First Passage in Real Systems

The first-passage properties of the finite and the semi-infinite interval are relevant to understanding the kinetics of a surprisingly wide variety of physical systems. In many such examples, the key to solving the kinetics is to recognize the existence of such an underlying first-passage process. Once this connection is established, it is often quite simple to obtain the dynamical properties of the system in terms of well-known first-passage properties

This chapter begins with the presentation of several illustrations in this spirit. Our first example concerns the dynamics of integrate-and-fire neurons. This is an extensively investigated problem and a representative sample of past work includes that of Gerstein and Mandelbrot (1964), Fienberg (1974), Tuckwell (1989), Bulsara, Lowen, & Rees (1994), and Bulsara et al. (1996). In many of these models, the firing of neurons is closely related to first passage in the semi-infinite interval. Hence many of the results from the previous chapter can be applied immediately. Then a selection of self-organized critical models [Bak, Tang, & Wiesenfeld (1987) and Bak (1996)] in which the dynamics is driven by extremal events is presented. These include the Bak–Sneppen (BS) model for evolution [Bak & Sneppen (1993), Flyvbjerg, Sneppen, & Bak (1993), de Boer et al. (1994), and de Boer, Jackson, & Wetting (1995)] directed sandpile models [Dhar & Ramaswamy (1989)], traffic jam models [Nagel & Paczuski (1995)], and models for surface evolution [Maslov & Zhang (1995)]. All of these systems turn out to be controlled by an underlying first-passage process in one dimension that leads to the ubiquitous appearance of the exponent −3/2 in the description of avalanche dynamics.