J. Phys. I France
Volume 2, Numéro 11, November 1992
Page(s) 2089 - 2096
DOI: 10.1051/jp1:1992269
J. Phys. I France 2 (1992) 2089-2096

Mean-field solution of a block-spring model of earthquakes

Didier 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 $d(\Delta)\sim\Delta^{-3/2}$ of bursts of size $\Delta$, with a cut-off $\Delta_{\max}\sim(\sigma_{\rm r}-\sigma)^{-1}$ as the stress $\sigma\to\sigma_{\rm r}$ and 2) a run away occurring at a well-defined stress threshold $\sigma_{\rm r}$. The total number of bursts of size $\Delta$ up to the run away scales as $D(\Delta)\sim \Delta^{-5/2}$. 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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