**A case of attempted idea theft**

**Uhlenbeck’s Problem**

This document presents the history of my Ph. D.
dissertation: *Contributions to the theory
of non-equilibrium thermodynamics *[1], written as a thesis in 1969 and
published as two journal articles in 1970 [2, 3]. However, the key ideas and
results that make up the thesis were conceived in 1967, now 40 years ago. My
purpose in writing this piece has several components. It is a good time to look
back and appreciate the content of the thesis. This will emphasize the
important contribution of George Uhlenbeck that I shall refer to as
“Uhlenbeck’s problem.” It will also provide an opportunity to reflect on
certain aspects of scientific research, of a human nature, that make science
more difficult to do and enjoy than one might anticipate. I hope these features
of the piece will be of special interest to young scientists. I also hope to
set the record straight regarding some recent priority claims about general
ideas associated with the applicability of hydrodynamic fluctuations theory to
modern day nanotechnology.

I entered The Rockefeller University in 1965 as a graduate
student in mathematics and physics. At that time there were about 100 graduate
students in total, of whom about 5 were in physics and mathematics. The rest
were in the life sciences, many on the medical side, and everyone had a
^{rd} student out of a final
total of 45. This is a large number of students for a theoretical physicist,
even today.

* At the time there
was a foreign language requirement that demanded competency in two languages. I
qualified in French, by translating three pages of mathematics wherein the
formulae and technical terms made it easy even though I had never studied
French. I also qualified in the reading of hieroglyphics on the basis of Sam’s
course and the exam was given at the museum of natural history where obelisks
had to be translated. I think the language requirement no longer exists.

George Eugene Uhlenbeck was born on 6 December, 1900 in
Batavia, Java (now Jakarta, Indonesia), and died 31 October, 1988 in Boulder,
Colorado, where his famous son, Olke, holds a faculty position and is a pioneer
in modern RNA research. George’s family moved to Holland when he was six, and
there he later earned his Ph.D. from Paul Ehrenfest. Ehrenfest was a student of
Ludwig Boltzmann, who had been a student of Jozef Stefan. This is one of the
royal lineages in thermodynamics and statistical physics. Other lineages
separately involve Maxwell, Gibbs, and Onsager.

My interest in working with George was to explore
irreversible thermodynamics. I was familiar with the work of Lars Onsager [4,
5], whom I had met several times in Miami (where he spent much time after
retiring from Yale, and played bridge with my parents) and at The Rockefeller
where he spent 6 months while I was a student, and I was also interested in
stochastic processes of the sort that George had written about in several
papers over two decades [6, 7]. However, George was in the habit of assigning
research problems to his students, and working on them in a very close
collaboration. He assigned to me the problem of off-diagonal long range order
in density matrices, an abstruse problem suitable for a talented mathematically
minded student. I did work on this problem for about a year (1965-1966) but
never really got into it. It was both difficult and not of my own choosing.
This situation began to be seen by the senior faculty as “Uhlenbeck’s problem.”
That is, **I** was the problem! During
1966-1967 I mostly explored the Onsager theory and especially the presentation
in the book by de Groot and Mazur [8]. I also read the stochastic papers of
Uhlenbeck et al. in the Dover volume edited by Nelson Wax [9]. During the
summer of 1967 I finally wrote up what I viewed was a completed picture of my
thinking regarding irreversible thermodynamics. By then George was rather
frustrated with me for lack of progress on his project. *Uhlenbeck’s problem* was what to do with me.

I
sought help from Mark Kac (one of the funniest and nicest men I have ever
known). Mark quickly saw that I had indeed found something new to say on a
subject that most experts thought was a finished subject. To be brief about it,
the issue had to do with time reversal properties of dynamical variables in the
theory. The Onsager theory was in two parts, one part for even variables and
one part for odd variables. I was trying to apply this theory to the
hydrodynamic Navier-Stokes equations in order to get the fluctuation formulae
proposed by Landau and Lifshitz [10]. There was a problem. The hydrodynamic
variables were neither even nor odd because of macroscopic dissipation. I found
a way to generalize Onsager’s theory so that it worked in this case too and
found a derivation for the Landau-Lifshitz formulae. Mark realized that this
was not merely window dressing and went to George on my behalf.

I mention in passing that the distinction about the
Landau-Lifshitz formulae and the proper derivation of them is still recognized
today. In a recent paper one finds the statement: “The fluctuating stresses for
the Navier–Stokes equations are proposed by Landau and Lifshitz at first, and theoretically verified by Fox
and Uhlenbeck.” [11] In anticipation of what comes below, this paper is an
example of the application of fluctuating hydrodynamics to a nanotechnology
problem.

George became convinced that I did have something new to
say and that I had not been living the good life as a prodigal graduate student
for the previous year. He quickly embraced my research project and made many
important suggestions. The most important suggestion was “*Uhlenbeck’s problem*”, the real problem!, proposed in the fall of
1967. I will now present a little history in order to place this problem into
context.

The theory of Brownian motion as a stochastic process was
proposed by Paul Langevin [12]. It was this theory that Onsager had generalized
into a theory for irreversible processes in general, in many variables. When
Onsager’s theory was applied to hydrodynamics, Landau and Lifshitz got the
fluctuating Navier-Stokes equations with explicit formulae for the fluctuations
based on the fluctuation-dissipation identity. It was this step that my work
made rigorous. Landau and Lifshitz got the correct formulae but not the correct
argument. The time symmetry was the heart of the issue. Nearly everyone refers
to the fluctuating Navier-Stokes equations as the Landau-Lifshitz equations.

George
wanted to “close the loop” in the reasoning. He wanted me to see if I could use
fluctuating hydrodynamics for a body in a fluid and derive the Langevin
equation and its stochastic properties from fluctuating hydrodynamics with
appropriate fluid boundary conditions (including the fluctuations) on the body.
This was part of a general picture about which he had written earlier in which
the idea of “contraction of the description” was a key element [13]. This idea
relates to the question of the origin of irreversibility in macroscopic physics
when the microscopic physics is time reversal invariant, and it had been the
main problem that Boltzmann confronted. The broader Uhlenbeck problem was to
work out the fluctuations for the Boltzmann equation and then show that the
Landau-Lifshitz hydrodynamic fluctuations could be derived from the Boltzmann
fluctuations by contraction of the description. This I did readily with an
extension of the Chapman-Enskog contraction method [1, 3].

The
next contraction was to get Langevin’s equation from the hydrodynamics. That
was more difficult but the reward would be that one would have come full
circle: Langevin equation-Onsager theory-Boltzmann equation
fluctuations-Landau/Lifshitz hydrodynamic fluctuations-Langevin equation. I
struggled with this problem for about three months and then one afternoon found
a beautiful boundary value identity that made it fall into place. This identity
is a central result in *Contribution to
irreversible thermodynamics I* [1, 2].

This result has a special significance for today’s
nanotechnology that only Uhlenbeck seemed to imagine at the time. From Langevin’s
equation and the work of Einstein on Brownian motion [14] it is possible to get
Einstein’s formula for the diffusion constant, D, of a Brownian particle in
terms of the radius, R, of the particle. The boundary condition derivation I
used based on fluctuating hydrodynamics put no lower limit on the size of the
radius R, other than the atomicity of matter, or its molecular combinations.
Thus the hydrodynamic fluctuations appeared to be valid even at the
sub-nanometer scale! Strong evidence for the correctness of this interpretation
is provided by light scattering [15]. From Rayleigh-Brillouin line-shape, as a
function of frequency, the measured diffusion constants, D, of molecules are
found to agree with the Stokes formula for the drag on a body [16], thereby
making the connection to the radius, R. While the wavelength of visible light
is in the hundreds of nanometers, the radius of the molecules is merely
nanometers.

In 1981-1982, Magdaleno Medina-Noyala and Joel Keizer
applied this thinking to the problem of the dynamic structure function for
neutron scattering [17]. They were able to compare their results with the
molecular dynamics calculations of Alley, Alder and Yip [18]. Remarkable
agreement was shown and the hydrodynamic fluctuation approach was much easier
to do than the time intensive and complex molecular dynamics simulations.
Interestingly, at the time the molecular dynamics results were viewed as the
benchmark for the theoretical results. This work underscored the applicability
at the nanoscale of the Landau-Lifshitz fluctuation formulae. Keizer made many
other applications in his monograph on irreversible processes [19], and
illustrated several more “contractions of the description.”

In 1999 my colleague Uzi Landman came to me armed with the
Landau-Lifshitz formulae, and wanted to know what I thought about the
applicability of hydrodynamic fluctuations at the nanoscale. He wanted to put a
bright new post-doc, Michael Moseler, on the problem and thought that it would
be potentially faster and cheaper to use fluctuating hydrodynamics equations
than to do molecular dynamics calculations. The question was would it be as
accurate as molecular dynamics. I assured him that it should work. His proper
concern for how to spend his resources on this post-doc’s efforts made him
consider: time spent, money spent in salary, computer time and equipment
dedication. To embark on this plan without feeling secure in the method would
have been potentially foolish.

I explained the contraction of the description ideas, and
the derivation of Langevin’s equation from hydrodynamics. The D and R
connection discussed above implied that fluctuating hydrodynamics does work at
the nanoscale. Uzi went forward and kept me appraised of the progress, that
turned out to be great. Post-doc Moseler did very fine work and was able to
construct the correct boundary conditions for a problem about nanojets. The
project succeeded and was featured on the cover of Science in 2000 [20]. There
is a reference in the paper to Landau and Lifshitz formulae [10] followed by a
reference to the Fox and Uhlenbeck paper [2]. In an acknowledgment, I am
thanked for bringing the authors’ attention to my paper. It can be said that if
Uzi had *not* spoken with me and had used the Landau and Lifshitz formulae
without my assurances that he and Moseler would have done the same work as they
did do. I agree with this assessment. The fact is that the application of the
Landau and Lifshitz fluctuation formulae is justified, whether one knows how to
justify it or not, and simply requires specification of the boundary conditions
for the particular problem at hand, e.g. nanojets. Thus, we would speak of the
“Landau and Lifshitz formulae” and refer to the boundary conditions as, say,
the “Landman-Moseler boundary conditions”, or better yet, the “nanojet boundary
conditions”. Who would be so hyperbolic as to make eponymous reference to the
equations (the Landau-Lifshitz equations) themselves, calling them the
Landman-Moseler equations? Well, Landman would and did. Enthusiasm for one’s
own work is a good thing, if it doesn’t go so far as to gloss over the
contributions of others. The fact is that Uzi did talk to me about this
problem, and at least for awhile, was encouraged by my assurances.

On
March 9, 2007 I read, by chance, the headline in the GT Newspaper, The Whistle
(March 5, 2007), that read: *Research suggests f**luid dynamics works on nanoscale in real world.* Of course this
title caught my eye. As I read, I learned of a struggle to understand how to
make stochastic hydrodynamic equations, a revelation about how to do it, the
claim that other scientists were negative about the possibility and the
eponymous nomenclature referred to above. Except for the self-named equations,
I thought someone was writing about me! Given my connection to this subject’s
history, I am the unique person who would truly know the history and resonate
with this news item. You, gentle reader, are not me, but must remember that it
is I who writes this. Try to imagine my reaction as I read the news item
(attached below). But alas, imagine my dismay on realizing that Uzi was making
these claims for himself. Claims of originality at the deeper level of
fundamentals rather than merely stating boundary conditions, the
Landman-Moseler boundary conditions, but instead coining the “Landman-Moseler
equations” for the well established Landau-Lifshitz equations. Claims about a
heroic struggle to find the correct path. *Claims,
of which even I would be proud!*

The paper by Kang and Landman [21] referred to in the news item attached below is about nanobridges. It is another application of the Landau and Lifshitz formulae but for new boundary conditions for this new problem. This work in another fine example of the correctness of the claim that fluctuating hydrodynamics works at the nanoscale. It is the news item that contains the disputed claims, not this paper. Even the part that clearly caught my eye is written by a science writer, not by Uzi, and is not in quotation marks. It could be that the author of the news item embellished the story and is to blame for the excessive claims. Then Uzi could pull the piece for editing. He hasn’t done so. The science writer told me that as a practice he has his pieces reviewed by the scientist before publication. Scientists should certainly be accurate in refereed journal publications, and they have a responsibility to be equally accurate in news reports for a general audience.

My
mind screamed **unfair**! So much
hyperbole that the *spin* became a *false claim*. Right in front of the one
person who has a right to these claims. But even more importantly, claims that
rightfully belong as well with Landau and Lifshitz, whose equations Uzi brought
to me in the first place. Since neither Landau nor Lifshitz can defend
themselves anymore, I am here to defend them! Leave aside any of my personal
claims! Landau and Lifshitz would have felt affronted by the news item still on
our
*on
their behalf*!

I
tried the Georgia Tech informal conflict resolution process in order to get the
item removed for editing. I tried direct contact with Uzi by email (see the
attached email). Since, to date, he has not requested that the item’s
appearances in several web sites be stopped for editing, it is manifest that he
stands by the news article and the explicitly made claims of priority. Nothing
was done by GT administrators or the ombudsperson. The conflict resolution
process failed. I have protested this procedural failure to GT President Wayne
Clough. At no time has anyone disagreed with my assessment of the facts of this
case.

Please
be sure to read below the attached news item and my email to Uzi dated March
10. Then re-read the text above.

The
applicability of hydrodynamic fluctuations at the nanoscale has been justified
theoretically by the derivation of Langevin’s equation from fluctuating
hydrodynamics 40 years ago. Use of the Landau and Lifshitz formulae with
appropriate boundary conditions in nanoscale projects works accurately and much
faster than molecular dynamics computations. Several examples of the success of
this approach now exist, including Rayleigh-Brillouin light scattering, neutron
scattering, nanoparticle suspensions, nanojets and nanobridges. It was the
insight of George Uhlenbeck that led to the derivation of Langevin’s equation
and to the justification of applicability of hydrodynamic fluctuations to the
nanoscale so many years ago. My solution to Uhlenbeck’s problem is the key to
why fluid dynamics works at the nanoscale.

**References**

[1] *Contributions to the theory of non-equilibrium thermodynamics,* R. F.
Fox, The

[2]
"Contributions
to Non‑Equilibrium Thermodynamics. I. Theory of
Hydro-dynamical Fluctuations", R. F. Fox and G.
E. Uhlenbeck, *Physics of Fluids* **13** 1893-1902 (1970).

[3]
"Contributions to Non‑Equilibrium Thermodynamics. II. Fluctuation
Theory for the Boltzmann Equation", R. F. Fox
and G. E. Uhlenbeck,
*Physics of Fluids* **13** 2881-2890 (1970).

[4]
“Reciprocal Relations in Irreversible Processes. I.”, L. Onsager, *Physical
Review* **37** 405-426 (1931). “Reciprocal Relations in
Irreversible Processes. II.”, L. Onsager, *Physical Review* **38**
2265-2279 (1931)

[5]
“Fluctuations and Irreversible Processes”, L. Onsager and S. Machlup,
*Physical Review* **91** 1505-1512 (1953). “Fluctuations
and Irreversible Process. II. Systems with Kinetic Energy.” L. Onsager and S.
Machlup, *Physical Review* **91** 1512-1515 (1953).

[6] “On the Theory of
Brownian Motion”, G. E. Uhlenbeck and L. S.
Ornstein,
*Physical Review* **36** 823-839 (1930).

[7] “On the Theory of
Brownian Motion II.”, M. C. Wang and G. E.
Uhlenbeck,
*Reviews of Modern Physics* **17** 323-342 (1945).

[8] *Non-equilibrium
Thermodynamics,* S. R. de Groot and P. Mazur (North-
Holland
Pub. Co., Amsterdam, 1962).

[9] “*Selected papers on
noise and stochastic processes*”, Edited by N. Wax
(Dover
Pub., New York, 1954).

[10] *Fluid Mechanics,*
L. D. Landau and E. M. Lifshitz, (Pergamon Press,
London, 1959), Chapter XVII.

[11] “Multiscale simulation
method for self-organization of nanoparticles in
dense
suspension.”, M. Fujita and Y. Yamaguchi, *Journal of Computational Physics*,
**223** 108-120 (2007).

[12] “Sur la theorie du
mouvement brownien”, P. Langevin, *Comptes rendus
Acad.
Sci. (
Paris
)*, **146** 530-533 (1908).

[13] *Lectures in
Statistical Mechanics,* G. E. Uhlenbeck and G. W. Ford,
(American
Mathematical Society, Providence, 1963).

[14] *Investigations on the
Theory of the Brownian Movement, *A. Einstein,
(Dover
Publications, New York, 1956).

[15] *Dynamic Light
Scattering with Applications to Chemistry, Biology and
Physics,
*B. J. Berne and R. Pecora, (John
Wiley ans Sons Inc., New York, 1976).

[16] See page 66, section 20, chapter II of reference [10].

[17]
“Spatially nonlocal fluctuation theories: hydrodynamic fluctuations for
simple fluids.”, J. Keizer and M. Medina-Noyola, *Physica*
**115A** 301-338 (1982).

[18]
“The Neutron Scattering Function for hard Spheres”, W. E. Alley, B. J.
Alder and S. Yip, *Physical Review* *A* **27**
3174-3186 (1983).

[19]
*Statistical Thermodynamics of Nonequilibrium Processes, *J. Keizer,
(Springer-Verlag, New York, 1987).

[20]
“Formation, Stability and Breakup of nanojets”, M. Moseler and U.
Landman, *Science* **289** 1165-1169 (2000).

[21]
“Universality Crossover of the Pinch-off Shape Profiles of Collapsing
Liquid Nanobridges in Vacuum and Gaseous
Environments”, W. Kang and U. Landman, *Physical Review Letters* **98**
064504 (2007).

**Links to
recent papers citing Fox and Uhlenbeck paper of reference [2] above:**

- Fujita and Yamaguchi, Journal of Computational Physics
- Fetzer et al., Physical Review Letters, 2007
- Camacho et al., Physica A, 2005

**Atlanta** (February 23, 2007) — In 2000, Georgia Tech
researchers showed that fluid dynamics theory could be modified to work on the
nanoscale, albeit in a vacuum. Now, seven years later they've shown that it can
be modified to work in the real world, too – that is, outside of a vacuum. The
results appear in the February 9 issue of Physical Review Letters (PRL).

A propane liquid nanobridge breaks up in a nitrogen gas environment. (Image: Georgia Tech/Uzi Landman) |

Understanding the motion of fluids is the basis for a tremendous amount of
engineering and technology in contemporary life. Planes fly and ships sail
because scientists understand the rules of how fluids like water and air behave
under varying conditions. The mathematical principles that describe these rules
were put forth more than 100 years ago and are known as the Navier-Stokes
equations. They are well-known and understood by any scientist or student in
the field. But now that researchers are delving into the realm of the small, an
important question arisen: namely, how do these rules work when fluids and
flows are measured on the nanoscale? Do the same rules apply or, given that the
behavior of materials in this size regime often has little to do with their
macro-sized cousins, are there new rules to be discovered?

It’s well-known that small systems are influenced by randomness and noise more
than large systems. Because of this, Georgia Tech physicist Uzi Landman
reasoned that modifying the Navier-Stokes equations to include stochastic
elements – that is give the probability that an event will occur – would allow
them to accurately describe the behavior of liquids in the nanoscale regime.

Writing in the August 18, 2000, issue of Science, Landman and post
doctoral fellow Michael Moseler used computer simulation experiments to show
that the stochastic Navier-Stokes formulation does work for fluid nanojets and
nanobridges in a vacuum. The theoretical predictions of this early work have
been confirmed experimentally by a team of European scientists (see the
December 13, 2006, issue of Physical Review Letters). Now, Landman and
graduate student Wei Kang have discovered that by further modifying the
Moseler-Landman stochastic Navier-Stokes equations, they can accurately
describe this behavior in a realistic non-vacuous environment.

"There was a strong opinion that fluid dynamics theory would stop being
valid for small systems,” said Landman, director of the Center for
Computational Materials Science, Regents’ and Institute professor, and Callaway
chair of physics at the Georgia Institute of Technology. “It was thought that
all you could do was perform extensive, as well as expensive, molecular dynamic
simulations or experiments, and that continuum fluid dynamics theory could not
be applied to explain the behavior of such small systems.”

A propane liquid nanobridge breaks up in a vacuum. (Image: Georgia Tech/Uzi Landman) |

The benefit of the new formulations is that these equations can be solved
with relative ease in minutes, in comparison to the days and weeks that it
takes to simulate fluid nano structures, which can contain as many as several
million molecules. Equally difficult, and sometimes even harder, are laboratory
experiments on fluids in this regime of reduced dimensions.

In this study, Landman and Wei simulated a liquid propane bridge, which is a
slender fluid structure connecting two larger bodies of liquid, much like a
liquid channel connecting two rain puddles. The bridge was six nanometers in
diameter and 24 nanometers long. The object was to study how the bridge
collapses.

In the study performed in 2000, Landman simulated a bridge in a vacuum.
The bridge broke in a symmetrical fashion, pinching in the middle, with two
cones on each side. This time, the simulation focused on a model with a
nitrogen gas environment surrounding the bridge at different gas pressures.

When the gas pressure was low (under 2 atmospheres of nitrogen), the
breaking occurred in much the same way that it did in the previous vacuum
computer experiment. But when the pressure was sufficiently high
(above 3.5 atmospheres), 50 percent of the time the bridge broke in a
different way. Under high pressure, the bridge tended to create a long thread
and break asymmetrically on one side or the other of the thread instead of in
the middle. Until now, such asymmetric long-thread collapse configuration has
been discussed only for macroscopically large liquid bridges and jets.

Analyzing the data showed that the asymmetric breakup of the nanobridge in a
gaseous environment relates to molecular evaporation and condensation processes
and their dependence on the curvature of the shape profile of the nanobridge.

"If the bridge is in a vacuum, molecules evaporating from the bridge are
sucked away and do not come back” said Landman. “But if there are gas molecules
surrounding the bridge, some of the molecules that evaporate will collide with
the gas, and due to these collisions the scattered molecules may change
direction and come back to the nanobridge and condense on it.”

As they return they may fill in spaces where other atoms have evaporated. In
other words, the evaporation-condensation processes serve to redistribute the
liquid propane along the nanobridge, resulting in an asymmetrical shape of the
breakage. The higher the pressure is surrounding the bridge, the higher the
probability that the evaporating atoms will collide with the gas and condense
on the nanobridge. Landman and Wei have shown that these microscopic processes
can be included in the stochastic hydrodynamic Navier-Stokes equations, and
that the newly modified equations reproduce faithfully the results of their
atomistic molecular dynamics experiments.

"Knowing that the hydrodynamic theory, that is the basis of venerable
technologies around us, can be extended to the nanoscale is fundamentally
significant, and a big relief” said Landman. “Particularly so, now that we have
been able to use it to describe the behavior of nanofluids in a non-vacuous
environment – since we expect that this is where most future applications would
occur.”

**E-mail
protest**

March 10, 2007

Dear Uzi,

It was with dismay that I read the piece about your recent PRL research in the March 5, 2007 Whistle.

When Michael Berry gave the Lilienfeld prize lecture for
1990, he looked at me in the audience (the APS March meeying met in

In 1999-2000, I shared with you, at your request, the ideas regarding applicability of hydrodynamic fluctuations at few angstrom scales. I mentioned the extension by Keizer et al. for neutron acattering, and the later extension by Alder et al. At the time you expressed gratitide for learning about this fact. Now I see you quoted as the originator of these ideas. Surely you could have been more generous with your citing of antecedents. No only was I slighted, but so were Keizer and Alder.

If you wish to achieve the stature of Sir Michael Berry and his ilk, you need to take advantage of opportunities to extol the wisdom of your colleagues.

Yours truly,

Ron