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Of overlapping Cantor sets and earthquakes: analysis of the discrete Chakrabarti–Stinchcombe model

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  • Bhattacharyya, Pratip

Abstract

We report an exact analysis of a discrete form of the Chakrabarti–Stinchcombe model for earthquakes (Physica A 270 (1999) 27), which considers a pair of dynamically overlapping finite generations of the Cantor set as a prototype of geological faults. In this model the nth 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 nth generation is found to assume the form of the binomial distribution for n Bernoulli trials with probability 13 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 12 for the success of each trial and the number k is determined by the magnitude of the earlier overlap. Although 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.

Suggested Citation

  • Bhattacharyya, Pratip, 2005. "Of overlapping Cantor sets and earthquakes: analysis of the discrete Chakrabarti–Stinchcombe model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 199-215.
  • Handle: RePEc:eee:phsmap:v:348:y:2005:i:c:p:199-215
    DOI: 10.1016/j.physa.2004.09.014
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    Cited by:

    1. Potirakis, Stelios M. & Zitis, Pavlos I. & Eftaxias, Konstantinos, 2013. "Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(13), pages 2940-2954.

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