Self-organized criticality

Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards criticality.
The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper 
published in 1987 in "Physical Review Letters", and is considered to be one of the mechanisms by which complexity arises in nature. Its concepts have been applied across fields as diverse as geophysics, physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology and others.
SOC is typically observed in slowly driven non-equilibrium systems with many degrees of freedom and strongly nonlinear dynamics. Many individual examples have been identified since BTW's original paper, but to date there is no known set of general characteristics that "guarantee" a system will display SOC.
Overview.
Self-organized criticality is one of a number of important discoveries made in statistical physics and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of complexity in nature. For example, the study of cellular automata, from the early discoveries of Stanislaw Ulam and John von Neumann through to John Conway's Game of Life and the extensive work of Stephen Wolfram, made it clear that complexity could be generated as an emergent feature of extended systems with simple local interactions. Over a similar period of time, Benoît Mandelbrot's large body of work on fractals showed that much complexity in nature could be described by certain ubiquitous mathematical laws, while the extensive study of phase transitions carried out in the 1960s and 1970s showed how scale invariant phenomena such as fractals and power laws emerged at the critical point between phases.
The term "self-organized criticality" was firstly introduced by Bak, Tang and Wiesenfeld's 1987 paper, which clearly linked together those factors: a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, pink (1/f) noise and power laws) in a way that could be linked to critical-point phenomena. Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, "self-organized" criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be "spontaneous"—and therefore plausible as a source of natural complexity—rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature.
Due to BTW's metaphorical visualization of their model as a "sandpile" on which new sand grains were being slowly sprinkled to cause "avalanches", much of the initial experimental work tended to focus on examining real avalanches in granular matter, the most famous and extensive such study probably being the Oslo ricepile experiment. Other experiments include those carried out on magnetic-domain patterns, the Barkhausen effect and vortices in superconductors.
Early theoretical work included the development of a variety of alternative SOC-generating dynamics distinct from the BTW model, attempts to prove model properties analytically (including calculating the critical exponents), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether conservation of energy was required in the local dynamical exchanges of models: the answer in general is no, but with (minor) reservations, as some exchange dynamics (such as those of BTW) do require local conservation at least on average. In the long term, key theoretical issues yet to be resolved include the calculation of the possible universality classes of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary algorithm displays SOC.
Alongside these largely lab-based approaches, many other investigations have centered around large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake size, and the Omori law describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; neuronal avalanches in the cortex; 1/f noise in the amplitude of electrophysiological signals; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). These "applied" investigations of SOC have included both modelling (either developing new models or adapting existing ones to the specifics of a given natural system) and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
In addition, SOC has been applied to computational algorithms. Recently, it has been found that the avalanches from an SOC process, like the BTW model, make effective patterns in a random search for optimal solutions on graphs. 
An example of such an optimization problem is graph coloring. The SOC process apparently helps the optimization from getting stuck in a local optimum without the use of any annealing scheme, as suggested by previous work on extremal optimization.
The recent excitement generated by scale-free networks has raised some interesting new questions for SOC-related research: a number of different SOC models have been shown to generate such networks as an emergent phenomenon, as opposed to the simpler models proposed by network researchers where the network tends to be assumed to exist independently of any physical space or dynamics. While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins.
These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.
Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model. However,
it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise.
This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models 
generally produce 1/fa spectra, with a<2.
Other simulation models were proposed later that could produce true 1/f noise, and experimental sandpile models were observed to yield 1/f noise. In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory, 
mean field theory,
the convergence of random variables,
and cluster formation. A continuous model of self-organised criticality is proposed by using tropical geometry.
Examples of self-organized critical dynamics.
In chronological order of development: