ONSAGER, LARS

(b. Oslo, Norway, 27 November 1903; d. Coral Gables. Florida, 5 October 1976)

chemistry, physics.

In 1968, when the news arrived that Lars Onsager had been awarded the Nobel Prize, the natural question was, “In physics or in chemisty?” Was it for his solution of the two-dimensional Ising model¹? Was it for explaining the electrical conductivity of ice²? Was it for flux quantization in superconductors³?was it for his theory of electrolytes⁴?As it happened it was for what was called the Onsager Reciprocal Relations, developed in two papers published in Physical Review in 1931⁶, and it was in chemistry.

Other major research for which Onsager is known includes the formula for the dielectric constant of polar liquids⁷. Isotope separation by thermal diffusion⁸, the energy spectrum of turbulence⁹, the statistical-mechanical description of vortices in two dimensions (negative absolute temperatures when they roll up⁵, quantization of the circulation of vortices in superfluid helium¹⁰, an explanation of the Wien effect¹¹, electron-ion recombination statistics¹², interpretation of the De Haas-Van Alphen effect¹³, the statistical interpretation of the dissipation function¹⁴, and a new definition of the Bose-Einstein condensation for interacting particles applicable to the lambda-point transition to superfluidity of liquid helium¹⁵.

Life. The son of Erling Onsager, a barrister, and of Ingrid Kirkeby Onsager, Lars was brought up in Oslo and graduated from secondary school there. He then studied chemical engineering at the Norges Tekniske Hogskole at Trondheim. While there, he worked through most of the problems in Whittaker and Watson’s Modern Analysis¹⁶—a difficult mathematics book with very challenging problems. At age twenty he discovered a correction to the just-published Debye-Hückel theory of electrolytes. In 1925. Onsager traveled to Zurich and told Debye about it in person. Debye took him on as a research assistant at the Eidgnossische Technische Hochschule, where he worked form 1926 to 1928. Years later, he told proudly of rowing in crew races on the Zürchersee.

In 1928 Onsager accepted a position as teaching associate in the chemistry department at Johns Hopkins University. His assignment was freshman chemistry. The students, however, complained that what he said in class was over their heads. Indeed, those who have heard Onsager lecture would agree that Onsager and freshmen were a mismatch. The challenges in his lectures, even to the expert, were formidable. In mathematical derivations, intermediate steps would be omitted, Onsager didn’t need them and preferred to overestimate his audience’s ability rather than he. The job ended after one term. During that spring he formulated the prize-winning Reciprocal Relations.

Later that same year Onsager moved to Brown University as an instructor in the chemistry department, but with no undergraduate teaching duties. He taught the graduate course in statistical mechanics and began a collaboration with Raymond M. Fuoss, a graduate student who made measurements on electrolytes, that was to last for over thirty years. During his first year in Providence, Onsager submitted an abstract (in Norwegian) for a meeting of the Scandinavian Physical Society, on simultaneous irreversible processes, that announced the Reciprocal Relations.

A couple of years later Onsager sent Debye a manuscript for publication in Physikalische Zeitschrift, of which Debye was an editor. It was another correction to a Debye formula, the one for the dielectric constant of a substance whose molecules have a permanent electric dipole moment. This time Debye did not yield so readily, and the “Onsager formula” was not published until 1936, when J. G. Kirkwood persuaded Onsager to rewrite the paper in English and send it to the Journal of the American Chemical Society⁷.

Another line of research that began during Onsager’s five years at Brown was thermal diffusion as a means of isotope separation. The Soret effect—a solution placed in a temperature gradient develops a concentration gradient—had been discovered in the nineteenth century. This is the kind of “coupled-flows” situation that the Reciprocal Relations are about. But it is also a practical application of abstract ideas in the kinetic theory of gases to the separation of an isotopic gas mixture. The influence of five years as a chemical engineering student appeared to be showing. Later, during World war II, separating the isotopes of uranium became a crucial step in the production of atomic-bomb material.

In 1933 Brown University, financially strapped, did not renew Onsager’s appointment. The distinguished electrochemist Herbert Harned wanted to bring him to Yale, which offered him a Sterling Fellowship, a prestigious postdoctoral appointment that could be held without teaching duties. Thus began a career at Yale that was to last until his retirement. The summer before Yale he spent in Europe. He had been corresponding about questions in electrochemistry with Hans Falkenhagen, then at the University of Cologne, and they spent some weeks together. During this time Onsager met Falkenhagen’s sister-in-law, Margarethe Arledter. They were married on 7 September 1933, before Onsager returned to the United States. They had three sons and a daughter.

Shortly after Onsager’s arrival in New Haven, there was an administrative snag. His appointment was to a postdoctoral fellowship, but he had no doctorate. Actually, he had submitted a version of the Reciprocal Relations to his alma mater, the Norges Tekniske Hogskole in Trondheim, as a doctoral dissertation, but apparently it had not been in the proper form. Harned hit on the obvious solution: Yale could award Onsager a Ph.D. Course requirements could be waived, but a dissertation was necessary. Onsager, the story goes, pulled a stack of papers out of a file, smiled, and asked, “Will this do?” The work, “Solutions of the Mathieu Equation of Period 4? and Certain Related Functions,” looked more like a mathematics than a chemistry dissertation. In fact, it seemed to be built on some of the densest parts of Whittaker and Watson. The local expert on that branch of mathematics, Einar Hille of the mathematics department, read it and suggested a Ph.D. in mathematics.

Apparently the chemists insisted on awarding the Ph.D. themselves. This was appropriate, since the problem had arisen in the chemical context of electrolytes⁴ as well as of ionized gases¹², where one must deal with Brownian motion. In working out the theory of random thermal motion of clusters of ions, Onsager obtained the Mathieu equation, which he knew from Whittaker and Watson. He used the results in the 1938 paper “Initial Recombination of Ions.” He never published the dissertation.

Onsager’s thirty-nine years at Yale included twenty-seven years as the J. Willard Gibbs professor of theoretical chemistry, Gibbs had been America’s first great theoretical physicist. He spent his working life as professor at Yale and is generally considered the father of statistical mechanics. That the Gibbs chair should be held by Onsager, the great twentieth-century practitioner of statistical mechanics, seems appropriate indeed. The appointment to that professorship followed what many consider Onsager’s spectacular achievement: the solution of the two dimensional Ising model. Onsager also held guest professorships at other universities. The sabbatical year 1951 to 1952 was spent as Fullbright professor at Cambridge University. His host was David Shoenberg, director of the Royal Society Mond Laboratory, the part of the Cavendish Laboratory dedicated to low-temperature physics. Onsager had been spending time with the low-temperature physicists at Yale, and in 1949 he had announced the quantization of hydrodynamic circulation in liquid helium. Shoenberg’s experimental research was on the De Haas-Van Alphen effect, the oscillations of magnetic susceptibility as magnetic field is varied. He knew that Onsager had been thinking about electrons in metals. Those thoughts culminated in a quantitative interpretation of the “susceptibility oscillations” in terms of the dimensions of the Fermi surface of the metal. Through a mixture of friendship, gamesmanship, and nagging, Shoenberg persuaded Onsager to write it down for publication shortly before leaving England—in three pages¹³.

Other visiting professorships came in 1961, at the then very new La Jolla campus of the University of California at San Diego; in the years 1967 to 1968, at the Rockefeller University in New York; and in 1968, at Göttingen. During the spring of 1970, Onsager was Lorentz professor at the University of Leiden.

The 1960’s saw the development of a theory of proton motion in ice. Threads from earlier work came together: conduction in electrolytes; dielectric properties; semiconductors; lattice combinatorics. Ice is a solid, so how can its protons move around? There is enough room in the crystal for molecular rotation, but a proton jump requires a defect. Onsager seized on Bjerrum faults¹⁷: “The positive ion moves by donating a proton, the negative by stealing one.” Ice doped with ions of an impurity becomes a protonic semiconductor.

In 1971 he turned 68, Yale’s official retirement age. Onsager had seven honorary doctorates: he had been awarded the Rumford Medal of the American Academy of Arts and Sciences (1953); the Lorentz Medal of the Royal Netherlands Academy of Sciences (1958); the American Chemical Society’s G. N. Lewis Medal (1962), J. G. Kirkwood Medal (1962), J. W. Gibbs Medal (1962), T. W. Richards Medal (1964), P. W. Debye Medal (1965); Yeshiva University’s Belfer Award in Pure Science (1966); the President’s National Medal of Science (1968); and the Nobel Prize in chemistry (1968). He had no intention of leaving New Haven. He had research grants and knew he could count on continued government research support, including a salary, past retirement. The chemistry department assured him an office in the Sterling Laboratory of Chemistry.

But Yale had a rule that an emeritus faculty member could not be principal investigator on a research grant. Onsager suggested that the rule be waived. The administration turned him down.

Onsager gave Yale several months to reconsider. He was too offended to enter into negotiations. When Yale did not change its position, he accepted an offer from the University of Miami’s Center for Theoretical Studies as Distinguished University Professor. He took along the research grants, which allowed him a group of postdoctoral research fellows and a secretary. The four years in Coral Gables saw the flowering of a number of seeds started earlier. The work on proton movement in ice, begun a decade earlier, was a natural sequel to over four decades of work on conduction of electrolytes. Were there analogues to transport mechanisms in membranes? At the Conference on Physical Principles of Biological Membranes (1968), Onsager’s “Possible Mechanisms of Ion Transit”¹⁸ revealed his considerable knowledge of biology as well as his legendary command of organic chemistry. Later he became an associate of the Neurosciences Research Program and attended its meetings.

For Onsager’s seventieth birthday the University of Miami Center for Theoretical Studies mounted a symposium of invited speakers, including many of his former students and postdoctoral fellows. His own contribution²⁷, “Life in the Early Days,” is a speculative paper on the origins of life on earth.

Reciprocal Relations and Fluctuation Statistics. Why did the Nobel committee choose the Reciprocal Relations for the award of the Prize? It might well have been because of their enormous reach into such a great diversity of phenomena. Wherever there are coupled flows—of matter, of energy, of electricity—reciprocal relations arise. In the first of his two 1931 papers, Onsager chose heat conduction in an anisotropic crystal as his example: coupled heat flows. He may well have made this choice because it illustrated a difficulty in defining macroscopic variables of state in terms of which the entropy would turn out to be a quadratic form. He overcame the difficulty by choosing the x, y, and z coordinated of the “center of mass” of the heat as the variables of state. No one had discussed heat conduction in those terms before; measurements are made in steady-state situations. Onsager needed a description of an irreversible process in terms of the relaxation of a nonequilibrium state to an equilibrium state. To define the rate constants (kinetic coefficients), he assumed a linear relationship between the relaxation rate and the departure from equilibrium. For small departures, linearity would be assured. The trick was in measuring the departure from equilibrium. Onsager’s “?-type” variables were a suitable measure of “how far away from equilbrium” a system was. Having the entropy function quadratic in the ?’s made possible a very general definition of “thermodynamic forces” tending toward equilibrium: the derivatives of the entropy function with respect to the ?’s. If the relation between the ?’s and their time derivatives is linear, then the relation between the thermodynamic forces and the corresponding “fluxes” (the d?/dt’s, also called J’s) is automatically linear. The coefficients in those linear relationships—the kinetic coefficients—are what the experimenter measures. They form a square matrix. Onsager’s theorem is that the matrix is symmetric.

In his Nobel address¹⁷ Onsager tells how his proof was born. He had been working on electrical conduction in electrolytes. The flows of the different species of ion interact because of the Coulomb forces between ions. The flow rates of the different ions (the fluxes) are linearly related to the gradients of their chemical potentials (the forces). The coefficients (he calls them L_ij) are essentially conductivities. The coefficient L₁₁ is the conductivity of ion species 1. It relates the current density of those ions to their chemical potential gradient, essentially a field. Similarly, L₂₂ is the conductivity of ion species 2. The coefficient L₁₂ measures how much a concentration gradient of species 2 affects the current of species 1 and L₂₁ measures how much a concentration gradient of species 1 affects the current of species 2. According to Onsager’s theorem, those two cross-coefficients have to be equal: L₁₂= L₂₁.

Onsager called the reciprocal of the L_ij = L₂₁. The R_ij matrix. It is essentially matrix of resistivity coefficients. The quadratic form in the fluxes ?_ijJ_iJ_j is called the dissipation function. It measures the rate at which entropy is produced in the system and tells how quickly free energy is degraded by the irreversible flows. Helmholtz had defined an analogous dissipation function in the nineteenth century, as had Rayleigh and Kelvin. They had recognized that for stationary flow—the steady state—the dissipation function is a minimum. Onsager’s insight was the recognition that this “principle of least dissipation” was equivalent to symmetry in past future. He speaks of “the principle of detailed balancing,” and in the proof of the theorem he very carefully defined and uses the somewhat more general “principle of microscopic reversibility.”

One of Onsager’s professors at Trondheim had been C. N. Riiber, who made optical-rotation measurements on sugar solutions and discovered that there were at least three modifications of galactose. Onsager writes:¹⁷

The possibility that any one of these might transform into either of the others gave rise to a little problem in mathematics. In analyzing it I assumed, as any sensible chemist would, that in the state of equilibrium the reaction 1 ? 2 would occur just as often as 2 ? 1, etc., even through this is not a necessary condition for equilibrium, which might be maintained by a cyclic reaction—as far as the mathematics goes; the physics did not seem reasonable, Now if we look at the condition of detailed balancing from the thermodynamic point of view, it is quite analogous to the principle of least dissipation.

The term “cyclic reaction” means that the rate of 1? 2? 3?1 reactions is faster or slower than the rate of 1? 3? 2? 1 reactions—exactly what detailed balancing forbids.

Reciprocal relations had been known well before Onsager’s proof. Perhaps best known is the Kelvin relation between the Seebeck e.m.f.¹⁹ and the Peltier heat²⁰. The Seebeck e.m.f. is the voltage developed in a circuit containing two junctions between dissimilar metals kept at different temperatures. The Peltier heat is the heat per unit charge transported from one junction to the other when a current flows in the circuit. Kelvin had proved his “second relation” fifty years before Onsager, using what is today called a pseudo thermodynamic argument.

Onsager’s proof rests on a postulate that connects irreversible behavior with fluctuation behavior: that the system does not remember its history, that it does not care whether it arrived at a given state as a result of a thermal fluctuation or as a result of an experimenter’s interference. Formally stated, the regression of fluctuations (the expected path from a nonequilibrium state “back” to equilibrium) is that predicted by the linear relation between the forces and the fluxes. In the language of stochastic processes, the postulate is that the fluctuations are linear Markov processes characterized by the kinetic coefficients that describe the system’s irreversible behavior.

Toward the end of the second 1931 paper, Onsager writes; “It is worth pointing out that the dissipation function has a direct statistical significance.” The hint lay untouched for twenty years. The statistical interpretation of the entropy function given by Einstein in 1910: the probability of a system’s being in a state ? as a result of a spontaneous thermal fluctuation is proportional to exp S(?)/k, where S is the entropy and k is Boltzmann’s constant. But what of the probability of a particular time sequence of states, a particular “path” ? (t)? Onsager’s Ph.D. student Stefan Machlup derived¹⁴ a probability functional for fluctuations about the equilibrium state. It is the exponential of a time integral of the function

where the “thermodynamic forces” X₁ are the derivatives of the entropy function with respect to the ?₁. The integral is defined by specifying the value of the ?’s at discrete points in time and choosing the minimum value consistent with those constraints. This is now called the Onsger-Machlup functional. The “slalom gate” definition of the integral is made complete by taking the minimum value of the integral subject to a constraint at each gate, that is, each specified point in time. The corresponding theorem in the frequency domain is known as the fluctuation-dissipation theorem.

Solution of the Ising Model . Ordered systems are most easily described in terms of a lattice model. The phase transition between the states of magnetic order and disorder (Curie point) in a crystal is typical of second-order phase transitions in general. The Ising model is a primitive attempt to find a mathematical model that will show such an order-disorder transition. Lattice points (atoms) are in one of two possible states. The usual convention is to call the states “up” and “down,” as for spin ½. Interactions are between nearest neighbors only. Though primitive, the hope was that the model would be mathematically tractable, that one could calculate thermodynamic functions for it. The real-life transition does occur at a well-defined temperature, called the critical temperature (T_c). In one dimension—a long chain of elementary magnets—the Ising model shows no phase transition; T_c is absolute zero.

In 1940, Wannier²¹ used the idea of the transfer matrix for an approximate treatment of the two-dimensional Ising lattice, showing a nonzero T_c. In 1942 Onsager presented a contributed paper²² announcing that he had calculated the partition function for the two-dimensional (square) Ising lattice exactly. It had a second-order transition.

Onsager writes²³:

It was a sort of investigation where you got a good lead, and certainly you had to pursue that: and before you reached the end of that lead, up opened another, and this was, if anything, even more fascinating, … It took a few months, though, to verify the guess, but it was doable…. Well, after I got out a paper on that Ising model, there was a young lady, Bruria Kaufman…. First of all, she insisted on working on the Ising model. Now, that was the kind of a task that I would never want to impose on a student.

He continues²⁴:

Unable to talk her out of the idea, I suggested that she explore the structure of W as well as the effect of joining crystal ends on a torus with a twist, etc. She made good progress; but as she got her bearings she was more intrigued by the ubiquitous trigonometric relations and decided to look for a possible connection with spinor theory. Why not? … By the summer of 1946 she had a beautifully compact computation of the partition function, bypassing all tedious detail.

So the thermodynamic properties of the Ising model were solved for zero magnetic field. But what of the spontaneous magnetization? What followed in the next two years involves terms like spinor algebra, the Milne integral equation, the Wiener-Hopf technique, elliptic integrals, and finally the evaluation of Toeplitz (infinite) determinants. The result was that the spontaneous magnetization near T_c is proved to be proportional to (T-T_c)^?. The approximate theories had given an exponent ½; the critical exponent ? was novel indeed. Onsager announced that result at a conference at Cornell in the summer of 1948, and again during a discussion period at a conference of the Italian Physical Society held in March 1949 at Florence. It appeared in the printed “proceedings” of the conference²⁵, but he never published the calculation. C. N. Yang published the result in 1952²⁶.

The “critical exponent” ? is quite close to experimental values for a number of real systems that have transitions of an essentially two-dimensional type. The three-dimensional Ising model has never been solved exactly. Thus Onsager’s solution stands as a monument.

Style. His colleagues were fond of saying Onsager was thirty years ahead of his time. “Irreversible thermodynamics”—the heart of the 1931 reciprocal relations—became fashionable only in the 1960’s. Critical exponents for phase transitions came in the late 1960’s. Negative absolute temperatures, first discussed by Onsager in 1949⁵, were not talked about until the invention of optical pumping and lasers²⁸.

Onsager published on the order of one journal article per year. A number of his discoveries were announced in the discussion periods at scientific meetings, sometimes cryptically; some he never tried to publish. For example, a complete manuscript exists of the band theory of electrons in metals, written just before the publication of the Bloch theory.

Onsager’s graduate students, who were never numerous, tended to come from physics rather than chemistry. His interests were so wide that students were often in entirely different scientific areas. There was never a “group,” and Onsager did not found a “school.”

With students and colleagues Onsager generally talked about what he was working on at the time, always in very few words. Questions were welcome and were patiently answered. He preferred working things out himself to reading other people’s solutions and urged students to do the same. His erudition was phenomenal, though occasionally exasperating. The low-temperature physicist C. T. Lane once said to him, “Lars, you don’t know everything.” Onsager answered, “But I’m learning.”

NOTES

1. L. Onsager, “Crystal Statistic I. A Two-Dimensional Model with an Order-Disorder Transition,” in Physical Review, 65 (1944), 117–149; Bruria Kaufman, “Crystal Statistics 11. Partition Function Evaluated by Spinor Analysis,” ibid., 76 (1949), 1232–1243; Bruria Kaufman and Lars Onsager, “Crystal Statistics III. Short-Range Order in a Binary Ising Lattice”, ibid., 1244–1252. See also Elliott W. Montroll, Renfrey B. Potts, and John C. Ward, “Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model,” in Journal of Matltenrcatical Physics, 4 (1962), 308–322.

2. L. Onsager and M. Dupuis, “Electrical Properties of Ice”, in Rendiconti S.I.F. “Enrico Fermi”, corso X, Varenna, Electrolytes, supp. to Nuovo Cimento (1960), 294–315; L. Onsager. “The Electrical Properties of Ice,” in Vortex, 23 (1962), 138–141; L. Onsager and L. K. Runnels. “Mechanism for Self-diffusion in Inc”, in proceedings of the National Academy of Sciences50 (1963), 208–210; Lars Onsager, Mou-shan Chen. Jill C. Bonner, and J. F. Nagle, “Hopping of lons in Ice,” in Journal of chemical Physics, 60 (1974), 405–419.

3. L. Onsager, “Magnetic Flux Through a Superconducting Ring”, in Physical Review Letters, 7 (1961) 50.

4. L. Onsager, “Zur Theorie der Electrolyte I”, in Physikalische Zeitschrift, 27 (1926), 388–392, and “…II”, ibid., 28 (1927), 277–298; L. Onsager and R. M. Fuoss, “Irreversible Processes in Electrolytes”, in Journal of Physical Chemistry, 36 (1932), 2689–2778.

5. L. Onsager, “Statistical Hydrodynamics”, in supp. to Nuovo Cimento, 9th ser., 6 (1949), 279–287.

6. L. Onsager, “Reciprocal Relations in Irreversible Processes I”, in Physical Review, 37 (1931), 405–426, and “…II”, ibid., 38 (1931), 2265–2279.

7. L. Onsager, “Electric Moments of Molecules in Liquids”, in Journal of the American Chemical Society, 58 (1936), 1486–1493.

8. W. H. Furry, R. Clark Jones, and L. Onsager, “On the Theory of Isotope Separation by Thermal Diffusion,” in Physical Review, 55 (1939), 1083–1095; W. W. Watson, L. Onsager, and A. Zucker, “Apparatus for Isotope Separation by Thermal Diffusion”, in Review of Scientific Instruments, 20 (1949), 924–927.

9. L. Onsager, “The Distribution of Energy in Turbulence” (abstract), in Physical Review, 68 (1945), 286. This was independently discovered by A. N. Komogorov, C. R. Acad. Sci. USSR. 30 (1941), 301–305, and 32 (1941), 16–18.

10. Remark in the discussion following a paper by C. J. Gorter on the two-fluid model of liquid helium, Supp. to Nnoro Cimento, 9th set-., 6 (1949), 249–250.

11. L. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” in Journal of Chemical Physics, 2 (1934), 599–615.

12. L. Onsager, “Initial Recombination of lons”, in Physical Review, 54 (1938), 554–557.

13. L. Onsager, “Interpretation of the de Haas-Van Alphen Effect”, in Philosophical Magazine, 7th ser., 43 (1952), 1006–1008.

14. L. Onsager and S. Machlup, “Fluctuations and Irreversible Processes”, in Physical Review, 91 (1953), 1505–1512; S. Machlup and L. Onsager. “Fluctuations and Irreversible Processes II. Systems with Kinetic Energy”, ibid., 1512–1515.

15. O. Penrose and L. Onsager, “Bose-Einstein Condensation and Liquid Helium”, in Physical Review, 104 (1956), 576–584.

16. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 2nd ed. (Cambridge, 1915).

17. Lars Onsager, “The Motion of lons: Principles and Concepts”, in Science, 166 (1969), 1359–1364. See also Les Prix Nobel en 1968 (Stockholm, 1969), 169–182.

18. Lars Onsager, “Possible Mechanisms of lon Transit”, in F. Snell et al., eds., Physical Principles of Biological Membranes (New York, 1970), 137–141.

19. T. J. Seebeck in Annalen der Physik und Chemie, 6 (1826), 133, 263.

20. A. Peltier, “Nouvelles experiences sur la caloricité des courants éeltriques”, in Annales de chimie, 56 (1834), 371–386; Comptes rendus…de l’, Académie des sciences, I (1835), 360.

21. H. A. Kramers and G. H. Wannier, “Statistics of the Two-Dimensional Ferromagnet. Part I”, in Physical Review, 60 (1941), 252–262. See Stephen G. Brush, “History of the Lenz-Ising Model”, in Review of Modern, Physics, 39 (1967), 883–893 and T. Shedlovsky and E. Montroll, Math. Phys., 4 (1963), the introduction to the “Proceedings of Conference on Irreversible Thermodynamics and the Statistical Mechanics of Phase Transitions,” a conference to celebrate the thirtieth anniversary of the Reciprocal Relations.

22. L. Onsager, “Crystal Statistics” (abstract), in Physical Review, 62 (1942), 559.

23. Lars Onsager, “Autobiographical Commentary of Lars Onsager”, in R. E. Mills et al. eds. Critical Phenomena in Alloys, Magnets and Superconductors (New York, 1971), xxi.

24. Lars Onsager, “The Ising Model in Two Dimensions”, in R. E. Mills et al. eds. Critical Phenomena in Alloys, Magnets and Superconductors (New York, 1971), 3–12.

25. Remark in the discussion following a paper by G. S. Rush-brooke, in supp. to Nnoro Cimento, 9th ser-., 6 (1949), 261. Further along in the same discussion period Onsager talks about the phenomenon of critical slowing down—that it takes a long time to reach equilibrium near a critical point. Perhaps this is the earliest announcement of a now familiar kind of “divergence”.

26. C. N. Yang, “The Spontaneous Magnetization of a Two-Dimensional Ising Model”, in Physical Review, 85 (1952), 808–816.

27. Lars Onsager, “Life in the Early Days,” in S. L. Mintz and S. M. Widmayer, eds., Quantum Statistical Mechanics in the Natural Seicnces (New York, 1947), 1–14.

28. E. M. Purcell and R. V. Pound, “A Nuclear Spin System at Negative Temperature”, in Physical Review, 81 (1951), 279–280; N. F. Ramsey. “Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures”, ibid., 103 (1956), 20–28.

BIBLIOGRAPHY

H. Christopher Longuet-Higgins and Michael E. Fisher, “Lars Onsager”, Biographical Memoirs of Fellows of the Royal Society, 24 (1978), 443–471, is a detailed biography containing a complete list of Onsager’s publications as well as a bibliography of (numerous shorter) biographies.

Microfilms of Onsager’s letters, research notes, and papers are archived at the Sterling Library of Yale University.

Stefan Machlup