
Interview with Benjamin Widom,
20 March 2003

Interview recorded at Cornell University, Ithaca New York.
Interview conducted by PoS collaborators:
Babak Ashrafi and Sam Schweber.
Edited by A. Martínez and S. Schweber.
PoS
To begin, we would like to know your background and how you entered
science generally.
BW
Well, I had been interested in chemistry since boyhood, so that
was an early start, and I was a chemistry major in college. I
went to college at Columbia, was an undergraduate at Columbia.
PoS
BW
I grew up mostly in New York. I was born in Newark, New Jersey,
and spent a couple of years, maybe around the age of 10 or 11,
in Detroit, but then came back and lived in New York City, in
Brooklyn. I went to Stuyvesant High School, which was a place
where a lot of students interested in science went, a place where
a lot of people became interested in science, but I already had
that interest when I went there. And as I said, I graduated from
Columbia as a chemistry major, came here to Cornell as a graduate
student. I came in February '49, so I came in midyear, the middle
of winter, and trudged up from the train station downtown, uphill
with a couple of very heavy suitcases, wading through the snow,
and ultimately found some dormitory I had been assigned to. I
did physical chemistry as a major as a graduate student, but I
was working on a quantum mechanical problem, not anything having
to do with what your interview is about. I was working on inelastic
molecular collisions, vibrational energy transfer with Simon Bauer.
Simon Bauer is still here, still a presence in the department.
He's 91 years old, attends seminars and asks questions, writes
papers. He's an amazing guy. So it was under him that I did my
thesis, in vibrational energy transfer, gasphase molecular collisions.
PoS
What preparation was required for that kind of work in physical
chemistry?
BW
Well, the theoretical background would have been physics and
mathematics, so I took physics courses in mechanics and electromagnetism.
PoS
BW
It was with Phil Morrison, who was still here at Cornell, so
it was a course, a graduate level mechanics and graduate level
electrodynamics course, and then I had quantum mechanics, with
Hans Bethe. I remember taking the second term quantum mechanics,
one that was then called applied quantum mechanics, with Hans
Bethe, but I'm trying to remember with whom I took the first term.
It's possible it was Ed Salpeter. But I don't swear by that. Yes,
pretty sure it was Ed Salpeter.
PoS
Was it a quantum electrodynamics course?
BW
No, not quantum electrodynamics, classical electrodynamics. ...
And then I had a course that is still called mathematical methods
of physics, at that time taught by Mark Kac. Now, I understand
that that course had been invented by Richard Feynman, but that
was before I was here, so when I was here it was taught by Mark
Kac, and it was a great course. And I took some additional courses,
differential equation theory with Mark Kac later, a later course.
Did I have any other physics courses? I don't think I had any
other physics courses. Freeman Dyson was still here at that time,
and I sat in on a bit of some of his advanced quantum mechanics,
but I don't think I ever finished sitting in, and in any case
I wasn't a registered student and was just more curious than anything.
PoS
And Statistical Mechanics?
BW
And then Statistical Mechanics. Hans Bethe taught again, and
I took his statistical mechanics course.
PoS
So most of the courses were really taken in physics?
BW
Yes, most of the courses. I took a couple of courses here in
Chemistry, one in advanced inorganic chemistry, which was mostly
oxidation reduction theory.
PoS
Peter Debye was still here?
BW
Debye was still here, and I heard some lectures of his, but not
a full course, I never had a full course with him.
PoS
And John G. Kirkwood was gone?
BW
Kirkwood was gone by then. [Paul] Flory was here. In fact, when
I originally came as a graduate student, I was originally planning
to do my work with a young assistant professor, John Bragg, you
might have heard his name in connection with the ZimmBragg theory
of the helix coil transition in biopolymers. And what I would
have worked on with him was essentially an application, one can
see now in retrospect, an application of the onedimensional Ising
model. But Bragg left to go to a job at General Electric shortly
after I came, and the department thought that they still owed
it to me to allow me to do a theoretical thesis, because that
was my interest. So I had a choice of doing statistical mechanics
with Paul Flory or doing quantum mechanics with Simon Bauer. And
what did I know? I thought, well, quantum mechanics is a very
jazzy subject, and that sounded great, and statistical mechanics
sounded dull, so I chose to do quantum mechanics with Simon Bauer,
and that's how I did that thesis. It wasn't until I got to North
Carolina as a postdoc with Oscar K. Rice that I really learned
phase transition theory and got to appreciate thermodynamics and
statistical mechanics more than I had here.
PoS
While you were a graduate student here at Cornell, how unusual
was it to be so involved with physics and math?
BW
It didn't happen a lot, but it happened. Especially at that time,
our graduate school still had the requirement of two minor subjects
in addition to a major subject, and those minors, if I'm remembering
right, had to be outside the field of the major. And with the
result that most of the physical chemistry graduate students had
a major in physical chemistry and a minor in physics and mathematics.
That's not to say that, as I did, that almost all their coursework
was in physics and math, typically they had less coursework in
physics and math, and more in chemistry, so mine was a little
unusual, but not unprecedented.
PoS
And how many lab courses did you have to take?
BW
Lab courses. I didn't have to take any lab courses.
PoS
So you didn't take any lab courses?
BW
I didn't take any lab courses here, no.
PoS
Was that unusual for a student?
BW
That was not unusual. The experimental graduate students got
their laboratory experience when they started their thesis research.
PoS
Was Bauer your main mentor while you were here?
BW
Yes. He was my main mentor, and generally in the area of chemical
kinetics, and as I said, vibrational energy transfer was one aspect
of chemical kinetics.
PoS
Did you work closely with him, or were you mostly on your own?
BW
It was mostly on my own. I learned from him what the problems
are, and what we are aiming for, and what we wanted to do, and
why we wanted to do it, but the actual working out of the model
and the calculations and so on, that I did myself.
PoS
Who were the readers on your thesis?
BW
Well, the only reader was Simon Bauer, but my thesis committee
members were Hans Bethe and Mark Kac. And so they were the ones
who examined me, both the admission to candidacy exam, and my
final thesis exam.
PoS
Did you work closely with any other students?
BW
Not work closely with them. I knew them. I knew those for example
who were working with Simon Bauer on the experimental side. Simon
Bauer had basically two projects. One was a molecular structure
determination by electron diffraction, the other was chemical
kinetics. So I knew generally the people in his research group,
but I mostly knew those who were working on the chemical kinetics
side the experimental side of what I was doing the theory on.
PoS
And you had an office in Baker or Rockefeller?
BW
Yes, well, at that time the graduate students didn't really have
offices. I did most of my work in my room in Collegetown. Or in
the library here.
PoS
BW
I finished in '52, and then went to Chapel Hill as a postdoc
with Oscar Rice. The reason I knew of Oscar Rice was because of
his early work in energy transfer theory. So I went there with
the thought that I would be continuing along the lines of my thesis,
and when I arrived, he told me that I could work on anything I
wanted to work on, but if I wanted to talk to him while I was
there, I'd better learn something about phaseequilibrium and
critical points, because that's what he was working on at the
time. So I thought I better learn something about phaseequilibrium
and critical points, and that of course was the beginning of the
story.
PoS
So, from what sources did you learn about that?
BW
Well, I read Oscar Rice's papers. And at that time his main concern,
and that was really a central concern in the later development
of the subject as well, was an obvious discrepancy between the
shape of the coexistence curve, what is now called the critical
point exponent beta. An obvious discrepancy between that and experiment.
And even though that discrepancy was known very early, even in
[Johannes D.] Van der Waals' day, it didn't really impress
itself on people's consciousness until [E. A.] Guggenheim's corresponding
states paper in 1945, when he put the coexistence curves of rare
gases and small molecules, methane, nitrogen, and so on, all on
a common scale and showed that on such a common scale those coexistence
curves all fell on top of each other, and with an obvious exponent
much closer to 1/3 than to 1/2. By then, even though that had
already been seen in Van der Waals' day, probably in the
late 19th century, even though it had already been seen then,
it was so obvious and undeniable at the time of Guggenheim's paper
that one had to find some understanding of what's going on. So
that was very much in Rice's mind, and Rice was doing experiments
as well as theory. Doing experiments in binary liquid mixtures,
which we know from various kinds of transcriptions, lattice gas
models and so on related to the Ising model, we know that it's
essentially the same thing as the binary liquid mixture with its
phase separation, it is essentially the same thing as the Ising
model with permanent magnetization, liquidgas critical points.
We knew by then these were all the same thing. Especially since
the [C. N.] Yang – [T. D.] Lee papers came out in '52, or
the early '50s, and that demonstrated the equivalence of Ising
model and lattice gas.
PoS
This was all part of your toolkit, so to say?
BW
This was all part of my toolkit. These are things I learned while
I was a postdoc, and in fact at some point Oscar Rice assigned
me the project of learning the YangLee papers, and then reporting
on them in a seminar. Which I did. I can't remember if it was
a formal seminar or just a gathering. There were physicists there
as well as chemists, and I presented the YangLee papers.
PoS
And the physicist Fritz W. London was there at that stage?
BW
London was in Duke, not in Chapel Hill, and he wouldn't have
been present at that gathering. I think Eugene [Merzbacher] might
have been there. I don't remember who else.
PoS
Lars Onsager was certainly known?
BW
Onsager was known. Onsager was known, I'm trying to remember
when. That, of course, was a tremendous advance in the subject.
I'm trying to remember when I learned about it. It's conceivable
that, yes, I did know about Onsager while I was a postdoc.
PoS
And you knew about the earlier paper by G. H. Wannier and H.
A. Kramers, and these people?
BW
Yes, I must have known about all those things. I didn't study
those really until later. It wasn't until later that I learned
transfer matrix methods and things like that.
PoS
Lev Landau, Vitali Lazarevich Ginzburg, and their work?
BW
Landau/Ginzburg much later. In fact, this is now another thread.
Well, I was already interested in interfacial tensions while I
was a postdoc with Oscar Rice. Among the experiments he was doing,
the measurements of interfacial tensions at approach to the critical
point. So I knew there was a critical point exponent associated
with surface tension interfacial tension. It wasn't until I
was already here as a faculty member that I learned about the
Van der Waals surface tension work, and the Van der Waals functional,
which in fact is the Landau/Ginzburg function, but it was preceded
by 70 or 80 years by Van der Waals. So I knew of the Van der Waals
freeenergy functional long before I knew of the Landau/Ginzburg
Hamiltonian, but once I saw that, of course, I saw that they were
essentially the same thing.
PoS
So when you were studying these papers in Chapel Hill, were you
working on a problem?
BW
Yes, it was a problem of trying to understand these deviations
of the critical point exponents from their mean field values and
to try to construct some kind of an equation of state that would
incorporate these and so on. That's something that I didn't accomplish
till later, but I was already interested in the problem at that
time. I remember even doing something, have you heard about the
socalled parametric model in critical phenomena? It's just a
way of expressing the equation of state in the neighborhood of
a critical point in a way that incorporates nonclassical critical
point exponents. It was developed by people like Peter Schofield.
We're now talking about a somewhat later stage in the history
of the subject. But I remember that I was already trying to do
such things when I was a postdoc at Chapel Hill, ideas that were
the same idea as lay behind these parametric models, but I never
got that far. I never succeeded.
PoS
Can you tell us a little more about what approach you were taking
at that time? Did you use works like nonclassical mean field?
BW
I don't think we used the expression mean field, because the
expression mean field came from the magnetization side, and while
we knew the connection to magnetism, we weren't particularly interested
in that. So we didn't use the expression mean field, but we certainly
knew about expansions in integer powers in the neighborhood of
the critical point, what's now called Landau expansion, we thought
of it more as Fowler/Guggenheim expansion, but we now call that
Landau expansion. So we knew about expansions in the neighborhood
of the critical point. And that was one of the things Oscar Rice
was doing, but trying to incorporate nonclassical critical point
exponents. We analyzed data on critical isotherms, and we found
powers close to 4. We now know that closer to 5 is the correct
answer, but we already knew at that time that the critical point
exponent for the critical isotherm that is close to 4 is nonclassical,
because 3 was the classical Van der Waals answer. So we analyzed
data, Oscar Rice and I analyzed data, we found this nonclassical
critical point exponent. Oscar Rice had some ideas about how critical
point exponents ought to be related to each other, what we now
call scaling laws. He did not know that the critical point exponent
which we now call gamma, which is the divergence of the susceptibility,
compressibility, he did not know that that was not classical.
So in those days we still thought that gamma had the value 1,
and we now know that it's more like 1 1/4. So Oscar Rice had imagined
some connection, had seen from thermodynamic arguments that there
had to be some connection among critical point exponents. And
the connection that he came up with was a precursor to the ones
that we now know except that he thought that gamma had the value
1. But except for that, he had what is now one of those criticalpoint/exponent
relations, and that was the kinds of things that we were interested
in.
PoS
BW
Yes, he did thermodynamics, you can get a lot from thermodynamics.
In fact, in later years, people like Stanley Rushbrooke and R.
B. Griffiths using thermodynamic arguments found inequalities
connected with these critical point exponents, but those same
thermodynamic ideas, with a little optimism, led to equalities
rather than inequalities.
PoS
So you knew what was the problem that you were going to address.
BW
PoS
Can you tell us about where you thought nonclassical approaches
lay, or what approach did you take towards uncovering the nonclassical
exponents?
BW
To uncovering their values, you mean?
PoS
Yes, what push did you have to get away from the classical.
BW
Well, ultimately, I knew what the classical equation of state
looked like in the neighborhood of the critical point, so this
really went back to Van der Waals, and these power series expansions
which we now call Landau expansions, carried to loworder gave
what we now know as the classical behavior of the critical point.
So I took that classical equation of state, and that form of it
near the critical point, and I asked myself, what are the least
changes that one would make in this in order that these exponents
come out with what we know to be from experiment and Ising model
calculations, what we know to be their nonclassical values. So
I tried to make the least radical change in the form of the equation
of state, and I came up with something. I'm now talking about
1964 or 1965, so I had already been here for quite some number
of years as a faculty member, and this was long after my postdoc
days. And I came up with some form of this modified classical
equation of state which incorporated nonclassical critical point
exponents for the coexistence, which we now call beta, and for
the susceptibility and compressibility curve, which we now call
gamma, those exponents. And I then said that from thermodynamics
alone, given this form of the equation of state that I was working
with, I could calculate the heat capacity, and see what that does
in the neighborhood of the critical point, and I calculated the
heat capacity and I found it diverging logarithmically. That was
pretty astonishing, because I already knew from Onsager that the
heat capacity of the twodimensional Ising model diverged logarithmically.
So I asked myself, what was it about this particular equation
of state, what were the features of this particular equation of
state that allowed me to do that calculation and that led to that
answer? And I saw that the particular features of that equation
of state, the very special one that I was working with, a highly
specific one, that those of its features that allowed me to incorporate
the nonclassical critical point exponents, and allowed me to
do that calculation for specific heat were a certain homogeneity
of form that is now called scaling. And so I said that if one
imagines that instead of the highly specific one that I was working
with, which I had no reason to think was correct, if I said that
I'll just abstract that crucial feature of it, that homogeneity,
and imagine that that's what does it, then I again calculated
the heat capacity and again found a logarithmic divergence. And
the reason the logarithmic divergence is at the values of beta
and gamma that I was working with were such that alpha plus two
beta plus gamma equals two gave me alpha equals zero because I
had two beta plus gamma equals two. But I could then see that
if beta and gamma were slightly different from the values I assumed,
I would have an alpha which was not zero, a power law divergence
of the heat capacity.
PoS
Back in the '5254 period, the approach you were taking was a
minimal alteration of the equation of state?
BW
No, that came later. In '52 to '54, I was taking the thermodynamic
arguments of Rice, which suggested connections among critical
point exponents, and studying experimental data to see if those
connections held. It's interesting, also, that one can in retrospect
I didn't see this until many years later one can see that
Bob Scott in the chemistry department at UCLA had already realized
that there must be connections among what we would now call connections
among nonclassical critical point exponents. What he knew was
that the specific heat of beta brass showed a Lambda point in
the neighborhood of its order/disorder transition. And he wanted
to see if he could understand that, so he also knew that what
we now call the critical point exponent beta had a nonclassical
value, and by arguments that one sees in retrospect we didn't
know at the time but in retrospect, were somewhat like the arguments
that Oscar Rice was using, thermodynamic arguments. But again
without recognizing that what we now call the critical point exponent
gamma had a nonclassical value, he was able to connect the nonclassical
beta to a nonclassical alpha, and see that that alpha actually
corresponded to a heat capacity divergence. That was really very
early.
PoS
BW
I could look it up for you? Want me to take a minute and look
it up for you?
PoS
BW
It's [Robert] Scott and it would have been probably in the late
40's.
PoS
Let me rephrase the questions. What I hear you saying is that
your thinking is primarily thermodynamics.
BW
It certainly was at that time, and I guess…
PoS
And what about the microscopic/macroscopic interplay?
BW
I never myself analyzed a microscopic model to determine its
critical point exponents other than models that I was able to
transcribe into the Ising model and make use of what had already
been known about the Ising model. I worked a lot with lattice
gas models, and with lattice liquid mixtures, and so on, so I've
done a lot with those, but always making the transcription to
the Ising model and making use of what others had found for the
values of critical point exponents.
PoS
What is the difference in thinking that you brought in? Granted,
having been exposed to physics courses, but having come out of
a community of chemists and physical chemists, the way you think
about the problems in contrast to physicists…
BW
Yes, well, I think the answer would be, I think about them thermodynamically,
and I also make use of the physical chemistry literature, that
is, liquid mixtures rather than magnets, are what I have in mind.
So in later years, for example, when I learned about tricritical
points from Bob Griffiths, to some extent also from Michael Fisher,
but I worked with Griffiths directly on that. The literature that
we ultimately uncovered was phase equilibrium, three or four component
fluid mixtures, so the idea of tricritical point arising for example
in certain magnets, or higher order critical points in general,
or in mixtures of the helium isotopes, helium3, helium4, that
was something the physicists knew about and I learned from the
physicists. But my own thoughts and activities were always centered
around liquid mixtures, things you could pull off the shelf in
a laboratory.
PoS
Did you work closely with O.K. Rice after you read the literature
he suggested?
BW
Oh yes, he and I worked closely together and we wrote some things
together.
PoS
Were there other people there that you worked with or talked
to a lot?
BW
No, he had one or two experimental students at that time, experimental
postdocs whom I knew quite well and talked with, but I wouldn't
say there was any advance, I don't think we influenced each other
very much in our thinking. I mean, we were friends, but I don't
think that our scientific interactions were that important to
either of us.
PoS
Were you still in touch with Bauer?
BW
With Simon Bauer? Well, I was really there only two years, and
then came back, so it was a relatively short time, so in a sense
I never lost touch with Simon Bauer.
PoS
So you come back to Cornell in '54?
BW
Came back here in '54 and I joined the chemistry faculty, first
as an instructor, the instructor grade existed at that time, then
assistant professor and so on.
PoS
At that time, is your work primarily in the area of phase transitions?
BW
Yes, since that time my work has been primarily in the area of
phase transitions. I did a few things also on chemical kinetics
that came from my graduate student days with Simon Bauer, but
most of what I did had to do with phase transitions. Some work
of later years has to do with the interfaces between coexisting
phases and interfacial structure and interfacial tension.
PoS
So when you came here, were you doing the same problem with the
same techniques, that is, looking at the data to find evidence
of thermodynamic relations …?
BW
To some extent, but mostly I was trying to construct nonclassical
equations of state. An equation of state that would replace the
Van der Waals, or what we now call mean field theory equations
of state to incorporate critical point exponents. And that I ultimately
did around '64, '65.
PoS
So why the change in method? Do you remember?
BW
Well, as I said, even while I was in Chapel Hill as a postdoc,
I was still trying to do that, I just didn't make much progress.
So I kept thinking about it and kept trying. I had a correspondence
with Michael Fisher in those days, in the 1950's.
PoS
So you knew about Cyril Domb's work and all of that?
BW
PoS
BW
The time that I really learned about the work of Michael Fisher
and Domb and so on was in '61, '62, when I was on sabbatical leave
in Amsterdam. And I was sitting in the group of [J.] De Boer,
in statistical mechanics. I wasn't working with people there,
I knew what they were doing, people like Eddie Cohen and Hans
Van Leeuwen were there at that time, so I knew what those people
were doing, and I knew about that work in cluster expansions that
they were developing, and diagrammatic expansions. So I knew about
all that stuff, but that's also where I was learning about what
Michael Fisher did with correlation functions and deviations from
OrnsteinZernike theory, and where I mostly appreciated the Onsager
work. I knew about it long before, but I got a greater appreciation
of it then. And I was still very excited about nonclassical coexistence
curves, and I saw them everywhere.
PoS
How much did you keep track of things which are happening in
liquid helium, for example? Such as at the University of Nottingham…
BW
PoS
Alexander Voronel's work on liquid transition?
BW
On heat capacity divergence through the critical point. Yes,
I'm not entirely sure when I first learned about that, but certainly
I knew about it from some fairly early stage on.
PoS
BW
PoS
we would like to know a little bit about what you were doing
between '54 and '61. What were you teaching, for example? Were
you teaching about phase transitions, was that something that
would show up in the courses?
BW
No, it wouldn't have. That's because that would have been a much
more advanced topic. I began teaching statistical mechanics from
a fairly early stage, I was teaching undergraduate physical chemistry,
and in fact my first teaching assignment was undergraduate physical
chemistry laboratory, where I knew as much as the students did.
And so I was teaching undergraduate physical chemistry, and in
the graduate courses I was teaching our beginning graduate quantum
mechanics and our beginning graduate statistical mechanics.
PoS
BW
In chemistry, yes, so that's what I was teaching. If you want
to know what I was doing, I can look at my publication list and
see if I can remind myself what I was doing then. So the late
50's and early 60's, that was your question?
PoS
Between '54 and the sabbatical.
BW
I guess the first thing that I did I should say that going
back to my time in Chapel Hill as a postdoc, there were theories
by Joe Mayer, I don't know if you've come across his name in connection
with your study, but essentially looking at cluster expansions
and virial expansions, and trying to learn something possibly
about phase transitions from the convergence or divergence of
these theories. There were some peculiar theories developed at
the time by Joe Mayer and also by Oscar Rice, the socalled Derby
Hat picture of phase coexistence. And that's all dead now. We
realize that none of that was right, and that was all very much
in the air. Also very much in the air was the question of the
convergence of virial expansions, and the possible divergence
being connected with phase transitions. So I looked at the virial
expansion for the ideal Bose gas, and analyzed the neighborhood
of its lambda transition, of its phase transition, the ideal Bose
gas. And I tried to estimate its radius of convergence. Had terrible
estimates, and never got anywhere close to it, but published the
paper. This was while I was still in Chapel Hill. I was doing
this paper called "The virial series in the BoseEinstein
gas," and in later years, you know, did you know Wolfgang
Fuchs here in mathematics? I got interested in the problem, and
in later years he found a much, much better estimate of the radius
of convergence and saw that it has a finite radius of convergence
that had nothing whatever to do with the phase transition. I mention
that only because that paper of mine came out in '54, and so that
was sort of toward the end of my postdoc years with Oscar Rice.
Then my first paper here at Cornell came about because of some
paper that I had read in the German literature on the vapor pressure
and heat of vaporization when you were near the critical point.
And so I wrote a little paper pointing out that the heat of vaporization
had to have the same nonclassical behavior as the coexistence
curve, and I had that paper translated into German by someone
in the German department here, and published. So that was not
my German, that was the German of my translator. So that was the
first thing that I did here. That was seeing that one had to have
the same critical point exponent for heat of vaporization as one
had for the coexistence curve. And then I published a paper, this
I'm sure that I was already working on in Chapel Hill, but it
didn't come out until I was here, called "The Structure of
the Configuration Integral: the Statistical Mechanics of Pure
Fluids." At that time I was very conscious of the work that
Lee and Yang had done on the zeroes of the partition function
lying on a circle in a complex fugacity plane of the lattice gas,
the equivalent of the Ising model. And so I took a Van der Waals
like a classical equation of state, and asked for the zeroes of
its partition function in a sort of thermodynamic plane. So that's
what I published. I don't think it was of any significance, but
I was very excited about it at the time. And then there is my
paper with Oscar Rice analyzing the critical isotherm in the neighborhood
of the critical point, that came out at the time when I was already
here. The work was done in Chapel Hill, but it came out when I
was here. Then I did a paper that went back to my work with Simon
Bauer on inelastic molecular collisions called "The Energy
Dependence of Inelastic Molecular Collision Probabilities." Another
paper, "Statistical Mechanics of LiquidVapor Equilibrium." So
this was coming back to the statistical mechanics, thermodynamics
side. Again, it wasn't very deep or a very important paper. It's
not one I'm especially proud of. And then I did some more on inelastic
molecular collisions. This is really off the subject, so I don't
know if you want to hear about that. You're aware of the fact
that going back to [James Clerk] Maxwell's days, it was recognized
that if molecules interacted with interaction potential of one
over r to the fourth. So force was one over r to the fifth, and
then what is in effect the Boltzmann equation, these transfer
equations could be solved exactly. And so I thought that one should
be able to do the same with the Schrodinger equation with one
over r to the fourth potential and this might be of some significance
in this problem of inelastic molecular collisions. So I worked
on that, and it turns out that with the 1/r^4 potential, the Schrodinger
equation becomes equivalent to a Mathieu equation, one of these
equations, or a Hill type equation. So I studied something about
these, again, exponents they're called, and associated them with
Hill type equations in connection with the 1/r^4 potential. But
that was quite a departure from anything we've really been talking
about. I did some more work on collision theory, relaxation of
string oscillator, onedimensional inelastic collisions with hard
rod interactions. Collision theory and kinetics of dissociation
of diatomic molecules, mean first passage times, bimolecular
reactions.
PoS
Do you ever talk to the physicists at that time, Bethe, and others?
BW
No, uh, Bethe I had known from my graduate student days. I had
a partial research assistantship because of a collaboration between
Simon Bauer and Arthur Kantrowitz. So I learned about energy transfer
on surfaces, and learned some little fluid dynamics, I remember
HugoniotRankine equations, I'm not sure I could tell you what
they are, but I remember the word. And then I did a wonderful
calculation, again, this is in the energy transfer area, not critical
phenomena. I did a wonderful calculation of the rotational relaxation
of rough spheres. I remember I had to calculate the spectrum of
a collision kernel, and that was really an amazing calculation.
It was one I particularly enjoyed a lot, and again one I don't
think was of particular importance, but it was a lot of fun. Then
my first experimental paper in critical phenomena, I had a visitor,
DeForest Rudd, who at that time, unfortunately he's no longer
living, at that time he was a professor at Lincoln University
in Pennsylvania. It's an all or mostly black college. And he was
on sabbatical leave here, and he wanted to work with me on an
experimental project. I told him that there were known cases of
lower critical solution points, that is, where you achieve the
critical point, not on increasing temperature, going through a
twophase region on increasing temperature until the point where
the two phases become identical at a critical point, but there
were known cases in which you had lower critical solution points,
where you go through the twophase region on lowering temperature,
and come to a point on decreasing temperature at which the two
phases become identical. Sometimes the socalled closed loop coexistence
curves, where the same system would have both an upper and lower
solution point. And I wanted to be sure that at the lower critical
solution point the critical point exponent had the same value
as an upper solution point. From what we now know, it's obvious,
but at the time it wasn't obvious, so I wanted to check that.
So I had DeForest Rudd work on a mixture of water and ethylene
glycol mono isobutyl ether, which had a known critical solution
point. And he did a careful study of the coexistence curve and
he found a critical point exponent that was close to the one third
that we already knew to be the case in the upper critical solution
point.
PoS
You had a laboratory at that state?
BW
Yes, well, we scrounged up a little laboratory space. In later
years I actually had a laboratory of my own.
PoS
BW
That was 1960. Then came a couple more papers…
PoS
May I ask how Rudd knew that he'd like to come here and work
with you?
BW
I'm not a hundred percent sure. I think I've just forgotten.
I think he came to the department as a visitor and then looked
around to see who's doing what, and then decided he'd like to
work with me. As I recall, that was the case. He might have known
about me before he came, but it would be doubtful. I don't see
why he would have.
PoS
And by 1960, you have tenure here and you have graduate students
and everything else.
BW
Yes, I've had very few graduate students. In the early days,
there were a couple of graduate students I did have who were great.
Well, my first graduate student worked with me on energy transfer
problems. His name is Jerome Weinstock. Then in somewhat later
years, I had John Wheeler, who's now at La Jolla, and John Zollweg,
who went to the University of Maine from here, and is now back
at the Cornell Theory Center. And John Wheeler and I worked on
phase transition problems, critical point problems, and John Zollweg
was my first experimental graduate student, and he worked on critical
points in threecomponent liquid solutions, this is the early
60's in the neighborhood of what's called the plait point, which
we recognize to just be another critical point but in a slightly
different context. You heard from Michael Fisher about what he
has called the renormalization of critical point exponents. If
you display a phase diagram, for example, in a space of what we've
come to call densities rather than fields, using later language
of Griffiths and Wheeler, then critical point exponents, the way
that the phenomena are represented, this is not a different kind
of critical point, but a different kind of representation, leads
to critical point exponents that are what Fisher later called
renormalized. And we had already recognized in a particular example,
earlier, we had a fluid model that we transcribed to an Ising
model. We had already perceived this socalled renormalization
where the socalled coexistence curve, for example, in the normal
representation we'd have a critical point exponent beta. In this
representation where you were dealing entirely with density variables
and no field variables, you'd have a critical point exponent beta
divided by one minus alpha, that is what Fisher later called renormalization.
But we had already seen this in the connection we made between
our fluid model and the Ising model, and we wanted, in general,
to see if this socalled plaitpoint model, threecomponent liquid
solution, where you achieve a critical point just by varying the
concentration and not temperature all at a single fixed temperature
that again had the same critical point exponent. And we found
a slightly larger exponent that corresponded to this socalled
renormalization. So except for this work by this visitor DeForest
Rudd, that was my first experimental work in critical phenomena,
and that was with my thengraduate student John Zollweg. So at
the same time I was working with John Wheeler as a theoretical
student and we were working on various kinds of solution models,
again with critical phenomena of liquid mixtures.
PoS
So in this period you go from Chapel Hill, to Cornell, then to
Amsterdam?
BW
Yes, on sabbatical leave.
PoS
In that period, you've given us a list of the topics that you've
worked on. Can you think about how the methods that you were using
changed? You started by doing quantum mechanics and energy transfer,
then did statistical mechanics at North Carolina, and what about
in this six year period?
BW
Again, except for the continued work on energy transfer and chemical
kinetics, I was working largely with are we talking after I
came back from sabbatical leave?
PoS
BW
Before the sabbatical. Well, again it was some statistical mechanics
and thermodynamics.
PoS
So no big changes in the methods you were using? No huge inputs?
BW
No big changes in the methods I was using, that's right.
PoS
What about your network of contacts, how has that changed in
the six year period? You started first by talking mostly to Bauer,
then second talking mostly to O.K. Rice, and then in the next
six years, who were the influences and what kind of conversations
were you having that affected your work?
BW
I can't think of people. Well, I was in continued contact with
Simon Bauer on the energy transfer side, continued contact with
Oscar Rice, so I maintained those contacts.
PoS
Did any correspondences emerge out of your German publication?
BW
Well, probably, probably with the person who wrote that original
paper that that instigated my paper, I think I had some correspondence
with him.
PoS
BW
No, you see, Michael Fisher, whom I got to know extremely well
later, and Domb and so on, I think that I learned about that stuff
mostly while I was on sabbatical.
PoS
What about people working in Amsterdam?
BW
That's where I really studied that literature and learned about
how Michael Fisher, for example, found that a calculation that
Onsager had done on the correlation function implied an exponent
gamma equals seven quarters for the twodimensional Ising model
and so on. So I learned about all those things in '61/'62, while
I was in Amsterdam.
PoS
Let me rephrase the question. You're now in Amsterdam talking
to people who do statistical mechanics, deriving equations of
state. Is there a bridge being made between you work, attempting
to find different formulations of equations of state and a microscopic
approach? How, from a statistical mechanics point of view, how
do you get deviations from what you call the classical?
BW
Except for having known of the Onsager papers and the work of
the Domb group, I didn't myself…
PoS
How about Onsager and Oliver Penrose and?
BW
Onsager and Penrose and the Long Range OffDiagonal order. No,
that I didn't become conscious of until much later.
PoS
Did we finish setting the stage?
BW
No, I'm trying to think of the answer to your question.
PoS
BW
Well, I knew about all that stuff.
PoS
But were you talking to people?
BW
PoS
So your contacts were local?
BW
PoS
Of the local group, who were the contacts? Between '54 and '61.
BW
No, I was essentially alone.
PoS
BW
Because that was a known center of modern statistical mechanics.
But not particularly because of anything to do with phase transitions
or critical phenomena, it's just to be in an environment of statistical
mechanics.
PoS
How long where you there?
BW
For nine months, that was my sabbatical leave.
PoS
BW
Well, the head of the Institute was [J.] De Boer. And then Eddie
Cohen was what we would now call an assistant professor, at that
time he had a different title, in Dutch, I think, Lecturer. So
he was there, and his interests were nonequilibrium statistical
mechanics. Theory of transport, ChapmanEnskog equations, Boltzmann
equations, things like that. Hans Van Leeuwen was a senior graduate
student in that group. He later came to do some postdoc work with
me at Cornell, oh, so that was a local contact that would have
been in the early 60's. I remember it before 61. No, I was really
pretty much alone.
PoS
What did you learn in Amsterdam? How did it affect your work?
BW
Oh, that's where I became fully aware of the kinds of things
that were being done in the Domb group and the things that Michael
Fisher did relating correlation functions to susceptibility and
so on. So that's where I learned that literature and that was
very important.
PoS
Not from seminars or conferences, but from reading the literature?
BW
Yes, from reading the literature.
PoS
And you were working alone?
BW
In Amsterdam, yes. Again, I knew the people, and was talking
to them about the sorts of things they were doing, a lot of work
in cluster expansions, diagrammatic expansions, and I knew something
of things on liquid helium that Eddie Cohen and Hans Van Leeuwen
were doing. But I didn't do anything in those directions.
Continue reading part II of the Widom interview.


