Sliding blocks with random friction and absorbing random walks

Abstract

With the purpose of explaining recent experimental findings, we study the distribution $A(\lambda)$ of distances $\lambda$ traversed by a block that slides on an inclined plane and stops due to friction. A simple model in which the friction coefficient $\mu$ is a random function of position is considered. The problem of finding $A(\lambda)$ is equivalent to a First-Passage-Time problem for a one-dimensional random walk with nonzero drift, whose exact solution is well-known. From the exact solution of this problem we conclude that: a) for inclination angles $\theta$ less than $\theta_c=\tan(\av{\mu})$ the average traversed distance $\av{\lambda}$ is finite, and diverges when $\theta \to \theta_c^{-}$ as $\av{\lambda} \sim (\theta_c-\theta)^{-1}$; b) at the critical angle a power-law distribution of slidings is obtained: $A(\lambda) \sim \lambda^{-3/2}$. Our analytical results are confirmed by numerical simulation, and are in partial agreement with the reported experimental results. We discuss the possible reasons for the remaining discrepancies.