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Analytical SOC Models
[chapter]

Markus Aschwanden

2010
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Self-Organized Criticality in Astrophysics
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There is something universal about SOC systems that does not depend on some particular physical parameters, because they all exhibit powerlaw distributions, a necessary but not sufficient condition. There is also something universal about a fractal dimension, because it is manifested in many different physical systems. In the previous chapter we reviewed cellular automaton models, which are based on mathematical redistribution rules and do not require any specific physical model. In this
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... del. In this chapter we develop an analytical theory of SOC phenomena that is "physics-free", and thus can be applied to sandpiles, earthquakes, or solar flares equally. Essentially, our analytical approach provides an understanding of SOC phenomena in terms of the universal statistics of nonlinear systems, which in the limit of linear system behavior degenerates to the statistics of random processes. The powerlaw distributions of SOC phenomena always represent the statistics of relatively rare events, the high-end or fat tails of probabilistic distributions, which are discernible in a log-log representation only. At the low-end, events are most frequent and can often be described by standard probability theory, such as Poisson statistics or Gaussian normal distributions. Monte-Carlo simulations can simulate such event statistics for any arbitrary physical process, in form of discretized event parameters that can be sampled in binned histograms. In the continuum limit, such distributions can sometimes be described by analytical functions. In analogy, we develop an analytical SOC theory that describes the continuum limit of numerical SOC simulations, as they have been reviewed in the previous chapter. The power of analytical models is the prediction of exact distributions, which can significantly deviate from exponential or powerlaw functions, often not evident from observations due to the limited statistics of rare events. Analytical models thus can then be forward-fitted to the statistical distributions of observed SOC events. The analytical theory will provide us also a rigorous mathematical definition of SOC processes.

doi:10.1007/978-3-642-15001-2_3
fatcat:ru4ojvl4xvffhnzlzm2lzbv6ae