Abstract
Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.
Highlights
About 25 years ago, the concept of self-organized criticality (SOC) emerged (Bak et al 1987), initially envisioned to explain the ubiquitous 1/f -power spectra, which can be characterized by a powerlaw function P (ν) ∝ ν−1
There exist some previous similar concepts in complexity theory, such as phase transitions, turbulence, percolation, or branching theory, the SOC concept seems to have the broadest scope and the most general applicability to phenomena with nonlinear energy dissipation in complex systems with many degrees of freedom. There is no such thing as a single “SOC theory”, but we rather deal with various SOC concepts, which in some cases have been developed into more rigorous quantitative SOC models that can be tested with real-world data
Computer simulations of the BTW type provide toy models that can mimic complexity phenomena, but they generally lack the physics of real-world SOC phenomenona, because their discretized lattice grids do not reflect in any way the microscopic atomic or subatomic structure of real-world physical systems
Summary
About 25 years ago, the concept of self-organized criticality (SOC) emerged (Bak et al 1987), initially envisioned to explain the ubiquitous 1/f -power spectra, which can be characterized by a powerlaw function P (ν) ∝ ν−1. While white noise represents traditional random processes with uncorrelated fluctuations, 1/f power spectra are a synonym for time series with nonrandom structures that exhibit long-range correlations These non-random time structures represent the avalanches in Bak’s paradigm of sandpiles. Bak’s seminal paper in 1987 triggered a host of numerical simulations of sandpile avalanches, which all exhibit powerlaw-like size distributions of avalanche sizes and durations. These numerical simulations were, most commonly, cellular automata in the language of complexity theory, which are able to produce complex spatio-temporal patterns by iterative application of a simple mathematical redistribution rule. An introduction and exhaustive description of cellular automaton models that simulate SOC systems is given in Pruessner (2012, 2013), and a review of cellular automaton models applied to solar physics is given in Charbonneau et al (2001)
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