J. Phys. I France 2 (1992) 2089-2096
Mean-field solution of a block-spring model of earthquakesDidier Sornette
Laboratoire de Physique de la Matière Condensée CNRS URA 190, Université de Nice-Sophia Antipolis, B.P. 71, Parc Valrose 06108 Nice Cedex, France
(Received 26 June 1992, accepted in final form 17 July 1992)
A mean field version of the Burridge-Knopoff block-spring stick-slip model of earthquake faults is mapped onto a cycled generalization of the democratic fiber bundle model (DFM). This provides an exactly soluble model which describes the set of earthquakes preceding a major earthquake. We find the coexistence of 1) a differential Gutenberg-Richter distribution of bursts of size , with a cut-off as the stress and 2) a run away occurring at a well-defined stress threshold . The total number of bursts of size up to the run away scales as . The exponent 5/2 reflects the occurrence of larger and larger events when approaching the run away instability (Omori's law for foreshocks). The Gutenberg-Richter and Omori power laws are not associated with a stationary criticality but to fluctuations accompanying the nucleation of the run away. Introducing long range correlations in the model lead to a continuous dependence of the above exponents as a function of the correlation exponent.
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