# Onsager, Lars

# 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 model^{1}? Was it for explaining the electrical conductivity of ice^{2}? Was it for flux quantization in superconductors^{3}?was it for his theory of electrolytes^{4}?As it happened it was for what was called the Onsager Reciprocal Relations, developed in two papers published in *Physical Review* in 1931^{6}, and it was in chemistry.

Other major research for which Onsager is known includes the formula for the dielectric constant of polar liquids^{7}. Isotope separation by thermal diffusion^{8}, the energy spectrum of turbulence^{9}, the statistical-mechanical description of vortices in two dimensions (negative absolute temperatures when they roll up^{5}, quantization of the circulation of vortices in superfluid helium^{10}, an explanation of the Wien effect^{11}, electron-ion recombination statistics^{12}, interpretation of the De Haas-Van Alphen effect^{13}, the statistical interpretation of the dissipation function^{14}, and a new definition of the Bose-Einstein condensation for interacting particles applicable to the lambda-point transition to superfluidity of liquid helium^{15}.

**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 Analysis*^{16}—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 Society ^{7}*.

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 4π 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 electrolytes^{4} as well as of ionized gases^{12}, 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 pages^{13}.

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 faults^{17}: “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 contribution^{27}, “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 “α-type” variables were a suitable measure of “how far away from equilbrium” a system was. Having the entropy function quadratic in the α’s made possible a very general definition of “thermodynamic forces” tending toward equilibrium: the derivatives of the entropy function with respect to the α’s. If the relation between the α’s and their time derivatives is linear, then the relation between the thermodynamic forces and the corresponding “fluxes” (the *dα/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 address^{17} 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 L_{11} is the conductivity of ion species 1. It relates the current density of those ions to their chemical potential gradient, essentially a field. Similarly, *L*_{22} is the conductivity of ion species 2. The coefficient *L*_{12} measures how much a concentration gradient of species 2 affects the current of species 1 and *L*_{21} 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: *L*_{12}*= L*_{21}.

Onsager called the reciprocal of the *L _{ij} = L*

_{21}. The

*R*matrix. It is essentially matrix of resistivity coefficients. The quadratic form in the fluxes

_{ij}*Σ*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.”

_{ij}J_{i}J_{j}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 heat^{20}. 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 α as a result of a spontaneous thermal fluctuation is proportional to exp *S(α)/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” *α (t)?* Onsager’s Ph.D. student Stefan Machlup derived^{14} a probability functional for fluctuations about the equilibrium state. It is the exponential of a time integral of the function

where the “thermodynamic forces” *X*_{1} are the derivatives of the entropy function with respect to the *α*_{1}. The integral is defined by specifying the value of the *α’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, Wannier^{21} 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 paper^{22} announcing that he had calculated the partition function for the two-dimensional (square) Ising lattice exactly. It had a second-order transition.

Onsager writes^{23}:

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 continues^{24}:

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})^{⅛}. The approximate theories had given an exponent ½; the critical exponent ⅛ 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 conference^{25}, but he never published the calculation. C. N. Yang published the result in 1952^{26}.

The “critical exponent” ⅛ 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 1949^{5}, were not talked about until the invention of optical pumping and lasers^{28}.

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 Sciences***50** (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

# Lars Onsager

# Lars Onsager

Lars Onsager (1903-1976) made significant contributions to chemistry, including his developments in the Debye-Hückel theory of electrolytic dissociation and his work with non-reversable systems. He received the 1968 Nobel Prize in Chemistry.

Born in Norway, Lars Onsager received his early education there before coming to the United States in 1928 to do graduate work at Yale University. After receiving his Ph.D. in theoretical chemistry he stayed on at Yale and ultimately spent nearly all of his academic career at that institution. Onsager's first important contribution to chemical theory came in 1926 when he showed how improvements could be made in the Debye-Hückel theory of electrolytic dissociation. His later (and probably more significant) work involved non-reversible systems —systems in which differences in pressure, temperature, or some other factor are an important consideration. For his contributions in this field, Onsager received a number of important awards including the Rumford Medal of the American Academy of Arts and Sciences, the Lorentz Medal of the Royal Netherlands Academy of Sciences, and the 1968 Nobel Prize in Chemistry.

Lars Onsager was born in Oslo (then known as Christiania), Norway, on November 27, 1903. His parents were Erling Onsager, a barrister before the Norwegian Supreme Court, and Ingrid Kirkeby Onsager. Onsager's early education was somewhat unorthodox as he was taught by private tutors, by his own mother, and at a somewhat unsatisfactory rural private school. Eventually he entered the Frogner School in Oslo and did so well that he skipped a grade and graduated a year early. Overall, his early schooling provided him with a broad liberal education in philosophy, literature, and the arts. He is said to have become particularly fond of Norwegian epics and continued to read and recite them to friends and family throughout his life.

In 1920, Onsager entered the Norges Tekniski Hoslashgskole in Trondheim where he planned to major in chemical engineering. The fact that he enrolled in a technical high school suggests that he was originally interested in practical rather than theoretical studies. Onsager had not pursued his schooling very long, however, before it became apparent that he wanted to go beyond the everyday applications of science to the theoretical background on which those applications are based. Even as a freshman in high school, he told of making a careful study of the chemical journals, in order to gain background knowledge of chemical theory.

One of the topics that caught his attention concerned the chemistry of solutions. In 1884, Svante Arrhenius had proposed a theory of ionic dissociation that explained a number of observations about the conductivity of solutions and, eventually, a number of other solution phenomena. Over the next half century, chemists worked on refining and extending the Arrhenius theory.

The next great step forward in that search occurred in 1923, when Onsager was still a student at the Tekniski Hoslashgskole. The Dutch chemist Peter Debye and the German chemist Erich Hückel, working at Zurich's Eidgenössische Technische Hochschule, had proposed a revision of the Arrhenius theory that explained some problems not yet resolved—primarily, whether ionic compounds are or are not completely dissociated ("ionized") in solution. After much experimentation, Arrhenius had observed that dissociation was not complete in all instances.

Debye and Hückel realized that ionic compounds, by their very nature, already existed in the ionic state *before* they ever enter a solution. They explained the apparent incomplete level of dissociation on the basis of the interactions among ions of opposite charges and water molecules in a solution. The Debye-Hückel mathematical formulation almost perfectly explained all the anomalies that remained in the Arrhenius theory.

Almost perfectly, but not quite, as Onsager soon observed. The value of the molar conductivity predicted by the Debye-Hückel theory was significantly different from that obtained from experiments. By 1925, Onsager had discovered the reason for this discrepancy. Debye and Hückel had assumed that most—but not all—of the ions in a solution move about randomly in "Brownian" movement . Onsager simply extended that principle to *all* of the ions in the solution. With this correction, he was able to write a new mathematical expression that improved upon the Debye-Hückel formulation.

Onsager had the opportunity in 1925 to present his views to Debye in person. Having arrived in Zurich after traveling through Denmark and Germany with one of his professors, Onsager is reported to have marched into Debye's office in Zurich and declared, "Professor Debye, your theory of electrolytes is incorrect." Debye was sufficiently impressed with the young Norwegian to offer him a research post in Zurich, a position that Onsager accepted and held for the next two years.

In 1928, Onsager emigrated to the United States where he became an associate in chemistry at Johns Hopkins University. The appointment proved to be disastrous: he was assigned to teach the introductory chemistry classes, a task for which he was completely unsuited. One of his associates, Robert H. Cole, is quoted in the *Biographical Memoirs of Fellows of the Royal Society:* "I won't say he was the world's worst lecturer, but he was certainly in contention." As a consequence, Onsager was not asked to return to Johns Hopkins after he had completed his first semester there.

Fortunately, a position was open at Brown University, and Onsager was asked by chemistry department chairman Charles A. Krauss to fill that position. During his 5-year tenure at Brown, Onsager was given a more appropriate teaching assignment, statistical mechanics. His pedagogical techniques apparently did not improve to any great extent, however; he still presented a challenge to students by speaking to the blackboard on topics that were well beyond the comprehension of many in the room.

A far more important feature of the Brown years was the theoretical research that Onsager carried out in the privacy of his own office. In this research, Onsager attempted to generalize his earlier research on the motion of ions in solution when exposed to an electrical field. In order to do so, he went back to some fundamental laws of thermodynamics, including Hermann Helmholtz's "principle of least dissipation." He was eventually able to derive a very general mathematical expression about the behavior of substances in solution, an expression now known as the Law of Reciprocal Relations.

Onsager first published the law in 1929, but continued to work on it for a number of years. In 1931, he announced a more general form of the law that applied to other non-equilibrium situations in which differences in electrical or magnetic force, temperature, pressure, or some other factor exists. The Onsager formulation was so elegant and so general that some scientists now refer to it as the Fourth Law of Thermodynamics.

The Law of Reciprocal Relations was eventually recognized as an enormous advance in theoretical chemistry, earning Onsager the Nobel Prize in 1968. However, its initial announcement provoked almost no response from his colleagues. It is not that they disputed his findings, Onsager said many years later, but just that they totally ignored them. Indeed, Onsager's research had almost no impact on chemists until after World War II had ended, more than a decade after the research was originally published.

The year 1933 was a momentous one for Onsager. It began badly when Brown ended his appointment because of financial pressures brought about by the Great Depression. His situation improved later in the year, however, when he was offered an appointment as Sterling and Gibbs Fellow at Yale. The appointment marked the beginning of an affiliation with Yale that was to continue until 1972.

Prior to assuming his new job at Yale, Onsager spent the summer in Europe. While there, he met the future Mrs. Onsager, Margarethe Arledter, the sister of the Austrian electrochemist H. Falkenhagen. The two apparently fell instantly in love, became engaged a week after meeting, and were married on September 7, 1933. The Onsagers later had three sons, Erling Frederick, Hans Tanberg, and Christian Carl, and one daughter, Inger Marie.

Onsager had no sooner assumed his post at Yale when a small problem arose: the fellowship he had been awarded was for postdoctoral studies, but Onsager had not as yet been granted a Ph.D. He had submitted an outline of his research on reciprocal relations to his alma mater, the Norges Tekniski Hoslashgskole, but the faculty there had decided that, being incomplete, it was not worthy of a doctorate. As a result, Onsager's first task at Yale was to complete a doctoral thesis. For this thesis, he submitted to the chemistry faculty a research paper on an esoteric mathematical topic. Since the thesis was outside the experience of anyone in the chemistry or physics departments, Onsager's degree was nearly awarded by the mathematics department, whose chair understood Onsager's findings quite clearly. Only at the last moment did the chemistry department relent and agree to accept the judgment of its colleagues, awarding Onsager his Ph.D. in 1935.

Onsager continued to teach statistical mechanics at Yale, although with as little success as ever. (Instead of being called "Sadistical Mechanics," as it had been by Brown students, it was now referred to as "Advanced Norwegian" by their Yale counterparts.) As always, it was Onsager's theoretical—and usually independent—research that justified his Yale salary. In his nearly four decades there, he attacked one new problem after another, usually with astounding success. Though his output was by no means prodigious, the quality and thoroughness of his research was impeccable.

During the late 1930s, Onsager worked on another of Debye's ideas, the dipole theory of dielectrics . That theory had, in general, been very successful, but could not explain the special case of liquids with high dielectric constants. By 1936, Onsager had developed a new model of dipoles that could be used to modify Debye's theory and provide accurate predictions for all cases. Onsager was apparently deeply hurt when Debye rejected his paper explaining this model for publication in the *Physikalische Zeitschrift,* which Debye edited. It would be more than a decade before the great Dutch chemist, then an American citizen, could accept Onsager's modifications of his ideas.

In the 1940s, Onsager turned his attention to the very complex issue of phase transitions in solids. He wanted to find out if the mathematical techniques of statistical mechanics could be used to derive the thermodynamic properties of such events. Although some initial progress had been made in this area, resulting in a theory known as the Ising model, Onsager produced a spectacular breakthrough on the problem. He introduced a "trick or two" (to use his words) that had not yet occurred to (and were probably unknown to) his colleagues—the use of elegant mathematical techniques of elliptical functions and quaternion algebra. His solution to this problem was widely acclaimed.

Though his status as a non-U.S. citizen enabled him to devote his time and effort to his own research during World War II, Onsager was forbidden from contributing his significant talents to the top-secret Manhattan Project, the United State's research toward creating atomic weapons. Onsager and his wife finally did become citizens as the war drew to a close in 1945.

The postwar years saw no diminution of Onsager's energy. He continued his research on low-temperature physics and devised a theoretical explanation for the superfluidity of helium II (liquid helium). The idea, originally proposed in 1949, was arrived at independently two years later by Princeton University's Richard Feynman. Onsager also worked out original theories for the statistical properties of liquid crystals and for the electrical properties of ice. In 1951 he was given a Fulbright scholarship to work at the Cavendish Laboratory in Cambridge; there, he perfected his theory of diamagnetism in metals.

During his last years at Yale, Onsager continued to receive numerous accolades for his newly appreciated discoveries. He was awarded honorary doctorates by such noble universities as Harvard (1954), Brown (1962), Chicago (1968), Cambridge (1970), and Oxford (1971), among others. He was inducted to the National Academy of Sciences in 1947. In addition to his Nobel Prize, Onsager garnered the American Academy of Arts and Sciences' Rumford Medal in 1953 and the Lorentz Medal in 1958, as well as several medals from the American Chemical Society and the President's National Medal of Science. Upon reaching retirement age in 1972, Onsager was offered the title of emeritus professor, but without an office. Disappointed by this apparent slight, Onsager decided instead to accept an appointment as Distinguished University Professor at the University of Miami's Center for Theoretical Studies. At Miami, Onsager found two new subjects to interest him, biophysics and radiation chemistry. In neither field did he have an opportunity to make any significant contributions, however, as he died on October 5, 1976, apparently the victim of a heart attack.

Given his shortcomings as a teacher, Onsager still seems to have been universally admired and liked as a person. Though modest and self-effacing, he possessed a wry sense of humor. In *Biographical Memoirs,* he is quoted as saying of research, "There's a time to soar like an eagle, and a time to burrow like a worm. It takes a pretty sharp cookie to know when to shed the feathers and … to begin munching the humus." In a memorial some months after Onsager's death, Behram Kursunoglu, the director of the University of Miami's Center for Theoretical Studies, described him as a "very great man of science—with profound humanitarian and scientific qualities."

## Further Reading

*Biographical Memoirs of Fellows of the Royal Society,* Volume 24, Royal Society (London), 1978.

*Current Biography 1958,* H. W. Wilson, 1958.

*Nobel Lectures in Chemistry, 1963-1970,* [Amsterdam], 1972. □

# Onsager, Lars

# ONSAGER, Lars

(*b.* 27 November 1903 in Oslo, Norway; *d.* 5 October 1976 in Miami, Florida), physical chemist who received the 1968 Nobel Prize in chemistry for his derivation of a set of equations known as the *Law of Reciprocal Relations*, which is sometimes described as the *Fourth Law of Thermodynamics* because it makes possible a study of irreversible processes.

Onsager was the son of Erling Onsager, a barrister of the Supreme Court of Norway, and Ingrid Kirkeby. He had a liberal early education that was a combination of private school and his mother's tutoring, followed by high school in Oslo. The combination gave him an extraordinary background in literature and fine arts. Onsager then entered the Norwegian Institute of Technology as a student in chemical engineering. However, in his first year at the institute he became interested in the theory of electrolyte solutions, an interest that was the beginning of a scientific curiosity and ultimately led to his receiving a Nobel Prize. Onsager soon mastered the new theory, and detected a flaw in it. He studied in a stimulating environment under professors who encouraged his efforts in studying theory.

After graduation in 1925 Onsager went with one of his professors on a trip to Denmark and Germany that included a stay of a couple of months at Zurich University with Peter Debye, coauthor with Erich Hückel of the new theory of electrolyte solutions. Onsager later described his first encounter with Debye, which started with his solemn statement, "Professor Debye, your theory of electrolytes is incorrect." He based his comment on a puzzling discrepancy between the theory and the actual results in experiments. Debye and Hückel had assumed that a central ion in solution would move uniformly in a straight line while all the other ions were randomly undergoing Brownian motion. Within a few years Onsager had developed the theory to include mobilities of all ions in equal proportions.

Onsager immigrated to the United States in 1928, embarking on a dubious career as a college instructor, first at Johns Hopkins University, where he lasted only a year, and next at Brown University. Following that he was at Harvard, where students called his science courses "advanced Norwegian." Onsager was described by a colleague as a contender for the title of world's worst lecturer. At Brown University he did the classic work on irreversible processes that ultimately led to the Nobel Prize, without the advantage of a Ph.D. He presented a thesis for a Ph.D. on this work to Brown, where the highly theoretical work was not doubted but regarded as too incomprehensible.

Yale was kinder. Onsager went to Yale as the Sterling and Gibbs Fellow in 1933, was made an assistant professor of chemistry in 1934, and was granted a Ph.D. in 1935. His dissertation was on the mathematical background for his interpretation of deviations from Ohm's law in weak electrolytes. Interestingly, Brown awarded him an honorary doctor of science in 1962 when he had become widely acclaimed for his research, most of it at Yale. In the early 1960s he was also reworking his "incomprehensible" discovery of "reciprocal relations," which had been ignored at Brown but was recognized with a Nobel Prize in 1968.

At Yale, Onsager's research covered a wide range of subjects that included colloids, superconductivity, turbulence, ice, electrons in metals, and dielectrics. Onsager also found a firm statistical basis for the theory of liquid crystal ordering, which led the way for the research that ultimately produced liquid crystal displays, better known as LCDs. Although he served as visiting professor in many prestigious universities during the years that followed, he officially stayed with Yale until retirement in 1972. In 1968, the year he received the Nobel Prize, he was a visiting professor at the University of Göttingen in Germany. Onsager became a United States citizen in 1945; his Nobel Prize lists him as an American.

In the presentation speech for the Nobel Prize in 1968, Professor S. Claesson of the Royal Swedish Academy of Sciences noted that the award to Onsager was a more than usual application of a special rule of the Nobel Foundation. The rule reads: "Work done in the past may be selected for the award only on the supposition that its significance has until recently not been fully appreciated." Also recognized in the speech was the fact that Onsager's contributions on irreversible thermodynamics have numerous applications not only in physics and chemistry but also in biology and technology.

From 1954 to 1971 Onsager received a total of nine honorary doctorate degrees, plus many prestigious medals and prizes in addition to the 1968 Nobel Prize. In the 1960s two conferences were based on Onsager's work: an international conference at Brown University in 1962, "Irreversible Thermodynamics and the Statistical Mechanics of Phase Transitions," and in 1965 at the National Institute of Standards near Washington, D.C., "Phenomena in the Neighborhood of Critical Points."

Onsager had difficulty communicating with people of lesser intellect throughout his life. Not only did his students find him difficult to understand, but many of his colleagues had the same problem. He had the reputation of being enigmatic and verbally obscure. Onsager simply did not recognize this and would continue lectures and conversations with what biographers describe as the Onsager grin and twinkling Nordic blue eyes, as if he were fully understood by all.

He is also described as having had a mischievous sense of humor. The chair of the department at Brown suggested he do some experiments rather than spend all his time on theoretical work. Onsager replied that he would like to do that, but he must have the equipment he needed: a tube that would stretch from the basement to the third floor of the chemistry building. Also, the tube must be made of platinum. Actually, such an experiment had already been performed—as part of the Manhattan Project for the atomic bomb. The chair never mentioned doing experiments again to Onsager.

Onsager married Margarethe Arledter on 7 September 1933. They had four children. Onsager died, presumably from a heart attack, at age seventy-two. He is buried in Grove Street Cemetery in New Haven, Connecticut.

Bibliographies of Onsager include Nobel archives, *Lars Onsager—Biography* (1968); H. Christopher Longuet-Higgins and Michael E. Fisher, *Lars Onsager November 27, 1903–October 5, 1976* (1991); and Harriet Zuckerman, *Scientific Elite: Nobel Laureates in the United States* (1995). An obituary is in the *New York Times* (6 Oct. 1976).

M. C. Nagel

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