First-passage time: Lattice versus continuum

Abstract

The well known approach, based on Schrödinger's integral equation, to the problem of calculating the first-passage probability density in time for classical diffusion on a continuum is revisited for the case of diffusion by hopping on a discrete lattice. It turns out that a certain boundary condition central to solving the integral equation, invoked first by Schrödinger and then by others on the basis of a physical argument, needs to be modified for the discrete case. In fact, the required boundary condition turns out to be determined entirely by the normalization condition for the first-passage probability density. An explicit analytical expression is derived for the first-passage density for a three-site problem modeling escape over a barrier. The related quantum first-passage problem is also commented upon briefly.