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.

From the statistics of solar radio bursts, we learn that we can discriminate between three diagnostic regimes: (i) the incoherent regime where the radio burst flux is essentially proportional to the flare volume (with a power law slope of ), as it occurs for gyroemission, gyroresonance emission, gyrosynchrotron emission; (ii) the coherent regime that implies a nonlinear scaling between the radio flux and the flare volume ; as it occurs for the electron beam instability, the loss-cone instability, or maser emission; and (iii) the exponential regime that does not display a power law function, but rather an exponential cutoff as expected for random noise distributions. Thus, the power law slopes offer a useful diagnostic to verify the flux–volume scaling law and to discriminate between coherent and incoherent radio emission processes, as well as to distinguish between SOC processes and non-SOC processes. An additional diagnostic comes from the inertial range of power law fits: SOC-related power law size distributions should extend over multiple decades, while power law ranges of less than one decade are most likely not related to SOC processes.

Can we claim that the dynamics of the solar wind is consistent with a SOC system? Observationally we find that magnetic field and kinetic energy fluctuations measured in the solar wind exhibit power law distributions, which is consistent with a SOC system. What about the driver, instability, and avalanches expected in a SOC system? The driver mechanism is the acceleration of the solar wind in the solar corona itself, a process that basically follows the hydrodynamic model of Parker (1958), and may be additionally complicated by the presence of nonlinear wave–particle interactions, such as ion-cyclotron resonance. Then, the instability threshold, triggering extreme bursts of magnetic field fluctuations, the avalanches of solar wind SOC events, can be caused by dissipation of Alfven waves, onset of turbulence, or by the ion-cyclotron instability. Thus, in principle the generalized SOC concept can be applied to the solar wind, if there is a system-wide threshold for an instability that causes extreme magnetic field fluctuations.

The size distribution of solar energetic particle (SEP) events, which represent a more energetic subset than flare events, is mostly found to follow power law distribution functions, rather than Poissonian random distribution functions. However, the numerical value of the power law slope is generally flatter than the slopes of the flare size distributions in hard X-rays, soft X-rays, and EUV (Hudson 1978), which can be explained in at least four different ways: (i) normal flares and proton flares are produced by two fundamentally different acceleration mechanisms; (ii) proton flares behave differently than normal flares; (iii) the fractal dimensionality of SEP events is different from normal flares; (iv) proton flares are subject to a selection bias toward the most energetic events and thus are not a representative sample of large flares. Nevertheless, the standard fractal-diffusive SOC model can explain the observed slopes of SEP size distributions, but observations reveal deviations from straight power law functions, or broken power law slopes, and thus are not unique and need to be modeled in more detail.

We focus on the statistics of SOC-related solar flare parameters in soft X-ray wavelengths, including their size and waiting time distributions. An early SOC model assumed a linear increase of the energy storage, but this pioneering model is not consistent with the expected correlation between the waiting time interval and the subsequently dissipated energy. The Neupert effect in solar flares implies a correlation between the hard X-ray fluence and the soft X-ray flux, which predicts identical size distributions for these two parameters. Quantifying of thermal flare energies in soft X-ray emitting plasma needs also to include radiative and conductive losses. The intermittency and bursty variability of the solar dynamo implies a nonstationary SOC driver, which yields a universal value for the power law slope of fluxes, but the power law slopes of waiting times vary with the flare rate. While our focus encompasses primarily SOC models, alternative models in terms of MHD turbulence can explain some characteristics of SOC features also, such as size distribution functions, Fourier spectra, and structure functions.