Spinodals, scaling, and ergodicity in a threshold model with long-range stress transfer

Abstract

We present both theoretical and numerical analyses of a cellular automaton version of a slider-block model or threshold model that includes long-range interactions. Theoretically we develop a coarse-grained description in the mean-field (infinite range) limit and discuss the relevance of the metastable state, limit of stability (spinodal), and nucleation to the phenomenology of the model. We also simulate the model and confirm the relevance of the theory for systems with long- but finite-range interactions. Results of particular interest include the existence of Gutenberg-Richter-like scaling consistent with that found on real earthquake fault systems, the association of large events with nucleation near the spinodal, and the result that such systems can be described, in the mean-field limit, with techniques appropriate to systems in equilibrium.

Simulated dynamical weakening and Abelian avalanches in mean-field driven threshold models

“…This form of temporal course-graining has been central to previous analysis of slider-block models [9].…”

“…In addition, this method compounds analytical difficulties involving multiple failures of individual blocks in a single step [9].…”

“…However, the limiting case of mean-field models are beginning to yield to analytical understanding [8,9,10,11]. Mean field model simulations exhibit complex event histories and regimes of behavior, including a power law magnitude-frequency relation.…”

A simple metric to quantify seismicity clustering

“…This metric, originally developed to study liquid systems and glasses Thirumalai and Mountain, 1993), was applied to earthquake simulations (Ferguson et al, 1999) and to regional seismicity by Tiampo et al (2003Tiampo et al ( , 2007. The result was the identification of periods of metastable equilibrium in seismic activity.…”

Characteristics of long-term regional seismicity before the 2008 Wen-Chuan, China, earthquake using pattern informatics and genetic algorithms

“…The fundamental concept is built up from numerical simulations of driven threshold systems (Egolf, 2000;Ferguson et al, 1999;Klein et al, 2000;Rundle et al, 1995Rundle et al, , 2000a and the physical meaning is explained by phase dynamics. After dividing the research region by a planar box of regular size, seismic anomaly in a box during an interval is evaluated by comparing its past long-term average seismicity and seismicity in other boxes.…”