Sliding blocks with random friction and absorbing random walks

With the purpose of explaining recent experimental findings, we study the d istribution 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. Th e problem of finding A(lambda) is equivalent to a first-passage-time proble m for a one-dimensional random walk with nonzero drift, whose exact solutio n is well known. From the exact solution of this problem we conclude that ( a) for inclination angles theta less than theta(c)=tan([mu]) the average tr aversed distance [lambda] is finite, and diverges when theta-->theta(c)(-) as [lambda]similar to(theta(c)-theta)(-1); (b) at the critical angle a powe r-law distribution of slidings is obtained: A(lambda)similar to lambda(-3/2 ). Our analytical results are confirmed by numerical simulation and are in partial agreement with the reported experimental results. We discuss the po ssible reasons for the remaining discrepancies.