Self-organization in leaky threshold systems: The influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting

Threshold systems are known to be some of the most important nonlinear self-organizing systems in nature, including networks of earthquake faults, neural networks, superconductors and semiconductors, and the World Wide Web, as well as political, social, and ecological systems. All of these systems have dynamics that are strongly correlated in space and time, and all typically display a multiplicity of spatial and temporal scales. Here we discuss the physics of self-organization in earthquake threshold systems at two distinct scales: (i) The "microscopic" laboratory scale, in which consideration of results from simulations leads to dynamical equations that can be used to derive the results obtained from sliding friction experiments, and (ii) the "macroscopic" earthquake fault-system scale, in which the physics of strongly correlated earthquake fault systems can be understood by using time-dependent state vectors defined in a Hilbert space of eigenstates, similar in many respects to the mathematics of quantum mechanics. In all of these systems, long-range interactions induce the existence of locally ergodic dynamics. The existence of dissipative effects leads to the appearance of a "leaky threshold" dynamics, equivalent to a new scaling field that controls the size of nucleation events relative to the size of background fluctuations. At the macroscopic earthquake fault-system scale, these ideas show considerable promise as a means of forecasting future earthquake activity.