Interview with Benjamin Widom, 20 March 2003

PoS

To begin, we would like to know your background and how you entered science generally.

BW

Well, I had been interested in chemistry since boyhood, so that was an early start, and I was a chemistry major in college. I went to college at Columbia, was an undergraduate at Columbia.

PoS

Where did you grow up?

BW

I grew up mostly in New York. I was born in Newark, New Jersey, and spent a couple of years, maybe around the age of 10 or 11, in Detroit, but then came back and lived in New York City, in Brooklyn. I went to Stuyvesant High School, which was a place where a lot of students interested in science went, a place where a lot of people became interested in science, but I already had that interest when I went there. And as I said, I graduated from Columbia as a chemistry major, came here to Cornell as a graduate student. I came in February '49, so I came in mid-year, the middle of winter, and trudged up from the train station downtown, uphill with a couple of very heavy suitcases, wading through the snow, and ultimately found some dormitory I had been assigned to. I did physical chemistry as a major as a graduate student, but I was working on a quantum mechanical problem, not anything having to do with what your interview is about. I was working on inelastic molecular collisions, vibrational energy transfer with Simon Bauer. Simon Bauer is still here, still a presence in the department. He's 91 years old, attends seminars and asks questions, writes papers. He's an amazing guy. So it was under him that I did my thesis, in vibrational energy transfer, gas-phase molecular collisions.

PoS

What preparation was required for that kind of work in physical chemistry?

BW

Well, the theoretical background would have been physics and mathematics, so I took physics courses in mechanics and electromagnetism.

PoS

Where?

BW

It was with Phil Morrison, who was still here at Cornell, so it was a course, a graduate level mechanics and graduate level electrodynamics course, and then I had quantum mechanics, with Hans Bethe. I remember taking the second term quantum mechanics, one that was then called applied quantum mechanics, with Hans Bethe, but I'm trying to remember with whom I took the first term. It's possible it was Ed Salpeter. But I don't swear by that. Yes, pretty sure it was Ed Salpeter.

PoS

Was it a quantum electrodynamics course?

BW

No, not quantum electrodynamics, classical electrodynamics. ... And then I had a course that is still called mathematical methods of physics, at that time taught by Mark Kac. Now, I understand that that course had been invented by Richard Feynman, but that was before I was here, so when I was here it was taught by Mark Kac, and it was a great course. And I took some additional courses, differential equation theory with Mark Kac later, a later course. Did I have any other physics courses? I don't think I had any other physics courses. Freeman Dyson was still here at that time, and I sat in on a bit of some of his advanced quantum mechanics, but I don't think I ever finished sitting in, and in any case I wasn't a registered student and was just more curious than anything.

PoS

And Statistical Mechanics?

BW

And then Statistical Mechanics. Hans Bethe taught again, and I took his statistical mechanics course.

PoS

So most of the courses were really taken in physics?

BW

Yes, most of the courses. I took a couple of courses here in Chemistry, one in advanced inorganic chemistry, which was mostly oxidation reduction theory.

PoS

Peter Debye was still here?

BW

Debye was still here, and I heard some lectures of his, but not a full course, I never had a full course with him.

PoS

And John G. Kirkwood was gone?

BW

Kirkwood was gone by then. [Paul] Flory was here. In fact, when I originally came as a graduate student, I was originally planning to do my work with a young assistant professor, John Bragg, you might have heard his name in connection with the Zimm-Bragg theory of the helix coil transition in biopolymers. And what I would have worked on with him was essentially an application, one can see now in retrospect, an application of the one-dimensional Ising model. But Bragg left to go to a job at General Electric shortly after I came, and the department thought that they still owed it to me to allow me to do a theoretical thesis, because that was my interest. So I had a choice of doing statistical mechanics with Paul Flory or doing quantum mechanics with Simon Bauer. And what did I know? I thought, well, quantum mechanics is a very jazzy subject, and that sounded great, and statistical mechanics sounded dull, so I chose to do quantum mechanics with Simon Bauer, and that's how I did that thesis. It wasn't until I got to North Carolina as a postdoc with Oscar K. Rice that I really learned phase transition theory and got to appreciate thermodynamics and statistical mechanics more than I had here.

PoS

While you were a graduate student here at Cornell, how unusual was it to be so involved with physics and math?

BW

It didn't happen a lot, but it happened. Especially at that time, our graduate school still had the requirement of two minor subjects in addition to a major subject, and those minors, if I'm remembering right, had to be outside the field of the major. And with the result that most of the physical chemistry graduate students had a major in physical chemistry and a minor in physics and mathematics. That's not to say that, as I did, that almost all their coursework was in physics and math, typically they had less coursework in physics and math, and more in chemistry, so mine was a little unusual, but not unprecedented.

PoS

And how many lab courses did you have to take?

BW

Lab courses. I didn't have to take any lab courses.

PoS

So you didn't take any lab courses?

BW

I didn't take any lab courses here, no.

PoS

Was that unusual for a student?

BW

That was not unusual. The experimental graduate students got their laboratory experience when they started their thesis research.

PoS

Was Bauer your main mentor while you were here?

BW

Yes. He was my main mentor, and generally in the area of chemical kinetics, and as I said, vibrational energy transfer was one aspect of chemical kinetics.

PoS

Did you work closely with him, or were you mostly on your own?

BW

It was mostly on my own. I learned from him what the problems are, and what we are aiming for, and what we wanted to do, and why we wanted to do it, but the actual working out of the model and the calculations and so on, that I did myself.

PoS

Who were the readers on your thesis?

BW

Well, the only reader was Simon Bauer, but my thesis committee members were Hans Bethe and Mark Kac. And so they were the ones who examined me, both the admission to candidacy exam, and my final thesis exam.

PoS

Did you work closely with any other students?

BW

Not work closely with them. I knew them. I knew those for example who were working with Simon Bauer on the experimental side. Simon Bauer had basically two projects. One was a molecular structure determination by electron diffraction, the other was chemical kinetics. So I knew generally the people in his research group, but I mostly knew those who were working on the chemical kinetics side-- the experimental side of what I was doing the theory on.

PoS

And you had an office in Baker or Rockefeller?

BW

Yes, well, at that time the graduate students didn't really have offices. I did most of my work in my room in Collegetown. Or in the library here.

PoS

And when did you finish?

BW

I finished in '52, and then went to Chapel Hill as a postdoc with Oscar Rice. The reason I knew of Oscar Rice was because of his early work in energy transfer theory. So I went there with the thought that I would be continuing along the lines of my thesis, and when I arrived, he told me that I could work on anything I wanted to work on, but if I wanted to talk to him while I was there, I'd better learn something about phase-equilibrium and critical points, because that's what he was working on at the time. So I thought I better learn something about phase-equilibrium and critical points, and that of course was the beginning of the story.

PoS

So, from what sources did you learn about that?

BW

Well, I read Oscar Rice's papers. And at that time his main concern, and that was really a central concern in the later development of the subject as well, was an obvious discrepancy between the shape of the coexistence curve, what is now called the critical point exponent beta. An obvious discrepancy between that and experiment. And even though that discrepancy was known very early, even in [Johannes D.] Van der Waals' day, it didn't really impress itself on people's consciousness until [E. A.] Guggenheim's corresponding states paper in 1945, when he put the coexistence curves of rare gases and small molecules, methane, nitrogen, and so on, all on a common scale and showed that on such a common scale those coexistence curves all fell on top of each other, and with an obvious exponent much closer to 1/3 than to 1/2. By then, even though that had already been seen in Van der Waals' day, probably in the late 19th century, even though it had already been seen then, it was so obvious and undeniable at the time of Guggenheim's paper that one had to find some understanding of what's going on. So that was very much in Rice's mind, and Rice was doing experiments as well as theory. Doing experiments in binary liquid mixtures, which we know from various kinds of transcriptions, lattice gas models and so on related to the Ising model, we know that it's essentially the same thing as the binary liquid mixture with its phase separation, it is essentially the same thing as the Ising model with permanent magnetization, liquid-gas critical points. We knew by then these were all the same thing. Especially since the [C. N.] Yang – [T. D.] Lee papers came out in '52, or the early '50s, and that demonstrated the equivalence of Ising model and lattice gas.

PoS

This was all part of your toolkit, so to say?

BW

This was all part of my toolkit. These are things I learned while I was a postdoc, and in fact at some point Oscar Rice assigned me the project of learning the Yang-Lee papers, and then reporting on them in a seminar. Which I did. I can't remember if it was a formal seminar or just a gathering. There were physicists there as well as chemists, and I presented the Yang-Lee papers.

PoS

And the physicist Fritz W. London was there at that stage?

BW

London was in Duke, not in Chapel Hill, and he wouldn't have been present at that gathering. I think Eugene [Merzbacher] might have been there. I don't remember who else.

PoS

Lars Onsager was certainly known?

BW

Onsager was known. Onsager was known, I'm trying to remember when. That, of course, was a tremendous advance in the subject. I'm trying to remember when I learned about it. It's conceivable that, yes, I did know about Onsager while I was a postdoc.

PoS

And you knew about the earlier paper by G. H. Wannier and H. A. Kramers, and these people?

BW

Yes, I must have known about all those things. I didn't study those really until later. It wasn't until later that I learned transfer matrix methods and things like that.

PoS

Lev Landau, Vitali Lazarevich Ginzburg, and their work?

BW

Landau/Ginzburg much later. In fact, this is now another thread. Well, I was already interested in interfacial tensions while I was a postdoc with Oscar Rice. Among the experiments he was doing, the measurements of interfacial tensions at approach to the critical point. So I knew there was a critical point exponent associated with surface tension-- interfacial tension. It wasn't until I was already here as a faculty member that I learned about the Van der Waals surface tension work, and the Van der Waals functional, which in fact is the Landau/Ginzburg function, but it was preceded by 70 or 80 years by Van der Waals. So I knew of the Van der Waals free-energy functional long before I knew of the Landau/Ginzburg Hamiltonian, but once I saw that, of course, I saw that they were essentially the same thing.

PoS

So when you were studying these papers in Chapel Hill, were you working on a problem?

BW

Yes, it was a problem of trying to understand these deviations of the critical point exponents from their mean field values and to try to construct some kind of an equation of state that would incorporate these and so on. That's something that I didn't accomplish till later, but I was already interested in the problem at that time. I remember even doing something, have you heard about the so-called parametric model in critical phenomena? It's just a way of expressing the equation of state in the neighborhood of a critical point in a way that incorporates non-classical critical point exponents. It was developed by people like Peter Schofield. We're now talking about a somewhat later stage in the history of the subject. But I remember that I was already trying to do such things when I was a postdoc at Chapel Hill, ideas that were the same idea as lay behind these parametric models, but I never got that far. I never succeeded.

PoS

Can you tell us a little more about what approach you were taking at that time? Did you use works like non-classical mean field?

BW

I don't think we used the expression mean field, because the expression mean field came from the magnetization side, and while we knew the connection to magnetism, we weren't particularly interested in that. So we didn't use the expression mean field, but we certainly knew about expansions in integer powers in the neighborhood of the critical point, what's now called Landau expansion, we thought of it more as Fowler/Guggenheim expansion, but we now call that Landau expansion. So we knew about expansions in the neighborhood of the critical point. And that was one of the things Oscar Rice was doing, but trying to incorporate non-classical critical point exponents. We analyzed data on critical isotherms, and we found powers close to 4. We now know that closer to 5 is the correct answer, but we already knew at that time that the critical point exponent for the critical isotherm that is close to 4 is non-classical, because 3 was the classical Van der Waals answer. So we analyzed data, Oscar Rice and I analyzed data, we found this non-classical critical point exponent. Oscar Rice had some ideas about how critical point exponents ought to be related to each other, what we now call scaling laws. He did not know that the critical point exponent which we now call gamma, which is the divergence of the susceptibility, compressibility, he did not know that that was not classical. So in those days we still thought that gamma had the value 1, and we now know that it's more like 1 1/4. So Oscar Rice had imagined some connection, had seen from thermodynamic arguments that there had to be some connection among critical point exponents. And the connection that he came up with was a precursor to the ones that we now know except that he thought that gamma had the value 1. But except for that, he had what is now one of those critical-point/exponent relations, and that was the kinds of things that we were interested in.

PoS

From thermodynamics?

BW

Yes, he did thermodynamics, you can get a lot from thermodynamics. In fact, in later years, people like Stanley Rushbrooke and R. B. Griffiths using thermodynamic arguments found inequalities connected with these critical point exponents, but those same thermodynamic ideas, with a little optimism, led to equalities rather than inequalities.

PoS

So you knew what was the problem that you were going to address.

BW

Yes.

PoS

Can you tell us about where you thought non-classical approaches lay, or what approach did you take towards uncovering the non-classical exponents?

BW

To uncovering their values, you mean?

PoS

Yes, what push did you have to get away from the classical.

BW

Well, ultimately, I knew what the classical equation of state looked like in the neighborhood of the critical point, so this really went back to Van der Waals, and these power series expansions which we now call Landau expansions, carried to low-order gave what we now know as the classical behavior of the critical point. So I took that classical equation of state, and that form of it near the critical point, and I asked myself, what are the least changes that one would make in this in order that these exponents come out with what we know to be from experiment and Ising model calculations, what we know to be their non-classical values. So I tried to make the least radical change in the form of the equation of state, and I came up with something. I'm now talking about 1964 or 1965, so I had already been here for quite some number of years as a faculty member, and this was long after my postdoc days. And I came up with some form of this modified classical equation of state which incorporated non-classical critical point exponents for the coexistence, which we now call beta, and for the susceptibility and compressibility curve, which we now call gamma, those exponents. And I then said that from thermodynamics alone, given this form of the equation of state that I was working with, I could calculate the heat capacity, and see what that does in the neighborhood of the critical point, and I calculated the heat capacity and I found it diverging logarithmically. That was pretty astonishing, because I already knew from Onsager that the heat capacity of the two-dimensional Ising model diverged logarithmically. So I asked myself, what was it about this particular equation of state, what were the features of this particular equation of state that allowed me to do that calculation and that led to that answer? And I saw that the particular features of that equation of state, the very special one that I was working with, a highly specific one, that those of its features that allowed me to incorporate the non-classical critical point exponents, and allowed me to do that calculation for specific heat were a certain homogeneity of form that is now called scaling. And so I said that if one imagines that instead of the highly specific one that I was working with, which I had no reason to think was correct, if I said that I'll just abstract that crucial feature of it, that homogeneity, and imagine that that's what does it, then I again calculated the heat capacity and again found a logarithmic divergence. And the reason the logarithmic divergence is at the values of beta and gamma that I was working with were such that alpha plus two beta plus gamma equals two gave me alpha equals zero because I had two beta plus gamma equals two. But I could then see that if beta and gamma were slightly different from the values I assumed, I would have an alpha which was not zero, a power law divergence of the heat capacity.

PoS

Back in the '52-54 period, the approach you were taking was a minimal alteration of the equation of state?

BW

No, that came later. In '52 to '54, I was taking the thermodynamic arguments of Rice, which suggested connections among critical point exponents, and studying experimental data to see if those connections held. It's interesting, also, that one can in retrospect --I didn't see this until many years later-- one can see that Bob Scott in the chemistry department at UCLA had already realized that there must be connections among what we would now call connections among non-classical critical point exponents. What he knew was that the specific heat of beta brass showed a Lambda point in the neighborhood of its order/disorder transition. And he wanted to see if he could understand that, so he also knew that what we now call the critical point exponent beta had a non-classical value, and by arguments that one sees in retrospect --we didn't know at the time-- but in retrospect, were somewhat like the arguments that Oscar Rice was using, thermodynamic arguments. But again without recognizing that what we now call the critical point exponent gamma had a non-classical value, he was able to connect the non-classical beta to a non-classical alpha, and see that that alpha actually corresponded to a heat capacity divergence. That was really very early.

PoS

When?

BW

I could look it up for you? Want me to take a minute and look it up for you?

PoS

We can look it up too.

BW

It's [Robert] Scott and it would have been probably in the late 40's.

PoS

Let me rephrase the questions. What I hear you saying is that your thinking is primarily thermodynamics.

BW

It certainly was at that time, and I guess…

PoS

And what about the microscopic/macroscopic interplay?

BW

I never myself analyzed a microscopic model to determine its critical point exponents other than models that I was able to transcribe into the Ising model and make use of what had already been known about the Ising model. I worked a lot with lattice gas models, and with lattice liquid mixtures, and so on, so I've done a lot with those, but always making the transcription to the Ising model and making use of what others had found for the values of critical point exponents.

PoS

What is the difference in thinking that you brought in? Granted, having been exposed to physics courses, but having come out of a community of chemists and physical chemists, the way you think about the problems in contrast to physicists…

BW

Yes, well, I think the answer would be, I think about them thermodynamically, and I also make use of the physical chemistry literature, that is, liquid mixtures rather than magnets, are what I have in mind. So in later years, for example, when I learned about tricritical points from Bob Griffiths, to some extent also from Michael Fisher, but I worked with Griffiths directly on that. The literature that we ultimately uncovered was phase equilibrium, three or four component fluid mixtures, so the idea of tricritical point arising for example in certain magnets, or higher order critical points in general, or in mixtures of the helium isotopes, helium3, helium4, that was something the physicists knew about and I learned from the physicists. But my own thoughts and activities were always centered around liquid mixtures, things you could pull off the shelf in a laboratory.

PoS

Did you work closely with O.K. Rice after you read the literature he suggested?

BW

Oh yes, he and I worked closely together and we wrote some things together.

PoS

Were there other people there that you worked with or talked to a lot?

BW

No, he had one or two experimental students at that time, experimental postdocs whom I knew quite well and talked with, but I wouldn't say there was any advance, I don't think we influenced each other very much in our thinking. I mean, we were friends, but I don't think that our scientific interactions were that important to either of us.

PoS

Were you still in touch with Bauer?

BW

With Simon Bauer? Well, I was really there only two years, and then came back, so it was a relatively short time, so in a sense I never lost touch with Simon Bauer.

PoS

So you come back to Cornell in '54?

BW

Came back here in '54 and I joined the chemistry faculty, first as an instructor, the instructor grade existed at that time, then assistant professor and so on.

PoS

At that time, is your work primarily in the area of phase transitions?

BW

Yes, since that time my work has been primarily in the area of phase transitions. I did a few things also on chemical kinetics that came from my graduate student days with Simon Bauer, but most of what I did had to do with phase transitions. Some work of later years has to do with the interfaces between coexisting phases and interfacial structure and interfacial tension.

PoS

So when you came here, were you doing the same problem with the same techniques, that is, looking at the data to find evidence of thermodynamic relations …?

BW

To some extent, but mostly I was trying to construct non-classical equations of state. An equation of state that would replace the Van der Waals, or what we now call mean field theory equations of state to incorporate critical point exponents. And that I ultimately did around '64, '65.

PoS

So why the change in method? Do you remember?

BW

Well, as I said, even while I was in Chapel Hill as a postdoc, I was still trying to do that, I just didn't make much progress. So I kept thinking about it and kept trying. I had a correspondence with Michael Fisher in those days, in the 1950's.

PoS

So you knew about Cyril Domb's work and all of that?

BW

Oh yes.

PoS

When?

BW

The time that I really learned about the work of Michael Fisher and Domb and so on was in '61, '62, when I was on sabbatical leave in Amsterdam. And I was sitting in the group of [J.] De Boer, in statistical mechanics. I wasn't working with people there, I knew what they were doing, people like Eddie Cohen and Hans Van Leeuwen were there at that time, so I knew what those people were doing, and I knew about that work in cluster expansions that they were developing, and diagrammatic expansions. So I knew about all that stuff, but that's also where I was learning about what Michael Fisher did with correlation functions and deviations from Ornstein-Zernike theory, and where I mostly appreciated the Onsager work. I knew about it long before, but I got a greater appreciation of it then. And I was still very excited about non-classical coexistence curves, and I saw them everywhere.

PoS

How much did you keep track of things which are happening in liquid helium, for example? Such as at the University of Nottingham…

BW

Yes, I did.

PoS

Alexander Voronel's work on liquid transition?

BW

On heat capacity divergence through the critical point. Yes, I'm not entirely sure when I first learned about that, but certainly I knew about it from some fairly early stage on.

PoS

Early 60's?

BW

Yes. Must have been.

PoS

we would like to know a little bit about what you were doing between '54 and '61. What were you teaching, for example? Were you teaching about phase transitions, was that something that would show up in the courses?

BW

No, it wouldn't have. That's because that would have been a much more advanced topic. I began teaching statistical mechanics from a fairly early stage, I was teaching undergraduate physical chemistry, and in fact my first teaching assignment was undergraduate physical chemistry laboratory, where I knew as much as the students did. And so I was teaching undergraduate physical chemistry, and in the graduate courses I was teaching our beginning graduate quantum mechanics and our beginning graduate statistical mechanics.

PoS

In chemistry or physics?

BW

In chemistry, yes, so that's what I was teaching. If you want to know what I was doing, I can look at my publication list and see if I can remind myself what I was doing then. So the late 50's and early 60's, that was your question?

PoS

Between '54 and the sabbatical.

BW

I guess the first thing that I did-- I should say that going back to my time in Chapel Hill as a postdoc, there were theories by Joe Mayer, I don't know if you've come across his name in connection with your study, but essentially looking at cluster expansions and virial expansions, and trying to learn something possibly about phase transitions from the convergence or divergence of these theories. There were some peculiar theories developed at the time by Joe Mayer and also by Oscar Rice, the so-called Derby Hat picture of phase coexistence. And that's all dead now. We realize that none of that was right, and that was all very much in the air. Also very much in the air was the question of the convergence of virial expansions, and the possible divergence being connected with phase transitions. So I looked at the virial expansion for the ideal Bose gas, and analyzed the neighborhood of its lambda transition, of its phase transition, the ideal Bose gas. And I tried to estimate its radius of convergence. Had terrible estimates, and never got anywhere close to it, but published the paper. This was while I was still in Chapel Hill. I was doing this paper called "The virial series in the Bose-Einstein gas," and in later years, you know, did you know Wolfgang Fuchs here in mathematics? I got interested in the problem, and in later years he found a much, much better estimate of the radius of convergence and saw that it has a finite radius of convergence that had nothing whatever to do with the phase transition. I mention that only because that paper of mine came out in '54, and so that was sort of toward the end of my postdoc years with Oscar Rice. Then my first paper here at Cornell came about because of some paper that I had read in the German literature on the vapor pressure and heat of vaporization when you were near the critical point. And so I wrote a little paper pointing out that the heat of vaporization had to have the same non-classical behavior as the coexistence curve, and I had that paper translated into German by someone in the German department here, and published. So that was not my German, that was the German of my translator. So that was the first thing that I did here. That was seeing that one had to have the same critical point exponent for heat of vaporization as one had for the coexistence curve. And then I published a paper, this I'm sure that I was already working on in Chapel Hill, but it didn't come out until I was here, called "The Structure of the Configuration Integral: the Statistical Mechanics of Pure Fluids." At that time I was very conscious of the work that Lee and Yang had done on the zeroes of the partition function lying on a circle in a complex fugacity plane of the lattice gas, the equivalent of the Ising model. And so I took a Van der Waals-- like a classical equation of state, and asked for the zeroes of its partition function in a sort of thermodynamic plane. So that's what I published. I don't think it was of any significance, but I was very excited about it at the time. And then there is my paper with Oscar Rice analyzing the critical isotherm in the neighborhood of the critical point, that came out at the time when I was already here. The work was done in Chapel Hill, but it came out when I was here. Then I did a paper that went back to my work with Simon Bauer on inelastic molecular collisions called "The Energy Dependence of Inelastic Molecular Collision Probabilities." Another paper, "Statistical Mechanics of Liquid-Vapor Equilibrium." So this was coming back to the statistical mechanics, thermodynamics side. Again, it wasn't very deep or a very important paper. It's not one I'm especially proud of. And then I did some more on inelastic molecular collisions. This is really off the subject, so I don't know if you want to hear about that. You're aware of the fact that going back to [James Clerk] Maxwell's days, it was recognized that if molecules interacted with interaction potential of one over r to the fourth. So force was one over r to the fifth, and then what is in effect the Boltzmann equation, these transfer equations could be solved exactly. And so I thought that one should be able to do the same with the Schrodinger equation with one over r to the fourth potential and this might be of some significance in this problem of inelastic molecular collisions. So I worked on that, and it turns out that with the 1/r^4 potential, the Schrodinger equation becomes equivalent to a Mathieu equation, one of these equations, or a Hill type equation. So I studied something about these, again, exponents they're called, and associated them with Hill type equations in connection with the 1/r^4 potential. But that was quite a departure from anything we've really been talking about. I did some more work on collision theory, relaxation of string oscillator, one-dimensional inelastic collisions with hard rod interactions. Collision theory and kinetics of dissociation of diatomic molecules, mean first passage times, bi-molecular reactions.

PoS

Do you ever talk to the physicists at that time, Bethe, and others?

BW

No, uh, Bethe I had known from my graduate student days. I had a partial research assistantship because of a collaboration between Simon Bauer and Arthur Kantrowitz. So I learned about energy transfer on surfaces, and learned some little fluid dynamics, I remember Hugoniot-Rankine equations, I'm not sure I could tell you what they are, but I remember the word. And then I did a wonderful calculation, again, this is in the energy transfer area, not critical phenomena. I did a wonderful calculation of the rotational relaxation of rough spheres. I remember I had to calculate the spectrum of a collision kernel, and that was really an amazing calculation. It was one I particularly enjoyed a lot, and again one I don't think was of particular importance, but it was a lot of fun. Then my first experimental paper in critical phenomena, I had a visitor, DeForest Rudd, who at that time, unfortunately he's no longer living, at that time he was a professor at Lincoln University in Pennsylvania. It's an all or mostly black college. And he was on sabbatical leave here, and he wanted to work with me on an experimental project. I told him that there were known cases of lower critical solution points, that is, where you achieve the critical point, not on increasing temperature, going through a two-phase region on increasing temperature until the point where the two phases become identical at a critical point, but there were known cases in which you had lower critical solution points, where you go through the two-phase region on lowering temperature, and come to a point on decreasing temperature at which the two phases become identical. Sometimes the so-called closed loop coexistence curves, where the same system would have both an upper and lower solution point. And I wanted to be sure that at the lower critical solution point the critical point exponent had the same value as an upper solution point. From what we now know, it's obvious, but at the time it wasn't obvious, so I wanted to check that. So I had DeForest Rudd work on a mixture of water and ethylene glycol mono iso-butyl ether, which had a known critical solution point. And he did a careful study of the coexistence curve and he found a critical point exponent that was close to the one third that we already knew to be the case in the upper critical solution point.

PoS

You had a laboratory at that state?

BW

Yes, well, we scrounged up a little laboratory space. In later years I actually had a laboratory of my own.

PoS

When was that?

BW

That was 1960. Then came a couple more papers…

PoS

May I ask how Rudd knew that he'd like to come here and work with you?

BW

I'm not a hundred percent sure. I think I've just forgotten. I think he came to the department as a visitor and then looked around to see who's doing what, and then decided he'd like to work with me. As I recall, that was the case. He might have known about me before he came, but it would be doubtful. I don't see why he would have.

PoS

And by 1960, you have tenure here and you have graduate students and everything else.

BW

Yes, I've had very few graduate students. In the early days, there were a couple of graduate students I did have who were great. Well, my first graduate student worked with me on energy transfer problems. His name is Jerome Weinstock. Then in somewhat later years, I had John Wheeler, who's now at La Jolla, and John Zollweg, who went to the University of Maine from here, and is now back at the Cornell Theory Center. And John Wheeler and I worked on phase transition problems, critical point problems, and John Zollweg was my first experimental graduate student, and he worked on critical points in three-component liquid solutions, --this is the early 60's-- in the neighborhood of what's called the plait point, which we recognize to just be another critical point but in a slightly different context. You heard from Michael Fisher about what he has called the renormalization of critical point exponents. If you display a phase diagram, for example, in a space of what we've come to call densities rather than fields, using later language of Griffiths and Wheeler, then critical point exponents, the way that the phenomena are represented, this is not a different kind of critical point, but a different kind of representation, leads to critical point exponents that are what Fisher later called renormalized. And we had already recognized in a particular example, earlier, we had a fluid model that we transcribed to an Ising model. We had already perceived this so-called renormalization where the so-called coexistence curve, for example, in the normal representation we'd have a critical point exponent beta. In this representation where you were dealing entirely with density variables and no field variables, you'd have a critical point exponent beta divided by one minus alpha, that is what Fisher later called renormalization. But we had already seen this in the connection we made between our fluid model and the Ising model, and we wanted, in general, to see if this so-called plait-point model, three-component liquid solution, where you achieve a critical point just by varying the concentration and not temperature all at a single fixed temperature that again had the same critical point exponent. And we found a slightly larger exponent that corresponded to this so-called renormalization. So except for this work by this visitor DeForest Rudd, that was my first experimental work in critical phenomena, and that was with my then-graduate student John Zollweg. So at the same time I was working with John Wheeler as a theoretical student and we were working on various kinds of solution models, again with critical phenomena of liquid mixtures.

PoS

So in this period you go from Chapel Hill, to Cornell, then to Amsterdam?

BW

Yes, on sabbatical leave.

PoS

In that period, you've given us a list of the topics that you've worked on. Can you think about how the methods that you were using changed? You started by doing quantum mechanics and energy transfer, then did statistical mechanics at North Carolina, and what about in this six year period?

BW

Again, except for the continued work on energy transfer and chemical kinetics, I was working largely with-- are we talking after I came back from sabbatical leave?

PoS

No, before then.

BW

Before the sabbatical. Well, again it was some statistical mechanics and thermodynamics.

PoS

So no big changes in the methods you were using? No huge inputs?

BW

No big changes in the methods I was using, that's right.

PoS

What about your network of contacts, how has that changed in the six year period? You started first by talking mostly to Bauer, then second talking mostly to O.K. Rice, and then in the next six years, who were the influences and what kind of conversations were you having that affected your work?

BW

I can't think of people. Well, I was in continued contact with Simon Bauer on the energy transfer side, continued contact with Oscar Rice, so I maintained those contacts.

PoS

Did any correspondences emerge out of your German publication?

BW

Well, probably, probably with the person who wrote that original paper that that instigated my paper, I think I had some correspondence with him.

PoS

Domb's group?

BW

No, you see, Michael Fisher, whom I got to know extremely well later, and Domb and so on, I think that I learned about that stuff mostly while I was on sabbatical.

PoS

What about people working in Amsterdam?

BW

That's where I really studied that literature and learned about how Michael Fisher, for example, found that a calculation that Onsager had done on the correlation function implied an exponent gamma equals seven quarters for the two-dimensional Ising model and so on. So I learned about all those things in '61/'62, while I was in Amsterdam.

PoS

Let me rephrase the question. You're now in Amsterdam talking to people who do statistical mechanics, deriving equations of state. Is there a bridge being made between you work, attempting to find different formulations of equations of state and a microscopic approach? How, from a statistical mechanics point of view, how do you get deviations from what you call the classical?

BW

Except for having known of the Onsager papers and the work of the Domb group, I didn't myself…

PoS

How about Onsager and Oliver Penrose and?

BW

Onsager and Penrose and the Long Range Off-Diagonal order. No, that I didn't become conscious of until much later.

PoS

Did we finish setting the stage?

BW

No, I'm trying to think of the answer to your question.

PoS

Joe Mayer?

BW

Well, I knew about all that stuff.

PoS

But were you talking to people?

BW

No, no I wasn't.

PoS

So your contacts were local?

BW

My contacts were local.

PoS

Of the local group, who were the contacts? Between '54 and '61.

BW

No, I was essentially alone.

PoS

Ok. So why Amsterdam?

BW

Because that was a known center of modern statistical mechanics. But not particularly because of anything to do with phase transitions or critical phenomena, it's just to be in an environment of statistical mechanics.

PoS

How long where you there?

BW

For nine months, that was my sabbatical leave.

PoS

Who was there?

BW

Well, the head of the Institute was [J.] De Boer. And then Eddie Cohen was what we would now call an assistant professor, at that time he had a different title, in Dutch, I think, Lecturer. So he was there, and his interests were non-equilibrium statistical mechanics. Theory of transport, Chapman-Enskog equations, Boltzmann equations, things like that. Hans Van Leeuwen was a senior graduate student in that group. He later came to do some postdoc work with me at Cornell, oh, so that was a local contact that would have been in the early 60's. I remember it before 61. No, I was really pretty much alone.

PoS

What did you learn in Amsterdam? How did it affect your work?

BW

Oh, that's where I became fully aware of the kinds of things that were being done in the Domb group and the things that Michael Fisher did relating correlation functions to susceptibility and so on. So that's where I learned that literature and that was very important.

PoS

Not from seminars or conferences, but from reading the literature?

BW

Yes, from reading the literature.

PoS

And you were working alone?

BW

In Amsterdam, yes. Again, I knew the people, and was talking to them about the sorts of things they were doing, a lot of work in cluster expansions, diagrammatic expansions, and I knew something of things on liquid helium that Eddie Cohen and Hans Van Leeuwen were doing. But I didn't do anything in those directions.