Physics of Scale Activities

Kadanoff interview, part III
 

Interview with Leo P. Kadanoff, part III

III. Phenomenology and universality

SS

    Do you actually remember ever reading Patashinski and Pokrovsky?

LPK

    I read it once to find out whether it included the stuff that I invented. It had in it the scaling of every correlation function -- and correctly. There are other works around at the time which did scaling incorrectly, but Patashinski and Pokrovsky, in my opinion, the time I read it, did the scaling. Not the running coupling constants, and not based so much on the free-energy, and not all the machinery, but the whole scaling story, correctly. [Note added:] I read it in the last year. It contains original and correct phenomenological scaling information.

SS

    If one were to press you in terms of trying to clarify how they derived these scaling laws and where they proceeded, would you be able to?

LPK

    I would not.

SS

    Would the physical assumptions be clear on how to get the scaling out?

LPK

    I don't know. My recollection was that I looked at a series of papers, including theirs, to check whether that series of papers had gotten the physics right. They had gotten, as far as I was concerned, the right bottom-line answers. I do not remember whether their physical derivation was elegant or inelegant. And I remembered that I concluded that it was both early and right. I do not remember the included arguments which were necessarily physically compelling. I assume that it looked somewhat like the earlier paper that I identify with Migdal and Polyakov (probably correctly, but I'm not sure), in which the wrong k^3/2 answer was achieved. If you have to ask me to guess at this stage, I assume that they used some Green's function version of BBGKY hierarchy, and that they then did some self-consistent thing, but I don't remember.

SS

    Is it right by virtue of exhibiting Green's functions which have the homogeneity properties that you ascribe as being right? That's correct, but the justification of it is not quite so transparent.

LPK

    I had stored [it] in my mind as phenomenological and parallel to Widom, but containing different information.

SS

    OK, so the accent is on phenomenological?

LPK

    That's where it is stored in my mind. But a modern reader might see different virtues and defects in the paper.

SS

    So when you compare it to Widom, you read Widom, what strikes you there?

LPK

    Widom did something extremely remarkable. He wrote two papers, back to back in J. Chem. Phys. In one of them he did the homogeneity for the thermodynamic functions, and in the other he did the same thing for the surface tension. He would've had the generating function if you sort of glued the two papers together and extended. That is, I think he had much of the story in mind. The thing I was always impressed by was how much of it he had. He had homogeneity assertions, but I believe that Pokrovsky and Patashinski had similar but different homogeneity assertions for correlation functions. That's my recollection. Certainly when we wrote the review paper and listed the papers that contributed, we listed Pokrovsky and Patashinski, and it stuck in my mind as being correct.

SS

    When you say phenomenological and make assertions about homogeneity in the equation of state, which presumably then is a statement about thermodynamic properties, and without necessarily asserting how you go from the microscopic to the thermodynamic description, that could still be remarkable, right? That's what I would understand as phenomenological.

LPK

    I don't recollect at the moment. It has been 35 years. The word phenomenological would suggest thermodynamic, in the sense of making thermodynamic sense out of correlation functions. But who knows? So, the review paper was written, and came out, and also was a great success, and then we come to the fact that the numerics weren't working quite wonderfully well. The last 3 percent of accuracy for the relationship between the various critical indices wasn't there for the three-dimensional Ising model, although of course it was there for the two-dimensional Ising model. And so I spent the next bunch of years doing stuff which was somewhat unfruitful, trying to derive extensions of the satisfying extensions of the theory which would better fit the numerics.

    Oh, I should mention I talk about the scaling in the review paper. I should come back and talk about universality since that's sort of the more important thing in a certain sense than scaling. The review paper was structured around my understanding of that time of universality. That put magnets and fluids in a different category. However, that's turned out to be false. The magnets and fluids turned out to fall into the same universality class. It could have been different depending on whether the exact symmetry between particles and holes or between the different directions of magnetization, or between high-density and low-density fluids, whether that was important or not. However, there's a more controversial question, and one that I have not been able to satisfy myself about. And I do not believe you will be able to satisfy the people that reach into your archive. The question is roughly speaking what did we know when?

SS

    Can you, on that score, in 1966 you write the paper on reviewing everything. Can you make precise what you understood by scaling at that stage?

LPK

    It is my belief that our understanding of scaling is the modern understanding of scaling, namely, that we understood as you changed the cutoff, you would change none of the crucial properties of the system, but that you would change some of the coupling constants. The coupling constants reflecting the deviation from criticality.

SS

    And was it clear to you how to assign a new spin to the bigger block, the justification of the assignment of a new spin?

LPK

    This might be a somewhat controversial issue. I will allow myself, as I'm not going to do very often, to get into the old controversy on this, because in fact Wilson in 1971, I think, misread the 1966 physics paper. He suggested that the block spin involved simple summation, and did not understand that I was willing to do a renormalization at that stage. So I think he was a little unfair to the point of view that was taken in the 1966 paper, which I believe is essentially the fully correct Wilson-Wegner point of view.

    Maybe I'm answering the wrong question. I consider myself to be a child of Landau as well as Martin and Schwinger and Glauber. Landau was all into collective variables of various kinds, did Fermi liquid theory based on collective variables. I thought of this as a collective variable and then we construct a Schwinger-ish effective Hamiltonian or effective free energy based upon this collective variable.

SS

    Which would have the same structure as the old.

LPK

    Yes, it was asserted that the new free energy would have the same structure as the old free energy. Yes, it was asserted that that had something to do with the King's College lattice independence. Yes, it was asserted that it had something to do with the Pippard version of smoothness, which he gave a name that I don't recall at the moment. Yes, I think we understood universality. It didn't matter what variables you described the problem in, there were universal answers. However, did we understand that as sharply as it was understood by Wegner in 1972? No. And it is one of those things, there are many things in physics, science in general, that you understand because they are around you in the air. And you have not yet even focused your mind on the sharpest form of the question.

SS

    Can I ask the question in a different way? Given that you, so to say, attribute the same kind of variable to the block, was it clear that this is something that had to be justified, or was it intuitively clear that --

LPK

    It was not a question of intuition entirely. In fact, there was a lot more information available to me and I was implicitly using it. In some sense I knew everything there was to know about the correlation functions in the two-dimensional Ising Model. Because there is a relation between varying coupling constants and correlation functions, I felt I knew how the structure changed when I varied the form of the coupling, and in fact I felt I knew it did not. That's this relevant/irrelevant/marginal distinction that grew up in its full form later. But it was not a fully formalized thing.

    OK, now we're coming to a piece of history which is remarkable, in that starting from the moment I had finished the correlation function paper, there were elements of the problem which I understood better than anyone else in the world. Some relatively short period, maybe four or five years, but those four or five years that I could say that this problem was mine and I lived in it, and there was no one else living in this same world. It's why I objected so strongly when you asked me whether I lived in the Green's functions. No, I lived in critical phenomena, and it was my room, my place alone. And so I knew things about the correlation functions which I could then use to construct arguments like the universality.

    That continued until Wegner came to visit Brown in the 1970 or 1971 time frame. Wegner came and wanted to know what I knew. I told him, and he understood, and he wrote it down better and more elegantly than I knew it. And then my world no longer had just me in it. But there was this period, I represent this to you, specially, because it's a very important thing for my life, but it must also be something that happens from time to time in the history of science. Some people, and you all can reach for historical figures, have their world and they're the people that understood it. For a time. This can't last forever, it may last a week, it may last a month, it may last a few years. And then it becomes occupied by the rest of science, by the rest of the world, but there's this wonderful period in which the world, at least, your world, is your oyster.

BA

    So, just to be clear, can you describe this world, and what makes it distinctive from...

LPK

    It is an entire self-consistent world in which you have a space filled with coupling constants that you know what happens when you vary those constants, each and every one of them. That there are only a few ways that the answers to the problem change. And that you can explore the whole space of the problem by using what came to be called afterwards the conformal operators or the scaling operators which describe the field, and that you have a whole world described in terms of these collective variables, in which you know what every variable does to the problem, and you know how the problem changes if you change the coupling constant associated with each and every one of them.

SS

    Given that you had worked so very hard on the two-dimensional Ising model, and knew, as you say, probably more about it than anyone else at that stage, it would seem to me that here is a case where you have worked out everything, where you could indicate what happens as you make blocks, and really look at what happens, what are the properties of the correlation functions. Did that ever come up?

LPK

    There are a variety of people for whom the story could be told. The 1967 review paper was intended to tell the story for the experimentalists. And we indeed told that story in a way which I think conveyed it, to some extent, to the experimental community. After that, and you have a better view of the history than I do, there was a series of short papers which emphasized various different aspects of the underlying theory. There's, for example, the paper of 1971-72 with Wegner in which we looked at the Baxter model, the eight-vertex model, in which there were continually varying critical indices.

SS

    There were earlier papers where we come to people like DiCastro (1968), explicitly making claims about renormalization. I mean, Boguliubov-Shirkov renormalization group methods to prove homogeneity of equation of state, a là Widom.

BA

    May I suggest that we go back -- do you remember the Brandeis summer school?

LPK

    Yes, we always lived at the Castle. Yes, I remember the Brandeis summer school and living in the Castle. We lived in the first floor of the Castle, and there was a skunk that came to visit, but that isn't what you wanted to know about.

BA

    I'm suggesting we go back to the chronology and build back up.

LPK

    OK, I'm happy to do that. Although you'll notice, I think, that there are aspects of the history that you know better than I do. So please continue to ask questions. What do you want to know?

SS

    At the same time as Dyson you were talking about stability of matter, his work with Leonard, or you didn't go to any other lectures?

LPK

    I probably went to his lectures. You're telling me I could have learned about the renormalization group from Dyson at that time?

SS

    No, on the contrary, I think this is already preparatory to what goes into Reviews in Modern Physics and various things like that, because this is the time when DiCastro is listening to you and then gets involved.

LPK

    I would like to be able to say that I understood what happened in my field. I did not read all of the papers. I have not read to this day all of the papers of the field, and this particular thread that you're asking about I have no way of exploring with you till you fill me in on some information.

SS

    What does come up is this issue of DiCastro in 1968 writing a paper in Nuovo Cimento about use of renormalization group, and Boguliobov and Shirkov, to claim that we can show, using these methods, that you can derive an equation of state that has the homogeneity property that Widom claims. OK, for the magnetic system. Then he gives a set of lectures at Varenna.

LPK

    I was at Varenna, I believe, much later. 1970. My recollection was that I was at Varenna post-Wilson-major-contribution. Is that correct or false?

SS

    No, it's pre-, it's 1970.

LPK

    OK. Then I'm remembering conversations with Wilson from thereafter. OK, let me try to describe to you this period from my perspective, which is not the perspective of the history of science. I'm spinning my wheels at this period, I'm following up on failures of hyperscaling, which doesn't exist. I am not learning the things that I would have had to have learned to do the work that Wilson did. I never knew them, I never learned them. I'm beginning to move, for my own reasons, I believe, in the direction of urban studies. And beginning to be discouraged with the critical phenomena problem. I will continue to be discouraged until I understand and accept in my mind Wilson's contribution, which will take a little bit of time after Wilson does it. And in the mean time, I'm working on critical phenomena, I'm working on urban phenomena, I'm not keeping up entirely with the critical phenomena stuff.

BA

    Do you recall if you were aware of DiCastro and Jona Lasinio's work?

LPK

    I was certainly aware of it afterwards, but I think I saw it all through Wilson's eyes.

SS

    And you don't recall listening to them? I mean, at Varenna Fisher is there, actually, and evidently gives DiCastro a very hard time.

LPK

    Apparently I didn't take it personally, because I don't remember. I'm sorry, I'm giving you all insights into my solipsistic side, but I didn't take it personally, and I don't remember it.

SS

    There is something in between that is clearly important, and that is the operator algebra which you do.

LPK

    The operator algebra I already sort of described to you. That is, it is implicit. I described to you the notion that you construct, that you have a limited number of operators in the theory. That they describe the limited number of correlation functions. That they are connected with a varying coupling constant. As a neccesary corollary to that point of view, it must be true that when you multiply two nearby operators, you don't get something new, you get the summation of the old things. So the operator algebra is built in. I am struggling with that notion. There is an unsuccessful Phys. Rev. Letter in which I do not quite claim, or do not quite unclaim, to understand the critical indices of the two-dimensional Ising model through the use of an operator algebra point of view. But I am clearly a precursor to the conformal algebra that comes up later. And I claim credit in roughly this period for the operator algebra of short-distance expansion (Wilson and I called them one thing or the other, but I forgot who called it which), but I sort of thought at the time (and think still) that parallel credit is appropriate. But in any case, there are a whole bunch of little applications, based upon an understanding of the various parts of this world that I feel I can wander through alone. Now it may be that by 1968 DiCastro was there, I just didn't notice him. It may be, in fact, Pokrovsky and Patashinski had been there all along and I just never bumped into them. But I'm presenting my point of view.

BA

    Is there a connection to your work in urban studies?

LPK

    None. As far as I can tell. I use some conservation laws in urban studies, learned how to program a computer. It was useful to me later on, but the urban studies stuff took off in a different direction. In urban studies I was mostly involved in constructing large-scale urban models, originally from the point of view of arguing that work by a man by the name of Forrester at MIT who produced some models of urban change was unscientific in the sense that you could get any conclusion you wanted by changing your point of view only a little bit. In particular, he had reached a rather conservative point of view; I managed to reach a rather liberal point of view, and it didn't change the equations a hell of a lot. But once you argue that something is to a large extent bull, you then have a tough time justifying continuing working on it, which I believe the funding agencies noticed.

KH

    Is there any causal relationship between the move to Brown and the interest in urban studies?

LPK

    I felt that the urban studies could be better done at Brown, it being in a city, which is sort of true. And that it would help me start in a new direction, yes. And it sort of worked out that way. So I did a lot, there was two-thirds of the scientific program for a bunch of years that was directed to urban things.

SS

    We were talking before about your trips through Russia. There's now a second trip, which is presumably the one where you meet Migdal and Polyakov.

LPK

    I certainly met Migdal, Polyakov, the big shots, Azbel, whole bunches of people in various visits. There was something like four or five, or maybe even six visits to the Soviet Union. Exactly when they occurred or how long we're going to integrate over, but there's certainly a place in which I gave a talk on the 1965 physics paper. I'm pretty sure I'm remembering that correctly. There's also a immediately-post-1970 visit in which Wilson gives a brilliant lecture about his renormalization stuff, in which he does it almost without using a word of English. Or any other language. Just equations. I remember, OK, we can talk about the selectivity of memory. I can remember Martin telling me how hard it is to give a lecture immediately after that brilliant presentation. I can't remember whether it was my lecture he was complaining about or his, but I think it was probably mine. And he was just sympathizing with me. But I can't remember. But in any case, there was a lecture of mine, earlier, and then a brilliant lecture from Wilson that I remember. And I certainly met and talked to all of the major and many of the minor figures in Soviet science from that era, running from about 1965 certainly through about 1975, I was pretty well acquainted with those guys. Not with their up-to-date work, I was always a little behind, but I did hear from them. There was several trips to Moscow, a trip to Yerevan (Armenia), which was probably the last in that series.

SS

    I mean, no, it's trying to answer the question that you raised before of the new insight into universality that you get by talking to Polyakov.

LPK

    I don't think I said that, but I certainly understood that Polyakov and the followers of the Landau school were getting universal results by analyzing Green's functions. There is a point in which the word universality arose, and I should have reminded you and me. Once upon a time, long ago and far away, I was sitting at a dollar bar in Moscow with Migdal and Polyakov. We were talking, as we would talk, about physics, gossiping, and they were describing some property of some equation of theirs, using the English word universality. I then imported that word into the United States as the appropriate word for describing lots of things that were going on here.

SS

    Your Varenna lecture in 1970 makes reference to that. I think there's also a paper.

LPK

    I think there was, I made a reference to it also in the talk I gave in Brazil, which was published in Physica, which is the one that I remember. I do not believe I discussed that at Varenna, but I may have. That is, the dollar bar story was, I think, a lighter story I told on a lighter occasion.

BA

    So you said before that the notion of scaling was from the beginning in your recollection pretty much what we say it is now. Is the same true for universality?

LPK

    It is my self-serving point of view that the major element of universality was set out in the 1967 review paper. That it was a reflection of a point of view that had been broadly held in the field, and never articulated previous to that. It was then developed in its mathematical form by Wegner in his work in the Phase Transitions and Critical Phenomena, and maybe other places. And developed in its phraseology by such additions as the work that Wegner and I did on marginal, relevant, and irrelevant critical exponents, although that may very well have been earlier as well. So my self-serving point of view is that it was there fully in 1967, but then, illogically, polished further later on.

BA

    So how did the content of the concept of universality change over time?

LPK

    I think by the time that Wegner had put it all together, after Wilson, universality was the primary tool used to explain scaling. Whereas in 1967 scaling was the big thing, and universality a lesser thing that came along with it.

SS

    I mean, if I remember correctly, in your Varenna lecture, you actually point to a paper by Griffith, by yourself, and likewise references to the Domb school, pointing out about these universal features of critical exponents and various systems.

LPK

    At Varenna, Griffith argues that properties of smoothness of the free-energy surface were developed and invented by him. I do not disagree with him at that time, or now, and suggest that we try to reflect the other one's credit equally.

SS

    I'm not asking about attribution of credit, I'm going back to Babak's point of the changing meaning of universality. In 1969 or 1970 there's a phenomenological concept of universality pointing to properties of the critical exponents which then had to be explained, so to say.

LPK

    But of course, and I'm responding by telling stories, and by pointing to events in history, which one can tie down, and being much less sure, comfortable, and scientific about attitudes and points of view. I can tell you, with moderate accuracy, or rather with discernible, noticeable inaccuracy, about historical events. But attitudes, geez! How do you know?

SS

    When does it change from being phenomenological to...

LPK

    Oh, OK. Again, I'm going to tell stories, because that's the best I can do. One of the visits to Moscow, certainly before Wilson, I make a bet with Migdal Jr. of a bottle of whiskey, Scotch versus vodka, that the critical indices will be rational numbers. I'm on the rational numbers side. I figure I can't lose. Of course, how are you ever going to prove a number's irrational. He, however, has some two-dimensional example up his sleeve. Well, you know, it's perfectly reasonable. He says he has it up his sleeve, but I claim it's not relevant to the problem at hand. After the 1971 paper by Wegner and myself, I have to pay off on the bet, because we have a critical index that varies continuously, and I can't argue rationality. So I do pay off on some visit or another to Moscow. I believe when Migdal comes to visit with me at Brown, he arrives with a bottle of vodka that had been brewed in his father's and his brewing place, and then modified in his father's and his oven, so that it's all twisted and irrational. (laughter) And it sits on the shelf, after I come here, it's sits on the shelf, there, where the chess set is, for a whole bunch of years, until one day I walk into the office, and while I'm looking at it, it dies of it's own internal stresses, spilling vodka all over the office.

SS

    And you're disconcerted about the smell more than anything else?

LPK

    No, I'm not disconcerted about anything. I claim it is a parable about the historical truth.(laughter) So I don't know the answer to your question.

BA

    So when did you learn, and how did you learn, about Wilson?

LPK

    All along I knew Wilson was making good stuff. As I told you earlier, when I was a graduate student, I was convinced that he was pretty damn good. And he announced [his results], I found out about it (I don't remember where I was), [and] I was a bit annoyed. I stuck with my urban studies stuff, and didn't look at it very carefully until after a while, but it became clear that there was something very solid in it all. I suspect that talk in Moscow was not only impressive to the Russians, but also to me. Urban studies was beginning to wear on me a bit. So at some point, I spent six or eight months learning about what had been done with the renormalization group. I learned about the epsilon expansion, the 1/n expansion, and a whole variety of absolutely wonderful things. And my annoyance at myself for not having done it was ameliorated by several different things. First, the wonderful achievements that had been made based on it. Second, the fact that it was based upon at least two separate things which were not within my repertoire. One was the concept of a fixed point, which I was innocent of. And the other was renormalization group, which I was also innocent of, even if I had in my sort of naive way invented it for myself. And so I comforted myself with the view that I didn't have the basic stuff that was needed to build it. And knew that among the other things that I didn't have was Wilson's ability to think in a more general way than I was willing or able to do. That he was willing to think about hundreds of coupling constants at the same time. And I was certainly not willing to think about more than three. And so I found all of those things comforting to me. I realize this is a solipsistic response, but you asked a question which asked for it.

    At that time, there was relatively little to add in the K-space version of the thing; they had done all the right things. Also the expansions had been done by very good people, and many of them had done all the right things. I did add a little bit to the real-space version of it. In fact, there's a sense in which I was an independent inventor of it, but not by the strict standards of the usual scientific world. The strict standards of the usual scientific world would count the earliest publication (that's the only reasonable way of doing it), and Niemeijer and Van Leeuwen were the unique inventors of the real-space method under those circumstances. However, there was some stuff with was reflected by the work of Houghton and myself which was begun, probably, a bit earlier, and published a lot later. And we worked on that for a while with some nice accomplishments, but nothing world-shaking.

BA

    How did you meet Wegner?

LPK

    I think he came to Brown to work with me. I believe that his version of the story is the same as mine. I knew something that no one else knew, he wanted to know it, he came, he found it out, and since we no longer publish in crytograms, or no longer think that it's appropriate to hide our chemical results, [he learned what he came to learn]. It's something that is entirely appropriate in the scientific community. And he built upon my thing, and I was very pleased.

SS

    In 1973 there's a conference at Temple which is--

LPK

    1973, from my point of view, is a period in which I'm trying to get back into science after my urban studies days. Houghton and I have this, as I recall, this real-space renormalization calculation. Which is not entirely working. The reason it doesn't work is connected with your earlier question about block spins. Although I did understand how to renormalize the block spins, I didn't understand how to do decimation at that time, or in fact that one shouldn't do decimation. Decimation you erase a bunch of spins and hold on to others, and it's an unstable procedure. So Houghton and I tried to do a fundamentally wrong thing, but we were getting pretty good answers, when Niemeijer and Van Leeuwen are doing the fundamentally right thing. Getting a little weaker answers than ours, but it's right. And we all continue -- critical phenomena is a pretty good world -- we all continue to be in communication with one another. I ask Wilson what the hell we're doing wrong, and we talk about it, and he explains to me something about the difficulties of decimation which he had also followed, and I understand what he tells me, and go back and basically threw away the method we were using, went for another method. So, in Temple we're trying to get back in the subject. There are new boys on the block, David Nelson is coming into the picture, but we're going out of the golden age and entering the silver period.

SS

    Does 1973, after Wegner and this Temple conference, mark a closure of some initial stage? The picture's clear in terms of all these methods having converged more or less to at least a physics viewpoint, and to some extent a mathematical justification of it.

LPK

    It's all mopping up operation after that. Pretty much after that.

SS

    How does your own world view of physics change as a result of what you have learned up to 1973, 1975?

LPK

    The most important thing that happens to me, as I think I've told you, is this ability to wander in my own world and feel the ownership of it. It's a little world, and not the best world ever discovered, but it gives me a feeling of understanding for the other people who have discovered their own worlds too. And it's an important part of the joy of science, having that. Although I've been graced with a lot more of it than most scientists have been, and a lot less of it than the greatest scientists have.

BA

    It seems that this period that you described as "having your own room" corresponds to something you also describe as a kind of disenchantment with the study of critical phenomena. In the late 1960's, after 1967, you turn to urban studies?

LPK

    Well, yes, I understand what you're saying, and being alone is wonderful, but it's not a very human or satisfying condition.

SS

    What I hear you say is that there's a sense of exhiliration at having created a small world, having been there for a while, and all of a sudden recognizing that lots of other people are coming to it because of its importance, and that you were very much responsible for that. What I'm alluding to when I asked is less of a psychological and more of a philosophical issue. 1976, 1977, you write a paper for Reviews of Modern Physics on "The application of renormalization group techniques to quarks and strings", which implies that "there is a different attitude to the way we look at physics," and I'm trying to get a handle in terms of this new conceptualization...

LPK

    OK, I hadn't remembered that I said that then, but there is another thread which you may see on my bookshelf in Wolfram's A New Kind of Science, beginning earlier than the period of the critical phenomena, and continuing much later. There's a change in attitude towards what you do with computer programs, which is very important. It is something that I have exploited in my scientific career, but I don't think I do or should claim credit for it. Just as critical phenomena is a world which you can explore for itself, there are many other worlds which one can explore. Some of those worlds can be constructed on the computer. Sometimes, as in the case of quarks and strings, that world that you construct on the computer has a direct analogue in a mathematization of nature. Sometimes one expresses the world ab initio as a sort of computer algorithm. Diffusion-limited aggregation is invented by my colleague Tom Witten is such a world that you can explore for yourself, is mostly described by a computer algorithm. Wolfram is also describing the exploration of such a world. This vision of the possibility of such explorations is described by Fredkin. I want you to read about it, Three Scientists and their Gods. It's a very important thread in modern theoretical science. I'm not sure that's what you're referring to, but it's one of those things that is important in this period, and has been important to me because I have made use of such worlds in training the generations of graduate students.

    The other question which one asks is, "After critical phenomena, what should I do with my life?" That's the way I ask it. Other people have to ask similar questions. And of course the answer is to try and be as useful as possible. And exploring the intersecting worlds of nature and the computer provides one answer to such a question. Also the students that are thereby trained can be very useful people in the world, if they choose to be.

SS

    You've answered one aspect of the question which was in the back of my mind. The other aspect of the question was if you were to have redone in 1975 the Kadanoff-Martin paper, and instead of going from the macro to the micro, that is from the macro to the Green's function, but now to go from what you had learned from the Green's function, more generally the microscopic world, what would you say about building up the macro world from that conception? That's the other part of it.

LPK

    OK, the other part of my science, which is, I think, what we're getting at, is what have I been doing with hydrodynamic equations. In the last 15 years, I have been using hydrodynamics equations and their computer simulations to describe emergent phenomena that occur in such situations. I've been looking at singularity formation in hydrodynamic systems with a goal of exploring the universality classes of the hydrodynamic systems. And with the goal of understanding how emergence and chaos works in a particular physical context. And so in fact, I have started out with known microscopics, it's microscopics of PDE's and tried to build macroscopics, such things as turbulence and singularities out of the known microscopics. It is also a fruitful scientific endeavour. It is not a lonely endeavour. The basic concepts of what one does in such a calculation were probably set by Greenspan in the 1960's. But the working out has had to wait the return of hydrodynamics to physics, and hence the work of experimentalists, and also the development of decent computers so that one can explore it computationally. Analytics is not good enough for it.

Return to part I of this interview.