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
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
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.
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