Book Review: A Guide to First-Passage Processes

Sidney Redner. 312 pp. Cambridge U. P., New York, 2001. Price: $80.00 ISBN 0-521-65248-0, J. R. Dorfman, Reviewer.

from the American Journal of Physics, Vol. 70, No. 11, p. 1166, November 2002

Many phenomena in chemistry, physics, and the biological sciences are governed by ``first passage time processes". For example, an understanding of such processes is important for those studying the theories of chemical reaction rates, neuron dynamics, and escape-rates in diffusion processes with absorbing boundaries. These systems typically depend on a random variable (or one that can be adequately approximated as random) reaching some threshold value. The probability distribution for the first time at which the threshold value is attained is the central quantity of interest in first-passage time processes.

A treatment of these phenomena occupies a significant portion of most books on stochastic processes that students and researchers might consult, such as that by N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1997). S. Redner's new book describes a large and varied collection of such processes and explains various methods for solving the equations used to describe them, including absorbing boundary methods, adjoint equations, and techniques based upon electrostatic image methods, among others. Redner does not hesitate to refer to the use of Mathematica as a tool to simplify long calculations. There is an emphasis on continuous time processes, but several discrete time processes are treated as well.

The book is very well written and provides clear explanations of the techniques used to determine first passage probabilities and related quantities, under a variety of circumstances. In addition to the traditional discussions of the diffusion and Fokker-Planck equations in various geometries, the author has taken examples from several areas of current research. Thus we find discussions of first-passage processes as they appear in the theories for self-organized criticality, walks on fractal networks, reaction-diffusion and convection-diffusion equations, kinetics of spin systems, aggregation reactions, and so on. There is a good bibliography, with the useful feature that the titles of journal articles are included in the citations.

The book requires familiarity with several mathematical techniques: Fourier-Laplace transforms and asymptotic analyses of time dependent functions at large times from their transforms, solutions of the diffusion equation in various geometries, but nothing beyond the grasp of a typical physics graduate student or an advanced undergraduate. Unfortunately, there are no exercises for readers to test their understanding. The author might consider adding some in future editions, or on a web page devoted to the book. In any case, this book can be highly recommended to anyone interested in its subject, both for its clarity of presentation and for the wide range of problems treated.

J. R. Dorfman is a Professor of Physics at the University of Maryland. His research interests include statistical mechanics and dynamical systems theory. He is co-author with P. Gaspard of papers applying escape-rate methods to derive formulas for transport coefficients in classical chaotic fluids. Dorfman is also the author of a recent book, An Introduction to Chaos in Non-equilibrium Statistical Mechanics (Cambridge U. P., 1999).