We introduce a new model for an earthquake fault system that is composed of noninteracting simple lattice models with different levels of damage denoted by q. The undamaged lattice models (q=0) have Gutenberg-Richter scaling with a cumulative exponent β=1/2, whereas the damaged models do not have well defined scaling. However, if we consider the "fault system" consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or b value, can be understood.

We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the N th step is random and its length equals λ N−1 , with λ < 1. As λ increases past a critical value λc, the endpoint distribution in two dimensions, P (r), changes from having a global maximum away from the origin to being peaked at the origin. The probability distribution for a single coordinate, P (x), undergoes a similar transition, but exhibits multiple maxima on a fine length scale for λ close to λc. We numerically determine P (r) and P (x) by applying a known algorithm that accurately inverts the exact Bessel function product form of the Fourier transform for the probability distributions.

We investigate the role of equilibrium methods and stress transfer range in describing the process of damage. We find that equilibrium approaches are not applicable to the description of damage and the catastrophic failure mechanism if the stress transfer is short ranged. In the long range limit, equilibrium methods apply only if the healing mechanism associated with ruptured elements is instantaneous. Furthermore we find that the nature of the catastrophic failure depends strongly on the stress transfer range. Long range transfer systems have a failure mechanism that resembles nucleation. In short range stress transfer systems, the catastrophic failure is a continuous process that, in some respects, resembles a critical point.

Natural earthquake fault systems are highly non-homogeneous. The inhomogeneities occur because the earth is made of a variety of materials which hold and dissipate stress differently. In this work, we study scaling in earthquake fault models which are variations of the Olami-FederChristensen (OFC) and Rundle-Jackson-Brown (RJB) models. We use the scaling to explore the effect of spatial inhomogeneities due to damage and inhomogeneous stress dissipation in the earthquake-fault-like systems when the stress transfer range is long, but not necessarily longer than the length scale associated with the inhomogeneities of the system. We find that the scaling depends not only on the amount of damage, but also on the spatial distribution of that damage.

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