Leo P. Kadanoff
"Making a Splash, Breaking a Neck: The Development of Complexity in Physical Systems"
April 17, 2000
Work done by Michael Brenner, Peter Constantin, Todd Dupont, Leo Kadanoff, Albert Libchaber, Sidney Nagel, Robert Rosner, and many others
We study the motion of fluids. Our program is characterized by close cooperation among experimenters, theoreticians, and simulators. We aim to develop a fundamental understanding of fluid flow. Mostly our work involves solving particular problems, e.g., “how does heat flow in a pot of water heated over a flame?” But, in following these problems we soon get to broader issues: predictability and chaos and the natural formation of complex “machines.”1
One of the great concepts of our physics profession is the simplicity of the laws of physics. Maxwell’s equations or the Schrödinger equation or Hamiltonian mechanics can each be expressed in a few lines. The ideas that form the foundation of our worldview are also very simple indeed: The world is lawful. Different laws apply in different situations. But always and everywhere the laws are consistent with one another and can be derived one from the other.
Thus, everything is simple, consistent, and neat. The laws of physics are expressible in terms of everyday mathematics, usually partial differential equations. Everything is simple and neat—except, of course, the world.
Visualize waves in a stormy sea. Each wave is different from the last, bigger or littler, moving east or north, perhaps breaking and throwing spray far up into the air. This jumble of varying objects defines one kind of complexity. But also think about any living thing. Any organism contains many different kinds of working parts, all functioning together as some amazingly intricate machine. Living things show functional intricacy and apparent purpose, and thus reflect another, richer, kind of complexity. Economic, social, and ecological systems also seem to show a rich complexity together with a great degree of organization.
In our world, each event has a multitude of different causes. The present is determined by the past, but the chain of causation is often complicated. Furthermore, small changes in a hypothetical past can have effects which grow larger and larger as time progresses, making many long-term predictions beyond practical possibility. Things that behave like this are said to be chaotic. In general, complexity is the product of chaotic processes.
But now come back to our question: Why, if the laws are so simple, is the world so complicated?
Before understanding comes observation.
We shall look at the world partly through the eyes of Harold Edgerton,2 inventor of strobe photography and founder of EG&G. He took great pleasure in recording natural processes in “stop-action.” Figure 1 shows the behavior of a tiny microcosm, constructed from milk. A drop falls into a glass. The drop makes a crater. The crater comes up and produces a crown with points. The points rise and separate from the main mass of milk. The crater subsides. A single drop of milk escapes from the mass of fluid and rises high in the air. So a very complicated event arises from the simplest of starting points.
Figure 1. Strobe photos of the splash produced by a milk drop landing on the surface of a glass of milk. Notice the regularity of the initial shape and the irregularity of the last three pictures. Photo by Dr. Harold E. Edgerton, © Harold & Esther Edgerton Foundation, 2001, courtesy of Palm Press, Inc.
Notice the irregularity of the crown. Edgerton liked perfect pictures. He wanted to produce a setup in which he could photograph a perfectly regular and reproducible structure. So he was frustrated by the result shown here. Over the years since this photo was taken, science has developed the concept of chaos to explain outcomes like this one. The idea is that small changes in the situation at the start of the splash can have an influence that will be magnified into a very large effect by the time the last picture is taken. More specifically, little gusts of air at the beginning of the process can change the number of points in the crown and the sequence in which the drops come off the points. The usual metaphor is that the presence or absence of a butterfly beating its wings in Brazil can substantially change the weather in Chicago two weeks later.3
Here at Chicago, a group of us4 became concerned with the details of the process in which a mass of fluid breaks in two. Instead of looking at the points atop the Edgerton crown, we focused upon fluid dripping from a pipette. (See figure 2 for Sidney Nagel’s sequence of pictures of a falling drop.) Just before a drop breaks loose, a thin neck connects it to the main mass of fluid. We were interested in this neck for several, related, reasons. We had good reason to believe that the shape of the thin neck was universal, that is, independent of the details of the experimental setup. Its shape would not depend upon whether the drop was moving up or down, whether the fluid was milk or honey, or whether the process was observed one microsecond or ten microseconds before the final separation. Experience and theoretical analysis have both led us to expect universal behavior whenever there is a qualitative change in the behavior of any physical system. Here the qualitative change is one in which a single mass of fluid separates into two. The mathematics and physics of qualitative change is, in itself, of considerable interest. Universality5 makes any result one gets broadly applicable.
Figure 2. Stroboscopic pictures of a drop of water falling from a pipette. Photo by Sidney R. Nagel. See X. D. Shi, M. P. Brenner, S. R. Nagel, Science 265 (1994): 219.
But at the same time as we scientists believe in universality, we also believe in chaos. In most situations, and certainly in the milk crown, we believe that the final outcome is sensitively affected by the details of the starting situation. Universality and chaos seem to be in opposition—incompatible. We physicists love physical situations in which a collision of ideas, an incompatibility, can occur. We believe that such situations lead to a deeper understanding of nature.
Theorists predicted the characteristic shape of the neck just before drop-separation. Experimentalists measured this shape by looking in exquisite detail at pictures of the drop. (This shape can be seen, for example, in the fourth plate of figure 2 or in the just-separating drops in Edgerton’s figure 1.) The first measurements showed exactly the predicted shape. So has chaos disappeared? Not at all. Sometimes, the experiment showed a behavior considerably richer than the one shown here. Sometimes, a little piece of the neck elongated, producing a skinnier neck. That might stretch out too, producing a yet skinnier neck. In both theory and simulations, this stretching could occur several times, producing an unpredictable number of necks, each one however approaching the universal shape. Chaos and universality can coexist in a single system, often tied together in surprising ways.
Lessons from Complexity
At one time, many scientists believed that the study of complexity could give rise to a new science. In this science as in others, there would be general laws, with specific situations being studied as the inevitable working out of these laws of nature. The study of complexity has not gone in that direction. No universally applicable laws of complexity have emerged. Instead, the systems we study have taught us lessons, rather like the lessons for life our grandmothers told us. They are general ideas which apply broadly, but they must be applied with care and good judgment.
Different portions of the drop behavior have different degrees of predictability. The order of separation of the tips of Edgerton’s crown are almost completely unpredictable. The shape at separation is almost completely predictable, except that sometimes a multiplicity of necks appear. Can’t you hear yourself being told, perhaps after a first unhappy love affair, “Sometimes you can know how things will come out, sometimes you have to live it to find out. . . .” No, complexity is not a science in the usual terms.
Fluids Heated from Below
Albert Libchaber, now at Rockefeller University, did a series of very carefully controlled experiments on turbulent motion in fluids heated from below. I played a role in analyzing and interpreting these experiments.
Turbulence is ever-changing. The swirls of wind in Chicago never have the same pattern for two seconds in a row. The pattern changes and changes and changes again. Nonetheless, wind is in some sense always the same. We wanted to understand the flow patterns in a heated fluid. We expect to see some roughly fixed elements, appearing in a mosaic of different combinations, producing ever-changing patterns.
One structure that is found in all heated fluids is called a plume.In most situations, heated fluid is less dense than its cooler counterpart. For this reason, the hotter parts of fluids feel forces that push them upwards. As heated fluid rises, it pushes aside fluid above it and is in turn deflected by this pushing. The rising fluid produces a little stalk, like the base of a mushroom while the deflected fluid produces a structure very much like the cap of a mushroom. As the pushing and deflection continue, the top of the cap folds over. This very characteristic mushroom-like structure can be found in many different situations.
Figure 3. Plume produced by thermonuclear explosion “Joe 4,” August 12, 1953. From Richard Rhodes, Dark Sun: The Making of the Hydrogen Bomb (New York: Simon & Schuster, 1995), photo 51.
Look first at figure 3, which is a picture of a nuclear explosion. It shows a very large plume produced by rising gases. Yet larger plumes are depicted in figure 4, which is reproduced from a computer simulation of the surface of the sun done by Malagoli, Dubey, and Cattaneo. This picture shows many cold plumes falling downward into the sun. Yet another, more mundane, plume can be seen in Edgerton’s photo (figure 5) of the fluid rising from a candle flame. (Each scientist is proud of his/her art. Edgerton did not have to include the bullet and its shock wave in his candle picture, but he put it in to show the power of the strobe technique which he invented. Nagel and coworkers did not have to construct museum quality pictures of fluid in motion, but they were proud of their strobe technique (see figures 2 and 11) and wanted to show off, too.)
Figure 4. Computer simulation of the temperature pattern near the surface of the sun. The darker colors indicate lower temperatures. Notice the many falling plumes near the surface. Simulation by Andrea Malagoli, Anshu Dubey, and Fausto Cattaneo.
Figure 5. Plumes from candle flame. This strobe photo also shows a bullet and its associated shock wave passing through the flame. Photo by Dr. Harold E. Edgerton, © Harold & Esther Edgerton Foundation, 2001, courtesy of Palm Press, Inc.
But what did Libchaber and coworkers actually see? They worked with a small container heated carefully and uniformly from below and cooled from above. They maintained a fixed temperature difference between the hotter bottom and the cooler top of their container.
Figure 6. Picture of a fluid heated from below, Jun Zhang, S. Childress, and Albert Libchaber, Physics of Fluids 9 (1977): 1034. The bright lines show regions of rapid temperature variation. The fluid is seen to contain many plumes, especially near the walls. A counterclockwise flow can be seen.
Figure 6 is a picture of a flow they observed. As you can see, the container is filled with plumes. Hot plumes congregate in an upwelling jet of fluid near the righthand wall of the container. A similar, downward jet formed from cold plumes occurs on the left-hand wall. Large numbers of hot plumes are also found in left-to-right motion in a layer called the mixing zone near the bottom of the container. A similar layer on the top contains cold plumes, moving in the opposite direction. The central region contains a few plumes, hot and cold, in mostly random motion. These plumes have gotten loose from the main flow but nonetheless participate in an overall counterclockwise motion. In addition, there are very thin layers, not really visible in the present picture, near the top and bottom walls. These boundary layers actually contain the majority of the temperature drop between bottom and top of the container.
Figure 7. Cartoon view of Rayleigh Benard cell, by G. Zocchi, E. Moses, and A. Libchaber, Physica A 166 (1990): 397. This picture focuses on the bottom portion of the fluid and the mechanism for production of warm plumes. On the top, there is a similar mechanism for the production of cold plumes.
So the container is a complex “machine” containing many different working parts: boundary layer, mixing zone, central region, jets. Figure 7 is a “cartoon” drawing which shows how the machine works. Start at the lower left-hand corner of the cell. A flow is progressing from left to right. This flow is like a wind. Like the wind on Lake Michigan, it can help waves move. In this case, the waves are changes in the height of the thin boundary layer at the bottom of the cell. Waves move from left to right throwing up spray as they go. Because the spray is hot fluid, it tends to rise forming the swirls and plumes shown in the picture. These structures grow larger. A few come loose and move into the central region. Most hit the right-hand wall and move upward as a jet toward the top of the cell. As the flow hits the top of the cell it makes a splash. The splash makes waves. The waves move along the top, producing cold spray and thence cold plumes. The plumes form into a downward jet at the left-hand side, splash on the bottom, and produce hot waves. . . . So a complex motion takes place, having the seeming purpose of moving heat from the bottom of the container to the top.
So we have a story and a few lessons.
The story:A cell is filled with a fluid. The fluid is strongly heated from below. Buoyancy raises the heated material and a flow starts.
Lesson I:A nonequilibrium system can organize itself to produce the most amazing complexity, with many different working parts, each serving a different function. This self-organized machine reminds one of the processes occurring in biological systems. I believe that biological systems have arisen precisely because physical systems have a natural tendency to generate complexity.
Lesson II:Memory can exist in a noisy environment. The system will move for a very long time in one sense of rotation. Here we showed a counterclockwise rotation. But if we had started the system in the other direction, it would have continued to flow in that sense for a very, very long time.
Lesson III:Disasters happen. Once in a very long while this orderly flow in one direction stops because of some fluctuation in the system. And, in a very uncharacteristic and unusual pattern of movement, the flow reverses itself. Most complex things undergo large, rather unpredictable changes. Every once in a while we get a tornado in Illinois. Or an earthquake. Or an ice age. Complex systems do big, unexpected things.
Example: Fluids in Motion: A Square Dance
We have seen that fluids can exhibit extremely complex patterns of motion. Generally, matter in motion is described by equations of a type called partial differential equations (PDEs). These equations relate rates of change in space to rates of change in time. Spatial variations translate into motion. The particular equation used to describe fluid flow motion is called the Navier-Stokes equation—
ut + (u º ▼) u = - ( ▼p) / Ρ + ν ▼2 u
▼ º u = 0
—and says how the velocity, u, and the pressure, p, depend upon space and time. PDE are for the initiates. However, as we shall see in just a moment, the basic ideas that go into the derivation of these particular PDEs are very simple indeed. We shall then consider a square dance (or, equivalently, a computer program) which realizes these ideas. Finally, we shall see how the dance, with only two basic steps, can reproduce all the complexity of fluid motion.
There are three basic ideas in the PDE:
First: A fluid contains many particles in motion. These particles will be our dancers.
Second:“Conservation laws:” Some things are never lost, only moved around. The number of particles and momentum never change—they only move from place to place.
Third: All changes are local and isotropic. These are technical words saying that all communication in the system is over short distances and that the system has, to a sufficient degree, a symmetry under rotations of its compass directions.
The big idea: Do the above right (plus a little more) and you will construct a model system with behavior just like that of real fluids.
We shall first use the metaphor that our fluid is a game, then translate that game into a dance, and finally show that the game and the dance have movements just like a fluid. For simplicity, all this is to happen in two-dimensional space. The space is represented by a board in the shape of a triangular lattice (See figure 8). On the board there are a group of pieces. Each piece comes with an arrow, which we pick to point in any one of the six directions along the axes of the lattice. Each site on the lattice can contain several pieces with their arrows pointing in different directions.
Figure 8. The Dance. The dancers are represented by arrows, as shown in the left-hand frame. In the first step, each dancer moves one unit in the direction of his/her arrow. The result is the second frame. Next, if the dancers on each point have arrows that add up to zero (by vector addition), they twirl around to reach the configuration of the third frame. This motion then repeats, endlessly.
Of course, the pieces are the particles, and their arrows are the different possible directions of their velocity or momentum. (The two are proportional to one another.) Our game or dance, must be set up so that the pieces or dancers move with rules that satisfy the three basic requirements of the fluids.
So start the dance. Begin from the first panel of figure 8. The caller for this American country dance cries “Promenade,” and each dancer moves one step in the direction of his/her arrow. This brings us to the second panel of the figure. Neither the number of particles nor the total momentum has changed. So far, so good. But actual particles must be able to change their momentum. So now the caller cries out “Swing your partner” and sotto voce “if your total momentum is zero.” (This is, after all, a mathematical dance.) All pairs on a given lattice site with oppositely directed arrows and all triplets with arrows sixty degrees apart have vectors which add up to zero. As shown in figure 8, when these two cases occur, the dancers rotate together maintaining their total momentum to be zero, producing the last panel of the figure. These two steps satisfy all our rules. The dance continues with the caller crying first one step then the next, throughout the long night.
To make a fluid, take a huge board with many dancers. Go through the two steps many, many times. My contention is that the resulting pattern of motion will, if smeared or averaged over a moderately large region of the board, exactly and precisely reproduce the solution of the NavierStokes equation and, in equal measure, the motion of the fluid.
To justify this contention, look at the non-trivial motion of a fluid past an obstacle. Such a flow, for a real fluid moving past a long cylinder, is shown in figure 9. The flow is made visible with lines of smoke placed in the fluid. Far away from the obstacle, the fluid velocity is uniform. Over much of the field of view, the smoke lines are almost straight. But as the fluid moves close to and past the cylinder, its flow changes quite considerably. The region behind the obstacle develops a characteristic pattern of swirls, in which neighboring swirls sit on opposite sides of the centerline and go around in opposite senses. This real flow pattern will now be compared with a computer-generated square dance.
Figure 9. Fluid flows past a cylindrical obstacle and produces a characteristic flow pattern called a von Kármán street. Photo courtesy of Peter Bradshaw, reproduced in Milton Van Dyke, An Album of Fluid Motion (Stanford, Calif.: Parabolic Press, 1982).
Working with a large computer at Los Alamos National Laboratory, d’Humière, Pomeau, and Lallemand implemented the square dance model on a lattice with about two million sites and with perhaps six million dancers. They set the dancers into an average motion from left to right and made them move past a computer version of an obstacle. They had each of the dancers do the little dance steps many thousands of times. To see the resulting pattern, they then set up regions in their system containing about one hundred lattice sites and found the average particle velocity in each of these regions. Their result is shown in figure 10. Just as in the real flow, one sees behind the obstacle a set of moving vortices with alternative directions of flow. Examples like this one showed that the dance model got the qualitative properties of the fluid just right.
Figure 10. Computer simulation of von Kármán street. See D. d’Humières, Y. Pomeau, and P. Lallemand, “Simulation d’allées de von Kármán bidimensionelles à l’aide d’un gaz sur réseau,” Comptes rendus de l’Académie des sciences, Série II 301 (1985): 1391–1394. I thank D. Rothman of MIT for providing this picture.
At Chicago, Guy McNamara, Gianluigi Zanetti, and I did a more quantitative check. In our computer, we set up a narrow channel, like a pipe, and looked at a fluid in steady motion along this pipe. For comparison, we could look at a simple exact solution of the Navier-Stokes equation. The comparison showed that the mathematical solution and the computer simulation gave precisely the same results.
So we learn that the richly complex motion of fluids can be constructed from the quite simple steps of the dancers. There is a lesson from it all, viz., complexity can arise from simplicity. Simple events, linked together, and repeated sufficiently often, can produce complex outcomes. It is possible that the complexity seen in biological systems is nothing more than the natural tendency of oft-repeated physical events to produce richly structured outcomes.
A Cautionary Note
I have described the work above as the outcome of the interaction between three different kinds of scientific tools: laboratory investigation, mathematical/theoretical analysis, and computer simulation. The quality of scientific technique and ability in the first two areas has remained roughly constant in recent decades. Sid Nagel uses an old camera derived directly from an Edgerton design. My use of universality ideas to solve PDEs arises directly from techniques put together by Kolmogorov, Greenspan, Barenblatt, Zeldovitch, and others three, four, or five decades ago. Modern advances in electronics and computer technique help us out some in lab and theory, but they mostly do not change our art. On the other hand, our capability for doing simulations has immensely improved as computers have gotten faster.
In some areas, simulational methods seem to be, in large measure, displacing old-fashioned theory and experiment. For example, the Department of Energy in its studies aimed at permitting it to safely preserve our stockpiles of nuclear weapons seems to have largely given up small-scale experiment. Instead, it uses computer simulation to assess the possible outcomes of aging and accidents upon the behavior of nuclear weapons.
I argue that a sole reliance on simulations is quite risky in any class of situations that has not had a full exploration by theory and particularly by experiment. Recent advances in computer technique have enabled us to construct better and more reliable studies of things that were mostly understood beforehand. But physical systems can produce results that are quite surprising. The discovery of such unexpected outcomes has, in recent years, been largely a product of laboratory experiments and theoretical work.6 Smallscale experiments can survey a wide range of conditions and of situations much more rapidly and flexibly than can simulation. Theory can put together a whole class of understandings and thereby focus predictive thought. But simulations will mostly help us see in more detail what we already know.
Let me focus on this issue by looking at a particular experiment. Figure 11 shows a fluid—water—and an electrode shown in black. Below the surface of the water there is a flat metallic plate set horizontally. An electrical potential difference of twenty thousand volts is maintained between plate and electrode. As a result there is a very strong electric field in the neighborhood of the plate. Because of its electrical properties, the water is drawn toward regions of high electric field. It forms a bump which moves upward toward the plate. The top of the bump rises and rises, and then almost comes to a point.
Figure 11. Strobe photos of behavior of fluid placed in a strong electric field. At the top of each frame, one can see an electrode, charged to 20,000 volts. In the first frames, a mound of fluid moves toward the high electric field region near the electrode. The fluid comes to a point. An arc of charged material is produced. Finally, the arc falls apart into many tiny fluid droplets. Lene Oddershede and Sidney R. Nagel, “Singularity during the Onset of an Electrohydrodynamic Spout,” Physical Review Letters 85 (2000): 1234–1237.
As a theorist I am intrigued by this point. I specialize in seeing how new structures, like the point, are produced. I have worked with several different students and postdocs in setting up computer simulations which can give the rising bump and the approach to the point. Doing an accurate simulation is hard, and we have not yet reached complete success, but we are getting there.
But look at the last three frames. Just after the fluid comes to its point, there is some motion between fluid and electrode. In the next frame, the motion resolves itself, and we can see that it is essentially a bolt of lightning flashing between the fluid and the electrode. In the next frame, the lightning has ceased and been replaced by the production of fine droplets of the water (rain?!) over an extended region.
There is a lesson from this, too: Complex systems sometimes show qualitative changes in their behavior. (Here a bump has turned into lightning and rain.) Unexpected behavior is possible, even likely.
This particular unexpected behavior would not, by a long shot, have been predicted by the simulations available to us today. Experiment found it. Theory can perhaps shed light on exactly what is going on. After the qualitative facts are exposed, we can then design simulations to test the ideas developed. To uncover and understand such things, it is best to have a balanced and interdisciplinary program of research.
In the End...
So perhaps there is no science of complexity. Nonetheless an open-minded and balanced scientific program can help us learn things about specific complex systems and even provide some general lessons about complexity. “Watch out for surprises” is one lesson. The need for a balanced program of research is another. Still another is that you should listen to your grandmother.
- This paper is based in part upon a previous publication by Nigel Goldenfeld and myself: N. Goldenfeld and L. P. Kadanoff, “Simple Lessons from Complexity,” Science 284 (1999). Semitechnical expositions of the main subjects treated here are given in: The square dance machine: L. P. Kadanoff, Physics Today 39 (September 1986): 7. Breaking Necks: Blowups and Singularities: L. P. Kadanoff, “Reference Frames,” Physics Today 50 (September 1997): 11–12. Fluids heated from below: L. P. Kadanoff, A. Libchaber, E. Moses, and G. Zocchi, “Turbulence dans une Boîte,” La Recherche 22 (1991): 628–638.
- Edgerton’s work can be seen in Stopping Time: The Photographs of Harold Edgerton (New York: Harry N. Abrams, 1987).
- Edward N. Lorenz, The Essence of Chaos (reprint, Seattle: University of Washington Press, 1996). See also James Gleick, Chaos: Making of a New Science (New York: Viking, 1987): 9–32. The latter work presents quite a different view of the scientific study of complexity than the one given here. For another view quite different from mine, see Murray Gell-Mann. The Quark and the Jaguar (New York: Freeman, 1994).
- Sidney Nagel, Itai Cohen, and X. D. Shi led the experimental effort. Theory and simulation came from a group that included Peter Constantin, Todd Dupont, Michael Brenner, Jens Eggers, Andrea Bertozzi, and myself.
- Universality is a concept that physical scientists derived from many independent sources. The idea, therefore, has many parents. I was fortunate enough to pick up the word from A. Migdal and A. Polyakov in a dollar bar in the Soviet Union. I then imported it into the United States, where both the word and the concept found broad scientific usage.
- A discussion of some early very notable accomplishments of computational physics can be found in Leo P. Kadanoff, “Computational Physics: Pluses and Minuses,” Physics Today 39 (1986): 7. A recent major accomplishment of simulational methods is the prediction of neutrino flux from the sun, which (when compared with observation) exposed an unexpected weakness in the common assumptions about neutrino behavior.
About the Lecturer
Leo P. Kadanoff is the John D. MacArthur Distinguished Service Professor in the Departments of Physics and Mathematics, the James Franck and Enrico Fermi Institutes, and the College.
He received all of his academic degrees from Harvard University, completing his Ph.D. in 1960. In the 1960s, while on the faculty of the University of Illinois, he made innovative and original contributions to the understanding of phase changes, such as the change of water from liquid to ice. This work was recognized with the Buckley Prize of the American Physical Society in 1977, the Wolf Foundation Prize in 1980, and the Boltzmann Medal of the International Union of Pure and Applied Physics in 1989.
As a Brown University faculty member in the 1970s, Kadanoff and his colleagues extended and applied the phase-transition work and developed a research program in computer simulations of urban dynamics.
Kadanoff joined the University of Chicago faculty in 1978. Working with students, junior scientists, and colleagues, he helped construct a new field of knowledge called soft condensed-matter physics, which deals with such phenomena as the flow of fluids and the behavior of granular materials.
He was awarded the National Medal of Science in 1999 for research contributions that have led to applications in engineering, urban planning, computer science, hydrodynamics, biology, applied mathematics, and geophysics.
Kadanoff received the University’s Quantrell Award for Excellence in Undergraduate Teaching in 1990 and is a member of the National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society. His other honors include the Centennial Medal of Honor of Harvard University, the Onsager Prize of the American Physical Society, and the Grande Medaille d’Or of the Académie des Sciences de l’Institut de France.