Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models,

*Diffundere*, the Latin origin of the word “diffusion,” means “to spread out.” Diffusion is practically ubiquitous and takes place in solids, liquids, and gases. Because it is so pervasive in physical phenomena, students in all branches of science and engineering should be exposed to diffusion theory. And for that reason, *Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models* by Daniel Gillespie and Effrosyni Seitaridou is a welcome addition to the existing literature.

The authors intend the book to be a “self-contained, tutorial introduction to simple Brownian diffusion for scientists.” Familiarity with calculus is the only prerequisite for performing the mathematical calculations within. Chapters 2 and 7 cover some of the essential concepts of stochastic processes, which undergraduate students not majoring in mathematics may not normally be exposed to. I think most diffusion theorists would agree with the authors that numerical simulations provide deep insight into stochastic processes generally and diffusion in particular. The principles of such numerical-simulation algorithms and the methods of their implementation are presented in considerable detail throughout the book. Thus, in relation to the mathematical and numerical toolkit required to understand simple Brownian diffusion, this book is indeed self-contained.

In a lively tutorial style, the authors discuss some of the most widely used mathematical formulations of diffusion. They have endeavored to organize and present the subject matter “from a purely logical perspective.” They emphasize the basic physical assumptions and the conditions for the validity of each of the mathematical formalisms. No subtlety is bypassed, and no limitation of the theory is swept under the carpet.

Physical diffusion of matter, for example, is generally described in terms of a diffusion equation for the local density of matter. That equation combines Fick’s law with the equation of continuity that captures the conservation of matter. The one-dimensional examples worked out in chapter 1 provide physical insight into the nature of diffusion phenomena. Those examples also illustrate a few elementary mathematical techniques for solving the diffusion equation under different sets of initial and boundary conditions.

Historically, diffusion theory developed along two separate lines (see the article by T. N. Narasimhan, *Physics Today*, July 2009, page 48). The heat diffusion equation, first formulated by Joseph Fourier, is an example of physical diffusion. By contrast, the diffusion equation for a probability, developed by Pierre Simon Laplace, results in the continuum limit of a random walk. Einstein’s theory, which essentially unified the two approaches in the context of Brownian motion, is covered in chapters 3 and 4.

In chapter 5, the authors, by discretizing the diffusion equation, prepare the foundation for Markov jump processes; in chapter 6, they derive the corresponding master equations. In that chapter, readers also get a firsthand introduction to the powerful numerical technique called Gillespie’s algorithm. The book’s first author developed it many years ago for exact numerical simulations of the trajectories that follow from the master equation.

Three years after Einstein developed his theory of Brownian motion, Paul Langevin formulated an alternative approach based on a stochastic differential equation that extends Newton’s deterministic equation. The logical foundation of Langevin’s theory and its implications are discussed in chapters 8 and 9.

Chapter 10 presents, at an elementary level, the extension of the equations for diffusion in the presence of a time-independent conservative external force—in the chapter, the authors work out the example of a constant force. It is a pity, however, that the chapter only mentions in a footnote Kramers’s diffusion model of chemical reactions, which treats the decay of a metastable state.

*Simple Brownian Diffusion* will certainly be used to form the core content of my senior undergraduate course on diffusion and related phenomena. However, I would hesitate to recommend this excellent exposition as a standalone textbook for two main reasons: Few examples are explicitly worked out, and no exercises are given. Nevertheless, I do recommend this book to all serious students who seek to have a thorough understanding of the conceptual subtleties in the theory of diffusion. The authors emphasize that “developing that understanding is the limited aim of this book.” In that endeavor they have been remarkably successful.

**Debashish Chowdhury** is a professor of physics at the Indian Institute of Technology Kanpur. His research concerns diffusion, transport, and self-organization in complex systems.