Rate- and state-dependent friction law and statistical properties of earthquakes

– In order to clarify how the statistical properties of earthquakes depend on the constitutive law characterizing the stick-slip dynamics, we make an extensive numerical simulation of the one-dimensional spring-block model with a rateand state-dependent friction law. Both the magnitude distribution and the recurrence-time distribution are studied with varying the constitutive parameters characterizing the model. While a continuous spectrum of seismic events from smaller to larger magnitudes is obtained, earthquakes described by this model turn out to possess pronounced “characteristic” features. Recent studies have revealed that an earthquake can be regarded as a stick-slip frictional instability which a natural fault driven by steady motions of tectonic plates exhibits. Hence, an earthquake occurrence is governed by the physical law of rock friction [1,2]. Unfortunately, our present understanding of physics of friction is still poor. We do not have precise knowledge of the constitutive law characterizing the stick-slip dynamics of earthquake faults. The difficulty lies partly in the fact that a complete microscopic theory of friction is still not available, but also in the fact that the length and time scales relevant to earthquakes are so large that the applicability of laboratory experiments on rock friction is not necessarily clear. Detailed characteristics of the friction force are specified by the constitutive relation [1–4]. One fundamental question in earthquake studies might be how the properties of earthquakes depend on the constitutive law and other material parameters characterizing earthquake faults. To answer this question and to get deeper insight into the physical mechanism governing the stick-slip process of earthquakes, proper modeling of an earthquake is essential. Indeed, earthquake models of various levels of simplifications have been proposed in geophysics and statistical physics, and their statical properties have been extensively studied mainly by means of numerical computer simulations. In model simulations, one can easily control the constitutive parameters, which is almost impossible for natural faults. One of the standard model used in statistical studies of earthquakes is the so-called springblock model originally proposed by Burridge and Knopoff (BK model) [5]. The BK model, combined with several types of constitutive relations, have been extensively studied by numerical simulations [6–13]. In order for the model to exhibit a dynamical instability corresponding to an earthquake, it is essential that the friction force exhibits a frictional weakening property, i.e., the friction should become weaker as the block slides. One of the simplest form of the