Stochastic Processes for Physicists: Understanding Noisy Systems

Stochastic Processes for Physicists: Understanding Noisy Systems , Kurt Jacobs

Cambridge U. Press, New York, 2010. $43.00 (188 pp.). ISBN 978-0-521-76542-8

Probability theory is strongly connected to physics: Not only are many physical systems well-modeled as random, or “stochastic,” processes, but there are deep links to statistical mechanics, dynamical systems, and the asymptotic “laws of large numbers” governing random phenomena. Sadly, as mathematical physicist Raymond F. Streater wrote, “there are few mathematical topics that are as badly taught to physicists as probability theory. Maxwell, Boltzmann and Gibbs were using probabilistic methods long before the subject was properly established as mathematics.” Evidently, this section of physics pedagogy needs updating.

That’s what Stochastic Processes for Physicists: Understanding Noisy Systems attempts to do. Author Kurt Jacobs specifically addresses the kind of stochastic processes that arise from adding randomly varying noise terms into equations of motion. Suppose, for example, that we’re interested in how the bulk magnetization of a paramagnet responds to an external magnetic field being shut off. The relaxation will have a deterministic component, reasonably modeled with an ordinary differential equation. But the actual magnetization of the material will be a sum over the magnetic fields of many atoms that share energy thermally with each other and with the environment. Thus, we expect fluctuations around the deterministic relaxation of the magnetization. Assuming the magnet is far from the critical point, those fluctuations will be Gaussian. The fluctuations are not just measurement errors on our part—the material really is becoming more or less magnetized. Because of the presence of noise terms, the differential equations that describe the system are not solved by ordinary integration; rather, solutions to stochastic differential equations call for stochastic integration.

Jacobs lucidly tackles the field of stochastic differential equations as a fairly unified whole, which it is, rather than a collection of special cases. After two chapters of setup, the core of the book—chapters 3 through 6—introduces stochastic differential equations and Ito calculus, named for probabilist Kiyoshi Ito, who worked out the rules for manipulating stochastic integrals with Gaussian white noise. In chapters 7-9, the author discusses the Fokker-Planck equation, a linear partial differential equation that concerns probability distributions, and Poisson and Lévy noise, which are non-Gaussian (stochastic partial differential equations are not touched on). These chapters all have considerable merits. Jacobs is an enthusiastic, clear, and concise writer. He presents each theory by means of heuristic arguments and calculations. The examples, although not treated in depth because of space, are pedagogically valuable.

But the book also has its weaknesses. Every model presented is a Markov process: Given a system’s current state, its past is irrelevant to its future. (Oddly, the term “Markov process” does not appear in the book.) It would have been good to see some mention of the powerful general theory of Markov processes, or the existence of non-Markovian stochastic processes.

More serious are misstatements about some basic points of probability theory. For instance, the author repeatedly says that the sum of two random variables has a distribution that is the convolution of their individual distributions; that is only true for independent variables. Similarly, the author remarks on page 30 that we are “forced” (his italics) to use Gaussian noise “for the purposes of obtaining a stochastic differential equation that describes real systems driven by noise.” Then he contradicts himself by the entire chapters that discuss Poisson and Lévy noise!

There is a strange animus to probability theory as used by mathematicians, statisticians, economists, and control engineers. Chapter 10 introduces the jargon of modern probability theory based on measure theory, but in a spirit of prophylaxis against infection. The modern approach was introduced about 80 years ago, precisely to handle physically motivated problems, such as continuous fluctuations, in a coherent way. Mathematicians and statisticians swiftly abandoned the kind of probability theory still taught in physics courses because the new variety is much more powerful and clear and the math is not that hard. (Having received a PhD in theoretical physics, and having used and taught measure-theoretic probability, I am quite sure about this.) Plenty of books on group theory are available for physicists, and quite properly so — physicists have different needs and interests than mathematicians. But none of the ones I’ve seen end by saying what, in effect, chapter 10 says of measure theory: “What the algebraists have to say about groups is useless, so don’t bother with their ideas.”

To sum up, Stochastic Processes for Physicists may be useful as a solid part of a course on stochastic models in physics, provided the lecturer can correct some of its odd swerves. But I cannot recommend it unreservedly for self study.