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Register a new user25 Years of Self-Organized Criticality: Concepts and Controversies
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Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attacted much comment, ranging from the very positive to the polemical. The other papers in this special issue (Aschwanden et al, 2014; McAteer et al, 2014; Sharma et al, 2015) showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Baks own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner, 2012). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfelds original papers.
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Shortly after the seminal paper {sl Self-Organized Criticality: An explanation of 1/f noise} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {sl Avalanches and the Distribution of Solar Flares} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.
The well known Sandpile model of self-organized criticality generates avalanches of all length and time scales, without tuning any parameters. In the original models the external drive is randomly selected. Here we investigate a drive which depends on the present state of the system, namely the effect of favoring sites with a certain height in the deposition process. If sites with height three are favored, the system stays in a critical state. Our numerical results indicate the same universality class as the original model with random depositition, although the stationary state is approached very differently. In constrast, when favoring sites with height two, only avalanches which cover the entire system occur. Furthermore, we investigate the distributions of sites with a certain height, as well as the transient processes of the different variants of the external drive.
The shape of clouds has proven to be essential for classifying them. Our analysis of images from fair weather cumulus clouds reveals that, besides by turbulence they are driven by self-organized criticality (SOC). Our observations yield exponents that support the fact the clouds, when projected to two dimensions (2D), exhibit conformal symmetry compatible with $c=-2$ conformal field theory (CFT), in contrast to 2D turbulence which has $c=0$ CFT. By using a combination of the Navier-Stokes equation, diffusion equations and a coupled map lattice (CML) we successfully simulated cloud formation, and obtained the same exponents.
The concept of percolation is combined with a self-consistent treatment of the interaction between the dynamics on a lattice and the external drive. Such a treatment can provide a mechanism by which the system evolves to criticality without fine tuning, thus offering a route to self-organized criticality (SOC) which in many cases is more natural than the weak random drive combined with boundary loss/dissipation as used in standard sand-pile formulations. We introduce a new metaphor, the e-pile model, and a formalism for electric conduction in random media to compute critical exponents for such a system. Variations of the model apply to a number of other physical problems, such as electric plasma discharges, dielectric relaxation, and the dynamics of the Earths magnetotail.
In this thesis we present few theoretical studies of the models of self-organized criticality. Following a brief introduction of self-organized criticality, we discuss three main problems. The first problem is about growing patterns formed in the abelian sandpile model (ASM). The patterns exhibit proportionate growth where different parts of the pattern grow in same rate, keeping the overall shape unchanged. This non-trivial property, often found in biological growth, has received increasing attention in recent years. In this thesis, we present a mathematical characterization of a large class of such patterns in terms of discrete holomorphic functions. In the second problem, we discuss a well known model of self-organized criticality introduced by Zhang in 1989. We present an exact analysis of the model and quantitatively explain an intriguing property known as the emergence of quasi-units. In the third problem, we introduce an operator algebra to determine the steady state of a class of stochastic sandpile models.
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