Self-organized criticality

In physics, 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 and/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 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 and others.

SOC is typically observed in slowly driven non-equilibrium systems with extended 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.

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. However, the term Self-Organized Criticality was firstly introduced by Bak, Tang and Wiesenfeld's 1987 paper which clearly linked together these 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 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.

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 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 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 sizes and the Omori law describing the frequency of aftershocks, and where models that displayed SOC were proposed and analyzed prior to the BTW 87 paper;); 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; 1/f noise in the amplitude envelope 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 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.

In addition, SOC has been linked to computation. 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, by far the most extensive data concern the solvent-accessible surface areas of globular proteins, plotted for modular segments centered on a specific amino acid, against lengths L, with 9 <= L <= 35. Because this fractal linearity holds for all 20 amino acids over the same range, it exists independently of any model. It can be regarded as the basic mechanism of life. 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 mathematical mechanisms. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model. However, this model was subsequently shown to actually generate 1/f² noise rather than 1/f noise. 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:

• Bak–Tang–Wiesenfeld sandpile
• Forest-fire model
• Olami–Feder–Christensen model
• Bak–Sneppen model

See also

• 1/f noise
• Complex systems
• Critical brain hypothesis
• Critical exponents
• Detrended fluctuation analysis, a method to detect power-law scaling in time series.
• Dual-phase evolution, another process that contributes to self-organization in complex systems.
• Fractals
• Ilya Prigogine, a systems scientist who helped formalize dissipative system behavior in general terms.
• Power laws
• Red Queen hypothesis
• Scale invariance
• Self-organization
• Self-organized criticality control

Further reading

• Adami, C. (1995). "Self-organized criticality in living systems". Physics Letters A. 203 (1): 29–32. arXiv:adap-org/9401001. Bibcode:1995PhLA..203...29A. CiteSeerX 10.1.1.456.9543. doi:10.1016/0375-9601(95)00372-A.

• Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus. ISBN 978-0-387-94791-4.

• Bak, P. and Paczuski, M. (1995). "Complexity, contingency, and criticality". Proceedings of the National Academy of Sciences of the USA. 92 (15): 6689–6696. Bibcode:1995PNAS...92.6689B. doi:10.1073/pnas.92.15.6689. PMC 41396. PMID 11607561.CS1 maint: Multiple names: authors list (link)

• Bak, P. and Sneppen, K. (1993). "Punctuated equilibrium and criticality in a simple model of evolution". Physical Review Letters. 71 (24): 4083–4086. Bibcode:1993PhRvL..71.4083B. doi:10.1103/PhysRevLett.71.4083. PMID 10055149.CS1 maint: Multiple names: authors list (link)

• Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of $1/f$ noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. PMID 10035754.CS1 maint: Multiple names: authors list (link)

• Bak, P., Tang, C. and Wiesenfeld, K. (1988). "Self-organized criticality". Physical Review A. 38 (1): 364–374. Bibcode:1988PhRvA..38..364B. doi:10.1103/PhysRevA.38.364.CS1 maint: Multiple names: authors list (link) Papercore summary.

• Buchanan, M. (2000). Ubiquity. London: Weidenfeld & Nicolson. ISBN 978-0-7538-1297-6.

• Jensen, H. J. (1998). Self-Organized Criticality. Cambridge: Cambridge University Press. ISBN 978-0-521-48371-1.

• Katz, J. I. (1986). "A model of propagating brittle failure in heterogeneous media". Journal of Geophysical Research. 91 (B10): 10412. Bibcode:1986JGR....9110412K. doi:10.1029/JB091iB10p10412.

• Kron, T./Grund, T. (2009). "Society as a Selforganized Critical System". Cybernetics and Human Knowing. 16: 65–82.CS1 maint: Multiple names: authors list (link)

• Paczuski, M. (2005). Networks as renormalized models for emergent behavior in physical systems. Complexity. The Science and Culture Series – Physics. pp. 363–374. arXiv:physics/0502028. Bibcode:2005cmn..conf..363P. CiteSeerX 10.1.1.261.9886. doi:10.1142/9789812701558_0042. ISBN 978-981-256-525-9.

• Turcotte, D. L. (1997). Fractals and Chaos in Geology and Geophysics. Cambridge: Cambridge University Press. ISBN 978-0-521-56733-6.

• Turcotte, D. L. (1999). "Self-organized criticality". Reports on Progress in Physics. 62 (10): 1377–1429. Bibcode:1999RPPh...62.1377T. doi:10.1088/0034-4885/62/10/201.
• Md. Nurujjaman/A. N. Sekar Iyengar (2007). "Realization of {SOC} behavior in a dc glow discharge plasma". Physics Letters A. 360 (6): 717–721. arXiv:physics/0611069. Bibcode:2007PhLA..360..717N. doi:10.1016/j.physleta.2006.09.005.
• Self-organized criticality on arxiv.org