Paths to self-organized criticality
TL;DR: In this paper, a pedagogical introduction to self-organized criticality (SOC) is presented, which unravels its connections with nonequilibrium phase transitions, showing that SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density.
Abstract: We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system; two commonly used methods are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density. Other paths to SOC, in driven interfaces, the Bak-Sneppen model, and self- organized directed percolation, are also examined. We review the status of experimental realizations of SOC in light of these observations.
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TL;DR: In this article, a review of recent developments in non-equilibrium statistical physics is presented, focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail.
Abstract: This review addresses recent developments in non-equilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic non-equilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.
1,475 citations
Cites background from "Paths to self-organized criticality..."
...Recently, it was pointed out [313–315] that SOC is in fact closely related to ordinary phase transitions into (infinitely many) absorbing states (for a survey see [238])....
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...An interesting example is the activated random walk of n particles [238]....
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...[238], continuous phase transitions into absorbing states can only be observed if the inertia of particles can be neglected....
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...[238], any system with conserved local dynamics and a continuous absorbing-states transition can be converted into a SOC model by (1) adding a process for increasing the density in infinitesimal steps, and (2) implementing a process for decreasing the density infinitesimally while the system is active....
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TL;DR: In this article, a review of recent developments in nonequilibrium statistical physics is presented, focusing on phase transitions from fluctuating phases into absorbing states, and several examples of absorbing-state transitions which do not belong to the directed percolation universality class are discussed.
Abstract: This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.
1,081 citations
TL;DR: The results suggest that neural activity in vivo shows a mélange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step.
Abstract: In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore we analyzed highly parallel spike recordings from awake rats and monkeys, anaesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a melange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.
552 citations
TL;DR: The concepts of criticality and universality are discussed when applied to biological systems and it is suggested that in some cases these systems can extract functional advantages close to criticality.
Abstract: Close to a transition between different phases a substance can show universal behavior that is independent of the microscopic details and is characterized by power law correlations and critical exponents. In this Colloquium the concepts of criticality and universality are discussed when applied to biological systems and suggest that in some cases these systems can extract functional advantages close to criticality.
430 citations
Cites background from "Paths to self-organized criticality..."
...…sizes of such avalanches turn out to be distributed as power laws, i.e. the system becomes critical without any apparent need for fine tuning16 (Bak, 1996; Bak et al., 1987; Christensen and Moloney, 2005; Dickman et al., 2000; Jensen, 1998; Pruessner, 2012; Turcotte, 1999; Watkins et al., 2015)....
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TL;DR: An account of the mathematical and physical foundations of criticality is provided and recent experimental studies are reviewed with the aim of identifying important next steps to be taken and connections to other fields that should be explored.
Abstract: The neural criticality hypothesis states that the brain may be poised in a critical state at a boundary between different types of dynamics. Theoretical and experimental studies show that critical systems often exhibit optimal computational properties, suggesting the possibility that criticality has been evolutionarily selected as a useful trait for our nervous system. Evidence for criticality has been found in cell cultures, brain slices, and anesthetized animals. Yet, inconsistent results were reported for recordings in awake animals and humans, and current results point to open questions about the exact nature and mechanism of criticality, as well as its functional role. Therefore, the criticality hypothesis has remained a controversial proposition. Here, we provide an account of the mathematical and physical foundations of criticality. In the light of this conceptual framework, we then review and discuss recent experimental studies with the aim of identifying important next steps to be taken and connections to other fields that should be explored.
363 citations
Cites background from "Paths to self-organized criticality..."
...Theoretical arguments seem to suggest that self-organized criticality can be fully realized only in systems in which the control parameter is conserved (Dickman et al., 2000)....
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...The most direct evidence for a phase transition is certainly provided by a phase diagram (Figure 1) (Dickman et al., 2000)....
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References
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31 Dec 1985
TL;DR: The construction, and other general results are given in this paper, with values in [0, ] s. The voter model, the contact process, the nearest-particle system, and the exclusion process.
Abstract: The Construction, and Other General Results.- Some Basic Tools.- Spin Systems.- Stochastic Ising Models.- The Voter Model.- The Contact Process.- Nearest-Particle Systems.- The Exclusion Process.- Linear Systems with Values in [0, ?)s.
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TL;DR: The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html as discussed by the authors.
Abstract: This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html
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TL;DR: Superconductivity of Metals and Alloys as mentioned in this paper is an introductory course at the University of Orsay, which is intended to explain the basic knowledge of superconductivity for both experimentalists and theoreticians.
Abstract: Drawn from the author's introductory course at the University of Orsay, Superconductivity of Metals and Alloys is intended to explain the basic knowledge of superconductivity for both experimentalists and theoreticians. These notes begin with an elementary discussion of magnetic properties of Type I and Type II superconductors. The microscopic theory is then built up in the Bogolubov language of self-consistent fields. This text provides the classic, fundamental basis for any work in the field of superconductivity.
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"Paths to self-organized criticality..." refers methods in this paper
...The resulting flux density gradient, known as the Bean state [77], bears some analogy with sandpiles, as pointed out by De Gennes over 30 years ago [ 78 ]....
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01 Jan 1996
2,442 citations
"Paths to self-organized criticality..." refers background in this paper
...alanches in SOC sandpiles, even the in abelian case, where quite a lot is known about the stationary properties [64]. V. SOC AND THE REAL WORLD Since SOC has been claimed to be the way “nature works” [65], we would expect to find a multitude of experimental examples where this concept is useful. Originally, SOC was considered an explanation of power laws, that it provided a means whereby a system could...
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Book•
01 Jan 1989
TL;DR: In this paper, B. Mandelbrot introduced fractal geometry fractal measures methods for determining fractal dimensions local growth models diffusion-limited growth growing self-affine surfaces cluster-cluster aggregation (CCA) computer simulations experiments on Laplacian growth new developments.
Abstract: Foreword, B. Mandelbrot introduction fractal geometry fractal measures methods for determining fractal dimensions local growth models diffusion-limited growth growing self-affine surfaces cluster-cluster aggregation (CCA) computer simulations experiments on Laplacian growth new developments.
1,989 citations