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A Mathematician Doing Physics AMathematicianDoing Physics: Mark Kac’sWork on the Modeling of Phase Transitions Martin Niss Roskilde University 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 transi- tions, 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 math- ematicians’ 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 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 Perspectives on Science 2018, vol. 26, no. 2 © 2018 by The Massachusetts Institute of Technology doi:10.1162/POSC_a_00272 185 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00272 by guest on 30 September 2021 186 A Mathematician Doing Physics 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 transi- tions, 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 formu- lation 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 representa- tions. 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 under- stand the theoretical practice of these theorists, Kaiser’s perspective on cal- culational 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 Quinn 1993, p. 4). Moreover, the mathematical physicists often worked on “questions mo- tivated 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 “workinamorerigorous 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). Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00272 by guest on 30 September 2021 Perspectives on Science 187 new model for which they had an idea of what mathematics might be rele- vant 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 illus- trated 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, phys- icists 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 break- through” (Domb 1996, p. 165) came with George A. Baker’s introduction of Padé approximant to the low temperature expansions, which was “apiece 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 chal- lenges 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 stan- dard book on continued fractions contained a chapter on the Padé approxi- mant (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 di- rectly 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 theo- retical practice of modeling of phase transitions. Hence, I wish to explore the Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00272 by guest on 30 September 2021 188 A Mathematician Doing Physics 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 sec- tion looks at how the mathematicians attempted to resolve the mathemat- ical 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 contem- porary physicists. Finally, some concluding remarks about Mark Kac’sdoing physics and the influence of his mathematical background are given.
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