Self-organized criticality
Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, self-organized criticality (SOC) is a property of (classes of) dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s which have a critical point as an attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance
Scale invariance
In physics and mathematics, 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...

 characteristic of the critical point of a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

, but without the need to tune control parameters to precise values.

The concept was put forward by Per Bak
Per Bak
Per Bak was a Danish theoretical physicist who coauthored the 1987 academic paper that coined the term "self-organized criticality."- Life and work :...

, Chao Tang
Chao Tang
Chao Tang is a Chinese physicist and professor at the University of California at San Francisco.In 1987, as a post-doctoral research scientist in the Solid State Theory Group of Brookhaven National Laboratory, he and another fellow post-doctoral scientist, Kurt Wiesenfeld, along with their mentor,...

 and Kurt Wiesenfeld
Kurt Wiesenfeld
Kurt Wiesenfeld is an American physicist working primarily on non-linear dynamics. His works primarily concern stochastic resonance, spontaneous synchronization of coupled oscillators, and non-linear laser dynamics...

 ("BTW") in a paper published in 1987 in Physical Review Letters
Physical Review Letters
Physical Review Letters , established in 1958, is a peer reviewed, scientific journal that is published 52 times per year by the American Physical Society...

, and is considered to be one of the mechanisms by which complexity
Complexity
In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...


arises in nature. Its concepts have been enthusiastically applied across fields as diverse as geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...

, physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

, evolutionary biology and ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

, bio-inspired computing and optimization (mathematics)
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

, sociology
Sociology
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...

, solar physics
Solar physics
For the physics journal, see Solar Physics Solar physics is the study of our Sun. It is a branch of astrophysics that specializes in exploiting and explaining the detailed measurements that are possible only for our closest star...

, plasma physics, neurobiology and others.

SOC is typically observed in slowly-driven non-equilibrium
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...

 systems with extended degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 and a high level of nonlinearity
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...

. 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
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...

 and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of complexity
Complexity
In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...

 in nature. For example, the study of cellular automata, from the early discoveries of Stanislaw Ulam and John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 through to John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

's Game of Life
Conway's Game of Life
The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970....

 and the extensive work of Stephen Wolfram
Stephen Wolfram
Stephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...

, made it clear that complexity could be generated as an emergent
Emergence
In philosophy, systems theory, science, and art, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems....

 feature of extended systems with simple local interactions. Over a similar period of time, Benoît Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

'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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

s carried out in the 1960s and 1970s showed how scale invariant
Scale invariance
In physics and mathematics, 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...

 phenomena such as fractals and power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

s emerged at the critical point between phases.

Bak
Per Bak
Per Bak was a Danish theoretical physicist who coauthored the 1987 academic paper that coined the term "self-organized criticality."- Life and work :...

, Tang
Chao Tang
Chao Tang is a Chinese physicist and professor at the University of California at San Francisco.In 1987, as a post-doctoral research scientist in the Solid State Theory Group of Brookhaven National Laboratory, he and another fellow post-doctoral scientist, Kurt Wiesenfeld, along with their mentor,...

 and Wiesenfeld
Kurt Wiesenfeld
Kurt Wiesenfeld is an American physicist working primarily on non-linear dynamics. His works primarily concern stochastic resonance, spontaneous synchronization of coupled oscillators, and non-linear laser dynamics...

's 1987 paper linked together these factors: a simple cellular automaton
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...

 was shown to produce several characteristic features observed in natural complexity (fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 geometry, 1/f noise and power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

s) in a way that could be linked to 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.

Due to BTW's metaphorical visualization of their model as a "sandpile
Bak-Tang-Wiesenfeld sandpile
In physics, the Bak–Tang–Wiesenfeld sandpile model is the first discovered example of a dynamical system displaying self-organized criticality and is named after Per Bak, Chao Tang and Kurt Wiesenfeld....

" 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
Barkhausen effect
The Barkhausen effect is a name given to the noise in the magnetic output of a ferromagnet when the magnetizing force applied to it is changed...

 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
Critical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...

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
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

 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 behaviour and the question of whether it is possible to derive a general rule for determining if an arbitrary algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 displays SOC.

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-invariant
Scale invariance
In physics and mathematics, 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...

 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-invariant
Scale invariance
In physics and mathematics, 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...

 behaviour such as the Gutenberg-Richter law
Gutenberg-Richter law
In seismology, the Gutenberg–Richter law expresses the relationship between the magnitude and total number of earthquakes in any given region and time period of at least that magnitude.orWhere:...

 describing the statistical distribution of earthquake sizes, and the Omori law
Aftershock
An aftershock is a smaller earthquake that occurs after a previous large earthquake, in the same area of the main shock. If an aftershock is larger than the main shock, the aftershock is redesignated as the main shock and the original main shock is redesignated as a foreshock...

 describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics
Econophysics
Econophysics is an 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 dynamics...

); landscape formation; forest fires; landslides; epidemics; 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
Niles Eldredge
Niles Eldredge is an American paleontologist, who, along with Stephen Jay Gould, proposed the theory of punctuated equilibrium in 1972.-Education:...

 and Stephen Jay Gould
Stephen Jay Gould
Stephen Jay Gould was an American paleontologist, evolutionary biologist, and historian of science. He was also one of the most influential and widely read writers of popular science of his generation....

). Worryingly, given the implications of a scale-free
Scale invariance
In physics and mathematics, 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...

 distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. 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.

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.

Realistic applications of SOC rely on exponents derived from assumptions that are often oversimplified. A promising exception to this limitation is the discovery by Marcelo Moret and Gilney Zebende that the globular compaction of protein chains by hydrophobic interactions is describable in non-Euclidean terms, with fractal critical exponents inferred from the evolution of solvent-accessible surface areas in chain segments of increasing length. Protein amino acid chains are conventionally represented by strings of letters, and their similarities quantified by counting letter identities in protein "words" (segments). Replacing the (alpha) letters with (alphanumeric) exponents enables non-Euclidean (contextual) analysis of universal water-mediated protein interactions. Membrane viral fusion proteins are ideal targets for SOC analysis, as they exhibit up to ~60% (hundreds of) mutations in public data bases.

Examples of self-organized critical dynamics

In chronological order of development:
  • Bak-Tang-Wiesenfeld sandpile
    Bak-Tang-Wiesenfeld sandpile
    In physics, the Bak–Tang–Wiesenfeld sandpile model is the first discovered example of a dynamical system displaying self-organized criticality and is named after Per Bak, Chao Tang and Kurt Wiesenfeld....

  • Forest-fire models
  • Olami-Feder-Christensen model
    Olami-Feder-Christensen model
    In physics, in the area of dynamical systems, the Olami–Feder–Christensen model is an earthquake model conjectured to be an example of self-organized criticality where local exchange dynamics are not conservative...

  • Bak-Sneppen model
    Bak-Sneppen model
    The Bak-Sneppen model is a simple model of co-evolution between interacting species. It was developed to show how self-organized criticality may explain key features of the fossil record, such as the distribution of sizes of extinction events and the phenomenon of punctuated equilibrium...


See also

  • 1/f noise
  • Complex system
    Complex system
    A complex system is a system composed of interconnected parts that as a whole exhibit one or more properties not obvious from the properties of the individual parts....

    s
  • Fractal
    Fractal
    A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

    s
  • Power law
    Power law
    A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

    s
  • Scale invariance
    Scale invariance
    In physics and mathematics, 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...

  • Self-organization
    Self-organization
    Self-organization is the process where a structure or pattern appears in a system without a central authority or external element imposing it through planning...

  • Critical exponents
  • Ilya Prigogine
    Ilya Prigogine
    Ilya, Viscount Prigogine was a Russian-born naturalized Belgian physical chemist and Nobel Laureate noted for his work on dissipative structures, complex systems, and irreversibility.-Biography :...

    , a systems scientist who helped formalize dissipative system behavior in general terms.
  • Red Queen effect

Further reading

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