A physical model for earthquakes: 1. Fluctuations and interactions

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  • JRJohn B. Rundle
    John B. Rundle
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Abstract
This is the first of a series of papers whose purpose is to develop the apparatus needed to understand the problem of earthquake occurrence in a more physical context than has often been the case. To begin, it is necessary to introduce the idea that earthquakes represent a fluctuation about the long‐term motion of the plates. This idea is made mathematically explicit by the introduction of a concept called the fluctuation hypothesis. Under this hypothesis, all physical quantities which pertain to the occurrence of earthquakes are required to depend on a physically meaningful quantity called the offset phase, the difference between the present state of slip on the fault and its long‐term average. For the mathematical treatment of the fluctuation problem it is most convenient to introduce a spatial averaging, or “coarse‐graining” operation, dividing the fault plane into a lattice of N patches. In this way, integrals are replaced by sums, and differential equations are replaced by algebraic equations. As a result of these operations, the physics of earthquake occurrence can be stated in terms of a physically meaningful energy functional: an “external potential” WE. WE is a functional potential for the stress on the fault plane acting from the external medium and characterizes the energy contained within the medium external to the fault plane which is available to produce earthquakes. A simple example is discussed which involves the dynamics of a one‐dimensional fault model. To gain some understanding, a simple friction law and a failure algorithm are assumed. It is shown that under certain circumstances the model fault dynamics undergo a sudden transition from a spatially ordered, temporally disordered state to a spatially disordered, temporally ordered state. It is found that the latter states are stable for long intervals of time. For long enough faults the dynamics are evidently chaotic.

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