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| In [[physics]], '''self-organized criticality''' ('''SOC''') is a property of (classes of) [[dynamical system]]s which have a [[critical point (physics)|critical point]] as an [[attractor]]. Their macroscopic behaviour thus displays the spatial and/or temporal [[scale invariance|scale-invariance]] characteristic of the [[critical point (physics)|critical point]] of a [[phase transition]], but without the need to tune control parameters to precise values.
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| The concept was put forward by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] ("BTW") in a paper<ref name=Bak1987>
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| {{cite journal
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| | author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]]
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| | year = 1987
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| | title = Self-organized criticality: an explanation of <math>1/f</math> noise
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| | journal = [[Physical Review Letters]]
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| | volume = 59
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| | issue = 4
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| | pages = 381–384
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| | doi = 10.1103/PhysRevLett.59.381
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| | bibcode=1987PhRvL..59..381B
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| }}
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| Papercore summary: [http://papercore.org/Bak1987 http://papercore.org/Bak1987].</ref>
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| published in 1987 in ''[[Physical Review Letters]]'', and is considered to be one of the mechanisms by which [[complexity]]
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| <ref name=Bak1995>
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| {{cite journal
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| | author = [[Per Bak|Bak, P.]], and [[Maya Paczuski|Paczuski, M.]]
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| | year = 1995
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| | title = Complexity, contingency, and criticality
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| | journal =Proc Natl Acad Sci U S A.
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| | volume = 92
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| | pages = 6689–6696
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| | pmid = 11607561
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| | doi = 10.1073/pnas.92.15.6689
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| | issue = 15
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| | pmc = 41396
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| |bibcode = 1995PNAS...92.6689B }}</ref> arises in nature. Its concepts have been enthusiastically 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]]<ref name=LinkenkaerHansen2001>
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| {{cite journal
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| | author = K. Linkenkaer-Hansen, V. V. Nikouline, J. M. Palva, and R. J. Ilmoniemi.
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| | year = 2001
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| | title = Long-Range Temporal Correlations and Scaling Behavior in Human Brain Oscillations
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| | journal = J. Neurosci.
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| | volume = 21
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| | pages = 1370–1377
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| | pmid = 11160408
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| | issue = 4
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| }}</ref><ref name=Beggs2003>
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| {{cite journal
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| | author = J. M. Beggs and D. Plenz
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| | year = 2006
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| | title = Neuronal Avalanches in Neocortical Circuits
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| | journal = J. Neurosci
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| | volume = 23
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| }}</ref><ref name=Chialvo2004>
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| {{cite journal
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| | author =[[Dante R. Chialvo|Chialvo, D. R.]]
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| | year = 2004
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| | title = Critical brain networks
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| | journal = Physica A
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| | volume = 340
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| | issue =4
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| | pages = 756–765
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| | doi = 10.1016/j.physa.2004.05.064
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| |arxiv = cond-mat/0402538 |bibcode = 2004PhyA..340..756C }}</ref><ref name=Boettcher1999>
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| {{cite journal
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| | author = Stefan Boettcher
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| | year = 1999
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| | title = Extremal optimization of graph partitioning at the percolation threshold
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| | journal = J. Phys. A: Math. Gen
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| | volume= 32
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| | doi = 10.1088/0305-4470/32/28/302
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| | pages = 5201–5211
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| | issue = 28
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| | arxiv = cond-mat/9901353
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| |bibcode = 1999JPhA...32.5201B }}</ref><ref name=Fraiman2009>
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| {{cite journal
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| | author = D. Fraiman, P. Balenzuela, J. Foss and D. R. Chialvo
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| | year = 2004
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| | title = Ising-like dynamics in large scale brain functional networks
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| | journal = Physical Review E
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| | volume = 79
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| | issue = 6
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| | pages = 061922
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| | doi = 10.1103/PhysRevE.79.061922
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| |bibcode = 2009PhRvE..79f1922F |arxiv = 0811.3721 }}</ref><ref name=deArcangelis2006>
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| {{cite journal
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| | author = L. de Arcangelis, C. Perrone-Capano, and H. J. Herrmann
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| | year = 2006
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| | title = Self-organized criticality model for brain plasticity
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| | journal = Phys. Rev. Lett.
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| | volume = 96
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| }}</ref><ref name=Poil2012>
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| {{cite journal
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| | pmid = 22815496
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| |date=Jul 2012
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| | author = Poil, SS; Hardstone, R; Mansvelder, HD; Linkenkaer-Hansen, K
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| | title = Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks
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| | volume = 32
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| | issue = 29
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| | pages = 9817–23
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| | doi = 10.1523/JNEUROSCI.5990-11
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| | journal = Journal of Neuroscience
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| }}</ref><ref name=Kitzbichler>
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| {{cite journal
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| | author = Manfred G. Kitzbichler, Marie L. Smith, Søren R. Christensen, Ed Bullmore1
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| | editor1-last = Behrens
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| | year = 2009
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| | editor1-first = Tim
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| | title = Broadband Criticality of Human Brain Network Synchronization
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| | journal = PLoS Comput Biol
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| | volume = 5
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| | doi = 10.1371/journal.pcbi.1000314
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| | pages = e1000314
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| | pmid = 19300473
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| | issue = 3
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| | pmc = 2647739
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| }}</ref>
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| <ref name=Chialvo2010>{{cite journal
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| | author= [[Dante R. Chialvo|Chialvo, D. R.]]
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| | year= 2010
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| | title = Emergent complex neural dynamics
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| | journal= Nature Physics
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| | volume= 6
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| | pages= 744-750
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| | doi = 10.1038/nphys1803
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| }}</ref>
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| <ref name=Tagliazucchi2012>{{cite journal
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| | author= Tagliazucchi E, Balenzuela P, Fraiman D and Chialvo DR.
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| | year= 2012
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| | title = Criticality in large-scale brain fMRI dynamics unveiled by a novel point process analysis
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| | journal= Front. Physiol.
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| | volume= 3
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| | pages= 15
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| | doi = 10.3389/fphys.2012.00015
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| }}</ref>
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| <ref name=Haimovici2013>{{cite journal
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| | author= Haimovici A, Tagliazucchi E, Balenzuela P and Chialvo DR.
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| | year= 2013
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| | title = Brain Organization into Resting State Networks Emerges at Criticality on a Model of the Human Connectome
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| | journal= Physical Review Letters
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| | volume= 110
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| | pages= 178101
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| | doi = 10.1103/PhysRevLett.110.178101
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| | bibcode=2013PhRvL.110q8101H
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| }}</ref> and others.
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| SOC is typically observed in slowly driven [[non-equilibrium thermodynamics|non-equilibrium]] systems with extended [[degrees of freedom (physics and chemistry)|degrees of freedom]] and a high level of [[nonlinearity]]. 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.
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| == Overview ==
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| 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 Horton Conway|John Conway]]'s [[Conway's Game of Life|Game of Life]] and the extensive work of [[Stephen Wolfram]], made it clear that complexity could be generated as an [[emergence|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 transition]]s carried out in the 1960s and 1970s showed how [[scale invariance|scale invariant]] phenomena such as [[fractals]] and [[power law]]s emerged at the [[critical point (physics)|critical point]] between phases.
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| [[Per Bak|Bak]], [[Chao Tang|Tang]] and [[Kurt Wiesenfeld|Wiesenfeld]]'s 1987 paper linked together these factors: a simple [[cellular automaton]] was shown to produce several characteristic features observed in natural complexity ([[fractal]] geometry, [[1/f noise]] and [[power law]]s) in a way that could be linked to [[critical point (physics)|critical-point phenomena]]. Crucially, however, the paper demonstrated 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 behaviour (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 the lab (or lab computer) where it was possible to tune control parameters to precise values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, and important papers on the subject are among the most cited papers in the scientific literature.
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| Due to BTW's metaphorical visualization of their model as a "[[Bak–Tang–Wiesenfeld sandpile|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 exponent]]s), and examination of the necessary conditions 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 class]]es of SOC behaviour and the question of whether it is possible to derive a general rule for determining if an arbitrary [[algorithm]] displays SOC.
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| Alongside these largely lab-based approaches, many other investigations have centred around large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behaviour. 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 invariance|scale-invariant]] behaviour such as the [[Gutenberg–Richter law]] describing the statistical distribution of earthquake sizes, and the [[Aftershock|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 cortex <ref name="Beggs2003" /><ref name=Poil2012 />; 1/f noise in the amplitude envelope of electrophysiological signals <ref name=LinkenkaerHansen2001 />; and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "[[punctuated equilibrium|punctuated equilibria]]" put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). These "applied" investigations of SOC have included both attempts at 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.
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| 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.
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| == Examples of self-organized critical dynamics ==
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| In chronological order of development:
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| * [[Abelian sandpile model|Bak–Tang–Wiesenfeld sandpile]]
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| * [[Forest-fire model]]
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| * [[Olami–Feder–Christensen model]]
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| * [[Bak–Sneppen model]]
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| == See also ==
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| * [[Pink noise|1/f noise]]
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| * [[Complex system]]s
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| * [[Detrended fluctuation analysis]], a method to detect power-law scaling in time series.
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| * [[Fractal]]s
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| * [[Power law]]s
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| * [[Scale invariance]]
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| * [[Self-organization]]
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| * [[Critical exponents]]
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| * [[Ilya Prigogine]], a systems scientist who helped formalize dissipative system behavior in general terms.
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| * [[Red Queen hypothesis]]
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| * [[Self-organized criticality control]]
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| ==References==
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| <references/> | |
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| == Further reading ==
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| * [http://www.poil.dk/s/overview-of-self-organized-criticality/694 An overview of self-organized criticality with many references]
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| * {{cite journal
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| | author = [[Adami, C.]]
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| | year = 1995
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| | title = Self-organized criticality in living systems
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| | journal = [[Physics Letters A]]
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| | volume = 203
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| | issue = 1
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| | pages = 29–32
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| | doi = 10.1016/0375-9601(95)00372-A
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| |bibcode = 1995PhLA..203...29A }}
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| | |
| * {{cite book
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| | author = [[Per Bak|Bak, P.]]
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| | year = 1996
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| | title = How Nature Works: The Science of Self-Organized Criticality
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| | publisher = Copernicus
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| | location = New York
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| | isbn = 0-387-94791-4
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| }}
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| | |
| * {{cite journal
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| | author = [[Per Bak|Bak, P.]] and [[Maya Paczuski|Paczuski, M.]]
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| | year = 1995
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| | title = Complexity, contingency, and criticality
| |
| | journal = [[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the USA]]
| |
| | volume = 92
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| | pages = 6689–6696
| |
| | url = http://pnas.org/cgi/content/abstract/92/15/6689
| |
| | doi = 10.1073/pnas.92.15.6689
| |
| | pmid = 11607561
| |
| | issue = 15
| |
| | pmc = 41396
| |
| |bibcode = 1995PNAS...92.6689B }}
| |
| | |
| * {{cite journal
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| | author = [[Per Bak|Bak, P.]] and [[Kim Sneppen|Sneppen, K.]]
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| | year = 1993
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| | title = Punctuated equilibrium and criticality in a simple model of evolution
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| | journal = [[Physical Review Letters]]
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| | volume = 71
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| | issue = 24
| |
| | pages = 4083–4086
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| | doi = 10.1103/PhysRevLett.71.4083
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| | pmid=10055149
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| | bibcode=1993PhRvL..71.4083B}}
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| | |
| * {{cite journal
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| | author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]]
| |
| | year = 1987
| |
| | title = Self-organized criticality: an explanation of <math>1/f</math> noise
| |
| | journal = [[Physical Review Letters]]
| |
| | volume = 59
| |
| | issue = 4
| |
| | pages = 381–384
| |
| | doi = 10.1103/PhysRevLett.59.381
| |
| | bibcode=1987PhRvL..59..381B}}
| |
| | |
| * {{cite journal
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| | author = [[Per Bak|Bak, P.]], [[Chao Tang|Tang, C.]] and [[Kurt Wiesenfeld|Wiesenfeld, K.]]
| |
| | year = 1988
| |
| | title = Self-organized criticality
| |
| | journal = [[Physical Review A]]
| |
| | volume = 38
| |
| | issue = 1
| |
| | pages = 364–374
| |
| | doi = 10.1103/PhysRevA.38.364
| |
| |bibcode = 1988PhRvA..38..364B }} [http://www.papercore.org/PerBak1987 Papercore summary].
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| | |
| * {{cite book
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| | author = [[Mark Buchanan|Buchanan, M.]]
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| | year = 2000
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| | title = Ubiquity
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| | publisher = Weidenfeld & Nicolson
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| | location = London
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| | isbn = 0-7538-1297-5
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| }}
| |
| | |
| * {{cite book
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| | author = [[Henrik Jeldtoft Jensen|Jensen, H. J.]]
| |
| | year = 1998
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| | title = Self-Organized Criticality
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| | publisher = [[Cambridge University Press]]
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| | location = Cambridge
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| | isbn = 0-521-48371-9
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| }}
| |
| | |
| * {{cite journal
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| | author = Turcotte, D. L.; Smalley, R. F., Jr.; Solla, S. A.
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| | year = 1985
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| | title = Collapse of loaded fractal trees
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| | journal = Nature
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| | url = http://www.nature.com/nature/journal/v313/n6004/abs/313671a0.html
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| | doi= 10.1038/313671a0
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| | volume = 313
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| | issue = 6004
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| | pages = 671|bibcode = 1985Natur.313..671T }}
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| * {{cite journal
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| | author = Smalley, R. F., Jr.; Turcotte, D. L.; Solla, S. A.
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| | year = 1985
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| | title = A renormalization group approach to the stick-slip behavior of faults
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| | journal = Journal of Geophysical Research
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| | bibcode = 1985JGR....90.1894S
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| | doi = 10.1029/JB090iB02p01894
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| | volume = 90
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| | issue = B2
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| | pages = 1894
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| }}
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| | |
| * {{cite journal
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| | author = Katz, J. I.
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| | year = 1986
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| | title = A model of propagating brittle failure in heterogeneous media
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| | journal = Journal of Geophysical Research
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| | bibcode = 1986JGR....9110412K
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| | doi = 10.1029/JB091iB10p10412
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| | volume = 91
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| | issue = B10
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| | pages = 10412
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| }}
| |
| | |
| * {{cite journal
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| | author = Kron, T./Grund, T.
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| | year = 2009
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| | title = Society as a Selforganized Critical System
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| | journal = Cybernetics and Human Knowing
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| | volume = 16
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| | pages = 65–82
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| }}
| |
| | |
| * {{cite journal
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| | author = [[Maya Paczuski|Paczuski, M.]]
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| | year = 2005
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| | title = Networks as renormalized models for emergent behavior in physical systems
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| | journal = ArXiv.org
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| | pages = physics/0502028
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| | arxiv = physics/0502028
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| |bibcode = 2005cmn..conf..363P |doi = 10.1142/9789812701558_0042
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| | series = The Science and Culture Series – Physics
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| | isbn = 978-981-256-525-9 }}
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| | |
| * {{cite book
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| | author = [[Donald L. Turcotte|Turcotte, D. L.]]
| |
| | year = 1997
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| | title = Fractals and Chaos in Geology and Geophysics
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| | publisher = [[Cambridge University Press]]
| |
| | location = Cambridge
| |
| | isbn = 0-521-56733-5
| |
| }}
| |
| | |
| * {{cite journal
| |
| | author = [[Donald L. Turcotte|Turcotte, D. L.]]
| |
| | year = 1999
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| | title = Self-organized criticality
| |
| | journal = [[Reports on Progress in Physics]]
| |
| | volume = 62
| |
| | issue = 10
| |
| | pages = 1377–1429
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| | doi = 10.1088/0034-4885/62/10/201
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| |bibcode = 1999RPPh...62.1377T }}
| |
| * {{cite journal
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| | author = [[Md. Nurujjaman/A. N. Sekar Iyengar]]
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| | year = 2007
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| | title = Realization of {SOC} behavior in a dc glow discharge plasma
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| | journal = [[Physics Letters A]]
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| | volume = 360
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| | pages = 717–721
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| |arxiv = physics/0611069 |bibcode = 2007PhLA..360..717N |doi = 10.1016/j.physleta.2006.09.005 }}
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|
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| *[http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=cond-mat&level=1&index1=20771 Self-organized criticality on arxiv.org]
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| [[Category:Critical phenomena]]
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| [[Category:Applied and interdisciplinary physics]]
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| [[Category:Chaos theory]]
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| [[Category:Self-organization]]
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