25 Years of Self-organized Criticality: Concepts and Controversies

Abstract

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 attracted much comment, ranging from the very positive to the polemical. The other papers (Aschwanden et al. in Space Sci. Rev., 2014, this issue; McAteer et al. in Space Sci. Rev., 2015, this issue; Sharma et al. in Space Sci. Rev. 2015, in preparation) in this special issue 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 Bak's 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 Wiesenfeld's original papers

On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

“…I believe this to have been important, because, for example, Per Bak et al's paradigm of Self-Organised Criticality, in which stationary spectra and correlation functions play an essential role, could surely not have been positioned as the unique solution to the 1/f problem [44] if it had been widely recognised just how different Mandelbrot's two existing routes to 1/f already were. In addition, I think that the route that Mandelbrot took from the fractional renewal models to multifractality (see e.g.…”

The Dependence of Solar Wind Burst Size on Burst Duration and Its Invariance Across Solar Cycles 23 and 24

“…As well as the probability density function (PDF) observations, Uritsky et al () found a power law dependence between the size and duration of bursts in A E ; similar power laws, with different scaling exponents, have since been found by Wanliss and Uritsky () for SYM‐H and by Moloney and Davidsen () for the ε parameter for extremely high burst thresholds. These observations, and theoretical considerations, led to a large body of work exploring self‐organized criticality (Bak et al, ; Watkins et al, ) and turbulence as possible mechanisms to produce the ubiquitous scale‐free nature of bursts (Aschwanden, ; Boffetta et al, ; Chapman et al, ; Crosby et al, ).…”

“…high burst thresholds. These observations, and theoretical considerations, led to a large body of work exploring self-organized criticality (Bak et al, 1987;Watkins et al, 2016) and turbulence as possible mechanisms to produce the ubiquitous scale-free nature of bursts (Aschwanden, 2011;Boffetta et al, 1999;Chapman et al, 1998;Crosby et al, 1992).…”

Avalanching Systems with Longer Range Connectivity: Occurrence of a Crossover Phenomenon and Multifractal Finite Size Scaling

“…According to Sornette [2], in a broad sense, SOC occurs when a system slowly driven from the outside spontaneously self-organizes into a globally-stationary state, which is characterized by self-similar distributions of relaxation event sizes and by fractal geometrical features. Quoting Watkins et al [3], the concept of SOC applies to "Slowly driven, avalanching (intermittent) systems with non linear interactions, that display non-trivial power law correlations (cutoff by the system size) as known from ordinary critical phenomena, but with internal, self-organized, rather than external tuning of a control parameter (to a non-trivial value)". However, it is now understood that some tuning is generally needed for some so-called SOC systems and the low frequency flicker noise behavior, which Bak and colleagues predicted does not really occur for the original model that they suggested.…”

“…However, it is now understood that some tuning is generally needed for some so-called SOC systems and the low frequency flicker noise behavior, which Bak and colleagues predicted does not really occur for the original model that they suggested. On the other hand SOC is a concept that has many controversial interpretations and even nowadays is not a well-defined physical process in the context of out-of-equilibrium dynamical systems (see [3] and the references therein).…”