Of overlapping Cantor sets and earthquakes: Analysis of the discrete Chakrabarti-Stinchcombe model

Abstract

We report an exact analysis of a discrete form of the Chakrabarti-Stinchcombe model for earthquakes [Physica A \textbf{270}, 27 (1999)] which considers a pairof dynamically overlapping finite generations of the Cantor set as a prototype of geological faults. In this model the $n$-th generation of the Cantor set shifts on its replica in discrete steps of the length of a line segment in that generation and periodic boundary conditions are assumed. We determine the general form of time sequences for the constant magnitude overlaps and hence obtain the complete time-series of overlaps by the superposition of these sequences for all overlap magnitudes. From the time-series we derive the exact frequency distribution of the overlap magnitudes. The corresponding probability distribution of the logarithm of overlap magnitudes for the $n$-th generation is found to assume the form of the binomial distribution for $n$ Bernoulli trials with probability 1/3 for the success of each trial. For an arbitrary pair of consecutive overlaps in the time-series where the magnitude of the earlier overlap is known, we find that the magnitude of the later overlap can be determined with a definite probability; the conditional probability for each possible magnitude of the later overlap follows the binomial distribution for $k$ Bernoulli trials with probability 1/2 for the success of each trial and the number $k$ is determined by the magnitude of the earlier overlap. Though this model does not produce the Gutenberg-Richter law for earthquakes, our results indicate that the fractal structure of faults admits a probabilistic prediction of earthquake magnitudes.