Book contents
- Frontmatter
- Contents
- Preface
- Errata
- 1 First-Passage Fundamentals
- 2 First Passage in an Interval
- 3 Semi-Infinite System
- 4 Illustrations of First Passage in Simple Geometries
- 5 Fractal and Nonfractal Networks
- 6 Systems with Spherical Symmetry
- 7 Wedge Domains
- 8 Applications to Simple Reactions
- References
- Index
4 - Illustrations of First Passage in Simple Geometries
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Errata
- 1 First-Passage Fundamentals
- 2 First Passage in an Interval
- 3 Semi-Infinite System
- 4 Illustrations of First Passage in Simple Geometries
- 5 Fractal and Nonfractal Networks
- 6 Systems with Spherical Symmetry
- 7 Wedge Domains
- 8 Applications to Simple Reactions
- References
- Index
Summary
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.
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- A Guide to First-Passage Processes , pp. 115 - 167Publisher: Cambridge University PressPrint publication year: 2001