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# Elliott Montroll

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In footnotes or endnotes please cite AIP interviews like this:

Interview of Elliott Montroll by Stephen Heims on 1983 September 26,

Niels Bohr Library & Archives, American Institute of Physics,

College Park, MD USA,

www.aip.org/history-programs/niels-bohr-library/oral-histories/31774

For multiple citations, "AIP" is the preferred abbreviation for the location.

### Abstract

Topics discussed include: his education at the University of Pittsburgh, his work with Joe Mayer, his Sterling Fellow post at Yale and work with the Ising model, his work with Kirkwood on wave theory during the war, and his work with ring diagram, cluster expansion theory, Feynman diagrams, Honsagger, phase transitions at Columbia, statistical mechanics problems, and his time at IBM.

### Montroll:

There is a fellow you may not think about, that's John Howe, who still exists, Lassiter and Howe. I see him fairly regularly in La Jolla, and I think, see, he was a little out of the network. All the other individuals knew each other, saw each other, and sometimes it's very hard to think of who did what first, because it was such a net, whereas he was completely separate.

### Heims:

That's interesting.

### Montroll:

Do you see how he —

### Heims:

— John Howe.

### Montroll:

John Howe, and he's in La Jolla. He is in the La Jolla Telephone Book. He was at General Atomics for a while. He went into nuclear energy business later. He's an interesting guy —

### Heims:

You're right, there's a paper —

### Montroll:

— paper by Lassiter and Howe —

### Heims:

— that’s in there.

### Montroll:

Yes, all the papers came out essentially the same month. The same month, I think, by [???] and my own and Lassiter and Howe. [???] was very close…

### Heims:

Maybe that's a good place to start, in the forties and fifties. The forties start out with: (names) You know, I used to see it in the papers. I know nothing more.

### Montroll:

I'd be glad to reminisce as best I can on this. I suppose the way to start is as a graduate student at the University of Pittsburgh, because that's when I first ran into [???]. He had come from Cambridge, and he was an instructor at the University of Pittsburgh at that time. If you ask me the year, I think it was 1937. Yes, September, 1937, he appeared in Pittsburgh as an instructor. He'd been at the Institute for Advanced Study the year before that and worked on the [???] wave functions. He had been at Cambridge the year before that working with Fowler. While I was an undergraduate I got interested in statistical mechanics, mainly through theory of electrolytes, which at that time I'd not been writing papers on but I wrote some on it later, and actually there's an interesting coincidence on the start of that, because [???]. My start with mathematical things turned out to be very close to what it was with Freeman Dyson's on a special book, DIFFERENTIAL EQUATIONS, when I was I guess a freshman or sophomore. There was a TA in my chemistry course who on his desk had [???’s] book on electrolytes. It was the first book I'd ever seen that had anything of statistical mechanics in it, and here were all these wonderful formulae, and we heard in freshman chemistry about theory of electrolytes. Nothing special in the mathematics but just the way you described it. I looked in the book, and I thought I knew a little bit of mathematics as a freshman or a sophomore, but here were all these strange things, these differential equations which I’d never seen before, so I took the book over to the calculus teacher, and said, "What are these things?” He said, “Oh, they're differential equations.” I said, “I want to study this subject.” He said, “All you have to do is read this book on differential equations," and took Piaggio's book on differential equations off his book case. “All you have to do is work all the problems in this book, and then you'll have no trouble reading the book on electrolytes." Well, when Freeman Dyson wrote his reminiscences recently in his autobiography, he mentioned a very similar experience. He found papers in relativity, and he went to this mathematics teacher, who said, “No problem, all you have to do is read this book and work all the problems in the book” which was Piaggio's DIFFERENTIAL EQUATIONS. So by a coincidence we both started fancier mathematics, from Piaggio’s DIFFERENTIAL EQUATIONS.

### Heims:

You must have picked up a lot of mathematics, more than physicists ordinarily do.

### Montroll:

Yes, as an undergraduate, after doing Piaggio, the professor said — his name was Culver [???], a most remarkable fellow, wrote practically no papers but — yes, Montgomery Culver. He was a great inspiration. He was a great expert on group theory and had an enormous number of volumes of special groups that he had worked out himself but never bothered to publish, and each month when the BULLETIN of the American Mathematical Society would come out, Culver would find a paper on special groups, and then held pullout one of his notebooks — “I did that 20 years ago, thought it wasn't worth publishing.” So he's essentially an unknown man, but a very good mathematician and an excellent inspiration as a teacher. So after I did these problems, he said “Well, why don't you come and take an examination and we'll give you credit for the course.” That looked like an easy way of taking mathematics courses because it always bored me to go and listen to people, write details down in their work, so as an undergraduate I took practically all the mathematics courses at the university, took most of the graduate ones, just by reading books and taking examinations. It was a great thing, because for five dollars one could take a course, by examination. So I learned a fair amount of mathematics that way, and started to get interested in statistical mechanics through electrolytes. There was a man by the name of Abraham Robinson who did experimental work but he was very good to talk to —

### Heims:

— in Pittsburgh —

### Montroll:

— about these things, and —

### Heims:

— Did you come across Pomfretter's work or was that later?

### Montroll:

I saw Hansauger's work and read some of it while I was an undergraduate. But there was no one there to talk to about statistical mechanics, so that was all independent study. Another thing I got interested in was integral equations at the time. I guess it was Lovett's book on theory of integral equations, so I read that and did all the problems, so I became rather expert, and then read many of the original papers, by [???] and started to take interest in the mathematics literature. Probably my specialty at that time was finding strange items in the mathematical literature, which is something I still use today. There's hardly a paper that I write today that doesn't have some obscure thing that I learned in just browsing mathematics journals at that time. But all of that was informal, and my first real discussions with anybody who knew any statistical mechanics occurred in 1937 when I was a first year graduate student. My vague memory is that either I took the — I think the same year that [???] appeared in Pittsburgh, he went to Westinghouse but he gave a quantum mechanics course at the university, so they were the first people that I got any interaction with in a systematic way on statistical mechanics. At that time Joe Mayer's papers came out on theory of condensation. I think it was 1937.

### Heims:

I think it was '38 (crosstalk )… old reprints…

### Montroll:

— old reprints to get the exact dates. Mayer and Harrison, Mayer and Ackerman came out in '37.

### Heims:

There was the conference in Amsterdam the next year where it was debated. Quite a bit.

### Montroll:

That's right. In September of '39, I guess, when I spent the year with Joe Mayer, I remember he got an enormous postcard from Amsterdam with the signatures of various people, [???] leading, saying “We agree with your theory of condensation.” I think the biggest signature was Sommerfeld's. But they all signed.

### Heims:

But Joe Hayer didn't go to that meeting.

### Montroll:

Joe’s lawyer was not at the meeting, no. Joe was still at Hopkins, and didn't go to the meeting himself.

### Heims:

He was a refugee from Germany?

### Montroll:

No, Joe Mayer's American. He was born in the United States. He lived in his youth in Canada a considerable part of the time. His father was a mechanical engineer. His father was born in Vienna. His mother was born in the United States. He learned statistical mechanics more or less by himself. He was a student of a physical chemist at the University of California, G.N. Lewis. Actually, the Mayer reminiscences, I don't know whether I have them here, I did have them in this pile, but I think I… He has his student reminiscences that were just published in I think ANNUAL REVIEWS OF PHYSICAL CHEMISTRY and I imagine you might be interested in that.

### Heims:

Yes, I'll look at them.

### Montroll:

He just barely brings it up to the period of condensation. I had that in this folder for another reason. Let's not interrupt now but I'll try to find that because you'll be interested.

### Heims:

Yes, right, if I could reference his… It would be handy, if you can locate it.

### Montroll:

When the Mayer papers came out, [???] said to me, "Why don't you read the papers?” That was perhaps a Wednesday or Thursday of some week. He said, "Why don't you report on it at the seminar next week." So in all innocence I said, “Yes, I'll read the paper and report on it at the seminar,” which was a disaster because those papers were tremendously difficult, and all the crustor expansion, and it was probably the worst seminar I've ever given. On the other hand, it told me that I ought to learn how to give lectures, which has intimidated me into giving fairly good lectures since then. The thing that I noticed —

### Heims:

You were then in Pittsburgh still?

### Montroll:

I was in Pittsburgh, a graduate student. But the one thing I noticed was these crustor integrals which were multiple integrals, and I quickly realized that when you have the chain integral, that you went from let's say X1 to X2, X2 to X3, X3 to X4, R1 to R2, and when one did the chain, and did the integrals over all these Mayer F function over all these points, that it was just an integrated turnal of an integral equation so all you’d have to do was find the Agon values of the integral equation and that would tell you what the crustal integrals were. So that was the basic idea of looking at Agon values and connecting them with chain problems.

### Heims:

Yes.

### Montroll:

Now, at that time I'd written my first paper on using integral equations, but in trying to find the distribution of primes, it turns out the distribution of primes can be expressed as a solution of an integral equation. Since I was all hepped on number theory at the time, I knew about integral equations, and actually the paper that I wrote, that was rejected by the BULLETIN of the American Mathematical Society, had all the basic things that are in Selburg’s theory. The important problem at the time was, was it possible to find the distribution of primes without using complex analysis? The way I solved the integral equation is that I turned it into a differential equation, and I differentiated the distribution function. The referee said that it's a discontinuous function, therefore you can't. But there are a dozen ways of solving that integral equation which I planned on continuing, but when [???] suggested reading these Mayer papers, then I saw the connection, the interrated kernels and the crustal integrals, and so I sort of got off on —

### Heims:

It must have been one of the few instances in physics where prime number theory connected with a —

### Montroll:

— yes, well, the connection wasn’t very close, except I had to learn about integral equations in order to solve that particular —

### Heims:

You wrote some papers with Mayer.

### Montroll:

Yes. I'll come to those next. So I used this problem on the interrated kernels and also you remember that what you get is, if you have the nth interrated kernel, it's the sum of the Agon values raised to the nth power, so if you start with three point, you get the lambda cubed, and if you have four points, it’s the lambda to the 4th, so it's clear you could sum up all the ring diagrams. Then in summing all the ring diagrams one would get a 1 over 1 — the eigen values to the [???], with no power, so that's the first summation of the special (crosstalk) theories of diagrams, the summing of the ring diagrams. So that was my PhD thesis. My PhD thesis first had to do with the interrated kernels and then summing the diagrams.

### Heims:

I see. Was it published?

### Montroll:

Yes. It was published actually in a paper with Joe Mayer. That’s how Mayer entered this. So that was basically what the PhD thesis was on — a number of things about integral equations which there's no point in going into because its disconnected from this, although there is a Teplett’s theorem on determinants which later appeared but basically I had many of the ideas on that in the thesis. That was before Segell had done the things — but they're very much like Segell's theorem. But the important point was the summing of the rings, first the realization that the interrated kernels gives sums of Eigen values raised to powers — now if you have a very high power, very big ring, then it's only the largest Eigen value that counts. Now, in curstel integral theory that's not so important because you have integrals over rings of all sizes, so you have to add them all up.

### Heims:

Yes, so you're summing some of the diagrams.

### Montroll:

Some of the diagrams. On the other hand, if you do something like the [???] problem where you have layer by layer, then you're interested only in the limit, where you have many of them together, and so therefore, it becomes connected with the largest.

### Heims:

What’s the connection between your use of values in summing these rings, and the use of it in connection with the icing model?

### Montroll:

I'll come to that. OK, so the thesis was on these Mayer crustal integrals, and what happened is, I guess I finished the thesis at the end of my, during the summer of my second year at University of Pittsburgh, and suddenly realized that I was really done. There was no point staying there any longer. And the question was, to look around for a place to post-doc, and I wrote around to a number of people in the summer. It was already very late to post-doc in September, but this same friend, Toby Dockleburger that I mentioned who — that Faulkenhagen's book — appeared one day at the university and said, “I’m driving to New York. Do you want to come with me and we'll visit some friends in New York?” So we drove to New York, and Mayer had just moved to Columbia from Johns Hopkins. “While I'm in New York why don't I go visit Joe Mayer and tell him what I've done?” So I visited Joe Mayer and told him what I'd done and he was telling me what he was interested in at the time, and he invited me to stay for the year. He had no money for me at Columbia, but I had $600 that I’d saved as a TA, so I went to New York to live with my $600, and then we got on and wrote these papers, one on molecular distributions and then the one with the cruster integrals and with the interrated kernels. The problem I was really interested in at that time was the electrolyte — but I thought, one has to take the Fourrier transform of the Mayer F function, you know, — the electrolyte problem is essentially taking the Fourrier transform of 1 over R and having studied mathematics too hard as a student, it was clear that that integral diverged, so I played and played and played with that, but I didn’t know what to do about its Fourrier transform. So I sketched the whole business of summing ring diagrams, how that should appear in the electrolytes, but asn’t able to finish it because of that, taking the Fourrier transform of the 1 over R. Now, ten years later, Mayer got interested in the same problem. We talked about it at that time. He didn’t see much about that. He just put in the factor E-R, lambda R divided by R, and then you take the Fourrier transform (crosstalk) [???] so that became his paper in 1950.

### Heims:

(???) I have a Texas, am I confused —?

### Montroll:

He went to Texas. He was literally fired from University of Pittsburgh for a very peculiar reason. It turned out that there were nationality rooms in the cathedral of learning at University of Pittsburgh. On the first floor there were these very elegant rooms. You know, the Cathedral of Learning is this 35 story building, skyscraper, which was called in those days the University of Pittsburgh. And the various ethnic groups in the Pittsburgh neighborhood collected money for what were called nationality rooms — the Scottish room, the Belgian room, the Dutch room, a Chinese room and so on. Now, [???] gave a course in atomic physics, graduate course at University of Pittsburgh, and one of the students was a graduate student in chemistry, a Chinese fellow who was a very bright fellow who, at the same time as a graduate student he was running an import-export Chinese business, and he was one of the people who helped organize the sponsoring of that Chinese national room. [???] flunked him in his course, and there was a great fuss made and he went to the chancellor saying that he was very busy with his business, that he really would study his chemistry better and pass it the next time. [???] was rather firm with this, and they didn’t — it wasn't that he was fired, but in those days, instructors had a year to year contract and so his wasn’t renewed. And then he went to Texas. Then from Texas he went to, I'm trying to remember, I think he went to Leyden. No, he went to Leyden and then he went to Texas after he was in Leyden. OK, when he went to Leyden that's when he got involved with Kramers, and —

### Heims:

Kramers was very much an Ehrenfest student, right?

### Montroll:

Kramers was an Ehrenfest student. I have something I'm going to give you which is the family tree of the Vienna school of statistical mechanics which is a little paper. This was really an after dinner talk that I've just written up, which is fine in this connection — in there, so I'll give you a copy of that. Kramers had been interested in the [eisen?] problem of phase transition. He was as an undergraduate a student of Ehrenfest, with a wonderful group, around 1915-16. The other members of that class were [???] and Jan Bergers and Volker so that whole group was undergraduates.

### Heims:

Now, Bergers did turbulence theory?

### Montroll:

He later did turbulence theory.

### Heims:

And Fokker — the Fokker?

### Montroll:

— Bronk and [???] did the differential geometry, and Kramers, they were all undergraduates with Ehrenfest during that period. Then Kramers went to work with Niels Bohr. He actually wrote his PhD thesis with Niels Bohr, and then that led to the famous Bohr-Slater-Kramers dispersion. (crosstalk) Then he returned to Holland where he was a professor at Utrecht. Now, there was considerable interest in statistical mechanics at Utrecht — Brownian motion(?) — there’s Uhlenbech and what's the other name that goes with it? No, there's the famous Baker Ulenbech…

### Heims:

Kahn? Kahn was a student?

### Montroll:

Kahn was a student of Uhlenbech, certainly. I have it here. It is the Uhlenbech Ornstein. Ornstein was at Utrecht, and Ehrenfest still had tremendous weight in statistical mechanics, so Kramers just after he returned to Holland went back to statistical mechanics. That's where he wrote his paper on electrolytes, because the first, the Gausian model, you see, that later was used as the —

### Heims:

— the predecessor of the spherical model?

### Montroll:

It was the predecessor of the spherical model. You'll find the Gaussian model was first discussed in Kramers' paper on electrolytes. The game was, he said, "Let's not look at an individual electrolyte with a fixed charge E, let’s talk the Gaussian distribution of charges, and then integrate over the charge distribution before integrating over the positions. “Then you wind up with a certain determinant, just by some dimensional analysis you can analyze, and finally get the Debuye-Hickle results, so that was an important thing because that was the beginning of using kind of a Gaussian model kind of idea, and later Sid Berlin and I generalized that to a spherical model, where you take the sums of the squares of the charges and you average over all charges so that the sums of the squares come out to get the right —

### Heims:

— total number N, yes.

### Montroll:

The total number N times the single, single charge. So Kramers had already started in that direction.

### Heims:

And your work with Berlin, was that in Chicago?

### Montroll:

No, that started when he was at Michigan but it went into detail when he went to Hopkins. He was at Hopkins for a while. That's when we worked together. Berlin was never at Chicago. He was at Michigan and then he went to Hopkins after Michigan. He was again part of the Vienna school, I like to call it, because he studied theoretical physics with Uhlenbech, even though he wrote his thesis with [???] on chemical binding, but Uhlenbech was his inspiration to get him interested.

### Heims:

There is a reminiscence about the spherical model by [???] I don't know if you've seen that?

### Montroll:

I haven't seen that.

### Heims:

It was printed in PHYSICS TODAY long ago, and — but I think it's his personal story, and I’d like to hear your — later, that's right.

### Montroll:

He was really the inventor of it for the theory of cooperatives? Well, getting back to [???] and Kramers, [???] went to Leyden and Kramers got him interested in electrolytes, I mean, in the Eisen model, and then they had their basic ideas there and then [???] came back to America to go to Texas, and he stopped at University of Pittsburgh and we had a discussion one afternoon, where he had some differential equations that corresponded to the one dimensional Eisen model, and it was in this blackboard discussion that I said, “All you're doing is multiplying matrices,” and then went back to the same idea that I had for the integrated interrated kernals, so if you read the Kramers-[???] paper, the fact that the partition function really comes from multiplying the matrices together, and that the trace of that, the nth power of the matrix then gives the partition function, where but the trace is just the lambda 1 to the N plus lambda 2 where those are the Eigen values, so the matrix, so —

### Heims:

— for you there was the direct link from the (crosstalk)

### Montroll:

— direct, because —

### Heims:

— oh, that's lovely, to have a connection between —

### Montroll:

— putting these things on the blackboard, and I said, "Just like the interrated kernels in the integral equations, all you have to do is find the —” They had Eigen values that came naturally out of their difference equations, but they didn't have it written as this direct matrix way, so while the basic (change tape) …piece of analysis, and so, especially from a blackboard discussion, and what we did have though was the, done very elegantly, is the rotation of the critical points, and then [???] moved on to Texas. I guess we spent a couple of days together discussing these things. And that was — after I spent the year with Mayer, and I was supposed to go to Yale to be a Sterling Fellow post-doc with (Unsold? Onsager?) I’d written up these papers, and —

### Heims:

— Let’s see, the Sterling, that's the foundation connected with Yale.

### Montroll:

Yes, one of these foundations. It turned out that Mr. Sterling had nothing to do with Yale, but he made a great deal of money in the oil business, if I remember correctly, and there was a question what to do with it, and someone encouraged him to give the money for Yale for the Sterling Chemistry Lab or Sterling Library, Sterling Fellowships, and it was Joe Mayer who wrote a very nice letter to (Unsold? Onsager?) recommending me for one of these fellowships. Now, the original idea of my going to Yale, from Yale’s view, was that Onsager had a lot of unfinished work in his filing cabinet, on the order of a dozen or more papers, but Onsager never got around to finishing any of these, and so I was considered as an attractive young man for them, because I could then take his folders in the filing cabinet and I would finish the papers. But when I got to Yale, I told Onsager about the (icing? Izing?) problem, and what [???] had been doing and what I'd been doing on this, and Onsager got interested in that and we devoted most of that —

### Heims:

I see, you were really interested in —

### Montroll:

I got involved in it. The problem he was interested in at that time was the entropy of ice, calculating the Pauling's number, which had to do with the entropy of ice, the sort of thing that Weed later worked out in great detail, and then Onsager and the fellow at Carnegie Mellon who was a student with [???], did detailed work, and that was the thing he was doing at the time. But when he heard about the icing problem, he started to think about that, and —

### Heims:

What was it like to work with Onsager?

### Montroll:

Oh, the most remarkable experience. One couldn't really work with Onsager but he spoke almost every day of these problems, and his technique was to first [???] the problem and then he did a line of three, then he did four and five and started to notice the systematics and literally get the answer for the, for a line of N, hunting for a two dimensional problem, but it took a tremendously long time to do the detailed analysis, in fact a couple of years. That wasn't finished in the year or so that I was at Yale.

### Heims:

You mentioned how he wrote something on a blackboard a couple of years before hand.

### Montroll:

Yes.

### Heims:

At that point did you think he had the proof, or —?

### Montroll:

He had the proof but it was tremendously complicated and almost impossible to bring out. The first statement of the partition function formula was at a meeting at the New York Academy of Science, I think it was '43.

### Heims:

'42 or '43. Did you attend that?

### Montroll:

I attended that meeting. Debye was the one who organized the meeting and this I'm not absolutely sure of, but you can check on the program, I think [???] gave a paper at that meeting. If it wasn't he, someone gave a paper on phase transitions, and after the paper was delivered, Onsager got up and made his usual remark, "Incidentally the partition functions to the icing problem is,” and he wrote it down on the blackboard.

### Heims:

The phase transitions were not a very active field of —

### Montroll:

— it wasn't a very active field, although it was starting to become rather active, with the discussion, of these first papers on the matrix method, plus Mayer's clusteretical(?) theory. The other thing interesting —

### Heims:

— this was in the middle of the war?

### Montroll:

Yes.

### Heims:

And there wasn't too much physics happening outside of Los Alamos and places like that.

### Montroll:

No. The situation there was, for myself and for what it was in New York City, is that I was supposed to stay on at Yale sort of indefinitely, or at least for a couple of years, but the war started, or at least it was getting so close to the war, many of the people got involved in war work. Kirkwood got involved in under water explosions, and he was trying to recruit people to work with him on theory of under water explosions, so I went from Yale to work with Kirkwood on [???] wave theory, and then I later went from there to the Manhattan Project in New York. I think I was with Kirkwood for a year, then Princeton for six months, and then the December that I was at Princeton, a fellow by the name of Manson Benedict who was a former-classmate of Kirkwood's was put in charge of the study of the isotope diffusion, the diffusion method for isotope separation. He was then at M.W. Kellogg Company and so he tried to recruit people to work on the theory of isotope separation, and then finally it was M.W. Kellogg offshoot, Kellax, which designed the K 25 isotope separation plant, which is still the one that's running at Oak Ridge.

### Heims:

So that took you away from —

### Montroll:

That took me away from the traditional statistical mechanics things, but [???] hired first Art Sprires, a student of Kirkwood's who worked on experimental problems, as thesis and then he invited me to join him.

### Heims:

Onsager was at Yale?

### Montroll:

He was at Yale. Onsager was very much interested in isotope separation problems and had written papers on thermal diffusion which, while I was at Yale I had read, and in fact some of the Onsager papers which were unfinished in the filing cabinet were on isotope separation. Those were the things I was supposed to look at that we never got around to.

### Heims:

Did he get into war work, World War II?

### Montroll:

He didn't because I think there might have been clearance problems because his wife had relatives that had been in Germany. His wife was Austrian but I think she had a brother who was in a Nazi… and so Onsager would be called in on something but I think he generally wasn’t involved in it.

### Heims:

There were very few, I mean, the whole solution of the icing model happened in the middle of the war and a very few other things in theoretical physics —

### Montroll:

Yes, that's right.

### Heims:

That happened during that time.

### Montroll:

Onsager had done the work and I learned about, I studied his calculations, which were tremendously difficult. Now, the one thing that did carry on during the war was the Wednesday night colloquium, theoretical physics colloquium at Columbia, and Graham, Rabi, Halpern from NYU, most of the theoreticians that were in Now York went to that Wednesday night colloquium, and phase transitions were discussed, and I remember I gave a series of lectures on the Onsager solution of the icing problem at that time. That was probably the first seminar series ever given on the Onsager problem, and Willis Lamb got very much interested in it at that time. Lamb then wrote some papers on phase transitions, Askin and Lamb, Askin and Teller, those all came out at that time. So all of these people were involved in war work at Columbia except Teller had gone to Los Alamos, but the rest of the group that had been working on isotope separation problems were spending some time — Nuremberg was also there, so it turned out there was a lot of interest and discussion. And at that time [???] Kaufman appeared. She was a graduate student at Columbia, and I think it was Graham or Nordsik that said, "We’re too busy now with other things.” The Columbia group in the physics group was working on magnetrons, and the radar business. They made the best small magnetrons. Whereas the group that was more connected with chemistry were more involved in the Manhattan Project. They had the Manhattan Project group, so called FAM group that was at Columbia, but I was involved with the one at Kellogg’s.

### Heims:

Yes. So [???] Kaufman was sent to Onsager?

### Montroll:

She was sent to Onsager, and she had studied group representation theory in the papers of Vile and Brower, and she realized that what Onsager had been doing was very closely connected with the Brower-Vile theory of group representation, and so that's where the Kaufman-Onsager papers stemmed.

### Heims:

What became of her?

### Montroll:

She lives in Israel now, in a kibbutz, and does practically no physics. She plays the piano, most of the time. Her husband, Selig Harris, I think is still professor of linguistics at the University of Pennsylvania. She probably still visits America occasionally. I used to see her frequently but I haven't seen her in recent years. She did work at Wiseman Institute for a while, on polymer chains, but I think in recent years she's not done any physics. A very clever woman.

### Heims:

That's the main thing she’s known for.

### Montroll:

That was the main thing, that's right. She also worked at the Institute later. I think she was Einstein's assistant for a while and worked on the two body relativistic problem.

### Heims:

What time is it? (omit… )

### Montroll:

Is this the kind of thing you want me to reminisce on?

### Heims:

Yes. There’s a lot of things, I wonder —

### Montroll:

— you may want to ask specific questions too.

### Heims:

Yes. I'm still trying to link things up. I'm glad you connected the summation of rings diagram, and this is maybe jumping too far ahead — I wonder how, later, the cluster expansion theory connected with Feynman diagrams, in terms of Feynman diagrams to deal with the phase transition problem, and how those linked up.

### Montroll:

Yes. Well, they started independently and certainly Mayer clusters, those were the first diagrams in theoretical physics. Now, Feynman knew about cluster integrals. I don't know whether they really influenced him in making the Feynman diagram. I mentioned that I was at Princeton in that interim between Cornell and going to New York. Feynman was then working, just starting to work on his Feynman diagrams at that time, and the thing that he was most excited about then was the idea that you could have a particle with the arrow one way and the anti-particle going the other way, and I don't think he really had thought through the diagram business then yet as a calculating device, so his original thoughts were more in ways of representing complex processes pictorially than as calculating devices. That came later, but the thing that he did notice at the time was again in connection with integral equations, which I had also noticed in that period. That is, he took the [???] function of the Schrodinger equation connected with —

### Heims:

Yes, but later on the methods seemed to be more taken from the point of view of your theory.

### Montroll:

That's right.

### Heims:

And I don't know if there's a connection to cluster.

### Montroll:

Yes, well, you can do the cluster integrals by Feynman diagrams just as well, and I think John Ward and I in our paper on Fermi gas and so on, the first direct connection was made.

### Heims:

I see that's you and John Ward, Fermi gas.

### Montroll:

Yes, around late 1950's. Now, there are so many different paths that one can take, I don't know (crosstalk)

### Heims:

…review article, [???] and Montroll REVIEW article.

### Montroll:

Yes, why don't we continue on the icing (Izing?) path. Now, Onsager had done all his work by brute force, and then Kaufman put systematics into it, with the —

### Heims:

She was a post-doc with Onsager?

### Montroll:

She was first, her PhD thesis was actually on the icing model and most of the work that was the Onsager-Kaufman work was in the PhD thesis. Then she was originally to be a student of either [???] or Nordsek but they said, “Onsager knows so much more about this, why don't you do it with him?" So that worked out.

### Heims:

She wanted to do —?

### Montroll:

She had recognized that there was a connection between Onsager's work and the Brower-Vile theory of group representation. Now, since I never discussed with Onsager precisely how much he did and how much she did, I think his influence, she, I believe, did the partition function that way first, but then I think he was the one who used the same methods to try to do the correlations, and that simplified his analysis of the correlations because by the methods that he used originally, it was just very, much more than the complicated and less systematic, than the Brower-Vile theory. Now, at the end of the war, Onsager got a letter from pointing out that what he was really doing was [???] algebra theory. Now, whether that letter still exists, I don't know, whether [???] has his own copy, whether there's a copy in Onsager's files somewhere, but if you can find the Onsager —

### Heims:

— isn't it that (Lee?) algebra is already implicit in the group idea?

### Montroll:

It's already in the group idea, but I think the [???] algebra still simplifies it further, so it was [???] who first noticed that what Onsager had done — I think first wrote a paper during the war on the cluster expansion, that is, the —

### Heims:

— temperature?

### Montroll:

— the low temperature series.

### Heims:

Yes. [???] is originally Dutch?

### Montroll:

He's originally Dutch.

### Heims:

But then worked in Germany?

### Montroll:

Well, he was in Switzerland for many years, in Zurich. And he's also a great man in history of science, and he wrote these famous books on group theory. I think his main interest now is history of science. Remember, he edited the collection of papers on quantum, on matrix mechanics. The one that has come out as a Dover book. And he also is a great expert on Babylonian mathematics. He’s a very learned man. So he was the first to do the low temperature series of expansions. The high temperature expansions had been done earlier. I think the first discussion of the combination of low and high temperatures appears in the [???] review paper.

### Heims:

Yes. I think I saw something by Kramers also, a paper, magnetism —

### Montroll:

— yes, Kramers and [???] did the high temperature expansions for the Brock spin wave model, and in that they have already the discussion of interaction of spin waves, so the first high temperature expansions I think were done by Kramers and [???], not with [???] but it’s very similar to –

### Heims:

Later on with the, later on in the fifties the series expansions because nearly a technology.

### Montroll:

Yes, that's right. They became very fashionable. They started with — I guess I have here some of the, you know, all the references that I pulled out — Ricebrook was very early with series expansions. Wakefield, who was a student of Ricebrooks, in the late 1940s, there are already series expansions, and then Cyril Dom was in them and then there was the Japanese group, Noguchi, Tanaka, and so it became a —

### Heims:

What do you have there, is that the —?

### Montroll:

Paper with Newell.

### Heims:

Oh — the one I was referring to, the Newell, yes.

### Montroll:

If we want to do this in some systematic order — once the Onsager-Kauffman papers came out, then it looked as though one could do much more, and in many places, the three dimensional Izing model was assigned to people as PhD theses. Nertzig after the war had moved to Illinois, and Gordon Newell, his graduate student, was designed the three dimensional Izing model. Now, in Holland, the Dutch – [???] was told to look at the three dimensional problem, and in England someone else was told. All over the world people were told to look at the three dimensional Izing problem by the Onsager method, which it turns out of course didn’t —

### Heims:

Were their efforts recorded?

### Montroll:

It didn't work. But many people tried.

### Heims:

Did they publish their efforts of the difficulties they encountered?

### Montroll:

No, there were no papers published. I'll get back to the closest to publication on that. But there were a few clever students who thought about the hexagonal lattice or triangular lattice, and so they forgot about their thesis assignment. Newell was one of them. He was at Illinois, and he thought he would try by the Onsager-Kaufman method the hexagonal lattice and the triangular lattice, which worked. [???] did it in Holland. He was a student of Kramers. Now, I don't think any of the others published. I think those are the only two whose — I can't remember whether — there may have been a Japanese who also recognized the connection, but the Japanese than quickly moved on to these strange lattices.

### Heims:

Lattice models.

### Montroll:

These various lattice models that were solvable models by the Onsager-Kaufman method. Then Newell came to Maryland, as the war was over. I went to the University of Pittsburgh. Then I was at Office of Naval Research for two years, head of the physics branch, when the Office of Naval Research had just started, so that was a very interesting experience, so there are all kinds of interesting sidelights for myself but (crosstalk)

### Heims:

— totally unrelated to this.

### Montroll:

Unrelated, but in the same way there were many interesting things about the Manhattan Project that I can tell you, about how it affected other later things that I did, but it has nothing to do with phase transitions, so I’ll stick to the phase transitions. Newell came, and at that time I was also interested in lattice vibrations because I was starting to get many graduate students, and there's the problem of thinking of interesting PhD theses that are on topics that not many other people are working on. So it looked like lattice vibrations were a good thing. Gordon Newell got interested in that.

### Heims:

In non-linear terms?

### Montroll:

Mostly the linear things but he worked with — what became fashionable was the distribution of normal modes — so this paper in 1947 I think on the two dimensional spring and mass model for doing — and that was the first place where singularities appeared in the normal mode distribution.

### Heims:

Yes. Yes. Was Mayer Doodin one of his students?

### Montroll:

Mayer Doodin came as a post-doc. He was a student of Frank’s at Bristol. While he was an American, he went under the Marshall Plan. And got a PhD with Frank.

### Heims:

There’s an old theory of melting, Lindemann’s theory of melting.

### Montroll:

Lindemann’s theory of melting.

### Heims:

Nothing ever was built on that?

### Montroll:

Oh, there are a number of people who have worked since on that, but somehow nothing has become so clear that it has become fashionable.

### Heims:

Yes, as lattice vibration.

### Montroll:

Yes. So there were a number of important papers in the seventies, on that problem, but nothing is really clearly worked out. And there are all ways of doing Lindemann theory in a more sophisticated manner. One thing that seems very doubtful, and that’s that you just do simple anharmonic vibrations. It’s highly likely that melting has to do with more cooperative effects between many molecules, making dislocations. My own view on how a proper theory of melting should come is that as one gets close to the critical point, that one starts to get an enormous number of dislocations formed, and these dislocations start to interact with each other, and it’s the dislocation, it's the excitation, rather than the anharmonicity, because the dislocation starts to involve many many atoms, and can start to lead to long range interactions. So I think it's a condensation of dislocations, all of which start to occur in a very small temperature range, rather than just anharmonicity.

### Heims:

Let's see, then you got involved with the spherical model.

### Montroll:

Why don't we finish with Newell? What appeared then is, while we were interested in lattice vibrations, there was a magnetism meeting here in Washington and I was supposed to review the theory of Izing model and phase transitions and so on, and at that time Kaufman had mentioned to me the letter about Lee(?) algebras, and so, she also sketched out what some of the ideas were. As a matter of fact, I think she visited Maryland and Gordon Newell and I listened to that, and then we decided that, in a review article, that we would rather present that method, instead of our version of the Kaufman-Onsager, even though he was the great expert on the Kaufman-Onsager, and at the same time, the Katz-Wert ideas came out, using the diagrams. Now, the basic ideas there I think go to Ward, the idea that you could write a determinant, I mean a participant function as a determinant, putting the diagram together from that , that the determinant generates all these closed diagrams that are equivalent to the ones you get in the series expansion, the basic idea of Ward.

### Heims:

Oh, this is the combinatorial way of deriving.

### Montroll:

The combinatorial, yes. Now… [???] mentioned that there were probably many interesting papers that were never published or researches that were never completed, on phase trans1tions, the Izing problem. During the early forties, many people worked on the problem who didn't succeed in getting it solved. For example, Pauli and [???] at the Institution for Advanced Studies.

### Heims:

Who's the second one?

### Montroll:

Joseph [???] worked very hard on the Izing problem without any success. Many PhD students were given the three dimensional Izing problem to solve by the Onsager method after it appeared. There was only one solution that almost reached publication. That's the work of John Maddox at Kings College in London. Do you know John Maddox? John Maddox was a student of [???] at Kings College in London. That was before Cyril [???] had moved there. [???] was interested in molecular spectroscopy and quantum mechanics of molecules, and wasn't especially learned in statistical mechanics. His student John Maddox got interested in the Izing problem and there is a little dramatic incident that I was involved in, concerning John Maddox. There was a meeting in Paris in the early 1950 and I can find the exact date for you if it would — it was '51 or '52. Maddox was reputed to have solved the three dimensional Izing problem by the Onsager-Kaufman method, was given a professorship at Manchester on the basis of this work, and was invited to the phase transition conference in Paris, to present his work. At that time John Maddox was a very Bohemian looking young English physicist with fiery red hair. He is now editor of NATURE. So he came to Paris, presented his analysis, which was 1mposiible to follow, and at the end came out with a formula which was just like Onsager's formula, but with three co-signs in it. Now, it turns out that several people, including myself, immediately after the publication of Onsager's two dimensional solution, guessed what the three dimensional would look like. We said, "You put three co-sines together with three hyperbolic tangents, and a different placed factor with 1 over 2 pi cubed in front and you have the three dimensional one." Now, how would you test it? The way you test it is to make the high temperature expansion. So I was very excited and I started to do it, because second order term was right, the third order term was right, the fourth order term, and one has to go to, if I remember correctly, about the seventh order term before it deviates, and then it deviates by only the smallest amount, so one thinks shat one has made a mistake. So you do it ten times, and you still discover that it deviates. So I knew that formula. When Maddox gave his — that's something that nobody had ever published, but nevertheless it was known to several people who had the disappointment of not having it come out right.

### Heims:

The Paris conference would be like '48?

### Montroll:

No, around '50, '50 or '51. I'll get the precise date. I can tell you what…

### Heims:

1951 in Paris.

### Montroll:

So immediately after the Maddox lecture, I pointed out that it wouldn't be right for the reasons that we mentioned, and Maddox had another review session. He went over his work and then discovered that in his calculation, he wanted to diagonalize the appropriate matrix, and he calculated the off-diagonal elements. In calculating the off-diagonal elements, one has to take sums of roots of unity. Now, if you take the second diagonal, it becomes a sum, there are certain sums over all roots of unity which then mash. Now, when you go to the next order one, it turns out that there's one root of unity that's left over next, you have — And in his summation, he didn't watch the summation limits, and he assumed that he just summed over all the roots of unity and got the vanishing of all the — so he found the mistake at the meeting. And then there was great discouragement. At the end of the meeting, we went back to London together. We took the night coach to Dover, and he was just very very shattered, because he'd gotten the job on the basis of this, and so I had a considerable time trying to encourage him, "Everybody makes mistakes." He's a very bright guy, and then I didn't see him for a while, and noticed that his name appeared as science correspondent of the MANCHESTER GUARDIAN, so he moved to the newspaper and became an absolutely outstanding science writer. And then he became editor of NATURE. So that's an example of one of the false starts of the time.

### Heims:

You mentioned Pauli. He made some efforts at the two dimensional…

### Montroll:

Yes — the two dimensional and also the three, he and [???]. But he had, with considerable disappointment, he and [???] didn't get anything new out of it. Pauli was very impressed with Onsager's work, which I mentioned — at the end of the wary when he was in correspondence with Casimir again and Casimir asked him "Was there any important theoretical work done in the United States during the war that they might not know about in Europe?" Pauli replied that the only thing really interesting was Onsager.

### Heims:

There's a collection of Pauli letters. Is that where this appears? There's a book of Pauli letters which I think Weisskopf has edited?

### Montroll:

I don't know. It's probably in one of those. It would be nice to get the original letter.

### Heims:

When would that be, about 1945, after the war?

### Montroll:

Yes.

### Heims:

That's worth looking for — to quote that — yes.

### Montroll:

— try to find the original letter.

### Heims:

Are there other events at conferences that stand out for you that might not be prominent in the Proceedings?

### Montroll:

The only other one is well known, that's Onsager's writing a formula for spontaneous magnetization on the blackboard first in Ithaca and then in —

### Heims:

Now, what were the issues in phase transitions? Did one try to do more with?

### Montroll:

That was the Florence meeting. The big question at the time — I think Kramers was the first to make a point on this — the big question was whether it was possible from the partition function to discuss both the ordered and disordered phase, because of an example, if one takes liquids and gases one makes one model for the solid, for solids and liquids, then you make another model for the liquid, and in each one, the partition function is calculated separately. So the basic question was, could you make a model that would have the full phase transition and have both the ordered and disordered phase in it, and that was what Kramers was looking for with the Izing problem. Now, incidentally, in Izing’s original thesis, he didn’t get a phase transition in two or three dimensions, and one of the reasons Heisenberg invented the Heisenberg model of ferromagnetism was, to find a model that would lead to a phase transition.

### Heims:

Yes. So was the Izing model seen as answering Kramers…

### Montroll:

Yes, the Onsager solution

### Heims:

— because you have the lattice structure built in, and part of the difference of the solid and liquid is that symmetry appears.

### Montroll:

Yes, symmetry appears, but the symmetry is more subtle through the relations between the low temperature and high temperature series expansions, the two partition functions, you don't need the series expansions. But as soon as Onsager gave the two dimensional model that's had both phases in it, then it with became clear that with the proper model one could get the full picture of the phase transitions.

### Heims:

Yes. It seemed the Izing model answered that so when later theorems of Van [???] and [???] it seems, well, of course they used the more realistic potential, but essentially people knew the answer by that time.

### Montroll:

People knew the answer by that time. They were convinced that —

### Heims:

— it just gave some insight into the mathematics, but —

### Montroll:

Yes. It was an interesting period then, because — have I mentioned this lunatic fringe? There was a lunatic fringe which would appear at Physical Society meetings, which would have their own small sessions discussing the Izing problem. (Come in…) The lunatic fringe used to come in and (crosstalk)… and some of the members of the lunatic fringe were Mark Katz and Arnold Seegert and [???] and [???] Kaufman, Yang and Lee, and —

### Heims:

Yourself?

### Montroll:

Myself and Onsager, if he was at the meeting, or Lewis Lamb if he was at the meeting. But at that time, the bulk of the members of the Physical Society had only a very minor interesting this question, because then —

### Heims:

The question being?

### Montroll:

Phase transitions. Then the excitement was in resonance work, in nuclear physics, in quantum electrodynamics, and so we were considered to be sort of a side group and very few papers were actually given at the Physical Society meetings on phase transitions. But in the early fifties — this was in the late forties — by the early fifties, the explosion occurred, and many people started to work on the problem. Almost everyone who was at the Institute for Advanced Study at that time in physics, in theoretical physics, devoted some attention to it. Yang and Lee were there at the time [???] was there, [???] was there. All of them became, all of them became involved in the phase transition problem, even if they didn't read or write papers on it.

### Heims:

So it sort of was —

### Montroll:

— Princeton Institute —

### Heims:

I was trying to think what distinguished these people.

### Montroll:

(crosstalk) …was also considered to be high class mathematics. It was a style that they would like to participate in. And there were the corresponding groups in Oxford. Christ had a number of people working on the Izing problem. In Holland Kramers had others working on the Izing problem. At University of Illinois [???] had people working on the Izing problem. It was not unusual at the time to have the three dimensional Izing problem assigned as a PhD thesis problem. Several people with that assignment did manage to get something out of it. One was Gordon Newell, who changed his own problem and made into calculating the participant function for the hexagonal and triangular lattices.

### Heims:

His thesis was with?

### Montroll:

With Arnold Nordsek. Now, Nordsek participated very much in the Columbia theoretical physics colloquium during the war, where phase transitions were discussed, and he was very much interested in the problem himself, even though he wrote no papers on it. Lamb, who was his office mate, did write the very nice paper on correlations in long range order.

### Heims:

Yes, Lamb was part of that group?

### Montroll:

He was part of that group.

### Heims:

And you also were, and some of the others? Yang was ?

### Montroll:

Yang wasn't in New York at the time. Peierls occasionally came to those.

### Heims:

Who were the other people interested in phase transitions their — at Columbia?

### Montroll:

There was the experimental group. Siegel, who measured specific heats of various alloys, and Andy Lawson. Andy Lawson was his student. Andy Lawson later went to Chicago. But Lawson and Siegel were very active on the experimental side of thermodynamic properties of alloys. Originally order-disorder theory with alloys was discussed in terms of the X-ray order lines, where the order disappears in the X-ray structure. But it was the Columbia group which was very early in actually measuring thermodynamic properties of alloy, the [???]-Lamb report of transitions. So the interest was both experimental and theoretical. Another survivor of the thesis assignment on the three dimensional Izing problem was [???] at Leyden, who converted his thesis topic to hexagonal and triangular lattices. His work was very similar to Newell's, using the Onsager-Kaufman method. Now, at that time the series expansion method became popular. That turned out to be more fashionable in England than in the United States, with Gong and with Russ his student, Wakefield, also the corresponding Japanese group. There the game was to make longer and longer series expansions. At first none of the people who were doing algebraic analytical work took much interest in the series expansion, and didn't pay much attention to Gong, especially Gong's analysis. In the end Gong was much more imaginative than was originally thought at the beginning, with his clever ways of extrapolating terms in a series and guessing how the phase transition would finally appear through those numerical analyses.

### Heims:

In phase numerical, did they give quite a bit of physical insight into the problem?

### Montroll:

Well, they didn't give too much critical insight but once one had the critical exponent for a two dimensional problem, one night that there were going to be strange exponents of the three dimensional problem, and so Dam was fully confident that there would be some kind of peculiar exponents, had his — Ackree at Kings College, Sacks and Dom and of course Fisher, Fisher moved out of, the series expansion business rather early, but Dom —

### Heims:

Fisher came from England too?

### Montroll:

Yes, Fisher was a student at Kings College. After [???] left Kings College, Dom became professor, the same chair that Maxwell had in the 19th century, so it was fitting that phase transitions be a topic discussed. Dom during the war worked with Christ on radar scattering problems, very good work. But it's remarkable how much information they finally developed, from the series, but without the original Onsager work, would never have guessed that there would be strange exponents in a problem. So that was probably as important an outcome of the Onsager work as any of the other, as the exact solution. Traditional theory, as you know, tended to have square roots.

### Heims:

Yes, the old —

### Montroll:

[???] and the original Landau [???] sets also and square roots. The [???] work had square roots. And so the fact that logarithms and strange powers appeared in the Onsager critical points, when the spontaneous magnetization, and for the specific heats, gave quite a new clue as to what the solution for three dimensions might be.

### Heims:

There's something else that became important. People tended to formulate things more and more in terms of correlation functions.

### Montroll:

That's right, yes.

### Heims:

And then later the Green functions were —

### Montroll:

Yes, and (crosstalk) dependents with hierarchies of Green functions and so on. (crosstalk) At that time there were two competing ways of looking at problems. One was series expansion and the other is hierarchies of differential equations or integral differential equations.

### Heims:

But that was usually just dependent on them, the Bozeman equation, and…

### Montroll:

Yes, eventually the derivations from the Bozeman equation came from that kind of analysis, but on the other hand, even in equilibrium theory, the Mayer cluster integral theory was series expansion whereas the Kirkwood Borne Green approach has the hierarchy of equations for correlation functions.

### Heims:

Yes.

### Montroll:

Now, one of the features that the hierarchy for correlation functions had is that one had to cut off the series somehow, somewhere. Now I think it's well understood that any subtle aspects of the phase transitions will disappear on terminating the series. One also knows now that by just summing certain crosses of diagram, one will probably not get proper understanding of the nature of the phase transitions. But at that time there was optimism on both sides that if you took the right diagrams and bummed them, you would be able to discuss the phase transition, or if you terminated a series by the [???] principle or some equivalent technique, that one could do it. It now is clear that the problem is more subtle. No, once the Izing model was successful, people looked for other models, especially the spherical model became of considerable interest. Katz, Mark Katz had the idea that one could consider the spins, instead of having possible values plus or minus 1 in the Izing model it could be a continuous variable with Gauss distributions. Whether Katz knew about Kramers' theory of electrolytes at that time I don't remember.

### Heims:

Katz really came from mathematics.

### Montroll:

He came from mathematics, but he had many discussions with George Uhlenbeck back at — George Uhlenbeck, who made a statistical mathematician out of him, and since George Uhlenbeck knew of the Kramers work, there is a slight chance but my memory of discussions with Kata at that time, which I had many of he didn't discuss the Kramers paper. I think I was the one rather than George Uhlenbeck who mentioned that the Gaussian model was very similar to Kramers. I'd known about Kramers' work, as I mentioned before by studying theory of electrolytes.

### Heims:

That was of interest because it could deal with the three dimensional.

### Montroll:

It could deal with the three dimensional but it also was clear that the Gaussian model wasn't suitable, and Katz then thought of the spherical model. Tee first analysis of the spherical model by Katz and Berlin was done in an extremely complicated, almost unreadable method that goes back to papers of von Neumann on certain problems in statistics. Now at times in the discussions, I found a much simpler method using techniques of Markhoff that were used in stochastic processes, and that were reviewed in Chandrasekhar's papers on stochastic processes" and I wrote a short note to the Proceedings of the Florence meeting on statistical mechanics at that time. I also had the — the Florence meeting of 1948.

### Heims:

The '48 meeting in Florence, and the — Katz-Berlin uses the saddle point integration, doesn't it?

### Montroll:

That was the final breakthrough that made it simple, and Katch and Berlin didn't write any papers on their original method. I wrote this paper on the Markhoff method (crosstalk) — saddle points weren't important in that method.

### Heims:

Yes. I haven't seen that. Do you have a reprint of that?

### Montroll:

Yes. The idea which I had then, which turned out not to work, was that if one had a large number of auxiliary conditions, one could break the set of lattice points up into subclasses, and one could take one subclass of lattice points and assume that the sums of the squares of the spins of those would become number N of that class, and then take the second class, and so instead of having one spherical condition, I thought, if you had little n spherical conditions — then when little n became large N, one would be able to get the Izing solution, which turns out not to be the case. It was a great disappointment. Later Berlin and I wrote a paper in which one assumed that each spin could be represented by a Fourrier, spin distributions by Fourrier transform with a delta function, for sigma squared, so sigma squared was plus or minus 1, but with delta functions. Then it was possible to express the participant function in terms of this strange determinant. This was the original idea of Kramers, to express the — with electrolytes — to express the participant function as a determinant, which looked perhaps, to be independent of dimensionality, so the motivations for doing, for making that kind of a form for the determinants was that the dimensionality didn't seem to be important. Now, in the Katz-Berlin paper that was finally published, the analysis was simplified tremendously from their original version, and I think that work was due to Berlin, but you can check, I'm 90 percent sure it was due to Berlin, that if one expressed the auxiliary condition, in sigma 1 squared plus sigma 2 squared and so on, had to be equal to N, one could write an appropriate delta function Fourrier transform representation, and then do the integrations. And so that led again to a determinant which could be evaluated.

### Heims:

Yes.

### Montroll:

So it was a period when Ted Berlin and I were busy trying to express partition functions as integrals over determinants, which would be independent of dimensionality, but that was again one of the failures that we discussed. It was very easy to give an elegant derivation of the one dimensional participant function, but we never succeeded in doing the two. On the other hand, it was that kind of formulation that was Wilson's first, Ken Wilson's first try to deal with, and it was that type of formulation that led him to his renormalization methods.

### Heims:

Formulation of the Izing problem.

### Montroll:

Formulation of the Izing problem as an integral over determinants, and using delta function representations for the individual spins.

### Heims:

Let's see — OK, that's —

### Montroll:

The next step then in development was that, one had the Onsager results, the Katz-Ward method had a number of combinatorial problems that were never solved, but if one were convinced that they could be solved, one would have a relatively simple way of presenting the solution to a group of students. The Lee algebra method was also not too difficult to give lectures and courses. On the other hand, the Onsager solution of the spontaneous magnetization and correlations was exceedingly difficult. Onsager as you know never published any papers on spontaneous magnetization. [???] Yang published a paper that was I think considered by most to be unreadable on the subject. So the next breakthrough for the Izing problem really came with the introduction of (off tape) After an idea started in the combinatorial problems associated with on the surface, how many —

### Heims:

— that was somewhat later, wasn't it?

### Montroll:

That was somewhat later. That was in the early sixties.

### Heims:

I'm really concerned, before '60 — yes, I wonder about the field theoretic method that came in — how they connect, as you experienced —

### Montroll:

Well, the field theory — there's another two points. One is the actual solution of the Izing problem by field theoretical methods, which came later, about the same time as the which was the work of Leeds and —

### Heims:

— Elliot Leeds?

### Montroll:

Elliot Leeds and the group at IBM who was in my theoretical group at IBM, but that comes later at the same time. Now, there were the field theoretical methods that came into statistical mechanics, but the Izing problem wasn't discussed at that time. The two problems that were discussed by field theoretic method were the electron gas problem and the hard core(?) gas. Now, you want me to give reminiscence on those, even though they’re not —?

### Heims:

Did people anticipate that sooner or later that would be useful for phase transitions?

### Montroll:

It was thought perhaps that it would be, but you remember, in the early and middle fifties, the problem of Einstein-[???] condensation was a tremendously important version of phase transitions. Also the problem of the —

### Heims:

— with the hard core —

### Montroll:

— with the hard core, and also the problem with the electron gas, was considered to be one of the unsolved problems, because at that time, its various integrals, the correlation integrals which Wigner tried to calculate diverged, and once one saw that you could do quantum electrodynamics and that certain of the traditional perturbation integrals could now be dealt with, there was a renewal of interest in the electron gas problem, thinking that there ought to be some kind of quantum theoretical methods that could be applied to that. Now, those came in while there was a considerable amount of interest in the early fifties, it was really not till '56, '57 that progress was really made, so there again was a situation where many people worked on the problems without any clear success. The closest to what you might call success was the [???] work on the plasma.

### Heims:

[???] did get in those days some low lying energy levels —

### Montroll:

— low lying energy levels —

### Heims:

— partition sum, that would be the first few terms —

### Montroll:

But the breakthrough again came in one very short period and the people were almost all together at the same time. Now, it was the summer of 19 — I think '57 at Brookhaven — where many of the points were discussed with these people… precisely… '57, summer of '57 which was an interesting summer. That summer Yang and Lee were at Brookhaven, [???] was at Brookhaven, Feynman was at Brookhaven, I was at Brookhaven and I think John Ward visited me at Brookhaven that summer or at least got interested in the same problem. Now, Feynman had been interested in the liquid helium problem really following the ideas of Onsager, but not making diagram expansions, whereas the others that I mentioned thought of making Feynman diagram expansions for the thermodynamic quantities, and Bruckner and Gell-Mann had thought about the diagrams being important, and I mentioned to Bruckner at the time that that's precisely what the basic idea was in the Debye-Hickle theory, so therefore there had to be a similar [???] similarity, and the interesting feature was that the, both in the quantum mechanical summation of ring diagrams one avoided the figure divergence in the correlation energy, and in the same way in the Debye-Hickle theory, one got peculiar exponents for the density dependence. Now, if one does dimensional analysis with a charged system, the density, the temperature and the charge all appear together, in the classical theory, as one single unit. In the quantum theory, H E and the density appear, if one goes to zero temperature, there's no temperature, so there's a different combination. Now, in the classical theory, a series expansion in the density responds to a low density expansion at the E, which is a perturbation parameter, appears an enumerator in the same way that the density does. If one looks for the corresponding dimensionless quantity [???] of the quantum gas, with H, E and rho together, the E being the perturbation parameter, if you think of the charge as perturbation parameter, appears an enumerator, but the density appears as the nominator. So what would correspond to a perturbation parameter in the classical theory, in Debye-Hickle's theory is a low density, because the density appears in the enumerator. Whereas in quantum mechanics — And so it was very clear from that that since both the Debye-Hickle theory and the summation of diagrams were in the perturbation parameter E, that there was similarity between the two, and the similarity came that the re-integrals gave the first contribution to both theories, and it was Gellman and Bruckner who worked out the first order correction to the, the first order terms, and then —

### Heims:

Terms for?

### Montroll:

For the correlation energy —

### Heims:

At high densities?

### Montroll:

At high densities. Then with their work done, the thesis was assigned to Dubois who was a Gellman student at Cal Tech, who then computed the higher order terms. He got the right functional forms for the higher order terms, but all the coefficients were wrong. I had a correspondence student, Barry Niman.

### Heims:

Let's see, that still doesn't get me to the critical —

### Montroll:

That's not the critical region.

### Heims:

This is low temperature.

### Montroll:

This is the low temperature high density. One also has the high density, I'm sorry, the low density quantum gas, low density high temperature. Now, with the low density high temperature, the first order, one has the Wigner lattice, so one does the lattice vibrations of the Wigner lattice, the small vibrations of the Wigner lattice, and one has the correlation energies for low densities. See, one is in a peculiar situation — in the quantum mechanical case high densities correspond to the lattice; low densities correspond to the gas. Now, if one calculated the anharmonic contribution to the Wigner latitude, one can get several high order terms in the series expansion for high densities, for low densities. One starts with the correlation terms at low temperature, I'm sorry, at low densities, at high densities, but one has to use the correct coefficient that Barry Niman derived in his thesis. So one has the series expansion at both ends, and now the question is, how do you get a phase transition from that? And I think, I'm prejudiced, but I think the only serious work that rotates the right critical point Bas the parameter R S, which is the measure of the dimensionless constants in measuring the density of the quantum electron gas was done by Ishahara and myself, and the method was to do a two point [???] approximation, where one takes all the terms in the low density series from the Wigner lattice, and all the terms that are known in the high density series from the correlation energies perturbation calculation, and then one finds a safe transition at R S approximately equal to about 15 or 16. In most metals —

### Heims:

This is using a [???] process —

### Montroll:

A two point [???] approximation. Now, in most metals, R S turns out to be about five or six, so it's not in that range. On the other hand it turns out that if one looks at white dwarfs, where actually the positive background and the electron gas are switched, and the positive charged background becomes the important part, it turns out that there are certain things known about white dwarfs which Van Horn at University of Rochester had studied, which seem to be consistent with this estimation.

### Heims:

I'm trying to recapture like the period —

### Montroll:

It was close to 1960. I saw…

### Heims:

I saw something that you, you know, there is a volume, lectures, University of Colorado, summer lectures that Uhlenbeck particularly gave, but you had an appendix —

### Heims:

— yes — yes — that's right — OK, so let's get back to Brookhaven, because this is what I started on and then got off on the specific problem. Now, at Brookhaven that summer, Feynman spoke about, Feynman was interested in the question of getting transport coefficients and he and Mike Cohn were interested in the…

### Heims:

— for helium —

### Montroll:

— for helium, but their methods weren't perturbation methods at the time. Yang and Lee became interested in the hard sphere model, and then developed the diagram procedure for doing the hard sphere gas; at the same time, Bruckner and Gellman had been doing the electron gas problem, looking only at the Green diagrams, and Ward and I were looking at what we thought was the general procedure of connecting Mayer crustal integrals with Feynman diagrams, and so there were three groups of us which accidentally that summer were starting to work on the Feynman diagram approach to statistical mechanics. Well, in the end, it occurred that we were all doing essentially the same thing. We were all doing it in our own language at the beginning, and while we were talking to each other, we didn't always realize that we were doing that.

### Heims:

But you did talk to each other.

### Montroll:

We talked to each other that summer, and somehow John Ward and I were doing our diagrams on cylinders, and Yang and Lee were making various lines which we didn't think had any connection with ours, and the ring integrals seemed to look different in the way Gellman and Bruckner were doing it. But as I say, in the end we were all doing very similar things. But it was a remarkable situation, that there were three teams there, none of whom at the beginning especially realized that the others were involved in that problem.

### Heims:

…three teams there… yeah…

### Montroll:

And at that time there was great optimism, that one would do all statistical mechanics, quantum statistical mechanics problems, by these series expansions. It turned out to be a great success, as far as making the next step with the correlation energies for the electron gas, and with regards to getting the first few terms. Also, Tai Wu was at Brookhaven that summer and worked with [???] and worked with Yang and Lee on the…

### Heims:

Eventually, the [???] method came to play a —

### Montroll:

— oh, there was a tremendous amount of activity, but it all finally reached a stone wall. And so while there was great activity that lasted up to about, into the 1970's —

### Heims:

You mean in the 1960's (crosstalk)

### Montroll:

— there were an enormous number of papers —

### Heims:

— do you recall your talk after presenting… you expressed —

### Montroll:

— oh, it looked as though everything was doable, and a time dependent —

### Heims:

(crosstalk) No, you weren’t that [???] at all —

### Montroll:

— oh, by then — that's right — The Folger lectures, one was starting to become, several of us were becoming —

### Heims:

I think you phrased it as “In spite of it all, we hope that things are right with the world,” something like that.

### Montroll:

Yes, that's right. There was the great excitement, but to get each higher order term became more and more complicated. However, progress in both the electron gas and the hard sphere gas did get beyond what the original Mayer cruster(?) integral theory progress was for normal gases, simply because the hard sphere potential and the Columb(?) potential were somewhat easier to deal with than the Van der [???] potential.

### Heims:

Somehow these came into Wilson's later work, didn't they, some of these?

### Montroll:

No, Wilson also got started on some of these things, but when he finally did his renormalization, there wasn't much sign of that kind of diagram. But he being a student of Gellman’s had learned all these methods.

### Heims:

Yes. Was he a student at that time?

### Montroll:

During the latter part of that period, yes. But the next step was looking at the non-equilibrium features, and what became clear by [???]’s analysis and now Green's analysis, and of course Feynman had done a little work on this, was that if you could consider some external potential as being the perturbation theory in that external potential, one could by diagrammatic methods correct the equilibrium values for various correlation functions and make the time dependent correlation functions and all —

### Heims:

— get the transport —

### Montroll:

— and all the transport, so there was tremendous excitement about that.

### Heims:

Actually you could get those in principle around the critical point as well.

### Montroll:

And you could get those around the critical point.

### Heims:

— although I don't —

### Montroll:

— Mel Green was especially concerned with —

### Heims:

— transport coefficients around the critical point —

### Montroll:

critical point, so there was a certain amount of success there, but it also changed to a stone wall.

### Heims:

What was the stone wall, the higher order terms?

### Montroll:

The stone wall came just when higher order terms and also, in actually doing some of these correlations and integrals, because you then had to find the correlation function from the (???) one technique was series expansion, but that became limited, in how many terms you could get. The other was by getting a hierarchy of —

### Heims:

— equations —

### Montroll:

— of integral differential equations, at various correlation functions, that would have to be terminated by some method like a [???] progression method, and it finally became clear that around critical points, that neither of those methods was suitable.

### Heims:

There was one other approach that some people did, Monte Carlo(?), using computers —

### Montroll:

Yes, there was a certain amount of computer work.

### Heims:

— that had a —

### Montroll:

— the most successful computer work was that of Bernie Alder on the, and his group, on the classical liquid situation, and that was done — you know, Bernie Alder at the meetings said, "It's a wonderful way to do statistical mechanics. All we do is use these big computers." But when one asked Bernie Alder how much the calculation would cost if one was not at Livermore, then the answer was a little embarrassing. It turned out that Bernie Alder had the advantage of having all the great computers that were going to be used for the analysis of nuclear test data at the time when the nuclear test ban appeared. You remember, they halted tests out in the Pacific, and Berkeley and Livermore were the places that were computerized with the most sophisticated computers to analyze the test data. And then suddenly when there were no tests, there was computer time, and Bernie had the great advantage of having those computers available —

### Heims:

Well, one thing, that happened after World War II, funds went into physics. That is, the PHYSICAL REVIEW grew and physics expanded a lot.

### Montroll:

Yes. I don't know how that affected the quality of what happened, did more remarkable things happen after that, or?

### Montroll:

Oh, I think more remarkable things happened. If one took the fraction of members of the physics community before the war who did remarkable work, that fraction was probably larger, but when the whole community expanded, the total number of people doing great work was larger, but the fraction may have remained smaller. On the other hand, so much of modern technology —

### Heims:

I'd say — in phase transitions?

### Montroll:

In phase transitions, the progress was small, and most of the work was done still by a small group, and even today the original ideas come from small groups of people. Now, I had the pleasure of being head of the physics branch at Office of Naval Research at the time when money was first put into physics, which was very exciting, in the late forties, '48,'49. I found it very exciting, and people at ONR had a kind of personal independence that doesn't exist in institutions like NSF. We were more or less left alone to think of what kinds of projects, pick good people, to try to keep the work with some connection of topics that the Navy, that might be important to the Navy, but there were no review committees, no peer analysis of proposals. One was supposed to use good taste, and while it's now considered unfashionable to say this, but the Old Boy network was very good. And at that time, the network was very strong, because a community of good people was — everybody knew each other, and this was an exciting time to know everyone. One hardly had to be in the community very long. One could be 25 years old and already know all the important people in his field. Now one's lucky if he's 35, to get the same kind of responsibility.

### Heims:

Meetings tend to be very large.

### Montroll:

Everything gets to be specialized. So that was an exciting period, and ONR for example spent the Americans to Europe to the early statistical mechanical meetings after the war, and helped bring the Europeans here. The London office of Office of Naval Research was an outpost where the news could be picked up. So the transfer of information and the transfer of people —

### Heims:

— there must have been kind of a rush after World War II.

### Montroll:

There was a rush after World War II.

### Heims:

— because people had been kind of isolated —

### Montroll:

— yes, the isolation, and on both sides of the Atlantic, people were tremendously hungry to meet either their old friends from before the war, or, for the younger people, to meet the other younger people whose names they recognized. So there was tremendous excitement and the esprit de corps that existed at that time, was wonderful, but again, the Office of Naval Research played a tremendous role in bringing people together, and at that time, for example, Navy helium was being sent to the European laboratories, for low temperature work (crosstalk) — to Leyden and to Oxford.

### Heims:

What sorts of things were you, was it related to the statistical mechanical problem?

### Montroll:

I was personally still working on statistical mechanics problems, when I had time to do them. That was the period when I was working with Ted Berlin and John Cochran on the spherical models. We did some papers on that. But then I came to the University of Maryland and worked with Gordon Newell and [???] Potts. Potts was a student of Cyril Dom’s. Just as there is the so-called Lance Izing problem, the Potts model should be called the Dom-Potts model, because Dom originally proposed the model. Potts wrote his thesis on it which was a very nice thesis. There were two aspects of the thesis. One had to do with certain series expansions that Dom was interested in. The other had to do with more analytical things, about the — what's now called the Potts model. Potts then, or Dom wasn't much interested in the analytical part, and no joint papers were written, so the papers that were written with Dom really didn't emphasize the model as such, so the name has become the Potts model instead of the Dom model. Or the Dom-Potts model. Now, Potts continued with his interest in statistical mechanics. He came as a post-doc to Maryland and we worked together on lattice vibrations with [???] and then later on the correlation coefficients and the Izing problem using the [???] methods. But after leaving Maryland, Potts went back to Australia.

### Heims:

Was that his origin?

### Montroll:

He was originally from Australia. I got interested in the traffic problem while he was — no, he went from the University of Maryland to Toronto. At that time, which was in the 1950s, about the same time as Ward and I were working on —

### Heims:

— you were working on shock waves at one time.

### Montroll:

(crosstalk) I worked on traffic with Bob [???]. There is an interesting story on the traffic business that goes back to working at Kelax during World War II. During World War II at Kelax, an important problem in the design of an isotope separation plant that would have an enormous number of stages was the stability of the isotope separation plant. If there was a small fluctuation in the plant, would it propagate and be amplified? Or would it be damped out? So one of the problems that I dealt with at that time was just the stability problem. Now, if you think about a separation plant with a large number of separated units that are connected with each other in a linear fashion, it's very much like the one dimensional lattice. So if the system is working at equilibrium, then all the units will have essentially the same characteristics. On the other hand, if there's a fluctuation, the fact that one unit is changing interacts with the neighboring units, so you can write down mathematical equations. They're not unlike the equations of lattice vibrations.

### Heims:

Right. Yes.

### Montroll:

Now, if you think of a defect problem — suppose one unit goes bad, and you ask, how does the influence go? The characteristic that one uses to describe that unit will be different from the neighbor's. That's precisely the kind of problem one gets in lattice vibrations, when one bas defects — you say, there's one mass that's different from the rest. And since all of this work was classified, in a sense it's still classified, no papers have been written on stability — although it would be possible now to write such a paper and it would be declassified, if one wants to go to the trouble to do it. But there was a question of... how could I… (off tape)… OK, now, if one is concerned with the problem of propagation of a stable fluctuation in an isotope separation plant, there are certain differential equations that one writes down that relate to the pressures and temperatures in one stage, with the next in the one, so a certain amount of analysis was done in that, and all of that analysis was very much used in the design of the piping, the pressure control and so on in the plant; and if it hadn't been used, the plants would have been a disaster, because no traditional chemical control plant expert had ever dealt with any plant with so many units, and their whole style of dealing with control of chemical plants was completely different from what had to be one in control that of isotope separation plants, and the original design that came from the chemical engineers would have led to unstable plants, which we corrected through our analysis, and the people involved in that were Joe Rainer, who became a great expert in numbers theory, Leon Hanken who was a symbolic logistician(?) and Manson Benedict, who after the war became chairman of the nuclear engineering department at MIT. So in any case, we'd done all this work, and there was the question of finding something publishable from it under a different name and part of it became "lattice vibrations with defects." Lipshitz had done some work of that sort independently, but I don't think he had anything to do with isotope separation plants. He was brought into the problem directly from lattice vibrations. Now, with Potts — my work with Bob Herman, who had moved to General Motors technical center, and who had asked the question,”What should one do there?" and was told he should work on traffic, so he made these models of traffic, and Potts then got interested in that, and moved from statistical mechanics to operations research and traffic when he went to Australia, and the whole Australian school of operations research, which has been highly successful in all branches of Australian life, from road service to banking to —

### Heims:

— all goes back to Potts —

### Montroll:

All goes back to Potts, not only in Australia but in Shanghai. Certain directors of the big Shanghai banks learned operations research from Len Potts.

### Heims:

Let me ask you about a different thing, which I probably — I'm thinking about from 1940 to 1960, phase transitions. We've talked about some aspects of that. If you ask, what the important points are, what are the important issues, important events, whether they eventually turned out right or wrong.

### Montroll:

Yes. I think that one [???] report and it turned out to be right, refers to the fact that one got strange exponents. From the Onsager theory, then from the Dom and Kings College analysis, and finally with the use of work of Baker's on [???] approximates, that one starts to get a pretty good picture of how strange exponents became very very important.

### Heims:

Yes. Why is the three dimensional strange, because it's different from the classical?

### Montroll:

The results are all so different. You remember the 18 power? for the spontaneous magnetization? The two thirds(?) power for the [???] — all these. So that was probably the most important outcome. On the other hand, another important outcome was that the traditional theories were right very close to the phase transition, so that if you actually looked at experimental data, the old square root laws looked almost right, until one got very close to the phase transition, and the subtleties appeared only at temperatures (crosstalk)

### Heims:

— you know, at the phase transition you had very large fluctuations.

### Montroll:

You had enormous fluctuations, so that was already known to [???] Kovsky and —

### Heims:

So how can one observe the, those exponents very accurately?

### Montroll:

Well, the important thing is, you have to have the systems come to equilibrium fast, even though there's — that was where the work of Fairbanks and Buckingham was so tremendously important, because the first phase transition specific heat that was measured with tremendous accuracy was that for liquid helium after the lambda point. Now, the remarkable thing about those experiments has that one had a block of an enormous amount of copper and essentially no helium in it, so when you have a block of copper, that's the main thing, with no helium, the fluctuations in the helium proves important and you can conduct the information from one part of the helium to another very easily through the copper. You remember that the cylinder the helium was in had ever so fine capillaries, of the helium, so the total mass of the helium was practically nothing, whereas the total mass of the copper was very large. On the other hand, the specific heat of the helium was infinite. So even though there was no mass, the specific heat was enormous, and then the corresponding experiments were done by a Russian [???] and that was the time when one thought that there should, on the basis of this experimental data, be a kind of universal theory of phase transitions, so that when one got very close to the phase transition itself, that the nature or the model was only of secondary interest. And that was what finally led Wilson and Fisher, at first Landau and Lipschitz and Ginsberg, to looking for more universal —

### Heims:

— there was the old Landau theory —

### Montroll:

The old Landau —

### Heims:

Was there subsequent Landau?

### Montroll:

Yes, Landau and Ginsberg theory, but it was mostly made for superconductors. There's a remarkable thing that's come out now, in reminiscences of Landau’s students. As you know, Landau essentially has written none of his papers. Lipschitz wrote practically all of Landau's papers.

### Heims:

I didn't know that.

### Montroll:

Landau had the ideas, but apparently when it came to writing, detailed papers, he almost got a kind of hysteria, whereas Lipschitz writes beautifully and understands everything exceedingly well, so if you read the reminiscences of Landau's students, its finally admitted that Landau discussed the work and Lipschitz wrote the papers, even when only Landau's name appears on the papers.

### Heims:

Another thing, there was the Pisa(?) thermodynamic theory. When was that, in the forties?

### Montroll:

In the late 1940s. That was modeled after the Landau-Lipschitz work, and there again that led to traditional kinds of exponents, and it was at the meeting at Cornell where Onsager discussed the Izing partition function, where the Pesaw work was jumped on by the Izing-Kogasaki at the time, so [???] Onsager and I all jumped on Pesaw on that.

### Heims:

I think you've sort of surveyed —

### Montroll:

— that period —

### Heims:

— for a first go-round.

### Montroll:

One should just mention one more thing, about the [???] because they became, and then I'm ready to stop.

### Heims:

OK. I'd like to sometime maybe — we'll have to find out —

### Montroll:

Yes. OK, but the [???] did become the way that the subject became simplified. There was a remarkable coincidence, that at the same day, Fisher and [???], received a preprint from Kostaline on the solution of the Dimer problem, the same day that Kostaline received a preprint from Fisher and Temperlee on the Dimer problem. Now, it turns out that there was a paper by M.S. Green, who was the colleague of Born in Born and Green's work, where he said that [???] are important in statistical mechanics, and wrote a paper, that isn't a very good paper but brought the attention of the public to [???], and both Temperlee, Fisher and Kostaline all saw that paper, and realized that one could do Dimers that way. Then Kostaline realized that you could reformulate the Izing problem as a Dimer problem, and that meant it went back to the old idea of Katz and Ward and the one I mentioned earlier with Kramers and with Berlin and myself, that it would be so nice if you could, evaluate partition functions as determinants, because it would seem as though it should be dimension independent, and so he was able to construct the partition function as a determinant and as a [???] which happens to be the square root of an anti-symmetric determinant, but the combinatorial problems that one has to do by that method fail in three dimensions, so one is in the same position. But at that time, Potts came to IBM when I was there, and we had done work on the defect in lattices effect of defect and it's possible when one does correlation coefficients with the Izing problem to express the correlation coefficients, as a partition function for a defective Izing lattice. So if one wants to find a correlation function between lattice point A and lattice point B, all you have to do is make a line between A and B and assume that the bonds along that line are defective bonds, with the temperature being related, with the coupling constants being related to an inverse of the coupling constant in a normal lattice, and then one has an expression for the correlation coefficient as a determinant. This is why I brought Potts and correlation functions in, because we immediately applied that technique to finding the correlation functions in spontaneous magnetization, and so that was the first proof I think of the Onsager —

### Heims:

Yes, was that Brown?

### Montroll:

We presented the paper at the Onsager celebration at the Brown meeting. Now, Kostaline was at that time working at Shell Laboratories in Amsterdam, and I first discovered the [???] from him, because I was Lawrence Professor at Leyden, which is a visiting professorship, and the person who was Lawrence Professor in a given year was supposed to give a seminar at Shell and one at Phillips, because Shell and Phillips gave a certain amount of money toward that, and when I was at Shell, Kostaline came to see me to tell me about his interest in [???], and that was an interesting coincidence because they were looking for a professor at Leyden at the time , and they were interviewing people from all over the world, and they asked me about someone who might be suitable. It turns out that Kostaline still was in Leyden, went to the Ehrenfest Colloquium every Wednesday night, and was a student of Kramers — everybody in Leyden knew him, but it never occurred to them that he would be a suitable person, because he was after all working off in industry. So on the basis of my recommendation, I said, "It's crazy you're bringing all these people from all over the world, and about three blocks away there's living this local fellow, " and on the basis then of his [???] he was made professor at Leyden. It's turned out very well, but it also has turned out to be probably the best way of solving the two dimensional Izing problem. As you know, there's the Wu(?) — book, which is just based on the… So you should talk to Wu.

### Heims:

Right.

### Montroll:

Now, you asked about the direct use of quantum mechanical methods, that is field theoretic methods, on the Izing problem. At the time that Potts and I were doing the correlations, and Ward also joined in on that calculation, I was at IBM, and Potts came for a year or two to IBM, and so I gave lectures on the [???] method, and Schultz, Maddus and Lee saw certain similarities between that and some quantum field theoretic methods, and so they went on a parallel effort, and then paper of Lee, Schultz and Maddus on field theoretic methods, where they actually found the correlation function as well. But then they learned —

### Heims:

— this was around — (crosstalk)

### Montroll:

— learned from the [???] matrices and applied it.

### Heims:

That was around late sixties?

### Montroll:

That was early sixties. '60, '61, '62. And then there is the story that you can read about the [???] matrices and the Sego theorem, which we have written in our paper. Sego's theorem about the calculation of matrices is very useful toward the calculation of the correlations, spontaneous magnetization. I know about the Sego theorem from Sego's papers, but realized that Mark Katz had done some work. He said, "Yes, he had worked on it." He may in fact even have turned me on to Sego's papers earlier, before we got onto this. But in Sego's papers he mentions that he was motivated to do this work by who was a mathematics professor at Yale. It turns out that got interested in this because Onsager had spoken to him about it, so Sego's theorems about matrices indirectly go back to Onsager, and then when we finally used the —

### Heims:

— to revise Onsager's derivations —

### Montroll:

— to revise Onsager's derivations. Onsager had three derivations; none of them were ever written for publication because he thought they were too complex and wanted to have them simplified before he published them, so there's no Onsager publication on correlations. End of story on this.

### Heims:

OK.