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.
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 first introduced in
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. An alternative view is that SOC appears when the criticality is linked to a value of zero of the control parameters.[10]
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.[1]
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[12][13]), 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 [clarification needed].
It has been argued that the BTW "sandpile" model should actually generate 1/f2 noise rather than 1/f noise.[14] 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.[15]
Other simulation models were proposed later that could produce true 1/f noise.[16]
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.
Self-organized criticality in nature
SOC has become established as a strong candidate for explaining a number of natural phenomena, including:
Despite the numerous applications of SOC to understanding natural phenomena, the universality of SOC theory has been questioned. For example, experiments with real piles of rice revealed their dynamics to be far more sensitive to parameters than originally predicted.[31][1] Furthermore, it has been argued that 1/f scaling in EEG recordings are inconsistent with critical states,[32] and whether SOC is a fundamental property of neural systems remains an open and controversial topic.[33]
Self-organized criticality and optimization
It has been found that the avalanches from an SOC process make effective patterns in a random search for optimal solutions on graphs.[34]
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.
^Laurson L, Alava MJ, Zapperi S (15 September 2005). "Letter: Power spectra of self-organized critical sand piles". Journal of Statistical Mechanics: Theory and Experiment. 0511. L001.