The typical undergraduate statistical-mechanics course does not usually delve into advanced topics that require probabilistic reasoning to understand, such as extreme-value statistics, anomalous diffusion, and random-matrix theory. Nevertheless, many of those topics are closely linked to ongoing research trends in various fields. Ariel Amir, a professor at Harvard University, aims to remedy that situation in Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications, a book of about 200 pages intended for advanced undergraduates and graduate students.

Brownian motion, or the random motion of particles suspended in a liquid or gas, was first observed in 1827 by botanist Robert Brown, who saw minuscule particles moving within pollen grains such as those pictured here.

Brownian motion, or the random motion of particles suspended in a liquid or gas, was first observed in 1827 by botanist Robert Brown, who saw minuscule particles moving within pollen grains such as those pictured here.

Close modal

Amir’s own work is at the vanguard of complex-systems theory, and his experience shows: One of Thinking Probabilistically’s main strengths is how it introduces many interesting and inspiring advanced topics, including barrier-escape problems, generalizations of the central limit theorem, and percolation. Each of those subjects has applications in several fields, which means that the book will appeal to students and researchers in a wide variety of disciplines.

Thinking Probabilistically assumes its readers have an undergraduate background in physics, that they are familiar with calculus and linear algebra, and that they have some background in probability theory and complex analysis. In that sense, the level of mathematical understanding required is on par with that of well-known textbooks such as Mathematical Methods of Physics (2nd ed., 1970) by Jon Mathews and R. L. Walker. Throughout, Amir is careful to prioritize physical intuition over mathematical rigor, and the author gives plenty of heuristic hints. For those who need help, the appendix contains brief reviews of probability theory, linear algebra, contour integration and Fourier transforms, some basic mechanics and statistical mechanics, and functional derivatives and Lagrange multipliers.

The book is a suitable springboard for self-study because it introduces a wide variety of topics and contains many references to current work. In the classroom, the book can function either as the basis for a course in special topics or as a source of material to spice up more traditional statistical-mechanics courses.

Chapters 1 through 3 cover random walks and the Langevin and Fokker–Planck equations. Because those topics lay the groundwork for the rest of the book, that portion should probably be read first. After that, the rest of the chapters can be digested independently. Many of the real-world examples presented in the later chapters come from physics, but Amir also includes interesting cases from other fields such as finance, biology, computer science, and even hydrology. It’s not often that you see topics like the Black–Scholes equation for option pricing and the Google PageRank algorithm discussed in a physics textbook, but Thinking Probabilistically covers both in depth.

Amir has chosen exercises inspired by research questions to support and deepen the arguments made in the main text. They are designed to invite readers to think outside—sometimes far outside—the box. For that reason, they are more open-ended than those one would find in standard textbooks.

There are two minor points of criticism that I feel compelled to mention. First is that the level of technical detail is a bit overwhelming at several points, including, for example, the discussion of random-matrix theory at the end of chapter 8. In fairness, Amir does warn the reader that things are going to get tricky before he delves into such topics, but the difficulty spike is still notable.

Second, when describing Brownian motion, the author propagates a common error found in many physics textbooks: that Robert Brown observed the motion of pollen grains. But pollen grains are far too large to exhibit visible Brownian motion. What Brown actually observed in 1827 was the motion of small particles within pollen grains. It is pedagogically important to correct that error because, as David Layton noted in a 1965 article in the Journal of Chemical Education, failing to do so “fosters a misleading impression of the scale of the phenomenon.” Making the correct statement also shows our friends in the life sciences that we physicists at least know a little bit of biology.

One of the book’s explicit goals is to bridge the gap between the world of active research and the sanitized treatment of random phenomena found in the typical textbook, thereby enabling readers to follow contemporary research papers. The book succeeds in that aim, but making that leap will require hard work on the part of a reader not already familiar with the techniques being presented. A necessary consequence of the book’s brevity is that the conceptual and technical jumps in reasoning are often quite large, and the reader will have to fill in many of the details.

Nevertheless, the methods for studying random phenomena introduced in Thinking Probabilistically will help readers understand reasoning techniques that may not be terribly familiar to physicists. Moreover, following the author’s arguments is a rewarding intellectual exercise in its own right.

Rob de Ruyter is a professor in the department of physics and the program in neuroscience at Indiana University Bloomington. His research focuses on the neural processing of sensory information in animals.