The Collected Works of Lars Onsager

The following sections are included:

• PREFACE

• CONTENTS

• BIBLIOGRAPHY — LARS ONSAGER

• PICTURE

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Among Onsager's papers on irreversible phenomena two articles stand out prominently. These are the two papers, [8] and [9], concerned with simultaneous irreversible processes, which were published in 1931. They belong to Onsager's earliest publications. Their essential result, the derivation of the famed reciprocal relations, which he had announced at two meetings, [6], [7], in 1929 and 1930, would ultimately lead to the award of the Nobel Prize in 1968.

The study of simultaneous, coupled, irreversible processes had already in 1930 a long history, which extends to the middle of the nineteenth century. The obvious question which arose in these studies is whether, in analogy to the reciprocal or symmetry relations of equilibrium thermodynamics and mechanics, similar relations exist for these coupled processes. Much effort was devoted during the second half of the nineteenth century, and the first decades of the twentieth, to show that symmetry relations do indeed exist for particular processes. However, the arguments and methods used to obtain these relations were quasi-thermodynamic in character and could not really be justified. But let us briefly review the situation around 1930 concerning the existence of reciprocal relations for simultaneous irreversible processes…

Onsager's work on the Ising model was a striking example of the successful application of sophisticated mathematical ideas and techniques to the exact solution of a physical problem. But its revolutionary conclusions set in motion one of the most fruitful areas of theoretical and experimental research in 20th century physics. The field of critical phenomena which his work vitalized, attracted researchers of great ability, and there were many different contributions of significance to the ultimately successful resolution of all of the major problems which had been posed.

Characteristically Onsager's publications were few in number: the classic 1944 paper published on his own [26], and two papers in collaboration with Bruria Kaufman [29], [32]. In addition a 1945 review paper by Wannier¹ contained new ideas of seminal importance due to Onsager, and a discussion remark at the Florence Conference² in 1949 reported on the result of an exact calculation of major significance. It was only twenty years later that Onsager revealed how the calculation had been performed [80].

In this brief commentary we shall endeavour to depict the pre-Onsager situation, to summarize the content of Onsager's publications, and to describe the challenges and consequences of Onsager's work…

About one third of Onsager's scientific papers are devoted to electrolyte theory. He started his scientific career in this field in 1925, just after finishing his diploma thesis on absorption spectra in Trondheim. The first problem he attacked was the theory of conductivity of electrolytes. He was then 22 years old.

We have to remember that in the first quarter of this century one of the great open questions in physics and chemistry was the discrepancy between the experimentally observed data for conductance and thermodynamic functions of solutions and the theoretical formulae which were then available. The theory of solutions was at that time based on the work of Arrhenius and Van't Hoff and on Planck's theory of ideal solutions (see Falkenhagen¹). As a freshman chemist in Trondheim, Onsager was introduced to this theory, according to which the. properties of electrolytes should be additive with respect to the constituent ions. However the experiments by Kohlrausch and other workers had shown strong deviations from the theoretical curves and, in particular, a characteristic scaling of the deviations from the ideal theory with the square root of the concentration. Many of the leading physicists and chemists of that time considered the so-called anomalies of electrolytes as a great challenge for theorists…

Of the small collection of Lars Onsager's papers on electrons in metals, one stands out as a really significant contribution, which in due course made possible the reliable interpretation of experiments on the de Haas-van Alphen effect: the oscillating variation, with magnetic field, of the magnetic moment of a metal crystal. This revealed how one metal differs from another in its electronic structure, as characterized by the shape of the Fermi surface. Onsager had clearly taken an interest in the effect as far back as 1948 [30], but seems then to have had no new theory in mind. By 1951, when he came to spend a year in Cambridge and shared an office with David Shoenberg, his occasional mysterious remarks on interpreting the data suggested his theory was fully formed; only as he packed up to leave did he hand to David the paper which was later published [35], having left it too late for any discussions.

The enthusiasm for semiconductor physics after 1945 had strengthened adherence to the algebraic description of electron dynamics, especially the assumption that the energy was a quadratic function of momentum. Shockley's letter¹ of 1950, to which Onsager refers, was probably the only attempt at the time to break away and analyse the behaviour of electrons in terms of the shape of the surfaces of constant energy in momentum space. But Shockley's treatment was classical, and Onsager was the first to publish a quantummechanical analysis based on the geometry of energy surfaces. The important result is that every section of the Fermi surface by a plane normal to the magnetic field contributes an oscillatory term to the magnetic moment, and the frequency is determined solely by the area of the section, not by any details of its shape. This was proved very economically by the application of semi-classical quantum theory. As the plane section is moved through the Fermi surface the area may go through maxima and minima, around each of which the frequency is stationary; it is these frequencies which dominates the observed oscillations. By studying how the frequencies change with the orientation of the magnetic field much information can be gathered on the shape of the Fermi surface — nearly enough to reconstruct it from the data alone, but not quite…

Lars Onsager wrote only one full-length paper on colloidal suspensions [31], yet he clearly had a keen interest in such systems for a considerable length of time. Apparently, he was the first to coin the phrase "potential of average force" in a lucid reassessment [12] of the work of Gibbs, Smoluchowski and Einstein on the application of statistical mechanics to mesoscopic particles. Kirkwood¹ was inspired in part by Onsager's presentation to develop a theory of liquids a few years later. The potential of mean force, of course, figures heavily in the modern theory of solutions, often starting with McMillan and Mayer's² formal analysis of 1945. For instance, Zimm's³ application to isotropic solutions of rods appeared only one year later. Actually, Mayer was already aware of the structure of a full formal theory at about the same time that Onsager published a very short preview of his virial theory and phase behaviour of rodlike colloids in 1942 [24]. Still, then and later [31], Onsager never did write a formal solution theory, rigorous beyond the second virial term.

Onsager's abstract [11] on colloid dynamics is tantalizing for it was never expanded into a publication proper. As early as 1932, he had figured out the correct Brownian component of the viscosity of a dilute suspension of ellipsoids, at least in a slender body approximation. Moreover, he shrewdly realized the implication of the theory for polymer physics, namely that chainlike molecules could not be perfectly straight if the experimental viscosities were to be understood. Around the same time, W. Kuhn⁴ reached similar conclusions in what are regarded as seminal papers although his computations are less exacting. Onsager's form for the viscosity was later rederived by Simha⁵ who gave an expression valid for less restricted aspect ratios. The precise mathematical physics of the dynamics of suspensions of anisometric particles has taxed the skills of a number of outstanding theoreticians. It continues to weigh heavily both in the craft of particle characterization and in the explanation of fascinating rheological phenomena witnessed in the laboratory…

Onsager's 1936 treatment of the dielectric constant e of a molecular fluid [16] yields a very different sort of expression than the one then in widespread use, which Debye had derived1 in 1912. For a model fluid of hard spheres, each bearing a scalar polarizability α and a permanent ideal dipole of moment μ, Debye's formula is…

First ideas about quantized circulation and vortices

The idea of quantized circulation in superfluid helium was first put forth to students and colleagues at Yale University by Lars Onsager beginning about 1946. Onsager enjoyed the drama of an important scientific announcement and made public his discovery in a remark following a paper by Gorter on the two-fluid model at the Conference on Statistical Mechanics in Florence in 1949. He said, in part, "Thus the well-known invariant called the hydrodynamic circulation is quantized; the quantum of circulation is h/m . In the case of cylindrical symmetry, the angular momentum per particle is a multiple of ℏ"¹. Enormous ramifications of this single remark have come about and it has been observed more than once that the ratio of scientific insight to length of announcement must be a record in the history of science…

In my first months at Bell Labs in 1949 I was fortunate enough to be required to share a room with G. H. Wannier. I learned much from Gregory, including an almost superstitious awe of Lars Onsager. This was little diminished when I actually met him in Japan in 1953, at the International Conference on Theoretical Physics, where his talks and discussion remarks gave fundamental insights into at least three subjects in the form of mysterious, oracular discussions, so many years ahead of their time as to bewilder most of their listeners. (The subjects were metallic diamagnetism, liquid helium, and superconductivity.)

I can't resist the temptation to indulge in one Onsager story of that trip. This was the first considerable scientific conference the Japanese had been able to manage, only a year after the peace treaty had been signed, and they were determined to make the most of it in terms of showing off their beautiful country as well as their science. It was also typhoon season, and we had waited through a typhoon in Osaka before being bussed to Nara. Not surprisingly, the bus slid off the terrible road (Japan then had no car industry, much less a road system) into a sinkhole, many miles from either city. Drivers, local farmers, and physicists stood around jabbering in several languages until Onsager, with a sigh, firmly took charge. He organized a work crew of local farmers to dismantle a log bridge over the ditch, arranged a system of levers, and with the muscle of 20 or 30 physicists and Onsager's direction and encouragement, we, to our astonishment, put the bus back on the road: all of this totally without communication from Onsager other than grunts, smiles and gestures. (I am told he owned a farm in New Hampshire — surely he came from Norwegian farm stock.)…

Onsager's contribution to turbulence theory is concentrated in a brief abstract and a short paper; it is dense, bold, and important; its full significance is only now beginning to be seen. It opened two important lines of inquiry, on the relation between the physical phenomenology of turbulence and the analytical properties of its equations of motion, and on the role of vortices in the dynamics of turbulence. Both topics are related to Onsager's other interests; the first to his work on irreversible processes, and the second to his work on electrolytes and superfluids.

Onsager's note on the distribution of energy in turbulence [28] is a one-paragraph rederivation of the Kolmogorov spectral law. Kolmogorov's work dates to the late thirties but was not known in the west until after the war; it is one of Kolmogorov's greatest achievements, and Onsager's independent derivation is no mean feat. The argument can be readily summarized: Turbulence is typically stirred on scales much larger than the scales where energy is dissipated; in three-dimensional space, the important "inertia!" range of scales that mediate between stirring and dissipation has a spectrum of universal form, which can be deduced from a few simple assumptions…

As is true in so many cases, Onsager's major contribution to reaction-rate theory, [17], represented an attack on a significant physical problem that at the same time required the solution to a non-trivial mathematical problem. The physical problem, as originally stated, is that of calculating the probability that an electron, initially removed from a charged particle, will ultimately recombine with that particle in the presence of a constant field. The interaction potential is assumed to be Coulombic.

Notwithstanding its very specific formulation, Onsager's results have been applied to a wide variety of problems in solid state physics^1-4, and physical chemistry⁵. In mathematical terms, the probability that the two particles do not combine is known as a splitting probability. A general discussion of such probabilities as calculated for systems whose evolution is described by the Smoluchowski equation is to be found in the text by Gardiner⁶…

The two chief facts of classical probability are: (1) the law of large numbers and (2) the central limit theorem. In their simplest form they state that, for a large number of statistically independent copies x_n, n = 1 , . . . , N of some single random variable x₀

One of Onsager's most sustained interests was unravelling the intricacies of crystalline water and related hydrogen bonded crystals. Onsager told me that he had been working on the ferroelectric transition in potassium dihydrogen phosphate in the late 30's before he shifted to the Ising model. However, it was not until 1960 that he published his first major paper on the electrical properties of ice [52]. This was rapidly (for Onsager) followed by another major paper in 1962 [58] and he also chose to summarize these papers in his acceptance speech for the G. N. Lewis award [57], consistent with ice being foremost on his mind at that time. His Nobel speech [75] also emphasized ice — he even presented an animated movie to illustrate the nature and motion of charged defects that carry the electrical current. This speech also indicates how ice fit naturally with his more widely recognized interests in electrolytes and irreversibility.

It is tempting to speculate why Onsager published his first major papers [52] and [58] where he did. These papers do contain quite a lot of review material and perhaps he thought they were not appropriate for primary research journals. This was a shame, because many researchers could not easily obtain these papers and perhaps thought that Onsaager did not value this work highly. Reprinting these papers here may be a belated remedy…

Onsager appreciated that biology would be a large part of the future of chemical and physical science. He attended a number of biological conferences at which he soaked up information, often with eyes closed, but, as has been noted many times, that was a rather counterintuitive indicator of his neural activity. Another indication of the level of his interest was that biological issues accounted for about 10% of his Nobel address [75] and for four papers [72, 73, 79, 83]. Even though these papers were short and preliminary, his well known reluctance for writing papers indicates that he placed a high priority on biological issues. On a personal note, I asked his advice in 1968 where to go for a sabbatical to explore biological issues and he strongly urged me to come back to Yale because he had an interesting idea.

That idea is mentioned at the end of [72] and [75] and is the principal content of [73] and [79]. The general issue was how ions get through membranes. His specific idea utilized general principles of protein folding, which were not that well established at that time, to argue that there should be chains of hydrogen bonds, involving the amino acid side chains. These chains would then provide a relatively low impedance pathway for ion conduction. As is clear from [72] Onsager knew better than to try to be completely general — indeed, he subtly warned that general irreversible thermodynamics would not likely resolve biological issues — and so he focussed on the sodium channel in nerve axons. However, in 1970 it became clear to us from TTX poisoning experiments¹ that the current carried per channel was far in excess of the current of sodium ions that his putative channels could carry, and he did not pursue this idea further…

In one of Onsager's favorite mathematics books, Whittaker and Watson's Modern Analysis¹, the chapter on Mathieu functions starts with the following:

"The preceding five chapters have been occupied with the discussion of functions which belong to what may be generally described as the hypergeometric type, and many simple properties of these functions are now well known.

In the present chapter we enter upon a region of Analysis which lies beyond this, and which is, as yet, only very imperfectly understood."

This appered in the 1915 edition, and is still true.

Onsager's idea of connecting solutions of the two equations obtained from separating variables in the equation (18) has not been studied systematically. Now that much more is known about the group-theoretical nature of certain partial differential equations, this is worth studying…

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Exact proton magnetic resonance eigenspectra of the spin Hamiltonian are calculated for a system containing one group of six identical protons and one group of two identical protons. The method of calculation considers each group of identical protons as a composite “particle” with fixed total spin, and does not require determination of the explicit form of the zero-order eigenfunctions. The calculated spectra are compared with experimental high-resolution spectra of propane at both 40 Mc and 60 Mc. Thevalues of ρ, the ratio of the spin coupling constant to the chemical shift, are found to be 0.415 and 0.277, respectively, at the two frequencies . The frequency-independent spin coupling constant is 7.26 cps.

The effect of realistic boundary conditions on the computation of the specific heat of an isotropic solid at low temperatures is investigated. Two cases are considered: the surface free of stress and the surface rigidly clamped. The first of these is the one of physical interest. For both cases a term in the specific heat arises which is proportional to the surface area and to T² and appreciably higher than Montroll's result in the case of the free solid. The effect of approximations introduced during the computation is estimated. Available experimental data are not adequate for a critical test of the theory.

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The partition function for a homogeneous plate of anisotropic elastic material is computed by solving an eigenvalue problem with realistic boundary conditions. The boundary contribution is separated by the approximation method which Dupuis, Mazo, and Onsager applied to the special case of an isotropic solid. The surface specific heat then is obtained as

The following sections are included:

• ONSAGER DISCUSSION REMARKS IN PRINT

• REPRINTED DISCUSSION REMARKS