Scaling of avalanches
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Abstract
Many systems are characterized by emergent macroscopic phenomena which cannot be elucidated from microscopic dynamics. Emergent collective activity such as avalanches are ubiquitous in complex systems. Examples of such phenomena include earthquakes, neural avalanches, and the Barkhausen effect, which describes the collective flipping of magnetic domains due to an applied external field. These macroscopic phenomena exhibit universal traits such as scale-invariant distribution functions and universal firing rates, which appear to be independent of the underlying microscopic details. Understanding the mechanism behind these emergent behavior is also of practical importance to researchers in seismology and is an important step to understanding the function of neural avalanches in the brain.
In this thesis, we present a generalization of the Fisher-Stauffer cluster scaling theory, which was originally developed for the study of percolation models and equilibrium thermal systems, such as the Ising model. We apply the generalized Fisher-Stauffer scaling theory to study avalanches in integrate-and-fire systems, which have been used to model earthquakes and networks of neurons. By using the Fisher-Stauffer scaling theory, we identify the critical point and the corresponding tuning parameters in the Olami-Feder-Christensen (OFC) model, which is a discrete time integrate-and-fire system. We also derive universal scaling functions and scaling identities, which relate the critical exponents in the OFC model. Additionally, we study the scaling behavior in the OFC model with spatial disorder to model earthquake phenomenology, such as Gutenberg-Richter scaling and Omori's law, which characterize temporal clustering of earthquakes. Lastly, we extend our scaling analysis to a simple model of neurons and discuss how the scaling laws for neural avalanches may be verified in future experiments.
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Attribution 4.0 International