Statistical Physics of Particles ,

Statistical Physics of Fields ,

Several years ago while Mehran Kardar and I were sitting outside a coffee shop in Berkeley, California, and discussing the behavior of a system that interested us, he conceived of and sketched a derivation of statistical field equations governing the system’s dynamics. Admiring what he was showing me, I asked him how he did it. He shared his logic, and it proved to be a useful lesson. Now, Kardar shares his approach with everyone who chooses to read * Statistical Physics of Fields* (and thus ends whatever undeserved advantage I held over some of my competition). The book is the second of two volumes—the other being * Statistical Physics of Particles*—that Kardar has written based on a two-semester statistical mechanics course he offers at MIT to physics graduate students. Over the past two decades I have admired Kardar’s contributions to theoretical physics, and now I admire his contribution to teaching physics.

Long ago, statistical mechanics was a tangential topic of physics, one that a single graduate textbook could summarize with reasonable completeness. That changed starting in the 1970s, when the importance and breadth of the field began to grow enormously. Statistical mechanics is now indispensable in virtually all of the natural sciences and beyond, from mathematical physics to molecular biology, from economics to social networking. One reason for its growing and unifying role is the development of renormalization group theory, which played out in the 1970s. Another is the ease of numerical simulation, which also began around the same time. Those and related developments changed the sociology of theoretical physics and placed statistical physics at its center.

Nowadays, no one- or two-volume text can cover the entire field. Nevertheless, a good text will provide a foundation for students who can then venture far beyond. I believe that Kardar’s two volumes serve that purpose. The discussions are succinct, focused, often elegant, and almost always demanding. Much of the physics is presented through solutions to exercises appended to all but the second volume’s last two chapters, and in using the books, students will learn by example. The first volume contains roughly 200 pages of standard text and about 100 pages detailing solutions to many, but not all, of the exercises; the second volume has about 250 pages of text plus about 100 pages of problem solutions. Students will not find the books easy going, but they will be substantially rewarded for their hard work. Precocious students might use the texts successfully without an instructor. They will likely need to be comfortable with mathematics typical of a graduate-level quantum mechanics course, and they will need to intuit the meanings of some notations.

Together, the two volumes cover many of the standard topics—ensembles, real gases, Bose–Einstein condensation, equilibrium Landau–Ginzburg theory, the Ising model, and so forth. Nonetheless, there are notable absences. Among the most significant omissions is a systematic treatment of linear response and the connections between measurements and correlation functions. Also, the book does not include a substantive discussion of numerical methods; it has only the briefest mention of the simplest of Monte Carlo schemes. What Kardar’s text does have—in the first eight chapters of the second volume—is a superb treatment of time-independent statistical field theory that starts with a discussion of elasticity in ordered solids and ends with the two-dimensional Coulomb gas and 2D melting. Several other good texts treat phase transitions by means of scaling, series expansions, and renormalization group theory, but Kardar’s coverage is special. Students getting their first exposure to the topic will obtain an excellent foundation through a remarkably compact yet reasonably complete and understandable presentation of the essentials of symmetries and physical reasoning and also the nuts and bolts of lattice sums, combinatorics, functional integrals, and field-theoretic perturbative calculations. Expert readers will enjoy seeing how Kardar does the job, and they might gain new insights.

The first volume includes cursory discussions of time dependence, but nonequilibrium phenomena are not otherwise discussed until the last two chapters of the second volume. Entitled “Dissipative dynamics” and “Directed paths in random media,” they survey several of the most interesting topics to which Kardar has contributed significantly. The approach taken in those chapters is closer to that of a review article than that of a textbook. Although they are a useful compendium, I prefer something more like the first eight chapters, with an assortment of exercises and illustrative problem solutions.

The first volume, * Statistical Physics of Particles*, is distinguished by its useful feature of teaching by example, but otherwise its presentations are similar to those in other excellent texts of similar level. On the other hand, the first eight chapters of * Statistical Physics of Fields* are stunning. With that volume Kardar has produced an excellent and unique textbook that will serve our community well for many years.