The role of heterogeneity in long-range interacting systems : from nucleation to earthquake fault systems

Abstract

The role of heterogeneity in two long-range systems is explored with a focus on the interplay of this heterogeneity with the component system interactions. The first will be the heterogeneous Ising model with long-range interactions. Earthquake fault systems under long-range stress transfer with varying types of heterogeneity will be the second system of interest.

First I will review the use of the intervention method to determine the time and place of nucleation and extend its use as an indicator for spinodal nucleation. The heterogeneous Ising model with fixed magnetic sites will then be reformulated as a dilute random field Ising model. This reformulation will allow for the application of spinodal nucleation theory to the heterogeneous Ising model by correcting the spinodal field and the critical exponent sigma describing the critical behavior of clusters in spinodal nucleation theory. The applicability of this correction is shown by simulations that determine the cluster scaling of the nucleating droplets near the spinodal. Having obtained a reasonable definition of the saddle point object describing the nucleation droplet, the density profile of the nucleating droplet is measured and deviations from homogeneous spinodal nucleation are found due to the excess amount of sparseness in the nucleating droplet due to the heterogeneity.

Earthquake fault systems are then introduced and a connection is shown of two earthquake models. Heterogeneity is introduced in the form of asperities with the intent of modeling the effect of hard rocks on earthquake statistics. The asperities are observed to be a crucial element in explaining the behavior of aftershocks resulting in Omori's law. A second form of heterogeneity is introduced by coupling the Olami-Feder-Christensen model to an invasion percolation model for the purpose of modeling an earthquake fault system undergoing hydraulic fracturing. The ergodicty and event size statistics are explored in this extended model. The robustness of the event size statistics results are explored by allowing for the dissipation parameter in the Olami-Feder-Christensen model to vary.