Sliding blocks with random friction and absorbing random walks

TL;DR

This paper models the distance a sliding block travels on an inclined plane with random friction, deriving exact solutions for the distribution of distances and identifying critical angles where behavior changes, supported by simulations.

Contribution

It introduces a simple random friction model for sliding blocks and provides exact analytical solutions for the distribution of traveled distances, linking it to first-passage-time problems.

Findings

Finite average distance below critical angle, diverging at critical angle

Power-law distribution of sliding distances at critical angle

Analytical results confirmed by numerical simulations

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.