Analytical treatment for a spring-block model

Abstract

The stick-slip process in a spring-block model is considered in terms of cellular automaton. The probability of the slip size is exactly calculated. It is shown analytically that there is a scaling relation D(\ensuremath{\Delta})\ensuremath{\propto}${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{\xi}}$ exp(-\ensuremath{\Delta}/${\mathrm{\ensuremath{\Delta}}}_{0}$) between the slip size \ensuremath{\Delta} and its probability D(\ensuremath{\Delta}) with a universal exponent \ensuremath{\xi}=-3/2, while the value of ${\mathrm{\ensuremath{\Delta}}}_{0}$ depends on the details of the system.