Introduction

We now develop the ideas of the previous chapter to determine basic first-passage properties for both continuum diffusion and the discrete randomwalk in a finite one-dimensional interval. This is a simple system with which we can illustrate the physical implications of first-passage processes and the basic techniques for their solution. Essentially all of the results of this chapter are well known, but they are scattered throughout the literature. Much information about the finite-interval system is contained in texts such as Cox and Miller (1965), Feller (1968), Gardiner (1985), Risken (1988), and Gillespie (1992). An important early contribution for the finite-interval system is given by Darling and Siegert (1953). Finally, some of the approaches discussed in this chapter are similar in spirit to those of Fisher (1988).

For continuum diffusion, we start with the direct approach of first solving the diffusion equation and then computing first-passage properties from the time dependence of the flux leaving the system. Much of this is classical and well-known material. These same results will then be rederived more elegantly by the electrostatic analogies introduced in Chap. 1. This provides a striking illustration of the power of these analogies and sets the stage for their use in higher dimensions and in more complex geometries (Chaps. 5–7).

We also derive parallel results for the discrete random walk. One reason for this redundancy is that random walks are often more familiar than diffusion because the former often arise in elementary courses. It will therefore be satisfying to see the essential unity of their first-passage properties. It is also instructive to introduce various methods for analyzing the recursion relations for the discrete randomwalk.