Self-Organized Criticality, Three Decades Later

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Background

Some thirty years ago, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper that provoked a paradigm shift in theoretical physics. While originally proposing an explanation of so-called 1/f noise, it soon transpired that their ideas, and the simple “sandpile” model they introduced, could have vast implications for understanding the natural and social world. The spontaneous appearance of scale-invariant or power-law distributions in phenomena as distinct as earthquakes, river networks and forest fires could now be connected to critical phenomena attending continuous phase transitions. The sandpile model, with its separation of time scales for driving and for relaxation, showed how such distributions could arise without tuning, which is necessary for these to be observable in nature.

Self-organized criticality (SOC) became a focus of intensive activity on several fronts: applications to various phenomena (Barkhausen noise, rain distributions, epidemics, neural activity...); studies of the precise scaling behavior of the sandpile and allied models; attempts to understand the mechanisms behind criticality in the absence of tuning. Experiment, observation, simulation and mathematical analysis were all brought to bear on these issues, so that the SOC paradigm spread into disciplines beyond physics. As the subject matured, it became clear that not all proposed instances of SOC were in fact realizations of this mechanism. At the same time, important distinctions emerged between different classes of model and their scaling properties, typically associated with symmetries or conservation laws. The notion of universality, so fundamental to critical phenomena, allowed different SOC models to be assigned to distinct classes. Controversies have lingered regarding precisely which models and phenomena possess SOC, and which do not exhibit asymptotic scaling behavior.

Although new research trends have claimed the attention of the statistical physics community, the SOC paradigm continues to attract interest, and its sphere of application grows steadily, as new areas such as neuroscience, data-processing algorithms, and solar physics discover its relevance. We therefore find this an appropriate moment to take stock of the SOC movement and the knowledge and controversies it generated, its current impact in the sciences, as well as to peer into upcoming developments and applications. While there are some existing reviews and monographs, these are more than five years old, and we feel that a fresh look and a different perspective would be useful.

Among the topics we propose to discuss in this volume are:
- Abelian and stochastic sandpile models;
- effects of global driving, as in earthquake models;
- conserved sandpiles;
- universality classes;
- applications of SOC in geophysics, astrophysics, atmospheric science and plasmas;
- evidences of SOC in biological physics and neuroscience;
- combinatorics;
- numerical simulation;
- applications to biological evolution and to financial markets.

We are confident that these explorations will stimulate the diffusion of SOC ideas in an even broader range of disciplines in the coming years.

Keywords: self-organized criticality, sandpile model, 1/f noise, statistical physics

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