Sliding blocks with random friction and absorbing random walks

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.