Field Theory of Non-Equilibrium Systems,

Cambridge U. Press
New York
, 2011. $80.00 (341 pp.). ISBN 978-0-521-76082-9

Our theoretical understanding of nonequilibrium statistical mechanics—a challenging subject—has been largely restricted to such near-equilibrium approaches as linear response theory and the fluctuation–dissipation theorem. There is a good reason for that: The richness and variety of nonequilibrium phenomena, it seems, means that it’s hard to identify universal constraints or construct generally applicable computational schemes.

One surprising exception is the so-called Keldysh technique, which is suitable for a broad range of phenomena (though there is also a large set of phenomena to which it doesn’t apply). The technique emerged from work done in the 1960s by Leonid Keldysh, Gordon Baym, Leo Kadanoff, and Julian Schwinger. It’s difficult to describe in nontechnical terms, but it can be viewed as an extension of Feynman’s path-integral technique to nonequilibrium statistical mechanics. By necessity, the evolution operators of nonequilibrium statistical mechanics involve both forward and backward time directions. The Keldysh technique, roughly speaking, centers on expressing those evolution operators as suitable quantum field theory integrals over closed time contours. The resulting integral expressions turn out to be computationally effective.

Field Theory of Non-Equilibrium Systems, written by theoretical condensed-matter physicist Alex Kamenev, is a lively pedagogical exposition of the Keldysh technique based on functional integration. At the start, the author uses the simplest possible system—a harmonic oscillator—to provide a detailed description of the method. He shows how the technique is applied in that familiar context and builds on that framework for more complex examples. The remaining early chapters, covering such areas as classical stochastic systems and fully interacting bosonic fields, outline the general procedure.

The later chapters, which can be read more or less independently, are devoted to various carefully chosen applications, based on such classical topics as the dynamics of collisionless plasma and electron–electron interactions in disordered metals. The author appears to have given a great deal of thought to the choice of material, with the goal of presenting a few key topics in strikingly clear pedagogical terms. For example, the 20-page discussion (in chapter 10) on the quantum transport of free fermions can easily be expanded into a monograph.

One shortcoming of the book is that experimental results are not discussed, possibly due to space limitations. Although that is understandable, readers may wish that the book had an accompanying volume with carefully chosen experimental results presented in similar pedagogical terms. Also, the author references the original papers from the vast literature on the Keldysh technique, but he provides almost no discussion of the developments in a historical context. That, too, is understandable; but given the author’s expertise and wisdom in choosing topics, an appendix that discussed the historical context of each chapter and that mentioned related developments not discussed in the book would have been invaluable to the intended readership.

The book is not, and was not intended as, an encyclopedic exposition of the Keldysh technique. It is meant for advanced graduate students and professionals who have not had prior exposure to the technique but would like to learn it. Experts in the field may also enjoy the diversity of the subjects covered and the clarity with which they are presented. Thanks to those features, Field Theory of Non-Equilibrium Systems is a welcome introduction to the field and could well become a classic.