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 Rockefeller University fellowship that came to $300 per month, 50% greater than the rate for an individually won NSF fellowship that I had before I arrived. The faculty in mathematics, physics and mathematical logic were remarkable, the result of recruitment by the visionary President of The Rockefeller, Detlev Bronk. Among the faculty were George Uhlenbeck, Sam Goudsmit (I took a course in Egyptology from Sam*), Mark Kac, Ken Case, Gian Carlo Rota, Henry McKean, Bram Pais, Saul Kripke, Robert Solovay, Hao Wang, Tony Martin, Baqi Beg, Nick Khuri and Eddie Cohen, only the last two of whom are still there. George was the senior statesman and a very wise man who attracted and inspired everyone. I wanted to work with him and was given the opportunity. As it turned out I was his 43rd 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 fluid 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 School of Physics web site because the Landua-Lifshitz equations have become the Landman-Moseler equations. I continue to protest the display of this news item 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.


[1] Contributions to the theory of non-equilibrium thermodynamics, R. F. Fox, The Rockefeller University , 20 March 1969 .

[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:

News item

Fluid Dynamics Works on Nanoscale in Real World

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).

Nanobrdige in a realistic atmosphere

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

300 dpi JPG = 1.09 MB

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.”

Nanobridge in a vacuum

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

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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 Atlanta that year) and thanked me for ”inseminating him with the idea” that led to Berry ’s phase, I was amazed and grateful. What a fine gentlemen. Many of my colleagues were there and Jerry Gollub sat next to me. Someday it may interest you as to what Jerry said after Michael’s remark.

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,