Introduction

This chapter is devoted to first-passage properties in spherically symmetric systems. We shall see how the contrast between persistence, for spatial dimension d ≤ 2, and transience, for d > 2, leads to very different first-passage characteristics. We will solve first-passage properties both by the direct time-dependent solution of the diffusion equation and by the much simpler and more elegant electrostatic analogy of Section 1.6.

The case of two dimensions is particularly interesting, as the inclusion of a radial potential drift ν(r) ∝ 1/r is effectively the same as changing the spatial dimension. Thus general first-passage properties for isotropic diffusion in d dimensions are closely related to those of diffusion in two dimensions with a superimposed radial potential bias. This leads to nonuniversal behavior for the two-dimensional system.

As an important application of our study of first-passage to an isolated sphere, we will obtain the classic Smoluchowski expression for the chemical reaction rate, a result that underlies much of chemical kinetics. Because of the importance of this result, we will derive it by time-dependent approaches as well as by the quasi-static approximation introduced in Section 3.6. The latter approach also provides an easy way to understand detailed properties of the spatial distribution of reactants around a spherical trap.

First Passage between Concentric Spheres

We begin by computing the splitting (or exit) probabilities and the corresponding mean hitting times to the inner and outer the boundaries of the annular region R− ≤ r ≤ R+ as functions of the starting radius r (Fig. 6.1).