![]() The numerical algorithms of cellular automata are extremely simple, basically a one-liner that defines the redistribution rule, with an iterative loop around it, but can produce the most complex dynamical patterns, similar to the beautiful geometric patterns created by Mandelbrot’s fractal algorithms (Mandelbrot 1977, 1983, 1985). 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. Consequently, 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 non-random time structures represent the avalanches in Bak’s paradigm of sandpiles. While white noise represents traditional random processes with uncorrelated fluctuations, 1/ f power spectra are a synonym for time series with non-random structures that exhibit long-range correlations. The term 1/ f power spectra or flicker noise should actually be understood in broader terms, including power spectra with pink noise ( P( ν)∝ ν −1), red noise ( P( ν)∝ ν −2), and black noise ( P( ν)∝ ν −3), essentially everything except white noise ( P( ν)∝ ν 0). 1987), initially envisioned to explain the ubiquitous 1/ f-power spectra, which can be characterized by a powerlaw function P( ν)∝ ν −1. ![]() 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.Ībout 25 years ago, the concept of self-organized criticality (SOC) emerged (Bak et al. 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. ![]() ![]() Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. 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. 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. ( 1987), the idea has been applied to solar physics, in “Avalanches and the Distribution of Solar Flares” by Lu and Hamilton ( 1991). Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/ f noise” by Bak et al.
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