Gibbs and Equilibrium Statistical Mechanics
In 1901, at the age of 62, Gibbs (1839-1903)
published a book called Elementary Principles in Statistical Mechanics
(Dover, New York). It was remarkable in several ways. First, it had as a
subtitle The Rational Foundation of Thermodynamics. Gibbs chose
this subtitle because he knew his theory did not agree with experiments,
as he emphasized in the preface to the book. Yet he believed there were no
other possible rational basis to thermodynamics. Second, the style of
writing of the book has an elegance akin to a long poem, which is quite
unique in the history of physics. Third, it was to become the fundamental
basis of twentieth century equilibrium statistical mechanics. A year
before Einstein's death in 1955, he was asked who were the most powerful
thinkers he had known. He replied (Ref. 1), ``Lorentz'', and added, ``I
never met Willard Gibbs; perhaps, had I done so, I might have placed him
beside Lorentz''.
Origin of Bose-Einstein and Fermi-Dirac Statistics
In the last chapter of Gibbs' book mentioned above, he found it necessary to insert a factor N! in the formula for the free energy:
In June 1924 Bose's paper was published. He derived Planck's radiation law by counting states of photons in a novel way. As soon as Einstein saw this paper, he generalized it to the counting of states of atoms, thereby predicting the phenomena of Bose-Einstein condensation, a most daring and insightful extrapolation which has only now been brilliantly experimentally confirmed.
A year and a half after Bose's and Einstein's papers, upon reading Pauli's article on the exclusion principle, Fermi realized in 1926 that he had now the concepts in hand to discuss the thermodynamics of a collection of electrons. The results were such fundamental concepts like the Fermi sea, the Fermi energy, etc.. According to Rasetti (Ref. 2), Fermi was not influenced by the earlier work of Bose and of Einstein.
Also in 1926, Heisenberg pointed out that the difference between the singlet and triplet energy levels of the (1s)(2s) states of He was due to the difference of the symmetry of the space wave function, which in turn was caused by the requirement of antisymmetrization of the total wave function of the two spinning electrons.
Finally in August 1926 Dirac developed the general theory of the symmetry of wave functions of Bose-Einstein and Fermi-Dirac particles.
Incidentally this chain of papers Einstein-Fermi-Heisenberg-Dirac reveals in a dramatic way the differences of styles of these four great physicists:
- Einstein's prediction of Bose-Einstein condensation of free particles was against all intuitive concepts of phase transitions at that time. To make such a prediction, without full mathematical rigor, based on a novel counting method extrapolated from photons to atoms, required a perception and a boldness that was uniquely Einstein's.
- Fermi's paper formed the basis of all subsequent theories about condensed matter physics. It has the hallmark of Fermi's physics: the ability to capture the fundamentals of the problem at hand and extract from it the essence that will affect all future developments. Solidity and imagination marched hand-in-hand in Fermi's work.
- Heisenberg's work produced the key idea that on the one hand linked the symmetry of wave functions to Pauli's exclusion principles and on the other hand resolved the great puzzle of how the spin alignment of two electrons can affect the Coulomb energy of the He atom. Furthermore the idea of the ``exchange integral'' which originated in this paper later produced another great achievement of Heisenberg's: the mechanism of ferromagnetism. However, Heisenberg's paper was long on originality but short on elegance and precision, a characteristic of all of Heisenberg's papers.
- In contrast, Dirac's papers were always elegant and precise. They also tend to be the final word in the problems that they address. In the case of his 1926 paper, very little can be added later to his masterly analysis.
Early Discussions of Phase Transitions in Statistical Mechanics
The development of quantum mechanics in 1925-1927 removed the difficulties that Gibbs had to face in checking his rational foundation against experimental facts. Thus was born quantum statistical mechanics. In the late 1920's and early 1930's, physicists and chemists applied the new quantum statistical mechanics to many problems in dilute gases and dilute solutions with great success.Then in the mid 1930's, because of the discovery of peaks in the C(p) versus T curve in alloys, the theory of order-disorder transformations became quite fashionable. An ``order parameter'' was introduced for the discussion of phase transitions. Such theories are now called mean field theories, which actually was first used by Weiss (Ref. 3) in 1907 for describing ferromagnetic transitions.
In 1937 J. Mayer attempted to formulate a theory of liquid-gas transition without introducing mean fields by examining the convergence properties of the virial series. There was a ``vigorous discussion'' of his ideas at the Van der Waals Centenary Congress (Ref. 4) on November 26, 1937 in Amsterdam, followed in the next months by elaborations/alternations of Mayer's theory. Incidentally, my own entry into statistical mechanics was related in a way to Mayer's theory. I was an undergraduate student in Kunming in 1941-1942 when Professor J.S. Wang gave several lectures on this theory of Mayer's and on subsequent developments. I did not then quite understand the complexities of the theories, but became fascinated with the subject (Ref. 5). That led to my working with Wang for my Master degree thesis on statistical mechanics. As to Mayer's theory, some ten years later in two papers (Ref. 6) Lee and I cleared up the confusion in this field.
Onsager and the ``Ising Disease''
In 1944, in an amazing paper, Onsager solved the two dimensional Ising model rigorously. It was the first in a field which undoubtedly will be covered in many presentations at this Conference. His paper was very difficult to read because he did not describe his strategy. He seemed to have a predilection for calculating the commutators of every other expression in sight without telling what he was aiming at. I still remember vividly today how I was frustrated in trying to understand that paper, first when I was a graduate student in China and then a graduate student in Chicago. It was many years later, in March 1965, that I finally learned (Ref. 7) how it had come about that Onsager was so fond of calculating those commutators.Young physicists today may find it surprising, even unbelievable, that in the 1950's the Ising model and similar problems were not deemed important by most physicists. They were considered arcane exercises, narrowly interesting, mathematically seducing, but of little real consequence. There was the phrase (Ref. 8), for example, of ``contracting the Ising disease''. In a recent article by Dyson in my Festschrift (edited by S.T. Yau, published by the International Press) he recalled how, in 1952, when he read my article about the magnetization of the Ising model, he was impressed by the beautiful complexity of the calculation and the beautiful simplicity of the result, but felt I was wasting my time.
The situation dramatically changed around 1960 because of several developments:(1) the experimental discoveries (Ref. 9) of divergences of specific heats near various phase transition points; (2) theoretical work on the critical exponents led gradually to the concept of universality and to some very useful inequalities among the critical exponents; and (3) the proposal of a scaling law (Ref. 10).
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