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Self-organized criticality (SOC) is a property of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s that have a critical point as an
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
. Their macroscopic behavior thus displays the spatial or temporal
scale-invariance In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical ter ...
characteristic of the critical point of a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
, 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 Tang Chao (; born 1958) is a Chair Professor of Physics and Systems Biology at Peking University. Education He had his undergraduate training at the University of Science and Technology of China, then went to the United States through the CU ...
and Kurt Wiesenfeld ("BTW") in a paper Papercore summary
http://papercore.org/Bak1987
published in 1987 in ''
Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journa ...
'', and is considered to be one of the mechanisms by which
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...
arises in nature. Its concepts have been applied across fields as diverse as
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
,
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
,
evolutionary biology Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life ...
and
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
,
bio-inspired computing Bio-inspired computing, short for biologically inspired computing, is a field of study which seeks to solve computer science problems using models of biology. It relates to connectionism, social behavior, and emergence. Within computer science, b ...
and
optimization (mathematics) Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
,
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
,
sociology Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation an ...
, solar physics,
plasma physics Plasma ()πλάσμα
, Henry George Liddell, R ...
,
neurobiology Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developme ...
and others. SOC is typically observed in slowly driven non-equilibrium systems with many
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
and strongly
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
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 Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...
in nature. For example, the study of
cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
, from the early discoveries of
Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
and
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
through to John Conway's Game of Life and the extensive work of
Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
, 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 Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
's large body of work on
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
showed that much complexity in nature could be described by certain ubiquitous mathematical laws, while the extensive study of
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
s carried out in the 1960s and 1970s showed how
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical ter ...
phenomena such as
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
and
power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
s emerged at the critical point between phases. The term ''self-organized criticality'' was first introduced in Bak,
Tang Tang or TANG most often refers to: * Tang dynasty * Tang (drink mix) Tang or TANG may also refer to: Chinese states and dynasties * Jin (Chinese state) (11th century – 376 BC), a state during the Spring and Autumn period, called Tang (唐) ...
and Wiesenfeld's 1987 paper, which clearly linked together those factors: a simple
cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tesse ...
was shown to produce several characteristic features observed in natural complexity (
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
geometry, pink (1/f) noise and
power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
s) 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 Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suffic ...
'' 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. 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.


Models of self-organized criticality

In chronological order of development: * Stick-slip model of fault failure *
Bak–Tang–Wiesenfeld sandpile The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, ...
* Forest-fire model * Olami–Feder–Christensen model * Bak–Sneppen model 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 Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing ...
s ), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
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 . 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. In addition to the nonconservative theoretical model mentioned above , other theoretical models for SOC have been based upon
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
,
mean field theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...
, the
convergence of random variables In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
, and cluster formation. A continuous model of self-organised criticality is proposed by using tropical geometry. Key theoretical issues yet to be resolved include the calculation of the possible
universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
es of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
displays SOC.


Self-organized criticality in nature

SOC has become established as a strong candidate for explaining a number of natural phenomena, including: * The magnitude of earthquakes (
Gutenberg–Richter law In seismology, the Gutenberg–Richter law (GR law) expresses the relationship between the magnitude and total number of earthquakes in any given region and time period of ''at least'' that magnitude. : \!\,\log_ N = a - b M or : \!\,N = 10^ wh ...
) and frequency of aftershocks ( Omori law) * Fluctuations in economic systems such as
financial markets A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial ma ...
(references to SOC are common in
econophysics Econophysics is a heterodox interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynam ...
) * The evolution of proteins * Forest fires * Neuronal avalanches in the cortex *
Acoustic emission Acoustic emission (AE) is the phenomenon of radiation of acoustic (elastic) waves in solids that occurs when a material undergoes irreversible changes in its internal structure, for example as a result of crack formation or plastic deformation due t ...
from fracturing materials 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. Furthermore, it has been argued that 1/f scaling in EEG recordings are inconsistent with critical states, and whether SOC is a fundamental property of neural systems remains an open and controversial topic.


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. An example of such an optimization problem is
graph coloring In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...
. The SOC process apparently helps the optimization from getting stuck in a
local optimum In applied mathematics and computer science, a local optimum of an optimization problem is a solution that is optimal (either maximal or minimal) within a neighboring set of candidate solutions. This is in contrast to a global optimum, which ...
without the use of any annealing scheme, as suggested by previous work on extremal optimization.


See also

*
1/f noise Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving ...
*
Complex system A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication sy ...
s * Critical brain hypothesis *
Critical exponents Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. * Critical Software, a company specializing i ...
* 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. *
Fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s * Ilya Prigogine, a systems scientist who helped formalize dissipative system behavior in general terms. *
Power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
s * Red Queen hypothesis *
Scale invariance In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
*
Self-organization Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suffic ...
* Self-organized criticality control


References


Further reading

* * * * * *
Papercore summary
* * * * * * * *
Self-organized criticality on arxiv.org
{{refend Critical phenomena Applied and interdisciplinary physics Chaos theory Self-organization