The Role of Heterogeneity in Long_Range Interacting Systems: From Nucleation to Earthquake Fault Systems
This event is part of the PhD Final Oral Exams.
Dissertation: William Klein, Pankaj Mehta,James Miller, Harvey Gould
In this dissertation the role heterogeneity in two long-range systems will be explored. 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 used to determine the time and place of nucleation and extend its use in a novel method that can be used 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 σ describing the critical behavior of clusters described in spinodal nucleation theory. The applicability of this correction is shown through the use of simulation work that obtains the percolation cluster scaling of snapshots of the Ising spin system. 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 the homogeneous spinodal nucleation begin to occur due to an excess amount of sparseness in the nucleating droplet.
Earthquake fault systems are then introduced and a connection is drawn connecting two such earthquake models. Heterogeneity is introduced to this model in the form of asperities with the intent of modeling the effect such hard rocks have on the earthquake faults. These 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. Lastly the dissertation is summarized and new directions of research in each of these systems is delineated.