I can cook anything, as long as the recipe starts with “Take it out of the freezer” and ends with “Put it in the microwave.”

Go ahead and scoff, but I’m proud of my culinary accomplishments. Furthermore, I submit that the art of microwaving frozen foods is not without intellectual challenge. Inferior technique could leave some bits of your burrito still frozen while other parts are overcooked. The underlying cause of this well-known problem has lately been explored in depth and detail through computer simulations done by Motohiko Tanaka and Motoyasu Sato of the National Institute for Fusion Science in Toki, Japan. They present their results in an arXiv preprint.

The first mystery of microwave cookery is that it works at all. To heat a substance, you have to agitate the molecules, augmenting their random motions. Radiation at visible or infrared wavelengths does a good job of this: A visible or infrared photon is absorbed by a single molecule, which then goes bouncing off in some unpredictable direction. But microwaves are orders of magnitude too large and slow and weak to stimulate individual molecules. (In this respect the prefix micro is misleading; the wavelengths range from about a centimeter up to 10 or 20 centimeters.) A microwave’s energy is spread out over many millions of molecules—a situation that doesn’t seem like a very promising way to get them all moving in different directions.

One clue to how microwaves induce heating is what you’re not supposed to put in the microwave oven: aluminum foil. Metals have an abundance of free electrons, which are accelerated by the electric field of the microwaves. This field changes polarity a few billion times per second, and so the electrons slosh back and forth through the foil at that frequency. This motion of the electrons does not in itself constitute heating because it is not random; on the contrary, it is a highly organized oscillation—an alternating electric current. But the oscillating electrons collide with imperfections in the metal, and soon the orderly movement degrades into heat.

So much for tinfoil; most of us eat little in the way of metallic food. And nonmetals do not have an abundance of electrons (or other charged particles) free to move under the influence of an alternating electric field. What foods have is water. Although the H₂O molecule is electrically neutral, it has positive and negative poles, where opposite charges congregate (most of the positive charge on the hydrogen atoms, most of the negative charge on the oxygen). When this dipole structure is immersed in an electric field, it takes up a preferred orientation, antiparallel to the applied potential. Since the microwave field changes direction at a frequency of a few gigahertz, the water molecules must continually flip or spin to keep pace. As with the sloshing of the electrons in tinfoil, this rotation of the water molecules can’t be considered heat because it’s too orderly; all the molecules are twirling in synchrony. But in liquid water the molecules are tightly packed, and as they spin they bump into one another like dancers on a crowded floor. These collisions randomize the motion, and so the temperature of the water rises.

Tanaka and Sato study this process in a simulated volume of water measuring about 40 Ångstroms on a side and containing about 2,700 water molecules. The molecules are attracted to one another through electrical forces, but there’s also a “hard core” repulsion that prevents them from getting too close. In the simulation, all of the forces acting on each molecule are recalculated every femtosecond (10^–15 second), after which the positions and orientations of the molecules are updated. The system is allowed to come to equilibrium at a specified temperature during an initialization phase that lasts for a simulated time of 100 picoseconds (100 × 10^–12 second), and then the microwave field is turned on for 500 picoseconds. Tanaka and Sato use an unrealistically intense field—about a million volts per centimeter—in order to speed up the process. (Although the simulated time is only about half a nanosecond, the actual running time on a cluster of four-processor Pentium workstations is 48 hours.)

In liquid water at 300 Kelvins (roughly room temperature), Tanaka and Sato find that microwave heating is quite efficient: The colliding, spinning molecules raise the temperature to about 350 Kelvins. But here’s the problem: In ice, unlike liquid water, Tanaka and Sato see almost no heating. The reason is that water molecules in an ice crystal are immobilized by strong electrostatic bonds, and microwaves have too little energy to break them free. In the oscillating microwave field, the ice molecules wobble back and forth a bit, but they cannot twirl, and thus they cannot collide. Tanaka and Sato don’t explicitly discuss the culinary implications of their work, but the inference is obvious: It’s because of this icy recalcitrance to microwaves that nuking a frozen burrito takes as much skill as baking a perfect soufflé or whipping up a sauce Bearnaise.

Interestingly, microwaves also lose much of their sizzle when water is superheated to 400 Kelvins. Under these conditions, the water molecules are easily set to spinning, but the bonds between them are so feeble that the rotation is not converted into random, thermal motion. I am tempted to see a certain philosophical significance in this curious behavior. There’s been much written about the specialness of the water molecule—most notably the geometric quirk that makes solid ice lighter than liquid water. If it were otherwise—if rivers and lakes and oceans could freeze from the bottom up—life would have had a hard time getting established on planet Earth. Now we know that modern slacker civilization also depends on a peculiarity of the water molecule. If we didn’t have this glorious interval of susceptibility to microwaves in a narrow window of temperatures near 300 Kelvins, I’d have to survive on Poptarts in the toaster oven.

It’s no secret that the way to win fame and fortune in physics is to invent a better refrigerator. Michael Faraday and James Prescott Joule and J. J. Thomson (Lord Kelvin) were all thinkers or tinkerers in refrigeration; the Kelvinator brand alludes to the last of those pioneers. Einstein and Leo Szilard held dozens of patents on refrigerator designs. And workers in cryogenics have won at least 10 Nobel prizes, starting with Heike Kammerlingh Onnes in 1913.

Here’s the latest cool idea on how to chill out: Let the hot atoms in a fluid escape through nanopores that block the lower-energy atoms. William J. Mullin of the University of Massachusetts in Amherst and Neal Kalechofsky of Oxford Instruments America Inc. suggests three ways this might work. The abstract of their recent paper:

We consider the possibility of adding a stage to a dilution refrigerator to provide additional cooling by “filtering out” hot atoms. Three methods are considered: 1) Effusion, where holes having diameters larger than a mean-free path allow atoms to pass through easily; 2) Particle waveguide-like motion using very narrow channels that greatly restrict the quantum states of the atoms in a channel. 3) Wall-limited diffusion through channels, in which the wall scattering is disordered so that local density equilibrium is established in a channel. We assume that channel dimensions are smaller than the mean-free path for atom-atom interactions. The particle waveguide and the wall-limited diffusion methods using channels on order of the deBroglie wavelength give cooling. Recent advances in nano-filters give this method some hope of being practical.

The plan sounds a little like Maxwell’s demon without the demon. The nanopores don’t have to be opened and shut for individual atoms; they discriminate naturally between various energy states of the atoms. Or at least that’s what Mullin’s and Kalechofsky’s calculations suggest; the crucial experiment has yet to be performed.

arXiv link: Theory of cooling by flow through narrow pores.

Update 2006-05-16: Oops. Lord Kelvin = William Thompson. Lord Kelvin ≠ J. J. Thompson.

Scientists are selfless seekers after truth, unswayed by worldly emoluments, immune to the tawdry enticements of fame, indifferent to prizes and honors. Thus I can’t quite imagine why anyone would bother ranking a collection of scientific papers by applying the algorithm that Google uses to decide which Web pages deserve the most prominent display. Nonetheless, if you’re an author of anything published in Physical Review or its various offshoots over the past century or so, P. Chen, H. Xie, S. Maslov and S. Redner have calculated the Google PageRank of your publications. The database they analyzed consists of 353,268 papers published in various American Physical Society journals between 1893 and 2003.

The best-known measures of “impact” for scholarly publications are based on counting citations; a paper’s impact rises when other papers refer to it. The PageRank algorithm applies this principle recursively. If paper A is cited by paper B, that fact will be weighted more heavily if paper B itself gets many citations. Chen et al. adapted this method of evaluation to the 3,110,839 citations within their network of Physical Review articles. They found that most of the papers with a high PageRank score are well-known works that also get high marks under other kinds of citation analysis. But there were some surprises—papers they call scientific gems. An example is a 1980 article by H. Rosenstock and C. Marquardt titled “Cluster formation in two dimensional random walks: Application to photolysis of silver halides.” Only three other articles in the database cite Rosenstock and Marquardt, and thus it remains an obscure publication from the point of view of most citation analyses; but it rises to 85th place in the PageRank standings because one of those three citing papers is itself a highly rated work. (The latter paper, by T. Witten and L. Sander, is “Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon,” with 680 citations as of June 2003. Chen et al. note: “The Witten and Sander article has only 10 references; thus a substantial fraction of its fame is exported to [Rosenstock and Marquardt] by the Google PageRank algorithm.)

Unfortunately, the database used by Chen, Xie, Maslov and Redner is not publically available, and so there is no convenient way for physicists to check their own PageRank standings. But, then again, physicists are so egoless, I’m sure they wouldn’t bother anyway.

See: Finding Scientific Gems with Google, on the arXiv.

Every morning I go fishing in the arXiv. Or at least that’s the way I’ve been thinking about this daily ritual: I cast my net over the waters and look to see what strange and wonderful creatures I’ve brought up from the deeps. Today it hit me that I have the metaphor backwards. I’m the fish, and what’s going on here is that the arXiv dangles lures in front of me to see if I’ll take the bait. Some days I’m just not biting. Today, however, I was snapping at one hook after another. I’ll be the first to admit that I sometimes choose a brightly colored bit of fluff in preference to a nutritious worm. (Although I did pass on today’s proof that P ≠ NP—or was it the converse?)

physics/0603229
Title: Laws of Graph Evolution: Densification and Shrinking Diameters
Authors: Jure Leskovec, Jon Kleinberg, Christos Faloutsos

How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing super-linearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log (log n))).

cond-mat/0603718
Title: Statistical Mechanics of Community Detection
Authors: Joerg Reichardt, Stefan Bornholdt

Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass…. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected….

physics/0603215
Title: Computer simulation of language competition by physicists
Authors: Christian Schulze, Dietrich Stauffer

… About every ten days a human language dies out, and in Brazil already more than half of the indigenous languages have vanished as a result of the European conquest. On the other hand, Latin has split in the last two millennia into several languages, from Portuguese to Romanian…. Thus similar to biology, also languages can become extinct or speciate into several daughter languages.

In contrast to biology, humans do not eat humans of other languages as regular food, and thus one does not have a complex ecosystem of predators eating prey as in biology. Instead, languages are meant for communication, and thus there is a tendency of only one language dominating in one region, like German in Germany etc. Will globalisation lead to all of us speaking one language in the distant future? For physics research, that situation has already arrived many years ago….

Thus in the history mankind we may have had first a rise, and later a decay, in the number of different languages spoken. In Papua New Guinea there are now 10³ languages, each spoken by about 10³ people; can this situation survive if television and mobile phones become more widespread there?

While we cannot answer these questions, we can at least simulate such ”survival of the fittest” among languages, in a way similar but not identical to biology….

physics/0510151
Title: Trainspotting: Extraction and Analysis of Traffic and Topologies of Transportation Networks
Authors: Maciej Kurant, Patrick Thiran

The knowledge of real-life traffic pattern is crucial for good understanding and analysis of transportation systems. This data is quite rare. In this paper we propose an algorithm for extracting both the real physical topology and the network of traffic flows from timetables of public mass transportation systems.

math.SP/0603630
Title: Sharp bounds for eigenvalues of triangles
Authors: B. Siudeja

The purpose of this paper is to prove the following theorem.

Theorem 1.1. Let T be a triangle in a plane of area A and perimeter L. Then the first eigenvalue λT of the Dirichlet Laplacian on T satisfies

The constants 9 and 16 are optimal.

math-ph/0603065
Title: Quasicrystals: algebraic, combinatorial and geometrical aspects
Authors: Edita Pelantová, Zuzana Masáková

math-ph/0603068
Title: A Spinorial Formulation of the Maximum Clique Problem of a Graph
Authors: Marco Budinich, Paolo Budinich

In this paper we propose a new representation of the maximum clique problem in complex space. After a brief review of this famous NP-complete problem, we show how the adjacency matrix of a graph can be expressed as the square of a symmetric complex matrix. The vectors forming this matrix have zero length and Cartan has shown that this geometry can be treated elegantly with spinors…. We finish with a formulation of the maximum clique problem in this formalism and show that each graph uniquely identifies a spinor whose properties surely deserve deeper studies.

Still more on reversible and zero-energy computing (see earlier bit-player posts here and here, and the American Scientist column):

M. Maissam Barkeshli of the University of California at Berkeley has a preprint titled “Dissipationless Information Erasure and Landauer’s Principle.” (The paper was first submitted to the arXiv last April, but I missed it then, and noticed it today only because it has just been updated.) Barkeshli’s “dissipationless erasure” does not challenge the basic premise that a reversible computer could operate (in principle) without energy loss, whereas an irreversible computer must dissipate at least some energy. He argues, however, that the energy-dissipating step in an irreversible machine need not necessarily be the erasure of a bit of information, as Rolf Landauer first suggested in 1961. Barkeshli describes a hypothetical computer technology in which erasing is free but writing a new value takes energy. The energy cost for the entire cycle remains the same.

When I was a kid, there were no toys I treasured more than magnets. I had dozens of them: horseshoes, bars, a couple of powerful alnico cylinders salvaged from old loudspeakers. The invisible but very palpable forces acting between these objects—pushing like poles apart, drawing unlike together—were signs to me that mystery still exists in the universe. And yet magnetism also taught me that mysteries can be unraveled and understood. In some book I borrowed from the public library (maybe it was a biography of William Gilbert?) I read an explanation of magnetism that seemed to make sense. Imagine an array of many thousands of magnetic compass needles, all packed tightly together, the book suggested. Each such needle will tug on its neighbors as the magnetic fields interact, and so the needles throughout the array will all tend to line up in parallel. This is how a permanent magnet (or ferromagnet) works, the book said; the imagined tiny compass needles represent individual atoms in magnetic materials such as iron.

There are two problems with this story. First, it fails to explain where the atomic magnetic fields come from. Are we to imagine inside each atom an array of even tinier compass needles? Second, the explanation is just plain wrong. The atomic-scale magnetic dipoles in a ferromagnet do not line up because of magnetic interactions like those between compass needles. The actual interatomic forces are quite different; they are short-range, quantum-mechanical interactions that have no direct counterpart in the world of macroscopic objects.

I have known the truth about ferromagnets for some time, but the debunking of the many-tiny-compass-needles story leaves another question still murky. If an array of compass needles is not a good model of a ferromagnet, what does happen when you bring a bunch of magnetic compasses close together? A recent paper by six authors in Japan, India and the U.S. answers this question in the most direct way possible—through experiments with real compasses. They used small, spherical, liquid-filled compasses meant for mounting on a car windshield.

Before reading on, you might try to guess the outcome of their experiments.

When the compasses are arranged in a line—a one-dimensional array—the needles do tend to line up head-to-tail, all parallel to the line. Even with as few as two compasses, this interaction is strong enough to overcome the influence of the earth’s magnetic field.

But a two-dimensional, square array behaves differently, and not at all like a ferromagnet. In fact, the square lattice of compasses is an antiferromagnet, with nearby elements pointing in opposite directions. Even more surprising, the antiferromagnetic lattice is twisted 45 degrees with respect to the underlying lattice of compass needles. If the edges of the compass array are parallel to the east-west and north-south axes, then the compass needles all point along the diagonals. Sets of four adjacent compasses form a sort of loop, with needles pointing northeast, southeast, southwest, northwest. The paper explains this curious structure as a superposition of two simpler antiferromagnetic states: In one of these states, alternating rows of the latttice are oriented east and west; in the other, alternating columns are directed north and south.

The paper, Ferroics: magnetic-compass lattice and optical phonon dispersions of dipolar crystals, is available from the arXiv and has been submitted to The American Journal of Physics. The authors are Takeshi Nishimatsu, Umesh V. Waghmare, Yoshiyuki Kawazoe, Benjamin Burton, Kazutaka Nagao and Yoshihiko Saito.