The best way to become acquainted with a subject is to write a book about it.
The subject of statistical mechanics dates back to the 19th century. It has a rich history, and the basics of the subject are taught in all undergraduate and graduate physics programs. Consequently there is a wealth of books that explain the elementary aspects of the subject which form the foundation for all thermal properties of condensed matter systems. The content of these books is all rather similar in that they cover thermodynamics, ensemble theory, one-body problems and the perfect Bose and Fermi gases. These topics are all considered to be closed subjects which are thoroughly understood.
This book is an outgrowth of the author's teaching of advanced courses in statistical mechanics which go beyond the topics covered in elementary courses and is aimed at introducing the reader to topics in which there is ongoing research. In contrast to the material in an elementary course almost all topics lead to open questions, and the aim of this book is to present these topics of ongoing research to as wide an audience as possible. Consequently in almost all chapters there are sections on open questions and what I call missing theorems where one's physical intuition suggests that results should be true but for which no proof yet exists. It is hoped that, by highlighting the many places where there are unresolved questions, this book can stimulate progress in the field.
The selection of topics in any advanced treatment of a subject is affected by the tastes of the author and so several comments about my selection of topics are in order. I have chosen to divide the subject somewhat arbitrarily into three parts: exact general theorems; series expansions and numerical results; and solvable models. Each of these divisions has an immense literature and within the confines of one book it is not possible to state all results and prove all known theorems. I have therefore adopted the procedure of stating and explaining many results but have only given the proofs of a selection of the theorems stated. There is no other alternative since there are many important theorems whose proof in the literature requires papers of 40–50 pages. For example the proof of the stability of matter is the subject of a book in its own right by Elliot Lieb; Rodney Baxter devotes an entire book to the free energy and order parameters of the six-vertex, eight-vertex and hard hexagon models; T.T. Wu and the present author devote an entire book to the Ising model. In this sense this present book can be considered to be an introduction and guide to, but is hardly a substitute for, the literature of the past 50 years.
(p.vi) The reader will almost instantly note that there are several well-known topics in statistical physics which are not covered in this book: namely the renormalization group and mean field theory. This omission is deliberate since both of these topics are well covered in many books and it is, in my opinion, superfluous to give one more account of these methods.
Constant progress is being made in statistical mechanics and it is certain that even at the time of publication some topics will have advanced beyond what is presented here. In particular I draw the reader's attention to several examples of recent work: the proof discussed in chapter 4 of Kepler's conjecture that no packing of hard spheres in three dimensions can be more dense than the face centered cubic lattice, and the discovery also presented in chapter 4 that for ellipsoids there are packings denser than the fcc lattice. There are several computations presented which were initiated in part by the desire to clarify and better understand the existing literature, in particular the computation of the tenth order virial coefficients in chapter 7, the diagonal susceptibility and the evaluation of the form factor integrals of the Ising model in chapter 10 and 12 and the treatment of the TQ equation of the eight-vertex model in chapter 14.
There are also many interesting and important problems which are omitted merely for lack of space. In particular there are many solvable models which have not been mentioned. Furthermore there is no discussion of the methods of the coordinate and algebraic Bethe's ansatz and the mathematics of quantum groups and conformal field theory. These topics require much more space than this book allows and are treated extensively by other authors.
However, it is hoped that in spite of the many necessary omissions that this book covers a sufficiently large number of topics so that the reader will gain an appreciation of the great breadth of the subject; the many areas of progress which have been made in the past 40–50 years, and the places where future advances will be made.
I am fond of saying that “you cannot say that you understand a paper until you generalize it.” This, of course, leads to the logical corollary that “no author can be said to understand his/her most recent paper.” This book is an excellent demonstration of the truth and meaning of this corollary. In every chapter there are open questions and topics that need further research and explanation. Some of the more obvious and unavoidable of these questions have been singled out for discussion but many, if not most, are quietly hidden away waiting for the reader to discover them. There are many derivations and computations presented and, barring misprints, the proofs should be sufficient to prove the conclusions. But in no place is it ever shown that the given proof is actually necessary for the conclusion and that the steps exhibited actually reveal the mechanism for the phenomena being discussed. The most glaring example of this problem is the evaluation of integrals done in chapter 12 by the use of MAPLE for which, at the time of writing, no analytic derivation exists.
There are many places in this book where I need to thank collaborators and friends for their help and suggestions: Nathan Clisby for the evaluations of virial coefficients in chapter 7; Jean-Marie Maillard for teaching me how to do the symbolic evaluation of integrals on the computer in chapter 12; Klaus Fabricius for collaboration on the Q matrices of the eight-vertex model of chapter 14; and Jacques Perk and Helen Au-Yang for collaboration on chiral Potts models and for figures 4 and 5 of chapter 15. The method (p.vii) used in chapter 6 to prove the Mayer expansion comes from a set of lectures given by Hans Groeneveld in the late 1960s who has given me much valuable advice in the preparation of that chapter.
I am most grateful to the National Science Foundation for partial support during much of the time when I was writing this book and to the Rockefeller Foundation for a one-month residency at their Bellagio Conference and Study Center where several chapters were revised and perfected.
In conclusion I must thank and acknowledge two remarkable people to whom I am deeply indebted and without whose encouragement and inspiration this book would never have been completed. The first is my late wife, Tun-Hsu Martha McCoy, who helped me every step of the way and put up with the innumerable frustrations I have had during the far too many years I have spent in writing. The other is my classmate of 51 years ago from Catalina High School in Tucson, Arizona, Margaret Hagen Wright, who has given me profound friendship in a time of great need.
Stony Brook, New York