Sliding blocks with random friction and absorbing random walks

Abstract

With the purpose of explaining recent experimental findings, we study the distribution $A(λ)$ of distances $λ$ traversed by a block that slides on an inclined plane and stops due to friction. A simple model in which the friction coefficient $μ$ is a random function of position is considered. The problem of finding $A(λ)$ 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 $θ$ less than $θ_c=\tan(\avμ)$ the average traversed distance $\avλ$ is finite, and diverges when $θ\to θ_c^{-}$ as $\avλ \sim (θ_c-θ)^{-1}$; b) at the critical angle a power-law distribution of slidings is obtained: $A(λ) \sim λ^{-3/2}$. 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.