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发信人: leader (kikizh), 信区: Physics 标题: Physical Review 100 years collection 上对统计物理的评述[转载] 发信站: BBS 珞珈山水站 (Wed Jan 2 22:49:54 2008) Physical Review 100 years collection 上对统计物理的评述[转载] Statistical Physics Joel L. Lebowitz Department of Mathematics and Physics, Rutgers University, New Brunswick, New Jersey 08903 Joel L. Lebowitz is the George William Hill Professor of Mathematics and Physi cs and Director of the Center for Mathematical Sciences Research at Rutgers Un iversity. He came to the United States in 1946 and was a student of Melba Phil lips, Peter Bergmann, and Lars Onsager. His work has been mainly in statistica l mechanics with an outsider's interest in foundational questions. His awards include the A. Cressy Morrison Award in Natural Sciences from the New York Aca demy of Sciences and the Boltzmann Medal from the IUPAP Committee on Thermodyn amics and Statistical Physics. The Physical Review is a relative latecomer to the world physics scene. Althou gh it started almost 120 years after the founding of the United States, its cl ientele was rather small in the beginning. It was not until the 1930s, when sc ientists (and others) with "wrong" ancestors and beliefs became hunted species in certain parts of Europe, that physics and other sciences began their explo sive stage of growth in the U.S. With this growth came the flowering of The Ph ysical Review. By the 1940s The Physical Review was the leading physics journa l in the world — a position it and its offspring, Physical Review Letters, ma intain to this day. Dyson1 describes Vol. 73 of The Physical Review, which con tains the issues from January to June 1948, his first year in the United State s, as follows: "Almost every paper in it is interesting, and many of them are memorable. In 1948 issues of the journal were thin enough to be read from cove r to cover. Many of us did just that.... The paper that impressed me most in 1 948 and still impresses me today is entitled 'Relaxation Effects in Nuclear Magnetic Re sonance Absorption,' a monumental piece of work, 34 pages long, by Bloembergen , Purcell, and Pound... . In the same volume of The Physical Review are many o ther wonderful papers on the most diverse subjects: Alpher, Bethe, and Gamow o n the origin of chemical elements; Wataghin on the formation of chemical eleme nts inside stars; E. Teller on the change of physical constants; Lewis, Oppenh eimer, and Wouthuysen on the multiple production of mesons; Foley and Kusch on the experimental discovery of the anomalous magnetic moment of the electron; Schwinger on the theoretical explanation of the anomalous moment; and Dirac on the quantum theory of localizable dynamical systems. This was a vintage year for historic papers and I mention these seven just to give the flavor of what physicists were doing in the early post-war years." Dyson's description illustrates the difficulty, nay impossibility, of selectin g a seminal sample from among 90 years of The Physical Review papers. It shoul d therefore be clear that this extremely space-constrained collection is not a scholarly compendium of either the "most important," "most original," "most c lever," "most memorable," or "most anything" papers that appeared in The Physi cal Review or Physical Review Letters prior to 1984. It is rather a very small sample of such papers collected from responses to the solicitation letter sen t out by H. Henry Stroke to a small sample of physicists, from conversations w ith a small sample of friends, from references in a small sample of books or r eviews I happened to have on my shelves, from some very time-limited sample vi sits to the library, and finally from inherently faulty personal recollections , judgments, and prejudices. In particular, I believe that mathematically rigo rous results are, in appropriate situations, not just "decorations on the cake "; they are essential for getting the physics right. Given the above, I apologize in ad vance for all sins of omissiona and commission and only hope that the selectio n gives the reader a feeling of how exciting good science can be. Having made my disclaimer, I take this opportunity to give a brief, highly per sonal, selective overview of developments in nonequilibrium and equilibrium st atistical mechanics in order to put the works contained here in some context. Nonequilibrium I am a bit younger and much less precocious than Dyson, so although I came to the United States a year or two before him, I did not start reading The Physic al Review until 1953, when I was a graduate student of Peter Bergmann's at Syr acuse University. It was also in that year that I first met Lars Onsager, whos e 1931 paper "Reciprocal Relations in Irreversible Processes II" is the earlie st one in this section of the book. (Note that this article was submitted on 9 November and published in the 15 December issue of that year.) Onsager's paper is the second of two works (the first one was too long for the book and is included in the CD) devoted to the deep underlying symmetry prope rties of the matrix connecting thermodynamic fluxes and forces. While particul ar examples of such symmetries had been known for a long time, it was Onsager who first understood that they are a general feature of transport in systems c lose to equilibrium. These symmetries are a direct consequence of the reversib ility of the laws governing the time evolution of the microscopic constituents of macroscopic matter and the additional (mild) hypothesis that the time evol ution of a macroscopic system following its preparation in a macrostate out of equilibrium is the same as that following a spontaneous fluctuation from equi librium to that macrostate. The "Onsager relations" regarding the symmetry of the transport coefficients Lij = Lji are, next to the second law of thermodyna mics, the only known results about nonequilibrium phenomena that have a genera lity resembling that of equilibrium thermodynamics. They are in fact related to Boltzma nn's probabilistic formulation of the second law connecting macroscopic entrop y with microscopic phase space volume. The connection between microscopic and macroscopic evolutions and the role that the large number of microscopic degre es of freedom play in determining universal features of macroscopic behavior w as further developed by Einstein in his beautiful quantitative theory of Brown ian motion and in his relation between equilibrium fluctuations and macroscopi c entropy. The works of Boltzmann and Einstein are the direct forerunners of O nsager's work. The fluctuation-dissipation theorem of Callen and Welton3 extends the Boltzman n-Einstein-Onsager connection between spontaneous fluctuations in equilibrium systems and the macroscopic response of the system when not in equilibrium. By considering the response to an externally imposed perturbation of the Hamilto nian describing the time evolution of the ensemble density in the Gibbs formal ism of statistical mechanics, Callen and Welton were led to a general expressi on for (some) frequency-dependent transport coefficients in terms of the Fouri er transform of the time evolution of equilibrium fluctuations. The same circl e of ideas leads also to the Green-Kubo4 formulas for transport coefficients a nd to the Onsager-Machlup5 formulation of probabilities for fluctuations in th e time evolution of macroscopic systems. The above general results apply to linear transport in systems close to equili brium. Their extension to systems far from equilibrium is still a very active research area. We should not be surprised, however, if it turns out that such general formulations may not be appropriate for some of the most interesting p henomena in such systems, including those most relevant to us, e.g., us, in ou r universe. These phenomena may not be capturable by general formulas; to quot e Dyson again, "God is in the details." Still, there have been major advances in our understanding of the potentialities inherent in nonlinear dynamical sys tems. These include the discovery of solitons in integrable systems represente d here by the paper of Zabusky and Kruskal.6 Another discovery, or more proper ly a belated realization by the physics (as opposed to the mathematics) commun ity, was that most dynamical systems, even those with only a few degrees of fr eedom, are very far from integrable — rather, the trajectories describing the ir time evolution are subject to sensitive dependence on initial conditions, parameters , perturbations; in short, they exhibit deterministic chaos.7 To put the discussion of chaos into this centennial framework, one should at l east mention the seminal work of Poincaré. It was Poincaré who first realize d the deep difference between linear and nonlinear evolutions with the inheren t tendency for complex, apparently chaotic, behavior of the latter. Following Poincaré, the torch of nonlinearity was mainly kept alive in what is now the Former Soviet Union (FSU). In the FSU, following a tradition established by Ly apunov, the work of mathematicians such as Kolmogorov, Bogolyubov, Rokhlin, an d members of their schools, helped develop many of the fundamental ideas of th e modern analytical theory of nonlinear dynamical systems. The works of Arnold , Chirikov, and Sinai from the FSU and of Birkhoff, Smale, Siegel, and Moser f rom the West have been particularly relevant for our understanding of determin istic chaos. (I never heard of Poincaré's work in any of my courses and first learned of it from Prigogine and Ford.) The analytic insights of Poincaré were subsequently visualized clearly throug h the use of that modern (it occurred in my lifetime) invention, the computer, by Hénon and Heiles, reprinted in Ref. 7. It was the computer too through wh ich Alder and Wainright8 discovered the first example of the now ubiquitous "l ong time tails" of correlations present in many-body dynamical systems. Before the advent of computer simulations, there was a general belief (if my memory serves me right) that correlations typically showed exponential decay in time — the power law decay of conserved densities satisfying diffusion equations b eing truly exceptional. One of the first uses of the computer in statistical m echanics was the computation by Fermi, Pasta, and Ulam9 (their famous Los Alam os report was never published in any journal) of the time evolution of a chain of 32 oscillators nonlinearly coupled via a cubic and/or quartic Hamiltonian. They found to their surprise that there was no equipartition of energy among the different (linear) normal modes on the time scale of their computation.b This was l ater understood to be a consequence of the celebrated Kolmogorov-Arnold-Moser (KAM) theorem.10 Another computer first from Los Alamos was the development of Monte Carlo methods by Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller.11 (The story has it that this was "cooked up" in the co urse of a dinner.) Some important items missing from The Physical Review and therefore from this collection are the papers by Lorenz,12 Ruelle and Takens,13 and Feigenbaum.14 Lorenz was the first to make clear the connection between the chaotic behavior of a dynamical system with a small number of degrees of freedom and the onset of turbulence in a driven fluid. Ruelle and Takens then showed that the onset of turbulence need not occur through an infinite sequence of bifurcations, as proposed by Landau, but can result directly from the interaction of three or more modes. The work of Ahlers and of Gollub and Swinney15 beautifully confirm s this. (I recall the excitement of going with Ruelle to Swinney's lab to watc h the experiment.) Feigenbaum, aided by a very modest computer, in what is for me still one of the best examples of the good uses of computers in science, d iscovered the existence of a simple "universal" (but far from obvious) feature in a large class of dynamical systems: period doubling leading to chaos. One of the first experimental papers to confirm this scenario clearly was by Libchaber and Maurer.16 Pomeau and Manneville17 found still another route to the onset of tu rbulence, that of intermittency — a route also confirmed by experiment. The situation is much less satisfactory with regard to fully developed turbule nce, perhaps the central problem of contemporary classical statistical physics . While the past 100 years have certainly seen important theoretical work on t his subject, including some deep insights by L. Richardson, Kolmogorov, Onsage r, Kraichnan, and others (mostly published in other journals), a full understa nding remains for the sesquicentennial celebration of The Physical Review. The celebrated Kolmogorov scaling law is represented in the book by a later indep endent short announcement by Onsager.18 (Apparently Onsager had this result fo r some years, but the war delayed its publication.19) Equilibrium Let me turn now to the simpler (but far from simple) equilibrium systems. Here too our story in this collection begins with Onsager. In an abstract of less than 70 words (which was in keeping with his generally sparing and precise utt erances), Onsager announced that he had obtained in closed form the partition function of the ferromagnetic Ising model in two dimensions and found that "Fo r an infinite crystal, the specific heat near T = Tc is proportional to -log\|T - Tc\|." This astonishing result can be said to mark the beginning of the mode rn era of equilibrium statistical mechanics. The full 1944 paper describing th is work is on the CD-ROM. Its precursor, the 1941 paper by Kramers and Wannier ,20 in which the exact value of Tc is found by duality, is also there.a The Ising model was the first many-body system with nontrivial interactions so lved exactly, and even after so many years and so many alternative derivations it is still an astonishing intellectual feat. I can still remember from my fi rst year of graduate school the admiration in Mark Kac's voice when speaking o f Onsager's solution in a colloquium at Syracuse University in 1953. He descri bed there his work with Berlin,21 on the solution of the much simpler spherica l model — "but the solution of the Ising model was a superheroic effort resis ting all assaults until Onsager threw at it his heavy machinery and the proble m collapsed under its weight." The formula for the spontaneous magnetization o f the Ising model was announced by Onsager at a conference at Cornell in 1948 and repeated in Florence in 1949, but no hint of its origin was given.19 This was supplied by Yang in 1952.22 Also in the same year Lee and Yang23 proved a remarkable theorem about the zer os of the partition function of the ferromagnetic Ising model in an external m agnetic field h: They must all be on the unit circle in the complex fugacity p lane z [in suitable units z = exp(h)]. The Lee and Yang paper was the companio n paper to one by Yang and Lee24 [which pointed out the relationship between p hase transitions and zeros (singularities) in the (logarithm of the) grand-can onical partition function]. Real singularities such as phase transitions occur only when the size of the system becomes truly macroscopic (formally in the i nfinite volume or thermodynamic limit), when these zeros can approach the real positive z axis and give rise to nonanalyticities in the thermodynamic functi ons. (The whole subject of existence of the thermodynamic limit and the equiva lence of ensembles in that limit is missing from this collection — the origin al papers by van Hove, van Kampen, Ruelle, Fisher, Dyson, and Lenard and other s did not appear in The Physical Review; see Refs. 25 and 26.) The Lee-Yang theorem about the location of the zeros implies that the free ene rgy of the Ising model with ferromagnetic pair interactions, considered as a f unction of the temperature T and magnetic field h, is an analytic function of h for . This is true in any dimension and extends also to continuous classical and even quantum spins with Ising-type symmetry, as long as there are only fe rromagnetic pair interactions — see Asano.27 Combining the Lee-Yang theorem w ith the Griffiths' (and other) inequalities, one can prove, cf. Ref. 25, that the magnetization is a real analytic function of h and T whenever , or T > Tc. For T < Tc the dependence of the magnetization on the field is discontinuous at h = 0, and the Ising system then has exactly two "pure phases," with sponta neous magnetization ±m(T). There are many other beautiful exact results know n and worth knowing about the Ising model, but space limitations have permitte d only a very few examples on the CD; see Refs. 25 and 28 for many more. The main reason for the continued interest in the two-dimensional ferromagneti c Ising model is that it is the paradigm of a spontaneous breaking of symmetry with an order parameter m(T), whose emergence (for h = 0) at T < Tc produces long-range correlations in the system. There are similar easily defined order parameters for the Ising antiferromagnet and the Heisenberg magnet, where the magnetization has a continuous symmetry, as well as for other phase transitio ns in which there is a classically describable symmetry breaking. The critical exponents and even the existence of a phase transition depend strongly of cou rse on symmetry and dimension, as shown in particular in the work of Hohenberg and of Mermin and Wagner.29 One thing that was unclear in the early 1950s, however, was what plays the rol e of the order parameter or what quantity develops long-range correlations in the purely quantum-mechanical normal-to-superfluid transition in He4 or in the superconducting transition. The answer came in the work of Penrose and Onsage r,30 who noted that the asymptotic value of the one-particle density matrix can, for interacting bosons, be interpreted as the square of the macroscopic w ave function of the condensate: a generalization to interacting systems of the Bose condensate in an ideal gas. Yang31 extended this idea to the superconduc ting transition in interacting fermions, where it shows up as "off-diagonal lo ng-range order" in the two-particle density matrix. (The seminal works of Land au and of Bogolyubov should also be noted here.) Following Onsager's 1942 work there was a 25-year hiatus in obtaining exact so lutions for nontrivial interacting systems. The next major breakthrough came w hen Lieb obtained the exact asymptotic number of configurations of a six-verte x model on the two-dimensional square lattice32 — a two-dimensional version o f a model introduced by Pauling to explain the residual entropy of ice. In thi s model the oxygen atoms sit at the vertices of the square lattice and there i s one hydrogen atom on each bond, which can be in one of two positions (closer to one or the other of the two oxygens connected by the bond). There is a con straint, however, which forbids any oxygen to have more than two close hydroge ns, and this makes the calculation of the number of configurations very nontri vial. The fact that the final answer found by Lieb for the square lattice, usi ng a Bethe-Ansatz trick, has the simple form of (3/2)log(4/3) for the entropy per oxygen atom, is to me still a mystery. (I repeat my offer of 1967 of a bot tle of champagne for a simple derivation.) Lieb's results, which include exact solution s of several other related models as well as of some one- dimensional quantum models,2,28 were soon extended by Baxter.33 While "doing his sums" on a boat r eturning from the U. S. to Australia, he obtained an exact solution of the eig ht-vertex model in which the constraint on the position of the hydrogen atoms is relaxed a bit, but there is an energy cost for different configurations. Th is work was the progenitor of several other exact solutions by Baxter and led in a natural way to the discovery of a whole class of solvable two-dimensional lattice models, including some with a continuously varying critical index; se e Ref. 28 for a review. This presented a challenge to the prevailing orthodoxy of universality, which held that critical indices depend only on the spatial dimension and symmetry of the order parameter. The discrepancy was resolved by Kadanoff and Wegner.34 The story of critical-point phenomena and their universality is one of the con tinuing sagas of statistical mechanics spawned in part by Onsager's solution o f the Ising model.c Prior to that it was generally accepted, despite some clea r experimental evidence to the contrary, that the behavior in the vicinity of a critical point is at least qualitatively correctly described by mean field t heory. These theories — van der Waals for fluids, Bragg-Williams for magnets, and Landau theory in general — give "superuniversal" classical critical expo nents: They are the same in all dimensions and for all symmetries. The two- di mensional Ising model, however, has exponents very different from the classica l ones and also from those found experimentally for three-dimensional systems. On the other hand, these exponents agree with experiments on highly anisotrop ic layered spin systems. Furthermore, Onsager's solution showed that while the critical temperature of the two-dimensional Ising model depended on the ratio of the coupling strengths in the x and y directions, the critical exponents did not. T his suggests strongly the kind of universality mentioned earlier. I remember F isher as the leading proponent of this view.36 It was he and others such as Do mb, who, by cleverly combining results from exact solutions, series expansions , thermodynamics, statistical mechanical inequalities, and experiments, starte d to bring order to this new universality; see the very nice collection of art icles in Ref. 37. [The series expansion method was pioneered by the King's Col lege group in London led by Domb; see Ref. 37. It was greatly aided by the Pad é technique; see Baker38 and Ref. 25(a) for the analysis of these series.] It was clear by the mid-1960s from both theory and experiment that the univers ality of critical phenomena cried out for a universal formulation and explanat ion.36 This came with the scaling theory of Widom39 and the Kadanoff block tra nsformation.40 It culminated in Wilson's renormalization groupd (RG) formulati on of critical phenomena.41 The RG was made into a practical tool for computin g critical exponents by the expansion about four dimensions developed by Wils on and Fisher.43 Computations were further facilitated by the development of f inite-size scaling methods and by the systematic use of field theory technique s (see Fisher and Barber44 and Ref. 25). Universality was given a deeper meani ng (at least for two-dimensional systems) through the use of conformal invaria nce by Polyakov45 and others in the FSU. This led to a "full classification" o f two-dimensional critical points by Friedan, Qiu, and Shenker.46 The Wilson RG had antecedents in field theory, Gell-Mann and Low,47 and in the application of those field theory ideas to statistical mechanics by Di Castro and Jona-Lasinio.48 Following Wilson's work it has spread and had enormous in fluence on almost all fields of science. It provides a method for quantitative analysis of the "essential" features of a large class of nonlinear phenomena exhibiting self-similar structures. This includes not only scale invariant cri tical systems, where fluctuations are "infinite" on the microscopic spatial an d temporal scale, but also fractals, dynamical systems exhibiting Feigenbaum p eriod doubling, KAM theory, singular behavior in nonlinear partial differentia l equations, and "chaos." Even where not directly applicable, the RG often pro vides a paradigm for the analysis of complex phenomena. The paper by Halperin, Hohenberg, and Ma49 brings the powerful machinery of the RG to bear on nonequ ilibrium phenomena near the critical point. For a recent review, see Refs. 42 and 50. On the cautionary side one should remember that there are still some serious op en problems concerning the nature of the RG transformation of Hamiltonians for statistical mechanical systems, i.e., for critical phenomena. A lot of mathem atical work remains to be done to make it into a well-defined theory of phase transitions.51 I will end this brief personal overview by noting that this chapter also conta ins Onsager's original abstract about the nematic phase for elongated molecule s and the tobacco mosaic virus and the paper by Halperin and Nelson52 on the h exatic phase. These papers, as well as the beautiful experimental papers on cr itical phenomena and low temperature fluids in this collection, deserve an int roduction of their own and much more space than they get here. I am not able t o discuss them or any of the other goodies, included and omitted, for lack of space, time, and competence. In any case, what is included is just the tip of the iceberg. There are many, many more where these came from, and the reader i s urged to settle down for a good read with any randomly picked volume of The Physical Review or Physical Review Letters. She or he will be amply rewarded f or the effort. Notes a In many cases some of the most basic papers in a given area were omitted ju st because of their length. For example, Onsager's solution of the Ising model is represented in the book by an abstract. In other cases a whole area had to be excluded, e.g., Kac potentials, one-dimensional quantum systems. Many of t hese can be found in the recent excellent compilation with comments by Daniel Mattis.2 b Note that even smooth trajectories can give rise to ergodic behavior of pro jections, as in the Lissajous figure generated by two incommensurate frequenci es. For systems with many degrees of freedom, any nonlinearity was expected to produce ergodicity or at least equipartition. c It is worth noting here that the only continuum system with short-range int eractions for which one can rigorously prove the existence of a phase transiti on is the Widom-Rowlinson model (Ruelle, Ref. 35). The situation is different for systems with long-range "Kac potentials," for which a modified form of mea n field theory can be proven in a suitable limit, but that is a different stor y, not included in this collection; see Ref. 25(a). d It is hard to give a precise definition of the RG. As noted by Benfatto and Gallavotti in their introduction to lecture notes on the subject, "The notion of renormalization group is not well defined. One usually means a theory in w hich scale invariance ideas are involved and appear technically as invariance or covariance properties, of various quantities, with respect to a noninvertib le transformation of coordinates. The noninvertibility is an essential feature . It is supposed to permit reducing effectively the difficulty of the problem. "42 References 1. F. Dyson, George Green and Physics, Phys. World 6, 33-38 (August 1993). 2. The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dim ension, edited by D. C. Mattis (World Scientific, Singapore, 1993). 3. H. B. Callen and T. E. Welton, Irreversibility and Generalized Noise, Phys . Rev. 83, 34-40 (1951). 4. M. S. Green, Markoff Random Processes and the Statistical Mechanics of Tim e-Dependent Phenomena. II. Irreversible Processes in Fluids, J. Chem. Phys. 22 , 398-413 (1954); R. Kubo, Statistical-Mechanical Theory of Irreversible Proce sses. I. General Theory and Simple Applications to Magnetic and Conduction Pro blems, J. Phys. Soc. Jpn. 12, 570-586 (1957). For a historical perspective see H. Nakano, Linear Response Theory: A Historical Perspective, Int. J. Mod. Phy s. B 7, 2397-2467 (1992). 5. L. Onsager and S. Machlup, Fluctuations and Irreversible Processes, Phys. Rev. 91, 1505-1512 (1953); Fluctuations and Irreversible Processes. II. System s with Kinetic Energy, ibid. 91, 1512-1515 (1953). 6. N. Zabusky and M. Kruskal, Interaction of "Solitons" in a Collisionless Pl asma and the Recurrence of Initial States, Phys. Rev. Lett. 15, 240-243 (1965) . 7. For a good overview, see introduction and articles in Universality in Chao s, edited by P. Cvitanovic (Hilger, London, 1984). For an excellent nontechnic al exposition see D. Ruelle, Chance and Chaos (Princeton University, Princeton , 1991). 8. B. Alder and T. Wainright, Velocity Autocorrelations for Hard Spheres, Phy s. Rev. Lett. 18, 988-990 (1967); Decay of the Velocity Autocorrelation Functi on, Phys. Rev. A 1, 18-21 (1970). 9. E. Fermi, J. Pasta, and S. Ulam, Collected Papers of E. Fermi (University of Chicago Press, Chicago, 1965), Vol. II, p. 978. For a historical perspectiv e see J. Ford, The Fermi-Pasta-Ulam Problem: Paradox Turns Discovery, Phys. Re p. 213, 272-310 (1992). 10. For a good exposition, see G. Gallavotti, Quasi-Integrable Mechanical Sys tems, edited by K. Osterwalder and R. Stora (Elsevier, Amsterdam, 1986), pp. 5 39-624. 11. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. T eller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phy s. 21, 1087-1092 (1953). 12. E. N. Lorenz, Deterministic Nonperiodic Flow, J. Atm. Sci. 20, 130-141 (1 963). 13. D. Ruelle and F. Takens, On the Nature of Turbulence, Commun. Math. Phys. 20, 167-192 (1971). 14. M. J. Feigenbaum, Quantitative Universality for a Class of Nonlinear Tran sformations, J. Stat. Phys. 19, 25-52 (1978); The Universal Metric Properties of Nonlinear Transformations, ibid. 21, 669-706 (1979). 15. G. Ahlers, Low-Temperature Studies of the Rayleigh-Bénard Instability an d Turbulence, Phys. Rev. Lett. 33, 1185-1188 (1974); J. Gollub and H. Swinney, Onset of Turbulence in a Rotating Fluid, ibid. 35, 927-930 (1975). 16. A. Libchaber and J. Maurer, Une Expérience de Rayleigh-Bénard de Géomé trie Réduite: Multiplication, Accrochage et Démultiplication de Fréquences, J. Phys. (Paris) Colloq. C3, 51-56 (1980). 17. Y. Pomeau and P. Maneville, Intermittent Transition to Turbulence in Diss ipative Dynamical Systems, Commun. Math. Phys. 74, 189-197 (1980). 18. L. Onsager, The Distribution of Energy in Turbulence, Phys. Rev. 68, 286 (A) (1945). 19. For an informal and informative description of Onsager's work, see H. C. Longuet-Higgins and M. E. Fisher, in Biographical Memoirs, National Academy of Sciences 60, 183-432 (1991). It and other historical material will appear in a forthcoming Onsager memorial issue of J. Stat. Phys. (January 1995). 20. H. A. Kramers and G. H. Wannier, Statistics of the Two-Dimensional Ferrom agnet. Part I, Phys. Rev. 60, 252-276 (1941). 21. T. Berlin and M. Kac, The Spherical Model of a Ferromagnet, Phys. Rev. 86 , 821-835 (1951). 22. C. N. Yang, The Spontaneous Magnetization of a Two-Dimensional Ising Mode l, Phys. Rev. 85, 808-816 (1952). 23. T. D. Lee and C. N. Yang, Statistical Theory of Equations of State and Ph ase Transitions. II. Lattice Gas and Ising Model, Phys. Rev. 87, 410-419 (1952 ). 24. C. N. Yang and T. D. Lee, Statistical Theory of Equations of State and Ph ase Transitions. I. Theory of Condensation, Phys. Rev. 87, 404-409 (1952). 25. (a) G. A. Baker, Quantitative Theory of Critical Phenomena (Academic, New York, 1990); (b) B. Simon, The Statistical Mechanics of Lattice Gases (Prince ton University, Princeton, 1993). 26. J. Lebowitz and E. Lieb, Existence of Thermodynamics for Real Matter with Coulomb Forces, Phys. Rev. Lett. 22, 631-634 (1969); E. Lieb and W. Thirring, Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter , ibid. 35, 687-689 (1975); E. Lieb, The Stability of Matter, Rev. Mod. Phys. 48, 553-569 (1976). 27. T. Asano, Lee-Yang Theorem and the Griffiths Inequality for the Anisotrop ic Heisenberg Ferromagnet, Phys. Rev. Lett. 24, 1409-1411 (1970); see also E. Lieb and A. Sokal, A General Lee-Yang Theorem for One-Component and Multicompo nent Ferromagnets, Commun. Math Phys. 80, 153-179 (1981). 28. R. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982). 29. P. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Ph ys. Rev. 158, 383-386 (1967); N. Mermin and H. Wagner, Absence of Ferromagneti sm or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Model s, Phys. Rev. Lett. 17, 1133-1136 (1966); ibid. 17, 1307(E) (1966). 30. O. Penrose and L. Onsager, Bose-Einstein Condensation in Liquid Helium, P hys. Rev. 104, 576-584 (1956). 31. C. N. Yang, Concept of Off-Diagonal Long-Range Order and the Quantum Phas e of Liquid He and of Superconductors, Rev. Mod. Phys. 34, 694-704 (1962). 32. E. Lieb, Exact Solution of the Problem of the Entropy of Two-Dimensional Ice, Phys. Rev. Lett. 18, 692-694 (1967). 33. R. Baxter, Eight-Vertex Model in Lattice Statistics, Phys. Rev. Lett. 26, 832-833 (1971). 34. L. P. Kadanoff and F. Wegner, Some Critical Properties of the Eight-Verte x Model, Phys. Rev. B 4, 3989-3993 (1971). 35. D. Ruelle, Existence of a Phase Transition in a Continuous Classical Syst em, Phys. Rev. Lett. 27, 1040-1041 (1971). 36. See M. E. Fisher, Correlation Functions and the Critical Region of Simple Fluids, J. Math. Phys. 5, 944-962 (1964). 37. Cooperative Phenomena Near a Phase Transition; A Biography with Selected Readings edited by H. E. Stanley (MIT, Cambridge, 1973); C. Domb and other art icles, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S . Green (Academic, New York, 1974), Vol. 3. 38. G. A. Baker, Jr., Application of the Padé Approximation Method to the In vestigation of Some Magnetic Properties of the Ising Model, Phys. Rev. 124, 76 8-774 (1961). 39. B. Widom, Equation of State in the Neighborhood of the Critical Point, J. Chem. Phys. 43, 3898-3905 (1965). 40. L. P. Kadanoff, Scaling Laws for Ising Models Near Tc, Physics 2, 263-272 (1966). 41. K. Wilson, Renormalization Group and Critical Phenomena. I. Renormalizati on Group and the Kadanoff Scaling Picture, Phys. Rev. B 4, 3174-3183 (1971); R enormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior, ibid. 4, 3184-3205 (1971); Feynman Graph Expansion for Crit ical Exponents, Phys. Rev. Lett. 28, 548-551 (1972). 42. G. Benfatto and G. Gallavotti, Renormalization Group, Lecture Notes, Rome (1993) (Princeton University Press, 1995, to be published). 43. K. Wilson and M. E. Fisher, Critical Exponents in 3.99 Dimensions, Phys. Rev. Lett. 28, 240-243 (1972). 44. M. E. Fisher and M. Barber, Scaling Theory for Finite-Size Effects in the Critical Region, Phys. Rev. Lett. 28, 1516-1519 (1972). 45. A. M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. 103B, 20 7-210 (1981); Quantum Theory of Fermionic Strings, ibid. 103B, 211-213 (1981). 46. D. Friedan, Z. Qiu, and S. Shenker, Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions, Phys. Rev. Lett. 52, 1575-1578 (1984). 47. M. Gell-Mann and F. Low, Quantum Electrodynamics at Small Distances, Phys . Rev. 95, 1300-1312 (1954). 48. C. Di Castro and G. Jona-Lasinio, On The Microscopic Foundation of Scalin g Laws, Phys. Lett. A 29, 322-323 (1969). 49. B. Halperin, P. Hohenberg, and S. Ma, Calculations of Dynamic Critical Pr operties Using Wilson's Expansion Methods, Phys. Rev. Lett. 29, 1548-1551 (197 2). 50. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Grou p, Frontiers in Physics Series (Addison Wesley, Reading, MA, 1992), Vol. 85. 51. R. B. Griffiths and P. A. Pearce, Position-Space Renormalization-Group Tr ansformations: Some Proofs and Some Problems, Phys. Rev. Lett. 41, 917-920 (19 78); A. C. D. van Enter, R. Fern醤dez, and A. D. Sokal, Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations: Scope and Limitations of Gibbsian Theory, J. Stat. Phys. 72, 879-1167 (1993); F. Ma rtinelli and E. Olivieri, Some Remarks on Pathologies of Renormalization-Group Transformations for the Ising Model, ibid. 72, 1169-1177 (1993). 52. B. Halperin and D. Nelson, Theory of Two-Dimensional Melting, Phys. Rev. Lett. 41, 121-124 (1978). -- ※ 来源:·珞珈山水BBS站 http://bbs.whu.edu.cn·[FROM: 159.226.37.*]
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标  题: Physical Review 100 years collection 上对统计物理的评述[转载]
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Physical Review 100 years collection 上对统计物理的评述[转载]

Statistical Physics
Joel L. Lebowitz
Department of Mathematics and Physics, Rutgers University, New Brunswick, New
Jersey 08903

Joel L. Lebowitz is the George William Hill Professor of Mathematics and Physi
cs and Director of the Center for Mathematical Sciences Research at Rutgers Un
iversity. He came to the United States in 1946 and was a student of Melba Phil
lips, Peter Bergmann, and Lars Onsager. His work has been mainly in statistica
l mechanics with an outsider's interest in foundational questions. His awards
include the A. Cressy Morrison Award in Natural Sciences from the New York Aca
demy of Sciences and the Boltzmann Medal from the IUPAP Committee on Thermodyn
amics and Statistical Physics.
The Physical Review is a relative latecomer to the world physics scene. Althou
gh it started almost 120 years after the founding of the United States, its cl
ientele was rather small in the beginning. It was not until the 1930s, when sc
ientists (and others) with "wrong" ancestors and beliefs became hunted species
in certain parts of Europe, that physics and other sciences began their explo
sive stage of growth in the U.S. With this growth came the flowering of The Ph
ysical Review. By the 1940s The Physical Review was the leading physics journa
l in the world — a position it and its offspring, Physical Review Letters, ma
intain to this day. Dyson1 describes Vol. 73 of The Physical Review, which con
tains the issues from January to June 1948, his first year in the United State
s, as follows: "Almost every paper in it is interesting, and many of them are
memorable. In 1948 issues of the journal were thin enough to be read from cove
r to cover. Many of us did just that.... The paper that impressed me most in 1
948 and still impresses me today is entitled 'Relaxation Effects in Nuclear Magnetic Re
sonance Absorption,' a monumental piece of work, 34 pages long, by Bloembergen
, Purcell, and Pound... . In the same volume of The Physical Review are many o
ther wonderful papers on the most diverse subjects: Alpher, Bethe, and Gamow o
n the origin of chemical elements; Wataghin on the formation of chemical eleme
nts inside stars; E. Teller on the change of physical constants; Lewis, Oppenh
eimer, and Wouthuysen on the multiple production of mesons; Foley and Kusch on
the experimental discovery of the anomalous magnetic moment of the electron;
Schwinger on the theoretical explanation of the anomalous moment; and Dirac on
the quantum theory of localizable dynamical systems. This was a vintage year
for historic papers and I mention these seven just to give the flavor of what
physicists were doing in the early post-war years."
Dyson's description illustrates the difficulty, nay impossibility, of selectin
g a seminal sample from among 90 years of The Physical Review papers. It shoul
d therefore be clear that this extremely space-constrained collection is not a
scholarly compendium of either the "most important," "most original," "most c
lever," "most memorable," or "most anything" papers that appeared in The Physi
cal Review or Physical Review Letters prior to 1984. It is rather a very small
sample of such papers collected from responses to the solicitation letter sen
t out by H. Henry Stroke to a small sample of physicists, from conversations w
ith a small sample of friends, from references in a small sample of books or r
eviews I happened to have on my shelves, from some very time-limited sample vi
sits to the library, and finally from inherently faulty personal recollections
, judgments, and prejudices. In particular, I believe that mathematically rigo
rous results are, in appropriate situations, not just "decorations on the cake
"; they are essential for getting the physics right. Given the above, I apologize in ad
vance for all sins of omissiona and commission and only hope that the selectio
n gives the reader a feeling of how exciting good science can be.

Having made my disclaimer, I take this opportunity to give a brief, highly per
sonal, selective overview of developments in nonequilibrium and equilibrium st
atistical mechanics in order to put the works contained here in some context.

Nonequilibrium

I am a bit younger and much less precocious than Dyson, so although I came to
the United States a year or two before him, I did not start reading The Physic
al Review until 1953, when I was a graduate student of Peter Bergmann's at Syr
acuse University. It was also in that year that I first met Lars Onsager, whos
e 1931 paper "Reciprocal Relations in Irreversible Processes II" is the earlie
st one in this section of the book. (Note that this article was submitted on 9
November and published in the 15 December issue of that year.)

Onsager's paper is the second of two works (the first one was too long for the
book and is included in the CD) devoted to the deep underlying symmetry prope
rties of the matrix connecting thermodynamic fluxes and forces. While particul
ar examples of such symmetries had been known for a long time, it was Onsager
who first understood that they are a general feature of transport in systems c
lose to equilibrium. These symmetries are a direct consequence of the reversib
ility of the laws governing the time evolution of the microscopic constituents
of macroscopic matter and the additional (mild) hypothesis that the time evol
ution of a macroscopic system following its preparation in a macrostate out of
equilibrium is the same as that following a spontaneous fluctuation from equi
librium to that macrostate. The "Onsager relations" regarding the symmetry of
the transport coefficients Lij = Lji are, next to the second law of thermodyna
mics, the only known results about nonequilibrium phenomena that have a genera
lity resembling that of equilibrium thermodynamics. They are in fact related to Boltzma
nn's probabilistic formulation of the second law connecting macroscopic entrop
y with microscopic phase space volume. The connection between microscopic and
macroscopic evolutions and the role that the large number of microscopic degre
es of freedom play in determining universal features of macroscopic behavior w
as further developed by Einstein in his beautiful quantitative theory of Brown
ian motion and in his relation between equilibrium fluctuations and macroscopi
c entropy. The works of Boltzmann and Einstein are the direct forerunners of O
nsager's work.

The fluctuation-dissipation theorem of Callen and Welton3 extends the Boltzman
n-Einstein-Onsager connection between spontaneous fluctuations in equilibrium
systems and the macroscopic response of the system when not in equilibrium. By
considering the response to an externally imposed perturbation of the Hamilto
nian describing the time evolution of the ensemble density in the Gibbs formal
ism of statistical mechanics, Callen and Welton were led to a general expressi
on for (some) frequency-dependent transport coefficients in terms of the Fouri
er transform of the time evolution of equilibrium fluctuations. The same circl
e of ideas leads also to the Green-Kubo4 formulas for transport coefficients a
nd to the Onsager-Machlup5 formulation of probabilities for fluctuations in th
e time evolution of macroscopic systems.

The above general results apply to linear transport in systems close to equili
brium. Their extension to systems far from equilibrium is still a very active
research area. We should not be surprised, however, if it turns out that such
general formulations may not be appropriate for some of the most interesting p
henomena in such systems, including those most relevant to us, e.g., us, in ou
r universe. These phenomena may not be capturable by general formulas; to quot
e Dyson again, "God is in the details." Still, there have been major advances
in our understanding of the potentialities inherent in nonlinear dynamical sys
tems. These include the discovery of solitons in integrable systems represente
d here by the paper of Zabusky and Kruskal.6 Another discovery, or more proper
ly a belated realization by the physics (as opposed to the mathematics) commun
ity, was that most dynamical systems, even those with only a few degrees of fr
eedom, are very far from integrable — rather, the trajectories describing the
ir time evolution are subject to sensitive dependence on initial conditions, parameters
, perturbations; in short, they exhibit deterministic chaos.7

To put the discussion of chaos into this centennial framework, one should at l
east mention the seminal work of Poincaré. It was Poincaré who first realize
d the deep difference between linear and nonlinear evolutions with the inheren
t tendency for complex, apparently chaotic, behavior of the latter. Following
Poincaré, the torch of nonlinearity was mainly kept alive in what is now the
Former Soviet Union (FSU). In the FSU, following a tradition established by Ly
apunov, the work of mathematicians such as Kolmogorov, Bogolyubov, Rokhlin, an
d members of their schools, helped develop many of the fundamental ideas of th
e modern analytical theory of nonlinear dynamical systems. The works of Arnold
, Chirikov, and Sinai from the FSU and of Birkhoff, Smale, Siegel, and Moser f
rom the West have been particularly relevant for our understanding of determin
istic chaos. (I never heard of Poincaré's work in any of my courses and first
learned of it from Prigogine and Ford.)

The analytic insights of Poincaré were subsequently visualized clearly throug
h the use of that modern (it occurred in my lifetime) invention, the computer,
by Hénon and Heiles, reprinted in Ref. 7. It was the computer too through wh
ich Alder and Wainright8 discovered the first example of the now ubiquitous "l
ong time tails" of correlations present in many-body dynamical systems. Before
the advent of computer simulations, there was a general belief (if my memory
serves me right) that correlations typically showed exponential decay in time
— the power law decay of conserved densities satisfying diffusion equations b
eing truly exceptional. One of the first uses of the computer in statistical m
echanics was the computation by Fermi, Pasta, and Ulam9 (their famous Los Alam
os report was never published in any journal) of the time evolution of a chain
of 32 oscillators nonlinearly coupled via a cubic and/or quartic Hamiltonian.
They found to their surprise that there was no equipartition of energy among
the different (linear) normal modes on the time scale of their computation.b This was l
ater understood to be a consequence of the celebrated Kolmogorov-Arnold-Moser
(KAM) theorem.10 Another computer first from Los Alamos was the development of
Monte Carlo methods by Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H.
Teller, and E. Teller.11 (The story has it that this was "cooked up" in the co
urse of a dinner.)

Some important items missing from The Physical Review and therefore from this
collection are the papers by Lorenz,12 Ruelle and Takens,13 and Feigenbaum.14
Lorenz was the first to make clear the connection between the chaotic behavior
of a dynamical system with a small number of degrees of freedom and the onset
of turbulence in a driven fluid. Ruelle and Takens then showed that the onset
of turbulence need not occur through an infinite sequence of bifurcations, as
proposed by Landau, but can result directly from the interaction of three or
more modes. The work of Ahlers and of Gollub and Swinney15 beautifully confirm
s this. (I recall the excitement of going with Ruelle to Swinney's lab to watc
h the experiment.) Feigenbaum, aided by a very modest computer, in what is for
me still one of the best examples of the good uses of computers in science, d
iscovered the existence of a simple "universal" (but far from obvious) feature
in a large class of dynamical systems: period doubling leading to chaos. One
of the first experimental papers to confirm this scenario clearly was by Libchaber and
Maurer.16 Pomeau and Manneville17 found still another route to the onset of tu
rbulence, that of intermittency — a route also confirmed by experiment.

The situation is much less satisfactory with regard to fully developed turbule
nce, perhaps the central problem of contemporary classical statistical physics
. While the past 100 years have certainly seen important theoretical work on t
his subject, including some deep insights by L. Richardson, Kolmogorov, Onsage
r, Kraichnan, and others (mostly published in other journals), a full understa
nding remains for the sesquicentennial celebration of The Physical Review. The
celebrated Kolmogorov scaling law is represented in the book by a later indep
endent short announcement by Onsager.18 (Apparently Onsager had this result fo
r some years, but the war delayed its publication.19)

Equilibrium

Let me turn now to the simpler (but far from simple) equilibrium systems. Here
too our story in this collection begins with Onsager. In an abstract of less
than 70 words (which was in keeping with his generally sparing and precise utt
erances), Onsager announced that he had obtained in closed form the partition
function of the ferromagnetic Ising model in two dimensions and found that "Fo
r an infinite crystal, the specific heat near T = Tc is proportional to -log|T
- Tc|." This astonishing result can be said to mark the beginning of the mode
rn era of equilibrium statistical mechanics. The full 1944 paper describing th
is work is on the CD-ROM. Its precursor, the 1941 paper by Kramers and Wannier
,20 in which the exact value of Tc is found by duality, is also there.a

The Ising model was the first many-body system with nontrivial interactions so
lved exactly, and even after so many years and so many alternative derivations
it is still an astonishing intellectual feat. I can still remember from my fi
rst year of graduate school the admiration in Mark Kac's voice when speaking o
f Onsager's solution in a colloquium at Syracuse University in 1953. He descri
bed there his work with Berlin,21 on the solution of the much simpler spherica
l model — "but the solution of the Ising model was a superheroic effort resis
ting all assaults until Onsager threw at it his heavy machinery and the proble
m collapsed under its weight." The formula for the spontaneous magnetization o
f the Ising model was announced by Onsager at a conference at Cornell in 1948
and repeated in Florence in 1949, but no hint of its origin was given.19 This
was supplied by Yang in 1952.22

Also in the same year Lee and Yang23 proved a remarkable theorem about the zer
os of the partition function of the ferromagnetic Ising model in an external m
agnetic field h: They must all be on the unit circle in the complex fugacity p
lane z [in suitable units z = exp(h)]. The Lee and Yang paper was the companio
n paper to one by Yang and Lee24 [which pointed out the relationship between p
hase transitions and zeros (singularities) in the (logarithm of the) grand-can
onical partition function]. Real singularities such as phase transitions occur
only when the size of the system becomes truly macroscopic (formally in the i
nfinite volume or thermodynamic limit), when these zeros can approach the real
positive z axis and give rise to nonanalyticities in the thermodynamic functi
ons. (The whole subject of existence of the thermodynamic limit and the equiva
lence of ensembles in that limit is missing from this collection — the origin
al papers by van Hove, van Kampen, Ruelle, Fisher, Dyson, and Lenard and other
s did not appear in The Physical Review; see Refs. 25 and 26.)

The Lee-Yang theorem about the location of the zeros implies that the free ene
rgy of the Ising model with ferromagnetic pair interactions, considered as a f
unction of the temperature T and magnetic field h, is an analytic function of
h for . This is true in any dimension and extends also to continuous classical
and even quantum spins with Ising-type symmetry, as long as there are only fe
rromagnetic pair interactions — see Asano.27 Combining the Lee-Yang theorem w
ith the Griffiths' (and other) inequalities, one can prove, cf. Ref. 25, that
the magnetization is a real analytic function of h and T whenever , or T > Tc.
For T < Tc the dependence of the magnetization on the field is discontinuous
at h = 0, and the Ising system then has exactly two "pure phases," with sponta
neous magnetization ±m*(T). There are many other beautiful exact results know
n and worth knowing about the Ising model, but space limitations have permitte
d only a very few examples on the CD; see Refs. 25 and 28 for many more.

The main reason for the continued interest in the two-dimensional ferromagneti
c Ising model is that it is the paradigm of a spontaneous breaking of symmetry
with an order parameter m*(T), whose emergence (for h = 0) at T < Tc produces
long-range correlations in the system. There are similar easily defined order
parameters for the Ising antiferromagnet and the Heisenberg magnet, where the
magnetization has a continuous symmetry, as well as for other phase transitio
ns in which there is a classically describable symmetry breaking. The critical
exponents and even the existence of a phase transition depend strongly of cou
rse on symmetry and dimension, as shown in particular in the work of Hohenberg
and of Mermin and Wagner.29

One thing that was unclear in the early 1950s, however, was what plays the rol
e of the order parameter or what quantity develops long-range correlations in
the purely quantum-mechanical normal-to-superfluid transition in He4 or in the
superconducting transition. The answer came in the work of Penrose and Onsage
r,30 who noted that the asymptotic value of the one-particle density matrix

can, for interacting bosons, be interpreted as the square of the macroscopic w
ave function of the condensate: a generalization to interacting systems of the
Bose condensate in an ideal gas. Yang31 extended this idea to the superconduc
ting transition in interacting fermions, where it shows up as "off-diagonal lo
ng-range order" in the two-particle density matrix. (The seminal works of Land
au and of Bogolyubov should also be noted here.)
Following Onsager's 1942 work there was a 25-year hiatus in obtaining exact so
lutions for nontrivial interacting systems. The next major breakthrough came w
hen Lieb obtained the exact asymptotic number of configurations of a six-verte
x model on the two-dimensional square lattice32 — a two-dimensional version o
f a model introduced by Pauling to explain the residual entropy of ice. In thi
s model the oxygen atoms sit at the vertices of the square lattice and there i
s one hydrogen atom on each bond, which can be in one of two positions (closer
to one or the other of the two oxygens connected by the bond). There is a con
straint, however, which forbids any oxygen to have more than two close hydroge
ns, and this makes the calculation of the number of configurations very nontri
vial. The fact that the final answer found by Lieb for the square lattice, usi
ng a Bethe-Ansatz trick, has the simple form of (3/2)log(4/3) for the entropy
per oxygen atom, is to me still a mystery. (I repeat my offer of 1967 of a bot
tle of champagne for a simple derivation.) Lieb's results, which include exact solution
s of several other related models as well as of some one- dimensional quantum
models,2,28 were soon extended by Baxter.33 While "doing his sums" on a boat r
eturning from the U. S. to Australia, he obtained an exact solution of the eig
ht-vertex model in which the constraint on the position of the hydrogen atoms
is relaxed a bit, but there is an energy cost for different configurations. Th
is work was the progenitor of several other exact solutions by Baxter and led
in a natural way to the discovery of a whole class of solvable two-dimensional
lattice models, including some with a continuously varying critical index; se
e Ref. 28 for a review. This presented a challenge to the prevailing orthodoxy
of universality, which held that critical indices depend only on the spatial
dimension and symmetry of the order parameter. The discrepancy was resolved by
Kadanoff and Wegner.34

The story of critical-point phenomena and their universality is one of the con
tinuing sagas of statistical mechanics spawned in part by Onsager's solution o
f the Ising model.c Prior to that it was generally accepted, despite some clea
r experimental evidence to the contrary, that the behavior in the vicinity of
a critical point is at least qualitatively correctly described by mean field t
heory. These theories — van der Waals for fluids, Bragg-Williams for magnets,
and Landau theory in general — give "superuniversal" classical critical expo
nents: They are the same in all dimensions and for all symmetries. The two- di
mensional Ising model, however, has exponents very different from the classica
l ones and also from those found experimentally for three-dimensional systems.
On the other hand, these exponents agree with experiments on highly anisotrop
ic layered spin systems. Furthermore, Onsager's solution showed that while the
critical temperature of the two-dimensional Ising model depended on the ratio
of the coupling strengths in the x and y directions, the critical exponents did not. T
his suggests strongly the kind of universality mentioned earlier. I remember F
isher as the leading proponent of this view.36 It was he and others such as Do
mb, who, by cleverly combining results from exact solutions, series expansions
, thermodynamics, statistical mechanical inequalities, and experiments, starte
d to bring order to this new universality; see the very nice collection of art
icles in Ref. 37. [The series expansion method was pioneered by the King's Col
lege group in London led by Domb; see Ref. 37. It was greatly aided by the Pad
é technique; see Baker38 and Ref. 25(a) for the analysis of these series.]

It was clear by the mid-1960s from both theory and experiment that the univers
ality of critical phenomena cried out for a universal formulation and explanat
ion.36 This came with the scaling theory of Widom39 and the Kadanoff block tra
nsformation.40 It culminated in Wilson's renormalization groupd (RG) formulati
on of critical phenomena.41 The RG was made into a practical tool for computin
g critical exponents by the  expansion about four dimensions developed by Wils
on and Fisher.43 Computations were further facilitated by the development of f
inite-size scaling methods and by the systematic use of field theory technique
s (see Fisher and Barber44 and Ref. 25). Universality was given a deeper meani
ng (at least for two-dimensional systems) through the use of conformal invaria
nce by Polyakov45 and others in the FSU. This led to a "full classification" o
f two-dimensional critical points by Friedan, Qiu, and Shenker.46

The Wilson RG had antecedents in field theory, Gell-Mann and Low,47 and in the
application of those field theory ideas to statistical mechanics by Di Castro
and Jona-Lasinio.48 Following Wilson's work it has spread and had enormous in
fluence on almost all fields of science. It provides a method for quantitative
analysis of the "essential" features of a large class of nonlinear phenomena
exhibiting self-similar structures. This includes not only scale invariant cri
tical systems, where fluctuations are "infinite" on the microscopic spatial an
d temporal scale, but also fractals, dynamical systems exhibiting Feigenbaum p
eriod doubling, KAM theory, singular behavior in nonlinear partial differentia
l equations, and "chaos." Even where not directly applicable, the RG often pro
vides a paradigm for the analysis of complex phenomena. The paper by Halperin,
Hohenberg, and Ma49 brings the powerful machinery of the RG to bear on nonequ
ilibrium phenomena near the critical point. For a recent review, see Refs. 42
and 50. On the cautionary side one should remember that there are still some serious op
en problems concerning the nature of the RG transformation of Hamiltonians for
statistical mechanical systems, i.e., for critical phenomena. A lot of mathem
atical work remains to be done to make it into a well-defined theory of phase
transitions.51

I will end this brief personal overview by noting that this chapter also conta
ins Onsager's original abstract about the nematic phase for elongated molecule
s and the tobacco mosaic virus and the paper by Halperin and Nelson52 on the h
exatic phase. These papers, as well as the beautiful experimental papers on cr
itical phenomena and low temperature fluids in this collection, deserve an int
roduction of their own and much more space than they get here. I am not able t
o discuss them or any of the other goodies, included and omitted, for lack of
space, time, and competence. In any case, what is included is just the tip of
the iceberg. There are many, many more where these came from, and the reader i
s urged to settle down for a good read with any randomly picked volume of The
Physical Review or Physical Review Letters. She or he will be amply rewarded f
or the effort.

Notes

a  In many cases some of the most basic papers in a given area were omitted ju
st because of their length. For example, Onsager's solution of the Ising model
is represented in the book by an abstract. In other cases a whole area had to
be excluded, e.g., Kac potentials, one-dimensional quantum systems. Many of t
hese can be found in the recent excellent compilation with comments by Daniel
Mattis.2
b  Note that even smooth trajectories can give rise to ergodic behavior of pro
jections, as in the Lissajous figure generated by two incommensurate frequenci
es. For systems with many degrees of freedom, any nonlinearity was expected to
produce ergodicity or at least equipartition.
c  It is worth noting here that the only continuum system with short-range int
eractions for which one can rigorously prove the existence of a phase transiti
on is the Widom-Rowlinson model (Ruelle, Ref. 35). The situation is different
for systems with long-range "Kac potentials," for which a modified form of mea
n field theory can be proven in a suitable limit, but that is a different stor
y, not included in this collection; see Ref. 25(a).
d  It is hard to give a precise definition of the RG. As noted by Benfatto and
Gallavotti in their introduction to lecture notes on the subject, "The notion
of renormalization group is not well defined. One usually means a theory in w
hich scale invariance ideas are involved and appear technically as invariance
or covariance properties, of various quantities, with respect to a noninvertib
le transformation of coordinates. The noninvertibility is an essential feature
. It is supposed to permit reducing effectively the difficulty of the problem.
"42

References

1.  F. Dyson, George Green and Physics, Phys. World 6, 33-38 (August 1993).
2.  The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dim
ension, edited by D. C. Mattis (World Scientific, Singapore, 1993).
3.  H. B. Callen and T. E. Welton, Irreversibility and Generalized Noise, Phys
. Rev. 83, 34-40 (1951).
4.  M. S. Green, Markoff Random Processes and the Statistical Mechanics of Tim
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