Statistical Physics: Statics, Dynamics, and Renormalization Leo P. Kadanoff World Scientific, River Edge, N.J., 2000. $78.00, $38.00 paper (483 pp.). ISBN 981-02-3758-8, ISBN 981-02-3764-2 paper
Even though I had few standards by which to gauge graduate students, it was clear that Leo Kadanoff, one of my first students, was special. No subject—from the mathematical properties of coherent states, which he studied with Roy Glauber, to the heating of nose cones, on which he consulted for a local defense contractor—was too abstract or too applied, and no calculation was too daunting. He took pride in finding answers speedily and cleverly. Spelling was not his forte, but there were few words between his equations. His talent and eclecticism were apparent, but the clear, elegant, economical, and idiosyncratic style that marks his lectures and writings was not yet evident.
With Gordon Baym, Kadanoff published his first book on statistical physics, Quantum Statistical Mechanics (Benjamin), in 1962. His recently published Statistical Physics: Statics, Dynamics, and Renormalization , is strikingly different. It is hardly surprising that books on statistical physics written almost 40 years apart should have little in common or that the newer of the two should be more inclusive or general.
In the most abstract sense, statistical physics encompasses all natural phenomena. Every process obeys the laws of physics and no system can be described with arbitrary precision. But this observation has no consequence. The significant fact is that the statistical mechanical techniques that were primarily used to explain the properties of gases and crystals fifty years ago, are now being widely applied to “squishy” matter (polymers, gels, and biomaterials), to living things, and even to such “unnatural” phenomena as financial systems and computational networks.
Readers of Quantum Statistical Mechanics might infer that microscopic dynamics and macroscopic thermodynamics are tightly linked and that techniques for calculating the static and dynamic behavior of complex systems are not beyond reach. Readers of Statistical Physics will immediately appreciate, from Kadanoff’s provocative allusions to glasses, turbulent fluids, chemical reactions, and vortices in superconductors, that (1) the links between thermodynamics and both hydrodynamics and microscopic dynamics are often very tenuous, and (2) the construction of relevant dynamical models is an art. Models that have no apparent connection with the underlying equations for fluid flow may be relevant both for turbulent flows and for stock price fluctuations.
The piquant hors d’oeuvres in the introduction to Statistical Physics may raise the expectations of some readers excessively. Apart from an interesting discussion of multiplicative random processes that explains why Gaussian distributions drastically underestimate the frequency of rare events, the ingredients of the main course are relatively conventional. The subjects, derivations, and examples that appear in the text (as distinct from the reprints)—in the half of the book that covers general statistical mechanics—are old friends: ensembles, Gaussian distributions, diffusion, Langevin and Fokker–Planck equations, conservation laws and hydrodynamics, correlations and fluctuations in systems at or near equilibrium, and noninteracting Bose and Fermi gases.
Some of these subjects are subtle, and Kadanoff spices his wide-ranging presentation with illuminating observations, illustrations, and instructive problems. The first half of the book concludes with reprints (altogether, reprints compose about 40% of Statistical Physics ) on diffusion-limited aggregation and self-organized criticality, two “big ideas” on which Kadanoff and his colleagues and collaborators have worked. Readers who haven’t experienced Kadanoff’s classes may well wonder what prompted the author to bind these reprints to the preceding text, interesting as they both may be. Readers may also lament the absence of what surely would be a valuable exposition of some aspects of the complex natural phenomena to which Kadanoff alludes. How about a second volume, Leo?
In the second half of the book (about 60% of which consists of reprints), Kadanoff discusses second-order phase transitions, an area in which his seminal contributions are legend. After an introductory chapter on mean-field theory, he discusses the phenomenological theory of scaling (a theory that he attributes to Ben Widom but that Kadanoff had developed independently before I referred him to Widom’s papers). Then he deals with fixed points, universality, operator expansions, duality, the Ising model, real space renormalization methods, and systems with two-dimensional continuous symmetries. One still marvels at the beauty and generality of the physical and mathematical concepts and the ingenious calculational techniques Kadanoff introduced during the late 1960s and early 1970s and lucidly reviews here, to say nothing of his work with Jack Swift on mode coupling and lattice hydrodynamics and its relation to the work of others on dynamic critical phenomena!
Nonetheless, should a text on applications of the renormalization group to continuous phase transitions say nothing about upper and lower critical dimensions? And should it dispense in a single sentence (“The most remarkable application of [Wilson’s] theory was the [Wilson and Fisher] development of an expansion about 4 dimensions.”) with the expansion techniques (involving dimension and the inverse of the number of components of the order parameter) that are widely used to calculate critical exponents?
In short, Statistical Physics is a collection of valuable essays and papers. Both the text and the reprints display Kadanoff’s ingenuity, imagination, and clarity. They’re worth having and reading, as are most of the classic papers of others, (including Tom Witten and Leonard Sander; Mike Kosterlitz and David Thouless; and David Nelson) that the book contains. Students who took courses in which Kadanoff discussed these papers were very well served.
But students not lucky enough to have heard Kadanoff in person may wonder why some classic materials are not included and wish that Kadanoff had bound the contents together with more pedagogic glue. They’ll also note that proofreading may still not be very high on Kadanoff’s priority list.
