Order statistics for first passage times in one-dimensional diffusion processes

Abstract

The problem of the statistical description of the first passage timet _{j, N} to one or two absorbing boundaries of the firstj of a set ofN independent diffusing particles in one dimension is revisited. An asymptotic expression for largeN of the generating function of the moments oft _{j, N} is obtained, and explicit expressions for the first two moments are presented. The results are valid for a specific but broad class of initial distributions of particles and boundaries. The mean first passage time of the first particle 〉t _{l, N} 〉 and its variance are compared with numerical estimates for an initial distribution in which all particles are placed at the midpoint of the diffusion region.