Statistical Mechanics Made Simple: A Guide for Students and Researchers , Daniel C. Mattis , World Scientific, River Edge, NJ, 2003. $48.00, $24.00 paper (252 pp.). ISBN 981-238-165-1, ISBN 981-238-166-X paper
Daniel Mattis’s Statistical Mechanics Made Simple: A Guide for Students and Researchers is an admirable piece of work by an outstanding expert in the field. But its title is rather misleading, and I question the author’s view on the appropriate audience for it. Mattis explains that the first four or five chapters are “suitable for an undergraduate course for engineers and physicists” and that the final five chapters “treat topics of recent interest to researchers.” His approach, though, leaves a big gap. It almost seems appropriate to review the first and second parts of the book separately. The first part is undoubtedly a difficult read for beginners; the second, however, is an engaging study of statistical mechanics for experts in the field.
The early chapters cover a good deal of elementary material, but on a fairly advanced level and from a different perspective than is normally taken. The author makes a number of clever observations or assumptions that lead to his obtaining nontrivial results in a few lines. However, compactness is not the same as simplicity, and my concern is that the intended audience of undergraduates might not be able to follow all the arguments.
The first chapter begins with a brief but excellent introduction to binomial and Poisson probability distributions. Mattis then introduces a multinomial distribution and presents its logarithm. Without further elaboration, he writes, “We then arbitrarily identify the first bracket on the rhs of the equation … with the negative of the energy of the system,” and the second bracket “with the product of its entropy and the temperature.” For experts, this approach is a clever way of pointing to the fundamental mathematical structure of the theory, but for beginning undergraduates, the author’s presentation will probably be completely mysterious.
Mattis’s introduction of the thermodynamic definitions of temperature, pressure, and chemical potential is likely, unfortunately, to present similar difficulties for students encountering the terms for the first time. Instead of showing readers that these thermodynamic definitions logically follow basic physical principles, Mattis makes them seem arbitrary. The same problem arises when the author introduces statistical mechanics in chapter 4. He begins by defining the quantum mechanical partition function as the trace over accessible states. Although such an approach is often done in other textbooks on statistical mechanics, I find it to be insufficiently justified for beginning students.
In other respects, the development in the first part of the book is clean, albeit terse. For example, despite its brevity, the introduction of Legendre transforms is remarkably readable and complete. Mattis discusses the Gibbs paradox along traditional lines by using an explanation and resolution of the problem for indistinguishable particles. Although I personally disagree with that interpretation, Mattis presents his argument with great clarity. He offers a good selection of introductory topics in the theory of fluids, including two-particle correlation functions, structure factors, and different phenomena occurring in one, two, and three dimensions.
Beginning with chapter 5, the level of treatment rises dramatically. The material becomes increasingly more interesting for experts and, in my opinion, less intelligible for students. The discussion in chapter 5 begins with a nice description of the raising and lowering operators in quantum mechanics. Many-body perturbations, phonons, photons, ferromagnons, Bose–Einstein condensates, the effects of interactions (both hard-core and weak), and the properties of superfluid helium are all covered in 21 pages. The chapter is fascinating to read, but I cannot imagine an undergraduate student successfully negotiating these pages without a great deal of assistance.
Chapter 6 on fermions also takes an advanced viewpoint; like chapter 7 on kinetic theory, it should be clear for experts. In chapter 8, Mattis treats transition matrices—including a derivation, in only a few pages, of the exact solution of the two-dimensional Ising model—in notable detail. Chapter 9, the final chapter, gives some interesting applications, again at a high level, of quantum field theory in condensed matter problems.
In view of the many topics covered in this slim volume, it may seem inappropriate to complain about the absence of some of my favorites. Nevertheless, I did miss a discussion of the renormalization group, and I regretted that molecular dynamics and Monte Carlo methods received only a passing mention.
A more accurate title for this book would have been Statistical Mechanics Made Compact, because topics are introduced and discussed very rapidly. Unless readers have a considerable amount of experience in statistical mechanics, they may become more confused than enlightened by using Mattis’s book. On the other hand, for advanced graduate students, researchers, and professors, Statistical Mechanics Made Simple contains a wealth of valuable material and unusual insights.
