PoS
What drew your attention to the literature by Domb's group?
BW
Well, I guess I was just reading literature in general on phase
transitions and critical phenomena from the interest that I had
developed as a postdoc. So I just followed that literature and
saw the papers as they came out.
PoS
Now, some of the people we talk to seem to say that they knew
what's going on, they knew who's vaguely doing what, in more or
less detail, but they claim now in hindsight that they did not
actually read the literature a lot. Other people seem to really
keep up with the literature meticulously.
BW
I would say that I kept up with the literature. There were many
surprises, I mean, in later years I discovered many papers that
I should have known about and didn't know about, so I don't mean
to say that I really knew the literature thoroughly, but basically
I was learning, basically I was following the subject through
the literature.
PoS
Being a member of the chemistry community, did you find a little
bit of a divide between chemistry and physics?
BW
No, I've had good relations with the Cornell physics department
and even in general, when I go to Europe…
PoS
I meant it in terms of the kinds of problems that you address.
BW
Yes, well, I would say the physicists might have been more interested
in magnets and liquid helium and the chemists were more interested
in classical liquid mixtures, but except for that…
PoS
How about the difference between microscopic and thermodynamic
approaches? Is thermodynamics much more of a crutch for you than
it is for the physicists?
BW
It might be. It's not as though I felt that. Let me put it this
way, if I could have worked out the properties of a microscopic
model the way Onsager did, I would have. I knew this would be
a wonderful thing to do. That I didn't do it wasn't because I
wasn't thinking along those lines, it's just that I never succeeded.
PoS
I also meant it in terms of having insight into structures which
combine by virtue of thinking thermodynamically in contrast to …
BW
I don't quite have an answer for that. After you guys are gone
I'll probably think of what I should have said.
PoS
You should email us. It's a bit much to ask, but can you try
and reconstruct for us, as you're reading Domb's literature in
Amsterdam, what you took the character of the field to be? What
were the problems and what were the promising ways to pursue them
in '60 or '61?
BW
Well, I could see that the work that they did in power series
expansions and so on was obviously fruitful. It was leading to
very believable answers. It wasn't yet the solution to the problem,
in so far as it was one problem with one solution, but it was
clear that it was a major step in understanding.
PoS
So what was the problem and what would the solution have been?
BW
Well, the problem would have been to do in three dimensions what
Onsager did in two dimensions. Basically. Or to do this even in
two dimensions and a non-zero magnetic field. What Onsager did
was zero external field. So these were the problems. It was clear
that those were the problems.
PoS
Ok, I can now phrase my question sharply. Coming out of thinking
thermodynamically and thinking about what Domb and his students
are doing, do you get a notion of universality, in other words,
that somehow for all of these different models that I'm looking
at…
BW
That was already very much in the air, even in my postdoc days.
For example, Rice and Bruno Zimm recognized that the coexistence
curve of a two-component mixture near critical solution point
had the same non-classical exponent as a liquid vapor coexistence
curve. And so that it would be universal exponents, so therefore,
equations of state of nearly universal structure, and all of that
was pretty clear.
PoS
What kind of conjectures would be put forward among the people
that you talk to that would account for that? What were the leads,
so to say, or what were plausible avenues which would explain
this kind?
BW
I understand your question, and I know an answer, but I'm afraid
that answer came from later years when it was already clear that
the correlation length or coherence length of density fluctuations
or composition fluctuations near a critical point was diverging
as one approached the critical point. But that that was the reason
for the universality, that because that long range of the correlations
meant that the detailed microscopics were not that important and
that is what would lead to universality, that was an idea that
came later, or at least that came to me later.
PoS
But, you said, well, it was clear that the next problems were
to be three-dimensional Ising models, etc, etc, and rather than
trying to answer the question, the problem is how do you account,
given that you're now aware thermodynamically that these properties
of the critical exponents, how do you account for that?
BW
So account for the deviation from classical exponents.
PoS
BW
That becomes the central problem, rather than solving, well,
when you say account for universality…
PoS
You're faced with the striking fact that all of these phase transitions
have common properties, and as a chemist, you're aware of the
diversity of systems which exhibit this.
BW
Yes, you see we already knew at that time about the relations
between various models, two-component liquid mixtures or one-component
liquid-gas equilibrium, or the magnetic model of permanent magnetization
below Curie point. We already knew that these are equivalent to
each other by, for example if no other, the Yang-Lee papers. So
we knew that whatever the mystery was, it was the same mystery.
So that we knew.
PoS
But there's a difference between being able to mathematically
transform one model and another and thereby understand that…
BW
The difference between that and then understanding any one of
them.
PoS
Right. Or understanding, it was really an issue already in '52
that universality was a phenomenon, not a coincidence.
BW
Well, we didn't think of it, that something,…
PoS
Was it something that needed explanation in '52? What we now
call universality. Already?
BW
We already knew in '52 that there were many different physical
contexts in which critical points were manifesting themselves,
and we already knew in '52 that whatever it was underlying any
one of them, that is the mathematical structure, is the same for
all of them. Because we knew that they could be related to each
other, so the problem was not to understand why a two-component
liquid mixture, or the one-component liquid vapor system, or the
magnetic systems, the problem was not to understand why they are
the same as each other, because that was already understood from
transcriptions among models, for example, as Yang and Lee had
shown. So that we already knew, so we knew that if in any one
of those physical contexts, one could understand the origin of
a non-classical critical point exponent, would then automatically
have understood it for that wider range of phenomena. But what
we did not know at the time is what is the real origin of these
non-classical critical point exponents. Now I have a feeling that
you're using the word universality in a somewhat different sense.
This was after the renormalization group theory, and one began
to think of the locations of fixed-points and appropriate flow
plane and to see that there would be some domain of starting points,
all of which would have the same fixed point, and all those that
were associated with the same fixed point would have the same
critical point exponents, and that they belong to one universality
class. That was a much later idea…
PoS
'67? When people recognize that there are universality classes,
that phase transition depends on the dimension, how many components
the order parameter has, and some symmetry. That's in '67.
BW
Yes, that's right and I remember hearing about these things from
[Leo P.] Kadanoff, though it wasn't in that language. What I did
recognize, or I guess, I don't know why, I took it as obvious,
that maybe because one could see that the critical point exponents
were independent of the substance, I took it as obvious that the
equation of state in the neighborhood of the critical point would
have some sort of universality of form, with parameters in it
that were substance dependent, but that the general form would
be universal. I don't think I ever used that word universal, but
it was clear that that had to be.
PoS
So that's the completion of the contrast you started to make
between two different uses of the word universality.
BW
Yes, much more, it was that idea, that whatever it was that was
happening had nothing to do with whether you were looking at Carbon
Dioxide or Nitrogen or Ethane, so that the basic features of the
equation of state had to have some universality of form.
PoS
And that awareness was present from '52 on?
BW
That awareness was present from '52 on, sure.
PoS
And to some extent you have that already in terms of laws of
corresponding states?
BW
Yes, yes. There were technical differences between that and the
law of corresponding states, but there are a lot of analogies
with the law of corresponding states. So that, and it was exclusively
as a verification of the law of corresponding states that Guggenheim
wrote that famous paper that I said made everybody aware of the
fact that the critical point exponent beta had a non-classical
value.
PoS
That's very interesting. So were you working on a specific problem
while you were in Amsterdam?
BW
PoS
Did you make contact with the Domb group?
BW
With the Domb group? No, not at that time. No, that was a few
years later, in '65, I again had a leave of absence, and I was
in Reading, where Guggenheim was, I went there to be with Guggenheim.
And while I was there, Guggenheim organized a little symposium,
and asked me who he should invite to it. And at that time I already
knew of the work of Michael Fisher, and Michael Fisher was at
King's College, and so we invited him and a few other people,
I'm not sure I remember who, and we talked about critical points
and phase transitions. That was '65.
PoS
So in '61, then you come back here. Did the reading that you
did then affect the direction of your work?
BW
Yes, very much. Let's see, maybe I should get to the next page
of my c.v., yes, now I should say also that during this time,
you asked about meetings and so on, and during this time my early
days as a faculty member, before '61 I did go to meetings, such
as Gordon conferences. There was a famous series of Gordon conferences
called Chemistry and Physics of Liquids, and I attended those,
they met in alternate years, and I attended those pretty regularly
in those years, so I learned a lot about what people were thinking
in liquid state theory.
PoS
They were primarily chemists?
BW
Chemists and physicists, but maybe primarily chemists. And so
I learned, I was hearing a lot about liquid state theory and hard
spheres, and these phase transition problems in general.
PoS
Do you remember reading, for example, Michael Fisher's lectures
at Boulder, in '63, '64, on phase transitions?
BW
Yes, I know what you mean. I'm just not sure. I certainly knew
of them later, whether I knew of them at the time I'm not sure.
PoS
'65 you write your papers on…
BW
Yes, that work on the equation of state, I did, I had been thinking
about it for many years, but it was coming together during the
time that I was in Reading, '65.
PoS
So that's the critical juncture?
BW
Yes, it was while I was in Reading that I found out that logarithmic
divergence of the heat capacity arising from one of the forms
of the equations of state that I had been postulating.
PoS
Let's go back to the chronology. You were telling us about the
Gordon conferences, and I wanted to ask if there was some particular
thing that you remember about what influenced you?
BW
Well, I remember a lot of stuff on the hard sphere phase transition
that [B. J.] Alder had found by computer Monte-Carlo simulation.
Alder and [T. E.] Wainwright and also [W. W.] Wood and [J. D.]
Jacobsen. And that was a very exciting topic, and the connection
to liquid state theory, Lee and Yang theory and so on, all of
that was being talked about at those times, and possible connections
to phase transitions being made.
PoS
Is there some point where we should ask you, either in this period
or later, about simulation and the use of computers in your own
work?
BW
I did very little with computers and essentially no computer
simulation. I knew the results of computer simulations, but we
didn't ourselves do any. So here at Cornell in '63, I still did
some work on collision theory, but then in '63, I published a
paper with the title "Some Topics in the Theory of Fluids." Now
this was sort of general statistical mechanics, and it didn't
have anything to do with phase transitions, it was a way of determining
the chemical potential by what is now called the insertion method.
So you would insert a ghost particle and watch its interactions
with other particles. And I applied that as a way of getting a
variety of approximate equations of state, and recognized that
what was happening in a dense liquid near its triple point, for
example, is very different from what was happening with a fluid
near the critical point, and that the long-range correlations
were essential near the critical point, but shorter range correlations,
primarily due to the strong repulsive forces between molecules
were important for a dense liquid, and made that distinction.
PoS
When, you talked about collision theory, you talked about phase
transitions in terms of equations of state.
BW
Yes, I was trying to think of them simultaneously, but there's
no connection.
PoS
BW
PoS
About correlations near the critical point.
BW
Yes, but that didn't have anything to do with the work on collision
theory. Sorry if I gave that impression, they're really very different
things. So I did, in this paper that was published in '63, I did
apply those methods for determining chemical potential by particle
insertion, I did apply those methods to determining approximate
equations of state near the critical point. I also applied it
to determining some properties of the two-dimensional, three-coordinate
Ising model, so the Ising model on a hexagonal lattice, where
some work had been done by [R. M. F.] Houtappel following Onsager,
and I was able to determine some properties of the correlations
between what in lattice-gas language would be correlations between
holes and occupied cells neighboring the holes, and for that hexagonal
lattice gas, using these methods, so I developed these methods
thinking of their possible applications to phase transitions,
but in fact, they've become extremely important in other people's
hands, not having anything to do with anything I did, as a technique
in computer simulation. It's used now almost universally for determining
chemical potentials in various computer simulations.
PoS
Let me ask a peripheral question. In '57 you have superconductivity
being solved by BCS and its corresponding phase transition. Eventually,
that work, which is so to say classical quantum statistical mechanics
gets translated into Green's functions methods and things like
that. How much are you aware of those particular approaches to
statistical mechanics?
BW
Aware of, but just very much on the sidelines. Aware of ‘cause
I was hearing the words and hearing people talk and so on, but
I myself was essentially ignorant of all those things.
PoS
Time-dependent Green's functions?
BW
None of that was in my consciousness.
PoS
You're in '63, and you wrote a paper on statistical mechanics.
BW
Great. Ok. So that paper, "Some topics in the theory of
fluids," was a sort of miscellany. Then I was interested
in the radial distribution function of fluids. I wrote a paper
called "On the Radial Distribution of Fluids" in 1964,
trying to understand something about the nature of the so-called
direct correlation function.
PoS
Published in what journal?
BW
Journal of Chemical Physics.
PoS
Now before we get to '65, is it possible to discern how your
time in Reading and Amsterdam has affected your work?
PoS
Are you doing more statistical mechanics?
BW
In 1962, I wrote a paper called "Relation between Compressibility
and the Coexistence Curve near the Critical Point," and I
think that might be. If I'm remembering right, I took the earlier
idea of Rice, which was now already conscious of non-classical
value for Gamma, and then wrote something on the connection between
the divergence between the compressibility and the shape of the
coexistence curve near the critical point. And that came, probably,
from my reading while I was in Amsterdam, maybe being conscious
of things that were done right by the Domb group, chiefly Michael
Fisher. Probably, if I'm remembering right, that would have been
how that went. So that paper was 1962.
PoS
Other than your invitation to Michael Fisher, in Reading, were
you in direct contact with anyone from Domb's group between Amsterdam
and Reading?
BW
I must have been in contact with Michael Fisher, certainly by
correspondence, I definitely remember Michael Fisher and me exchanging
correspondence.
PoS
Do you remember what it was about? Or what you learned from it,
detailed or vague?
BW
I guess we were discussing issues that were in the air and of
common concern. I'm sure I learned important things from Michael
and that correspondence, I know that however it happened, whether
from conversations with him or correspondence, I learned a lot
of very important things from Michael, but I'm not sure I could
tell you when each of these things happened.
PoS
Were you in contact with Melville Green at that stage?
BW
I wasn't in contact with Mel Green, and in fact this famous meeting
at the Bureau of Standards on critical points was a time that
I was in England. I think that was in '65. And I was in Reading
at the time, and I didn't come back for that meeting. So I missed
that, and so I also missed Mel Green's, I should say, did you
know Peter Debye was a member of this department? And he, if I
remember, was at that meeting, and gave a talk on correlation
functions. He had already many years in connection with his work
on phase transitions in polymer solutions made plots of scattering
as a function of the square of their S parameter, essentially
a measure of the scattering angle that were Ornstein-Zernike plots,
and he talked about that quite a bit. So the ideas about correlation
functions I already knew about and had talked some with Debye
about them.
PoS
You never talked to Bethe about alloys and long-range order?
BW
No, I knew of his work, and I knew from the work of the Domb
group that they talk about the Bethe lattice, a lattice on which
the Bethe approximation would be exact, so I knew about those
things, but I never talked to Hans Bethe about it.
PoS
How about Onsager? How much contact did you have with Onsager?
BW
Essentially zero, but one interesting contact I did have with
him, I don't remember whether it was while I was a postdoc or
when I had become a young faculty here. But I was at a meeting,
maybe of the American Physical Society. I knew of Onsager's work,
of course, and about his and Yang and Lee's determination of the
degree of the coexistence curve, the two-dimensional Ising model
that beta equals one eighth. And I saw him walking down the corridor
during this meeting, and I accosted him and I said, "Professor
Onsager, you found this power one eighth for the spontaneous magnetization
of the two-dimensional Ising model. Do you think there's any prospect
of our learning what that exponent is in three dimensions?" And
he stopped. He obviously had no idea who I was. And he stopped,
thought a moment, and said "The beast may not be algebraic," meaning
it might not be an algebraic number. And then walked on. And that
was essentially the whole of my contact with Onsager. In somewhat
later years, I worked with John Rowlinson on a model that we called
the penetrable sphere model, and again it was a model of liquid-vapor
equilibrium. It could be transcribed into a two-component liquid
equilibrium, but let's say a model of liquid-vapor equilibrium
that had many very interesting features. And I talked about that
at one of the Rutgers statistical mechanics meetings. This was
many years later than you would be talking about, and Onsager
was in the audience and asked a question in connection with my
talk at that statistical mechanics meeting. That and my running
into him in the halls, were the only contacts I had.
PoS
When do you get a hold of the proceedings of the National Bureau
of Standards Conference?
BW
I think I got them essentially immediately.
PoS
BW
It must have been when I came back.
PoS
BW
I knew that that conference was going on, and when I came back
I knew it had gone on, and I got the proceedings. I can't remember
if someone sent them to me or I asked for them.
PoS
Let me ask a sociological question. Around this time, from '60
to somewhat later, it's clear that phase transitions become an
essential interest to a much larger community.
BW
PoS
When do you have a sense that this is really happening, or getting
center stage?
BW
Well, I had the sense that it was happening in just the period
you said. We talked, early '60's.
PoS
And Michael Fisher would be responsible for what?
BW
Oh he was an essential figure.
PoS
So tell us about Reading.
BW
Well, I had already at that time known about non-classical critical
point exponents from earlier work, and Guggenheim was very much
interested in phase transitions and critical points, and he had
written that great paper on the coexistence curve. And I gave
a lecture series on phase transitions in critical phenomena there
in Reading. And Guggenheim liked what he heard, and he said we
had illuminated some things for him. We already at that time,
maybe largely from the work of the Domb group and Michael Fisher,
we already at that time knew very much about the dimensionality
dependence of critical point exponents. And Guggenheim was interested
to hear about that.
PoS
How old was Guggenheim at that stage?
BW
Guggenheim might himself have been in his 60's, maybe. I think
he retired not too long after that.
PoS
How long were you in Reading?
BW
Just a few months. Just a one term leave of absence that I took.
I took a special leave from here, I got some kind of fellowship.
I think I had a Guggenheim and NSF fellowship. Guggenheim and
Fulbright, to go over there.
PoS
And the rest of the time you spent where? You were just away
for one semester?
BW
Just away for one semester, yes.
PoS
But most of the time in Reading.
BW
The whole of the time of that semester I was in Reading.
PoS
What did you make of the proceedings of the Green conference?
BW
I thought it was terrific. I thought they were very interesting
papers, and very important advances.
PoS
BW
Well, it might have been, I'm remembering if it's there that
this work of [George] Benedek , with [Peter] Heller, was published,
and that was very striking. And Michael Fisher gave a summary
talk, brought together a lot of what was then known about critical
point exponents. There's the scattering stuff.
PoS
Can you say what was striking about those?
BW
Well, I was just the other day looking at the proceedings again.
And also very interestingly, [M. R.] Moldover and [W. A.] Little
showed the heat capacity of Helium-4 and Helium-3, and in case
of Helium-3, it was the heat capacity at the liquid-vapor critical
point. In the case of Helium-4, there were two heat capacity peaks.
There was that of the liquid-vapor critical point, and there was
the Lambda Transition. And that was a very striking figure, so
there were a lot of things that impressed themselves on my consciousness.
So here's the paper by Benedek. So there was all this stuff on
magnets and anti-ferromagnets. And it was cementing one's understanding
that there were close connections between that and critical points
and fluids. John Rowlinson had a very nice paper on critical points
and fluids. There was the stuff on logarithmic singularities so
that [W. M.] Fairbank and Moldover-Little had those papers. And
then there's the scattering, so Debye gave a talk, and then Fisher
talked on critical fluctuations, and then [Harry] Brumberger,
who had done a lot of experiments on scattering at near critical
points of polymer phase separation, with Debye. Then [Benjamin]
Chu, who had been a postdoc with Debye, talked about measurements
of critical opalescence. I paid less attention, to relaxation
phenomena. That turned out to be extremely important later, but
I didn't pay much attention to it at the time. There were the
measurements by [J. D.] Lorentzen on the effect of gravity on
the appearance of critical phenomena. And there was [Sam F.] Edwards’ paper
on statistical mechanics of a single polymer chain, which became
extremely important later, but at the time didn't have that much
of an impact on me. I was thinking in other directions. Yes, so
I could see that in this thing there were important summaries
that were already known and things that I hadn't known about that
I learned about from these papers. I thought it was a great conference.
PoS
Lots of people point to that conference as being very important.
So it'd be interesting to know if it served, well clearly it did,
as a place that brought people from different disciplines who
might have not have known of each other's work together.
BW
Yes, I think it did serve that purpose.
PoS
But did it, in addition, point you in new directions? Did it
offer new methods or new questions?
BW
Since I was already working and thinking in those directions,
I wouldn't say it pointed me in new directions, but I'm sure that
it helped cement my knowledge of things that I might have been
dimly aware of, so I'm sure it reinforced my understanding of
things. I'm sure it did that. But I wouldn't say it pointed me
in new directions, because I had already been working very much
in those directions.
PoS
Let me shift conversation slightly to another phase transition,
mainly getting Michael Fisher to come to Cornell.
PoS
BW
Yes, I think we were already trying in '65.
PoS
Let's not skip over your scaling paper. As you described earlier
on, when we were talking about North Carolina, I had a very continuous
picture of you trying to get these equations of state that would
give you non-classical results.
BW
Yes, or would incorporate non-classical results. Yes.
PoS
And you've described the varieties of work that you've done and
people you've talked to, and I wonder if something happened in
the period '63, '64, '65, that we hadn't talked about yet, that
enabled you to write that paper?
BW
I can't think of what might have, because it was something that
in more primitive ways I was already trying to do when I was a
postdoc in Chapel Hill.
PoS
So what was new in that paper? How come you could do it in '65?
BW
Well, I guess I had been, over the years, trying to construct
these equations of state to incorporate non-classical critical
point exponents, and what happened at that moment was that I finally
saw that one of these equations of state that I was working with
gave me a diverging heat capacity, and that was an astonishing
moment. It was really quite a revelation. And that was the point
at which I then was led to ask myself what aspects of that particular
equation of state led to these results. And so then I saw that
what was the important aspect of it was the homogeneity of form.
Not the particular equation of state which I was working with,
which I'm sure wasn't of any use or wasn't correct, I know it
had defects in it, but the successful part of that was its homogeneity
of form, that's when I came to realize it was this homogeneity
that allowed all these various non-classical aspects of critical
phenomenon to be brought together-- the non-classical thermodynamic
aspects of critical phenomena to be brought together.
PoS
BW
Yes, exactly. It's not that I wouldn't have wanted to have a
statistical mechanics model, and I'm sure that I along with the
rest of the statistical mechanics world tried hard, but I never
succeeded.
Continue reading part III of the Widom interview.
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