at large flare sizes , and thus, makes nanoflares less important for coronal heating. Such scaling laws are numerically simulated with cellular automaton codes and are applied to the time evolution of coronal loops, magnetic field line breading, and magnetic reconnection processes.

Among black-hole systems, we find a variety with applications of SOC, such as soft gamma-ray repeaters, magnetars, blazars, black holes in accretion disks, and galactic fast radio bursts. Gamma-ray bursts, soft gamma-ray repeaters, as well as black-hole objects, are found to be self-consistent with the theoretical prediction of the FD-SOC mode. Galactic phenomena that possibly have some characteristics in common with SOC models are: fractal galaxy distributions; cosmic ray energy spectrum; extragalactic fast radio bursts; and extragalactic background fluxes.

Research in “complex physics” or “nonlinear physics” is rapidly expanding across various science disciplines, for example, in mathematics, astrophysics, geophysics, magnetospheric physics, plasma physics, biophysics, and sociophysics. What is common among these science disciplines is the concept of “self-organized criticality systems,” which is presented here in detail for observed astrophysical phenomena, such as solar flares, coronal mass ejections, solar energetic particles, solar wind, stellar flares, magnetospheric events, planetary systems, and galactic and black-hole systems. This book explains fundamental questions: Why do power laws, as hallmarks of self-organized criticality, exist? What power law index is predicted for each astrophysical phenomenon? Which size distributions have universality? What can waiting time distributions tell us about random processes? This book is the first monograph that tests comprehensively astrophysical observations of self-organized criticality systems. The highlight of this book is a paradigm shift from microscopic concepts (such as the traditional cellular automaton algorithms) to macroscopic concepts (formulated in terms of physical scaling laws).

The generalized fractal-diffusive SOC model predicts the probability distribution functions for each parameter as a function of the dimensionality, diffusive spreading exponent, fractal dimension, and type of (coherent/incoherent) radiation process. The waiting time distributions are predicted by the FD-SOC model to follow a power law with a slope of during active and contiguously flaring episodes, while an exponential cutoff is predicted for the time intervals of quiescent periods. This dual regime of the waiting time distribution predict both persistence and memory during the active periods, and stochasticity during the quiescent periods. These predictions provide useful constraints of the physical parameters and underlying scaling laws. Significant deviations from the size distributions predicted by the FD-SOC model could indicate problems with the measurements or data analysis. The generic FD-SOC model is considered to have universal validity and explains the statistics and scaling between SOC parameters but does not reveal the detailed physical mechanism that governs the instabilities and energy dissipation in a particular SOC process.

The fractal nature in avalanching systems with SOC is investigated here for phenomena in the solar photosphere and transition region. In the standard SOC model, the fractal Hausdorff dimension is expected to cover the range of [1, 2], with a mean of for 2-D observations projected in the plane-of-sky, and the range of [2, 3], with a mean of for real-world 3-D structures. Observations of magnetograms and with IRIS reveal four groups: (i) photospheric granulation with a low fractal dimension of ; (ii) transition region plages with a low fractal dimension of ; (iii) sunspots at transition region heights with an average fractal dimension of ; and (iv) active regions at photospheric heights with an average fractal dimension of . Phenomena with a low fractal dimension indicate sparse curvilinear flows, while high fractal dimensions indicate near space-filling flows. Investigating the SOC parameters, we find a good agreement for the event areas and mean radiated fluxes in events in transition region plages.

The size distribution of waiting times are found to have an exponential distribution in the case of a stationary Poissonian process. In reality, however, the waiting time distributions reveal power law-like distribution functions, which can be modeled in terms of non-stationary Poisson processes by a superposition of Poissonian distribution functions with time-varying event rates. We model the time evolution of such waiting time distributions by polynomial, sinusoidal, and Gaussian functions, which have exact analytical solutions in terms of the incomplete Gamma function, as well as in terms of the Pareto type-II approximation, which has a power law slope of , where represents the linear time evolution, or with representing nonlinear growth rates, which have a power law slope of . Our mathematical modeling confirms the existence of significant deviations from ideal power law size distributions (of waiting times), but no correlation or significant interval–size relationship exists, as would be expected for a simple (linear) energy storage-dissipation model.

The occurrence frequency distributions (size distributions) are the most important diagnostics for self-organized criticality systems. There are at least three formats for size distributions: (i) the differential size distribution function, (ii) the cumulative size distribution function, and (iii) the rank-order plot. Each of the three formats (or methods) has at least three ranges of event sizes: (i) a range with statistically incomplete sampling; (ii) an inertial range or power law fitting range with statistically complete sampling; and (iii) a range bordering finite system sizes. Only the intermediate range with power law behavior should be used to determine the power law slope from fitting the observed size distributions. The establishment of power law functions in a given observed size distribution depends crucially on the choice of the fitting range, which should have a logarithmic range of at least 2–3 decades. Often the fitted distribution functions exhibit significant deviations from an ideal power law and can be fitted better with alternative functions, such as log-normal distributions, Pareto type-II distributions, and Weibull distributions.

Among stellar systems, we find many with applications of SOC, such as stellar flares or pulsar glitches. Stellar flares occur mostly in the wavelength ranges of ultraviolet, soft X-rays and UV, and in visible light. A breakthrough in new stellar data was accomplished with the Kepler spacecraft, which allowed unprecedented detections of exoplanets, while the same light curves could be searched for large stellar flares. Exploiting these promising new datasets, one finds that most stellar flare datasets exhibit dominant size distributions that converges to a power law slope of , regardless of the star type. The size distributions of pulsar glitches are mostly found outside of the valid range of the Standard FD-SOC model and thus require a different model. Power law fits are not always superior to fits with the log-normal function or Weibull function. This discrepancy between observed and modeled power law slopes in stellar SOC systems is mostly due to small-number statistics of the samples, incomplete sampling near the lower threshold, and due to ill-defined power law fitting ranges, which can cause significant deviations from ideal power laws.