## Abstract

After World War II, quite a few mathematicians were attracted to the modeling of phase transitions as this area of physics was seeing considerable mathematical difficulties. This paper studies their contributions to the physics of phase transitions, and in particular those of the by far most productive and successful of them, the Polish-American mathematician Mark Kac (1914–1984). The focus is on the resources, values, and traditions that the mathematicians brought with them and how these differed from those of contemporary physicists as well as the mathematicians’ relations with the physicists in terms of collaboration and reception of results.

## 1. Introduction

After World War II, quite a few mathematicians, including Mark Kac, John von Neumann, and Nobert Wiener, worked on the physical problem of phase transitions, i.e. changes in the state of matter caused by gradual changes of physical parameters such as the condensation of a gas to a liquid and the loss of magnetization of a ferromagnet above a certain temperature (called the Curie temperature). The significance of these mathematicians was not so much that they brought mathematical rigor to the theoretical description of the phenomena,1 but that they applied their mathematical skills and tools to solve concrete mathematical problems encountered in the microscopic modeling of phase transitions. The present paper deals with these mathematicians’ contributions to the physics of phase transitions, and in particular those of the by far most productive and successful of them, namely the Polish-American mathematician Mark Kac (1914–1984). As we shall see, Kac not only worked on mathematical problems identified by other scientists, but also participated actively in the formulation of models. In short, he was doing physics.

In his take on the practice turn in science studies, the historian of physics David Kaiser (2005a, 2005b) argues that the historian who wishes to make sense of the changes and developments in modern theoretical physics needs to take as a starting point that—at least since the middle of the twentieth century—the task of most theoretical physicists has been to calculate or get “the numbers out” (see also Schweber 1994). Kaiser continues: “They [the theorists] have tinkered with models and estimated effects, always trying to reduce the inchoate confusion of ‘out there’ […] into tractable representations. They have accomplished such translations by fashioning theoretical models and performing calculations” (Kaiser 2005b, pp. 42–3). Kaiser takes this calculational task of the theorist as the basis for his historiography of the dispersion of Feynman diagrams.

In the field of the modeling of phase transitions, the theorists likewise spent much of their time on calculational tasks; so if we want to understand the theoretical practice of these theorists, Kaiser’s perspective on calculational skills is equally relevant here. Indeed, as we shall see below the calculational tasks were formidable in this field and the mathematical issue put a major constraint on what could be done in terms of the modeling. In this field, mathematics is used to derive the thermal behavior of the models, such as the specific heat as a function of temperature, but the physicists encountered great obstacles in their mathematical analysis which was far from straightforward to solve because the standard mathematical tools in the physicists’ toolkit was often found wanting. In view of this, the physicists could act in two ways: Either they could try to identify some mathematical knowledge that might solve the mathematical problems and then apply this mathematics to an existing model, or they could set up a new model for which they had an idea of what mathematics might be relevant to solve the model; the mathematical obstacles were so great that their removal had to be taken into account when inventing new models. In both cases, the insight into what mathematics might be relevant for solving models was a crucial asset of a scientist working on phase transitions in the period under study.

This insight into relevant mathematics, which is based on aspects like mathematical knowledge and prior experiences with modelling, can be illustrated by George A. Baker’s application of the so-called Padé approximant in 1961 in the area of modeling of phase transitions. After World War II, physicists started to obtain approximate solutions to models by using so-called series expansions. This method worked for high-temperature expansions, but there were difficulties for low temperature expansions. “The breakthrough” (Domb 1996, p. 165) came with George A. Baker’s introduction of Padé approximant to the low temperature expansions, which was “a piece of mathematics which had laid dormant since the end of the nineteenth century” (Domb 1996, p. 22). In order to provide the breakthrough, Baker first had to be aware of the existence of this little known mathematical theory, and second, had to realize that it might solve some of the mathematical challenges in the field. Baker recalled how he became interested in statistical mechanical problems as a graduate student at Berkeley in the mid-1950s. After graduation Baker went to Los Alamos Scientific where there was a lot of interest in the computation of Coulomb wave functions, which is a problem unrelated to phase transitions, using continued fractions. The standard book on continued fractions contained a chapter on the Padé approximant (Baker 2010). So, Baker was led to the Padé approximant through another problem (the computation of wave functions) after which he realized that this piece of mathematics could help overcome some of mathematical obstacles of phase transitions.

The Baker example indicates that an individual’s particular arsenal of mathematical knowledge, problem solving skills, insights etc., enables one individual to solve problems that others cannot solve. While they saw no reason to state it explicitly, the mathematicians who got involved in the phase transitions business must have known this and fundamentally believed that they could contribute to the area due to their different and larger arsenal. In addition to their different mathematical background, it is safe to assume that because the mathematicians had not been trained directly within the worldview/paradigm of the physicists, they made other choices in their modeling than contemporary physicists.

The purpose of present paper is to use the case of the mathematicians to shed light on the role of mathematical knowledge and insight in the theoretical practice of modeling of phase transitions. Hence, I wish to explore the following questions: in what ways did the mathematicians help in this modeling enterprise? What resources, values, and traditions did they bring with them and how did they differ from those of contemporary physicists? What were the mathematicians’ relations with the physicists, both in terms of collaboration and how the physicists received the mathematicians’ results? These questions will be answered for the contributing mathematicians in general, but the main focus will be on Mark Kac.

Section two provides the background to the paper with a description of the physicists’ microscopic modeling of phase transitions as well as the two most important models of phase transitions used in the era. The next section looks at how the mathematicians attempted to resolve the mathematical difficulties encountered by the physicists. The rest of the paper focuses on the role of Mark Kac. In section four Kac’s biography as well as his approach to applied mathematics is given, then his contributions to the solution of the mathematical problems of others is described. Then comes the core of paper, namely Mark Kac’s road to the three models he proposed as well as how they were perceived by contemporary physicists. In addition to modeling, Kac also formulated a general theory of phase transitions; the development of this theory is described as well as its reception by contemporary physicists. Finally, some concluding remarks about Mark Kac’s doing physics and the influence of his mathematical background are given.

We draw on the accounts of the historical development of the physicists’ understanding given in Brush (1983), Hoddeson et al. (1992), and Domb (1996). Hoddeson et al. (1992) is an accessible and non-technical treatment, while Brush (1983) and Domb (1996) have a more technical scope. However, the work of the mathematicians does not feature prominently in these three accounts and is integrated into the overall narratives, in contrast to the perspective given here on how the practice of the mathematicians qua mathematicians deviate from contemporary physicists.

## 2. Microscopic Modeling of Phase Transitions

This section describes the basics of the theorists’ practice of microscopic modeling of phase transitions in the period. The philosopher Sang Wook Yi has given a useful definition of the notion of model in condensed matter physics, which captures the essence of the microscopic models of phase transitions, the area of the present paper:

What I mean by a model in this paper is a mathematical structure of three elements: basic entities (such as “spins”), the postulated arrangement of the basic entities (say, “spins are located on the lattice point”) and interactions among the basic entities (“spin-magnetic field interactions”). As a rough criterion, we may take a model to be given when we have the Hamiltonian of the model and its implicit descriptions that can motivate various physical interpretations (interpretative models) of the model. (Yi 2002, p. 82)

When such a microscopic model has been set up, the theorist applies the theoretical machinery of statistical mechanics to set up equations that can then be solved, and this in principle predicts the relations between the observable macroscopic properties of the system given only a knowledge of the microscopic constituents and forces between the components. In physicist parlance, they “solve a model,” i.e., they derive the thermal properties of the model such as the heat capacity or the average magnetization as a function of temperature.

Two microscopic models received particular attention from physicists interested in phase transitions after World War II: The Lenz-Ising model and the Mayer model of gas condensation.2 The Lenz-Ising model was proposed in 1924, i.e. prior to the advent of the modern theory of quantum mechanics in the latter half of the 1920s and was seen basically as a mathematical structure that can be given different physical interpretations; if we understand the variables of the model as representing electron spins, we have a model of a ferromagnet. If we interpret the variables of the model as denoting the presence or absence of a gas molecule at the site, the model represents a gas. In ferromagnetic parlance, the basic structure of the model is a lattice to which a spin is assigned to each site of the lattice; the spin can only point either up or down and two neighboring spins interact with an interaction energy—J if they are parallel and J if they are antiparallel. If the lattice of the model is three-dimensional, the model is said to be three-dimensional; this is the most important case, because most real materials are three-dimensional, but films are better described by a two-dimensional version of the model. The one-dimensional model was not seen as corresponding to something in the real world, but is useful because it can easily be solved; Ernst Ising had done so in 1924 (See Niss 2005).

The model was seen as a crude model of ferromagnets, gases, etc. However, a major attraction of the model was that it is so simple that there was hope of solving it, also in higher dimensions than one. This was exactly the motivation of the American-Norwegian scientist Lars Onsager (1903–1976) for his study of the model:

When the existing dearth of suitable mathematical methods is considered, it becomes a matter of interest to investigate models, however far removed from natural crystals, which yield to exact computation and exhibit transition points. (Onsager 1944, p. 118)

In the 1930s it dawned upon physicists, that it is immensely difficult to deduce the thermal properties for the model with the use of the machinery of statistical mechanics, but in 1944 Onsager succeeded in determining the transition temperature of the two-dimensional version of the model in what was considered to be a mathematical tour-de-force of complicated mathematics. Moreover, his solution showed that an approximate scheme, the so-called mean field method that was used to study microscopic models, failed to give correct answers. This meant that there was no easy way of circumventing the mathematical difficulties in the study of these models.

The other model was proposed by the American chemist Joseph E. Mayer (1904–1983) in 1937, when he published a paper on imperfect gases and their condensation (Mayer 1937). This paper led to an avenue of research followed by several physicists, including George E. Uhlenbeck (1900–1988), directed towards a theoretical understanding of the statistical mechanics of gases condensing to the liquid state. The generally accepted goal was to explain the general features of behavior of gases and liquids, in particular the existence of phase transitions (condensation and evaporation). Moreover, since phase transitions is a feature of most substances, the explanation should be quite general and be based on a model that does not assume a very specific interaction between the molecules (see, e.g., Kahn and Uhlenbeck 1938).

Mayer and his co-workers used a model of N identical molecules and the aim was to do “without arbitrary physical or mathematical assumptions” (Mayer and Harrison 1938, p. 87). However, they wrote, the system had to be limited to have specific properties if such a treatment were to be feasible. Consequently, they introduced three fundamental assumptions that were seen as reasonable. Mayer believed that his and coworkers’ work had shown the existence of the condensation of the gas for this model, i.e., a phase transition, even though it was not possible to evaluate all integrals. However, many others were critical of this conclusion. Kahn and Uhlenbeck wrote that Mayer’s reasoning “was far from convincing, especially from the mathematical standpoint” (Kahn and Uhlenbeck 1938, p. 399) and that his explanation of the condensation phenomenon was incomplete. They showed that for a well-studied system, the ideal Fermi-Dirac gas, Mayer’s reasoning led to an incorrect conclusion regarding the existence of the condensation phenomenon. Moreover, they identified the flaw in the reasoning to be an assumption that some coefficients in the definition of the theory are independent of the volume, which they argued was not the case.

Like for the Lenz-Ising model, the studying of the Mayer model was beset with mathematical difficulties. In the words of the chemist Oscar K. Rice,

Mayer’s and Harrison’s theory is highly mathematical, and the results depend upon assumptions regarding the behavior of certain integrals. These are exceedingly difficult to evaluate because, near the critical point the integrands have large numbers of positive and negative parts, which nearly cancel each other. (Rice 1947, pp. 315–16)

The attempt to resolve the mathematical difficulties of Mayer’s theory was a research program for quite a few physicists in the first two decades following Mayer’s paper.

## 3. The Contributions of the Mathematicians

By the end of World War II, the physicists’ handling of the Lenz-Ising model and the Mayer model, had taught that the mathematical difficulties involved in examining microscopic models were formidable. Consequently, a major concern was the solubility of the models. Moreover, the focus of most research in the area of phase transitions was not the formulation of new models, but the examination of existing models. Much effort went into numerical studies of the properties of three-dimensional Lenz-Ising model using series expansions and into obtaining rigorous mathematical results for this model and the Mayer model (see Hoddeson et al. 1992; Domb 1996).

At first, physicists tried to get mathematicians interested in these matters and this led to attempts by several mathematicians to help resolve mathematical difficulties. In 1941, the Dutch mathematician Bartel van der Waerden published a paper on the Lenz-Ising model that he dedicated to his Leipzig colleague Werner Heisenberg at the latter’s fortieth birthday. The article was a result of a conversation with the Göttingen physicist Richard Becker (Schneider 2011). Van der Waerden provided a minor result when he determined the average energy of the two-dimensional Lenz-Ising model as a function of temperature for all temperatures below the critical temperature.

Van der Waerden did not, however, find the quantity of real interest to physicists, namely the energy at the critical point. This was left to Lars Onsager, whose extremely advanced mathematical study of the two-dimensional Lenz-Ising model in 1944 was widely held as a major mathematical achievement. Shortly after Onsager’s solution, two American mathematicians, Fred Supnick and F. J. Murray independent of each other raced with physicists to get the solution of the three-dimensional Lenz-Ising model (see Supnick 1946, 1947; Murray 1952). Other mathematicians hoped that new ways of solving the two-dimensional model might shed light on the three-dimensional version; this is the case for a paper by Kac and J. C. Ward (1952), who gave a combinatorial solution of the two-dimensional Lenz-Ising model. All these attempts were in vain as the three-dimensional case has not yet been solved.

A few years later, another prominent mathematician John von Neumann was also involved in the problem. In 1947 he finished a manuscript on how to use computers and random numbers to evaluate the problem, complete with estimation of how long it may take to calculate using the method. In a note directed to Mayer, von Neumann wrote that “As you see, it is an attempt to answer part of the question you formulated when we last met at Aberdeen.”8 Unfortunately, no replies from Mayer exist and we don’t know why von Neumann choose not to publish it.

In the above examples, the mathematicians enter the process when the theories of models are already formulated. At this stage, the mathematicians tried to assist in the purely mathematical problem-solving aspect of the process. The interesting thing about these endeavors is that the mathematicians with their sophisticated mathematical skills were not more successful than the physicists when it came to resolving the mathematical difficulties encountered for the two models (the Lenz-Ising model and the Mayer model). So, in contrast to what the mathematicians themselves had expected, they could not contribute to this situation. In fact, as a testament to their level of mathematical problem-solving skills, scientists who considered themselves to be physicists and chemists rather than mathematicians were in fact the main providers of the most pertinent results.

## 4. Mark Kac’s Mathematical Background and His Approach to Applied Mathematics

We have already seen that Mark Kac9 was involved in the mathematics of phase transitions in various ways. We now take a closer look at his contributions, but first some remarks about his mathematical background and his approach to applied mathematics. Kac was born in Poland, where he studied pure mathematics with a focus on probability theory at the University of Lwów in 1937 under the auspice of Hugo Steinhaus. After receiving his Ph.D. on the concept of independent functions in probability theory in 1937, he spent a year as post-doctoral fellow at Johns Hopkins before continuing to Cornell. He stayed at Cornell until he joined the Rockefeller Institute (later the Rockefeller University) in 1961. Kac collaborated with, among others, the number theorist Paul Erdös. In one paper, they used probability theory to describe the behavior of number-theoretic functions in a powerful and highly original way. Later they invoked an invariance principle to give limit theorems in probability theory. Moreover, building on the work of the Soviet mathematicians Andrei N. Kolmogorov and Aleksandr Khinshin and the American mathematician William Feller, Kac was the first to establish a connection between certain stochastic processes and differential and integral equations. The editors of his selected papers have written that “it is characteristic of his style that he will explore these connections in a simple case, such as Brownian motion, if the special case, as usually happens, suggests the whole picture.”10 Kac was a frog, rather than a bird, in Freeman Dyson’s sense (Dyson 2009)—both when he worked on mathematics and physics. He was interested in the details of particular objects as well as solving problems rather than the surveying of the broad vistas of mathematics. In Kac’s own words: “To begin with, I was always interested in problems rather than in theories” (Feigenbaum 1985, p. 476).

His interest in physics, he has said, was aroused in his secondary school days. He continues:

It was kept alive while I was a student at the University, but there was little activity in theoretical physics, which I found extremely difficult to learn without guidance. I attended a seminar (conducted by [Leopold] Infeld) on quantum mechanics, but the experience was extremely frustrating because even though it was clear to me that I was in the presence of great ideas, these ideas were maddening elusive. (Kac 1979, p. xxii)

Around the time of World War II, Kac’s research interests began to include problems related to physics. The decisive event that led to a sharp change in the direction of his work, he has once stated, was his association with the group of George Uhlenbeck at MIT’s Radiation Laboratory (Kac 1979). This group focused on the propagation of electromagnetic waves and questions related to the detectability of weak signals. Since this is intimately connected to the theory of random noise and thus stochastic processes, Kac’s mathematical skills could be put to a good use in this field. This association with the physicist Uhlenbeck affected his scientific work profoundly:

For one thing, it brought me face to face with physics as it is done by physicists and not merely imagined by mathematicians. It hardened my outlook on mathematics, and it reinforced my natural tendency toward the concrete and away from the formal and abstract. (Kac 1979, pp. xxiv–xxv)

After World War II, Kac worked on both physical problems and purely mathematical ones.

Despite his work on physical problems, throughout his life Kac consistently saw himself as a mathematician and felt that he never “came close to becoming a physicist. I was too old for that, and the way my mind works, I am not really suited for tackling problems posed directly by nature” (Kac 1979, p. xiv). In 1972 when he gave a talk on phase transitions at the Symposium on the Development of the Physicist’s Conception of Nature in the Twentieth Century at Trieste, he elaborated on his outsider role: “I feel a bit strange in talking about the development of the physicist’s conception of nature because I am not a physicist by training, and I could therefore watch this development only from the sidelines. Although this position often affords a better view, it also distorts it and makes it easy to miss some of the essential plays” (Kac 1973, p. 514). At an institutional level, the physicists acknowledged Kac’s work: he was able to publish in the top-notch journals, including Physical Review, and he was invited to give talks at summer schools and at the prestigious International Union of Pure and Applied Physics on Statistical Physics.11

In a discussion remark, Kac himself was arguing that a mathematics background gives a particular view on the world:

Mathematics – and by the same token, physics for mathematicians – is a way of looking at a problem. Regardless of whether you have learned differential equations, or probability theory, or what have you, it is not the specific content of any specific mathematical discipline; it is something which, at the moment, I can’t define, which we call mathematics, which represents a point of view. This point of view cannot be divorced from abstraction, because abstraction is the essence of this very point of view. The same applies to Physics. I’m not interested in physics because I want to predict exactly the critical temperature of argon. Render unto Caesar what is Caesar’s. (Kac 1963, p. 98)

In some reflections on what applied mathematics ought not to be, he introduced an interesting image of dehydrated elephants:

I cannot resist referring once more to a wartime cartoon depicting two chemists surveying ruefully a little pile of sand amidst complex and impressive-looking apparatus. The caption read: “Nobody really wanted a dehydrated elephant but it is nice to see what can be done.” I am sure that we can all agree that applying mathematics should not result in creation of “dehydrated elephants.” (Kac 1972, p. 17)

Kac asked how mathematicians can contribute to physics. Mathematics, “it seems self-evident,” is not of much use in finding laws of nature because this demands too much empirical knowledge. Therefore, mathematicians enter when the law of nature is already there. At this stage, they can do two things:

(a) to help find ways in which the law can “best” be formulated and (b) to help in drawing conclusions, which hopefully will be significant to the further development of the subject. Of the two, the first role is fraught with more danger, since it tends to provide fertile grounds for breeding “dehydrated elephants.” This is especially true today because contemporary mathematics is geared to playing with “formulations” and physics, so full of vague, unprecise and unrigorous statements seems to cry out for being better “formulated” and fitted into familiar structures. (Kac 1972, p. 18)

While Kac advocated that mathematicians should be involved in applying mathematics outside of mathematics, he was critical of the role assumed by some contemporary mathematicians involved in the application of mathematics and he, for instance, detested attempts to make physical theory more formal. What the applied mathematician can do for physics, he believed, is to help with the application of the laws of physics to physical phenomena, i.e., he or she can participate in modeling physical phenomena.

## 5. Kac’s Formulation of Models

Kac contributed extensively to the theory of phase transitions. His first research excursion into statistical mechanics seems to be the discussion with Norbert Wiener about the validity of Mayer’s approach around 1940 mentioned above. His most important contribution to the area is his formulation of three models. This happened in a situation where most physicists studied a few existing models rather than proposed new ones.12

### 5.1. The Gaussian and Spherical Model

Kac has given an account of the development of how he arrived at the first two models, the Gaussian model and the spherical model, in a tribute to his collaborator Theodore Berlin (1917–1962). Kac was introduced to the Lenz-Ising model in 1947 by Uhlenbeck and the implications of Onsager’s solution of the two-dimensional case. This got Kac interested in the Lenz-Ising problem and he looked more into it. One of the first things he did was trying to solve the one-dimensional Lenz-Ising model with interactions between next-nearest neighbors as well as nearest neighbors13, but he did not publish anything. He later recalled about the three-dimensional Lenz-Ising model that “It soon became obvious that it was not a problem one solves on the spur of the moment, and, in the best mathematical tradition, not being able to solve the original problem, I looked around for similar problems which I could solve” (Kac 1964, p. 41). Note the reference to the mathematical tradition. This search led Kac to introduce the Gaussian model. Like the Lenz-Ising model, this model has spins situated at the sites of a lattice site, but the behavior of the spins differs in the two models. In the Lenz-Ising model each spin can take on only two values, but in the Gaussian model the spin values are continuous ones whose lengths are distributed according to the Gaussian distribution, hence the name of the model. Kac thought the model was “a shattering discovery” (Kac 1964, p. 41), but he and Uhlenbeck soon discovered that the model has a serious drawback at low temperatures, where the model leads to nonsensical results,14 so when they considered the model from a physical perspective they had to dismiss it as irrelevant. In the published words of Berlin and Kac (1952), who dealt with the model, it “become invalid for temperatures below a certain critical temperature” (Berlin and Kac 1952, p. 822).

In the following weeks, Kac thought about this problem and realized that another model might do the job. Again, we have spins on a lattice, but in the spherical model they can have any value subject to the condition that the sum of their squares remained equal to N, the number of lattice sites: σ12 + σ22 + … σN2 = N. This model is preferable to the Gaussian model as it does not suffer from the above drawback and it “moves us closer to the original problem” (Berlin and Kac 1952, p. 827). Kac himself was unable to solve this model, as he could only reduce it to an integral that he could not solve. He consulted his Cornell colleague Richard Feynman, who produced a solution for the one-dimensional case, but the method could not be generalized to higher dimensions. Kac gave a lecture in Ann Arbor hosted by Uhlenbeck on the model; Ted Berlin attended the lecture and quickly produced a solution in one, two, and three dimensions using the method of steepest descent. Kac relates:

I had hardly heard of the method of steepest descent, and what I had heard was mainly a warning against its use because of the difficulties of rigorous justification. Here, before my eyes on about ten handwritten pages, was one of the most advanced applications of the method. Only a combination of a physicist with the fantastic power and skill of a classical analyst could have produced the results. To say that I was immensely impressed would be a pale understatement of my reaction to that letter. (Kac 1964, p. 41)

So, the mathematician was impressed by the physicist’s solution to the mathematical problem arising for a model proposed by the mathematician.

Berlin and Kac first published their results at the 1948 Annual Meeting of the American Physical Society in New York, January 26–29, 1949 in a short note (Kac and Berlin 1949). A longer note in 1952 (Berlin and Kac 1952). We will examine the latter. They began this paper with a lengthy discussion of the mathematical difficulties of phase transitions and noted that the two-dimensional Lenz-Ising model and the Bose-Einstein gas are the only non-trivial examples of phase transitions which can be examined exactly. In such a situation, a particular kind of models is allowed:

We agree with Onsager that it is desirable to investigate models which yield to exact analysis and show transition phenomena. It is irrelevant that the models may be far removed from physical reality if they can illuminate some of the complexities of the transition phenomena. (Berlin and Kac 1952, p. 821)

Therefore, they concluded, we are justified in looking at the two “mathematical models”—the Gaussian and the spherical. They dismissed the former model because of the spurious effects described above, but the latter model, which was point of focus in the paper, is important: “The virtue of the spherical model of a ferromagnet is that its properties can be rather extensively discussed and that a three-dimensional lattice has ferromagnetic properties” (Berlin and Kac 1952, p. 827). However, Berlin and Kac did not harbor any illusions about the relationship between the model and the real world: “With respect to the physical ferromagnet, the model has nothing to say positively,” because the spins are not given the correct quantum mechanical representation (Berlin and Kac, 1952, p. 827). Moreover, the restriction on the squares of the spins means that the situation where the lengths of all but one spin are zero has non-zero probability, that is, unphysically. Finally, this constraint on the spins means that spins a large distance apart can in effect feel each other, i.e., there is a long-range interaction between the spins, yet another unphysical effect. Yet, they found that the spherical model might be more realistic than the Lenz-Ising model because the spins of the latter has an anisotropy (the spins point up or down), hence “It is more likely that the spherical model transition is closer to the actual transition behavior” (Berlin and Kac 1952, p. 828). Later on, in the paper on dehydrated elephants, Kac wrote about some models, including the spherical model that they “have all the earmarks of being ‘dehydrated elephants,’ and it may turn out that indeed they are” (Kac 1972, p. 26). He continued:

But what gives these models a hold on life—tenuous though this hold may be—is that in spite of admitted lack of realism, they are firmly rooted in reality, and they were conceived to deal with real questions. Without such rooting and without real questions to guide us, we may well find ourselves fighting windmills and triumphantly emerging from pyrrhic victories. (Kac 1972, p. 26, emphasis in the original)

Turning now to their results, they concluded that the one- and two-dimensional spherical models do not exhibit transitions, whereas the three-dimensional model does. This means that the three-dimensional model exhibits ferromagnetic properties, but the two-dimensional variant does not. Since this conclusion agrees with a result for low temperature from the so-called spin wave theory where the spins are given a more theoretically accurate representation, they believed the spherical model had some bearing on the nature of the transition. Once again, Kac was thus interpreting his results physically.

What was the reaction to the model? Both Berlin and Uhlenbeck liked the model and Berlin “had a strong belief in the model” while Uhlenbeck “attached a great deal of importance” to their investigations (Kac 1964, p. 42). Regarding a wider audience, Kac wrote in a letter to Berlin, 15 November 1948: “As I wrote you before, my talk on the spherical model created quite a stir. Feynman thinks we are extraordinarily close to the solution of the whole affair. He says that if he had time he would look into the matter himself.”15 By the solution to the whole affair, Feynman seems to be referring to a solution of the Lenz-Ising model in three dimensions and all these positive remarks concern this prospect. Berlin and Kac at first considered the potential relevance of the spherical model to be as an approximation to the Lenz-Ising model and many physicists shared this view (see, e.g., Lax 1955 and Domb 1960) along with Berlin and Kac’s dismissive attitude (at this time) towards the model as a realistic model of ferromagnets. Kac recalled in his Berlin tribute that physicists were quick to challenge the relevance of the model: “As soon as the results concerning the spherical model became known by word of mouth, questions were raised as to whether the singularity had any physical significance” (Kac 1964, p. 42). Cyril Domb seems to express a widespread sentiment in a letter to George E. Uhlenbeck, 10 February 1970: “Again for many years I considered the spherical model an academic mathematical exercise with little relevance to real transitions in physical systems.”16 Domb, a major protagonist in the development of the modern theory of phase transitions, definitely did not see the model as rooted in reality; in fact, he saw it as a dehydrated elephant. In view of the uses of the model in the literature in the 1950s and 1960s, Domb’s view seems to be far more widespread than Kac’s.

During the late 1960s, however, the understanding of the role of models changed among physicists working on phase transitions. One aspect of this change is reflected in Eugene Stanley’s discussion of the spherical model in his widely-read textbook. After embracing the well-known motivation for studying the spherical model, namely that it can be solved exactly, Stanley writes that, “This statement is not intended to deprecate model systems which as yet have no physical system as counterpart—there is generally much to learned from a theoretical model when considered in its own right” (Stanley 1971, p. 17) From the late 1960s onward, many physicists consider a key question of phase transitions to be on what features of the interaction do so-called critical exponents (i.e., exponents that describe physical quantities near phase transitions) depend? Candidates for such features were the dimension of the system, the range of the interaction, and the dimension of the spin. The spherical model provided one example for this classification. Moreover, in 1971, Kenneth Wilson used the Gaussian model as the first step in his renormalization group approach (Wilson 1971).

The physicists interested in the ferromagnets at the time Kac invented the spherical model took another route when modeling such magnets: They focused on the more realistic, but also more complicated Heisenberg model, which they tried to solve (see, e.g., van Vleck 1945). This model represents the spin correctly from a quantum mechanical point of view, a feature the physicists valued highly and they saw the Lenz-Ising model and even more the spherical model as unsatisfactory in this respect. While they accepted Kac’s argument that the spherical model could potentially show the way to a solution of the three-dimensional Lenz-Ising model, these physicists would probably not have invented a model that could be considered even further removed from the correct spins. Kac accepted models which were not acceptable to physicists at that time. Moreover, according to himself, Kac was led to propose the model due to his mathematical inclination, so it seems to be because he was a mathematician that made him freer than the physicists and allowed him to see some mathematical connections due to the mathematical tools at his disposal.

### 5.2. The Kac Model

The beginnings of the most successful of Kac’s models were modest. Again, Kac motivated the model, which came to be known as the Kac model, by appealing to the paucity of exact calculations. The model was a one-dimensional version of the well-known hard sphere model; in the hard sphere models the gas molecules behaves like small billiard balls moving around and bouncing into each other.17 In one dimension, the model consists of N hard rods of constant length moving on a line of finite length. Kac’s contribution was the way he modeled the interaction between the molecules, namely with an exponential decaying attraction of the distance x between the rods:
$Vx=∞forx<δ−αexp−γxforx≥δ$
Here δ denotes the radius of the spheres.

Kac introduced the model in 1957. His background in probability theory and statistics helped him to see that this model was soluble because he could see a connection between the model and a stochastic process:

The crucial feature of the potential responsible for the success of the calculation is that exp (−γ |t|) is the correlation function of a stationary, Gaussian, Markoffian process (the Ornstein-Uhlenbeck process). (Kac 1959, p. 8)

This identification allowed Kac to show that the mathematical problem of determining the thermodynamic properties could be rephrased as a purely mathematical problem, namely the determination of the maximum eigenvalue of a linear integral equation with a so-called positive definite, Hilbert-Schmidt kernel. To this problem, Kac could apply his knowledge of the branch of mathematics called functional analysis, including the Courant-Weyl lemma and properties of the Fredholm determinant. From this analysis Kac was capable of deriving some properties of the model, in particular that the gas does not condense; however, he did not obtain the explicit equation of state.

So, in this historical situation, plagued by mathematical problems, Kac’s mathematical background in the mathematical theories of stochastic processes and functional analysis could be put to good use. It was this background that allowed him to see a connection between this model and these two theories which in combination were crucial for the solution to the problem. Few contemporary physicists were well-enough versed in the theory of stochastic processes to see the connection that Kac saw and probably none could have combined the two mathematical theories. As the “Uhlenbeck” in the Ornstein-Uhlenbeck process, Kac’s collaborator George Uhlenbeck of course knew this stochastic process better than most, but since he lacked Kac’s background in the mathematical theory of functional analysis, he was unlikely to have been able to establish the connection. In Enigma of Chance, his autobiography, Kac wrote about the Kac model: “I later learned that Feynman had introduced the same model but his treatment was different from mine” (Kac 1985, p. 142). Indeed, Feynman (1972) discussed a similar model in his lectures on statistical mechanics given in 1961 at the Hughes Research Laboratories. Feynman also motivated the model with the low number of soluble models. Feynman’s mathematical analysis differed from Kac’s. Feynman set up a recursion equation relating the partition function of a gas of N molecules to one of N + 1 molecules. Using a number of variable transformations as well as Fourier and Laplace transformations, Feynman obtained an integral equation that “in principle” would yield the requested information, but he did not derive the properties of the model that Kac could obtain. Hence, in this case Kac’s mathematical resources enabled him to get further than contemporary physicists.

Kac’s solution meant that the Kac model was relevant to physicists as an exactly soluble model; in a letter of 19 January 1960, Uhlenbeck wrote Kac that the latter’s model “really interests me now very much.”18 Moreover, some physicists, including G. A. Baker, started to be interested in what would happen if the short-range interaction of such models as the Lenz-Ising model was replaced by a long-range interaction, such as in the Kac model. However, the Kac model became really interesting to physicists when Kac explored the model further in collaboration with Uhlenbeck and Per Christian Hemmer, a young visiting physicist from Trondheim.19 In a series of three papers, this trio modified Kac’s model slightly (but they still called it Kac’s model),20 and showed that this model not only exhibits a gas-liquid phase transition, but also that its equation of state is exactly given by the van der Waals equation.21 In 1873, the Dutch van der Waals had proposed his eponymous equation on somewhat unclear grounds. The equation predicts a phase transition from the liquid to the vapor state, and was seen as qualitatively correct, but was known to give quantitatively incorrect results. Still, it was considered to be important to determine whether the equation could be derived from some microscopic model (Kac 1985). Kac, Uhlenbeck, and Hemmer provided an affirmative answer to this question as well as determined many important properties of the model. The scientist turned historian, John S. Rowlinson, has given the following account of the development:

Thus, ninety years after its introduction, a model was found which led rigorously to the van der Waals equation, and so rehabilitated it in the eyes of theoretical physicists; it was now an exact equation for a well-defined, if unrealistic, model, rather than merely a crude representation of the behavior of realistic model. (Rowlinson 1988, pp. 58–9)

According to E.G.D. Cohen, a colleague of Kac and Uhlenbeck at Rockefeller University, their work was quite important for Uhlenbeck. In the 1950s, Uhlenbeck had realized that his previous approach to the condensation phenomena, using the so-called virial expansion of Kamerlingh Onnes, was wrong. The work of the trio gave him hope that the solution lay in combining van der Waals theory with Onsager’s solution. Kac has told Cohen that “when the Van der Waals equation came out [of the mathematical analysis of the Kac model], something stirred in Uhlenbeck and he really started to take the approach seriously and to immense himself completely in the problem” (Cohen 1990, p. 618). However, in the 1960s, with the advent of the modern theory of phase transition, the van der Waals theory was discarded (Domb 1996).

Kac’s proposal of this model was important because it initiated research which helped solve an important physical problem: to provide a microscopic foundation of the van der Waals theory. Contemporary physicists were happy and stressed the fact that the model “is particular [sic] amenable to analysis” (Helfand 1964, pp. III-41). However, they were not sure whether the model is capable of capturing the essential features of the system in question: “The paramount question whether these findings are general or artifacts of the model is still largely unsettled” (Helfand 1964, pp. III-41).

This model was thus much more successful than the spherical model which was not seen as relevant by contemporary physicists, at least not as a model in its own right. For physicists, the functions of the two models are fundamentally different and this affected how they perceived the model. The physicists probably would have favorably received any statistical mechanical model that rigorously leads to the van der Waals equation. It didn’t matter that such a model was an unrealistic representation of the interaction between the gas particles because the equation was anyway known to be an inaccurate description of real systems. It is much more important that no uncontrolled approximations are introduced when the properties of the model are deduced. Thus, what was sought was mathematical skills in showing that a model leads to the correct behavior rather than physical insight into workings of real systems.

For the spherical model and Gaussian model, on the other hand, the situation was different: these models were supposed to say something about real systems according to Kac and for a model to do so, the physicists have some requirements if they are to accept a model. Kac reached the same conclusion with regard to the Gaussian as the physicists and rejected this model. He held, however, a more favorable view of the spherical model and saw this model as much more than just an approximation to the Lenz-Ising model and of only academic interest. In fact, he perceived, in contrast to most physicists, this model as a better representation of the ferromagnets than the Lenz-Ising. The physicists could not accept the distortion of the spins in the model.

Kac was led to invent the Kac model because he could see some connections with probability theory and the theory of stochastic processes that no contemporary physicists were able to do. In a situation where most models were insoluble, the ability to guess that a model is soluble is a crucial skill. Moreover, in such a situation the details of the model are less interesting—the important thing is that the model can be solved. The fact that Kac was not steeped in the physical tradition may also explain that he operated more freely with respect to the fundamental physical theories, i.e., he had a different modeling practice from that of contemporary physicists. Thus, Kac’s background in mathematics gave him some skills and an outsider role which enabled him to play a crucial role in the solution of an important physical question. So, what enabled Kac to contribute to this area was neither his special mathematical skills and insight nor his particular modeling practice that made him make special modeling decisions, but the unique combination of the two that allowed him to see other solutions than his contemporaries.

## 6. The Mathematical Mechanism of Phase Transitions

So far, Mark Kac had pursued answers to research questions given by physicists. However, he now also gradually formulated a position concerning what it means to understand phase transitions and how mathematics could help in this respect. Kac tried to put the ideas derived from the study of such simplified models into a more unified description of phase transitions and found the mathematical mechanism responsible for phase transitions. The approach was based on an observation that, for the microscopic models, the phase transition was reflected in a double degeneracy of an eigenvalue of an integral operator. He was “thrilled” when he discovered this mathematical fact for the Kac model because “That nature should know so much mathematics bordered on the miraculous!” (Kac 1979, p. xxiii). In the published version of lectures given at the 1966 Brandeis Summer Institute in theoretical physics, Kac mentioned a host of models, including the Lenz-Ising model and the Kac model and concluded, “We have in fact seen that in all cases, the mathematical mechanics responsible for the phase transition is the asymptotic degeneracy of the maximum eigenvalue of a certain linear operator (though not in all cases were we able to prove this degeneracy rigorously)” (Kac 1968, p. 301, emphasis in the original). Kac used this observation to bridge the mean field theories and the Onsager solution of the Lenz-Ising model. In the lecture, Kac said that his approach of basing the general view of phase transitions on asymptotic degeneracy could be criticized because of “the lack of a clear-cut physical interpretation of the linear operators involved. The transfer matrix L and the kernel K are tricks devoid of physical significance” (Kac 1968, p. 303). Kac, however, answered this criticism by saying that mathematically speaking this matrix and kernel contain all the physical information about the system.

It later turned out that the observation of a degeneracy of the eigenvalue in the linear operator is not true for all relevant models and so it was simply not possible to build a theory of the mathematical mechanism on these grounds. Even before this became clear, this program of Kac’s did not receive much attention from most other scientists. Physicists were interested in the mathematical mechanism of phase transitions, but found, as Kac had predicted, that his approach was lacking physical interpretation. Kac’s ideas were neglected because they were not thought to yield any insight into the physics. The linear operators could not be given a physical interpretation. Consequently, it was not possible to go from such a mathematical mechanism to the physical mechanism of phase transitions. Moreover, during the 1960s, it was realized that the classical models, which Kac and Uhlenbeck tried to reconcile with the Lenz-Ising model did not capture the physics of phase transitions (see Niss 2011), so attention focused exclusively on Lenz-Ising like models and how they could explain the newly obtained scaling laws (see, e.g., Domb 1996).

## 7. Concluding Remarks

We have seen that for the case of microscopic modeling of phase transitions, the participating mathematicians were not successful when it came to solving the mathematical problems encountered by the physicists. Rather, they had to engage more directly in the formulation of models; in short, they had to do physics. This applies also to Mark Kac, who invested much time and effort into understanding phase transitions, and was by far the most successful of the mathematicians. Being trained in mathematics and starting his career with working 15 years or so on questions of pure mathematics, Kac consistently saw himself as a mathematician and was acutely aware that he was an outsider in physics.

Scientists engaged in formulating models of phase transitions at the time had to strike a balance between solvability of the model and physical realism of the constituents of the model (gas molecules, spins, etc.) and the interaction between them. In his formulation of the three models covered here, Kac was mainly led by mathematics considerations, such as the solvability of the models, whereas the physical realism of the models played quite a small role in considerations. Contemporary physicists were also preoccupied with solvability of the models, but they put much greater emphasis on the realism of the models than Kac did; consequently, Kac’s models were of little relevance to them as representations of the physical systems. Prior to the late 1960s, Kac followed the directions given by the physicists and he tried to answer questions posed by physicists. Later on, he started to formulate the direction himself by focusing on the mathematical mechanism of phase transitions, but this did not meet much interest simply because it was not possible to give a physical interpretation of the mechanism.

The question of the role of Kac’s mathematical skills compared to contemporary physicists is complex: on the one hand his background in mathematics, in particular his knowledge of probability theory and functional analysis allowed him to see some connections which physicists could not see and he consequently formulated models based on this. His study of the Kac model is a case in point, where he got deeper into the mathematical analysis of the problem than Feynman and Uhlenbeck due to his mathematics background. Moreover, Uhlenbeck relied on Kac for the resolution of mathematical questions. In a letter of 1960, Uhlenbeck wrote to Kac that “there are certainly quite definite mathematical questions which need your expert attention.” Similarly, the physicists Chen-Ning Yang and T. D. Lee also drew on his help in 1951 when they proved the so-called unit circle theorem for the Lenz-Ising model (published in 1952); Kac helped them with proving a special case when they got stuck. It is evidence of Kac’s mathematical prowess that he was able to help after John von Neumann and the Norwegian-American mathematician Atle Selberg had been unsuccessfully consulted (Yang 1983, p. 15). On the other hand, Kac needed the help of physicists to solve mathematical problems and often he conjectured something in lecture notes which was later proven by physicists to be wrong. It is interesting in this respect that it was the physicist, Theodore H. Berlin, who provided the solution using the methods of steepest descent.

Kac relied on Uhlenbeck for insight into the physics, both on the experimental and the theoretical side. In a letter of 1958, for instance, Uhlenbeck replied to what appears to be a question from Kac about the experiments pertaining to the order-disorder transition:

You are quite right that experimentally it is not certain at all that there is a discontinuity in the specific heat. You may have seen the curves in Mott and Jones (Theory of the Properties of Metals, p. 228) on the spec. heat of Nickel at the Curie point. Look also at the spec. heat curves in Nix+Shockley (Rev. of Mod. Phys. Vol. 10, p. 1, Figs. 4, 34, 35, 36) in order-disorder transitions. I spoke of the approximate theories (Weiss, Bragg. Williams, Bethe, Kirkwood) which all give a discontinuity. But they may be all wrong, and therefore I think your observation is really quite interesting. Also any further qualitative consequences on the shape are worth pursuing. Experimentally at best curves look like a Greek lambda (one speaks of Lambda points!), at the high temp. side the spec. heat got up all right, a cusp is quite possible (between A and B, often points are missing) but c [the specific heat] drops faster on the high T side than on the low T side.22

Here Uhlenbeck described the relevant experiments and theories as well as gave an assessment of whether they should be trusted.

As the collaboration with the physicists progressed, in particular with Uhlenbeck, Kac began to evaluate the experimental evidence himself. Through Per Christian Hemmer, Uhlenbeck and Kac learned about the measurements by the Norwegian physicist Hans Lorentzen. In a letter to Uhlenbeck, on April 15, 1963, Kac wrote,

In the realistic case the only limit is the thermodynamic limit and I fail to see how this would produce a critical region unless experiments are done in capillary tubes and somehow [?] to the partition function become important. This would mean that the appearance of the critical region may depend on the kind of experiment one performs. This is probably nonsense but the worry remains. And if Lorenzen is right and if our results agree qualitatively with his the puzzle will be greater than his! Well, this being Physics (experimental Physics at that) I can say that “dass ist schon ihre Sorge” and forget about the whole business. But being now a Professor of Physics I am not allowed to, so I’ll stick annoyed [?]. But it sure is strange.23

Kac had clear ideas about the role of the mathematician for the physics, but not everyone agreed. In a letter of June 13, 1972, Bob Hermann, another mathematician inclined towards physics, strongly opposed Kac’s view that mathematicians attempting to give mathematical reformulations of the theories of physics are producing dehydrated elephants:

I was disappointed by your article on the interaction between math and science in the Quarterly. You seem to be the typical blind man describing his part of the elephant. Surely [Saunders] Mac Lane is right that an important (and DO-ABLE) task for mathematicians is up-dating the description of the mathematical structures of contemporary science.24

Later in the letter, Hermann raised an important point:

Your outlook is no doubt very much conditioned by your experiences in statistical mechanics, a small and manageable part of the scene. If you look at other areas, where mathematics is used more as a speculative tool (elementary particle physics is the prime example of course, but there are others) mathematicians could play an enormously useful role if they combined Mac Lane’s program with an attempt at communication with the scientific types who are stumbling around in the dark.25

This should serve as a memento that mathematicians participating in other areas of physics might have other experiences with doing physics than Mark Kac.

## Notes

1.

Simultaneously with the development studied here, the new field of mathematical physics branched out of theoretical physics and mathematics, but the former development should not be seen as part of the latter development. While this new field shared a name with an earlier academic discipline, the new version was a profession in contrast to the previous one (Schweber 1987). Jaffe and Quinn have given the following description of the post-war version of mathematical physics: “The mathematical work that in some sense straddles the boundaries between the two [mathematics and physics] is commonly referred to as mathematical physics, though a precise definition is probably impossible” (Jaffe and Quinn 1993, p. 4). Moreover, the mathematical physicists often worked on “questions motivated by physics, but they retained the traditions and the values of mathematics” (Jaffe and Quinn 1993, p. 4). Out of the mathematical physics endeavor grew axiomatic (later constructive) quantum field theory and rigorous statistical mechanics, hence stressing the points of Roger Newton (2000) that mathematical physicists “work in a more rigorous mathematical mode” than the usual theoretical physicist and that they rely “less on their physical intuition than on strictly proving mathematical conclusions that others either take for granted or are unaware of” (Newton 2000, p. 59).

2.

For the history of the Lenz-Ising model see Niss 2005, 2009, 2011. For the history of the Mayer model, see Brush 1983.

3.

Wiener to Mayer, Feb. 6, 1940, Box 21, Folder 12, Joseph E. Mayer Paper, Mandeville Special Collection Library, USCD.

4.

Wiener to Mayer, August 17, 1940, Box 21, Folder 12, Joseph E. Mayer Paper, Mandeville Special Collection Library, USCD.

5.

Mayer to Wiener, MSS 47, Box 21, Folder 12, Wiener Papers, MIT, Wiener.

6.

Wiener is probably referring to papers that were published in 1946. See Bogolyubov & Sankovich (1994) for a description of Bogolyubov’s contributions to statistical mechanics.

7.

Wiener to Moe, November 19, 1945, Box 5, Folder 72, MC22, Nobert Wiener Papers, MIT Institute Archives.

8.

The note accompanies a letter addressed to Edward Teller. von Neumann to Edward Teller, May 22, Box 19, Folder 26, Joseph E. Mayer Paper, Mandeville Special Collection Library, USCD.

9.

Here we focus on equilibrium statistical mechanics rather than non-equilibrium statistical mechanics because the former was considered more important to pure physics than the latter. Kac made substantial contributions to non-equilibrium statistical mechanics. The models studied in Dresden’s paper on models in nonequilibrium statistical mechanics were almost exclusively given by Kac. In fact, Dresden called them Mark 1, Mark 2, and Mark 3 with the justification: “Since it will be necessary to refer many times to different models, and since “Kac model” is far from a unique designation it seems appropriate to denote this model by the first name of its inventor. In addition this gives an urgent ring to an old problem” (Dresden 1962, p. 316).

10.

Baclawski and Donsker summarized Kac’s approach: “To understand these one most focus on the underlying theme in most of Kac’s work: that of using the insights, intuition, and techniques of probability theory in other areas. The connections that Kac elucidated and emphasized, between probability and physics and between probability and other parts of mathematics, were also, of course, particularly stimulating to the development of probability theory itself: many of the most fruitful directions in which probability has gone can be traced directly to his pioneering efforts” (Baclawski and Donsker, 1979, p. xxiv).

11.

Other outlets were Journal of Fluids, Journal of Mathematical Physics, and Journal of Statistical Physics.

12.

A few new models were proposed, including the Ashkin-Teller model of Julius Ashkin and Edward Teller in 1943 and the Potts model proposed by Cyril Domb to his student Renfrey Potts for the latter’s thesis of 1951.

13.

Kac to Uhlenbeck, August 5, 1947, folder 4 box 1, 450K11 Kac papers, Rockefeller University Archives, RAC.

14.

The problem is that the energy attains complex values for low temperatures.

15.

Kac to Berlin, November 15, 1948, Box 1, Folder 7, RU 450K11 Mark Kac papers, Rockefeller University Archives, RAC.

16.

Domb to Uhlenbeck, February 10, 1970, Folder 20, Box 4, 450U60 George Uhlenbeck papers, Rockefeller University Archive, RAC.

17.

For the history of the hard-sphere model, see Brush 1983.

18.

Uhlenbeck to Kac, January 19, 1960, Box 30, Folder 6, 450K11 Mark Kac Papers, Rockefeller University Archives, RAC.

19.

It has not been possible to uncover the individual contributions of the trio.

20.

They replaced Kac’s α in the potential with α0γ and looked at the limit of γ → 0.

21.

The In order to show this, they had to employ a standard trick, namely to couple the analysis with the Maxwell equal-area rule to ensure that the isotherm is truly horizontal between co-existent phases.

22.

Uhlenbeck to Kac, January, 1948, Folder 2, Box 1, 450 K11 Mark Kac, Rockefeller University Archives, RAC.

23.

Kac to Uhlenbeck, April 15, 1963, Folder 4, Box 5, 450U600 George Uhlenbeck Papers, Rockefeller University Archives, RAC.

24.

Hermann to Kac, June 13, 1972, Box 12, Folder 1, 450K11 Mark Kac Papers, Rockefeller University Archives, RAC.

25.

Hermann to Kac, June 13, 1972, Box 12, Folder 1, 450K11 Mark Kac Papers, Rockefeller University Archives, RAC.

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