C. N. Yang made part of his early reputation in statistical mechanics, calculating, for example, the magnetization in the two-dimensional Ising model of a ferromagnet. He subsequently worked on particle physics, inventing modern gauge field theory ("Yang-Mills theory"), and winning the 1957 Nobel Prize for his formulation with T. D. Lee of tests for parity violation in weak interactions. Yang and Lee also did considerable work together on statistical and solid state problems. Yang's influence and the techniques he introduced still persist, e.g., in the "Yang-Baxter" equations which appear in many problems and were first introduced by Yang (Phys. Rev. Letters 19, 1312 (1967)).

The following is the written version of a short talk Yang gave on some historical developments in statistical mechanics in the twentieth century. It is well worth reading to get an overall picture of some of the principal developments as seen by a master of the subject (and of many other subjects), and for his commentary on some of the leading theorists of the era.

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:

1. 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.

2. 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.

3. 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.

4. 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.

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.

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).

With all these developments equilibrium statistical mechanics finally became respectable and was considered physically important in the 1960's.

1. A. Pais, Subtle is the Lord, the Scientific Biography of Albert Einstein, (1982).
2. R. Rasetti in Collected Papers of Enrico Fermi, vol. 1, p. 178 (University of Chicago Press, 1965).
3. P. Weiss, J. of Physics 6, 667 (1907).
4. M. Born and K. Fuchs, Proc. Roy. Soc. A166 , 391 (1938).
5. Chen Ning Yang, Int. J. Mod. Phys. B2 , 1325 (1988).
6. C. N. Yang and T.D. Lee, Phys. Rev. 87 , 404 (1952); T.D. Lee and C. N. Yang, Phys. Rev. 87 , 410 (1952).
7. Chen Ning Yang, Selected Papers 1945-1980 with Commentary, p. 11-13 (Freeman, 1983).
8. A. Pais, Nucl. Phys. 5 , 297 (1958).
9. W.M. Fairbank, M.J. Buckingham and C.F. Kellers, Proceedings of the Fifth International Conference on Low Temperature Physics (Madison, Wisconsin) p. 50. See also W.M. Fairbank and C.F. Kellers in Critical Phenomena, National Bureau of Standards Miscellaneous Publication 273 (1965); W.K. Robinson and S.A. Friedberg, Physical Review 117 , 402 (1960); M.I. Bagatskii, A.V. Voronel and V.G. Gusak, J. Exp. Theor. Phys. (in Russian), 43 , 728 (1962). [English translation: Soviet Physics JETP 16 , 517.]
10. B. Widom, J. Chem. Phys. 12C , 3898 (1965).

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