Nobel Prize Season

Update: I definitely wasn’t expecting this! It’s nice to see something applied win the prize.

So Nobel Prize season is upon us once again. It’s always fun to try to guess who the winners will be — and because I know nothing about Medicine, Peace, or Economics, I’ll stick to making predictions on the Physics and Chemistry prizes. I happened to pick 2007’s Physics winners correctly by sheer luck: maybe it’ll happen again?

Actually, I don’t know Chemistry as a field broadly enough to make any reasonable predictions, either. Just for fun, I’m going to guess Richard Zare will win this year, for the extensive work he has done using lasers for all sorts of spectroscopy — in particular, he developed laser-induced fluorescence spectroscopy. The only problem with picking Zare is that his work might (?) be too closely related to that of last year’s Chemistry picks (Shimomura, Chalfie and Tsien) for their discovery and development of green fluorescent protein.

This year I’m guessing that the Physics prize will go to Aharanov and Berry for their work on quantum topological and geometrical phases (see, for example, the Aharonov-Bohm effect or Berry phase). There are a number of reasons for this guess. Every physics student learns about how a charged particle moving in a region of zero electric and magnetic field is still affected by the potentially non-zero electromagnetic vector potential A – that is, its wave function picks up a phase shift given by integrating A along its path. This is an incredibly deep and fundamental result of quantum mechanics: unlike in classical electrodynamics, in quantum electrodynamics the effects of this potential can be felt. For example, as Aharonov and Bohm proposed in 1959, two charged particles going in opposite directions around a circle encircling a solenoidal magnetic field interfere when they are recombined — in particular, their phase difference (which can be measured) is directly proportional to the magnetic flux penetrating the circle, even though they feel zero magnetic field along the path they traverse!

Indeed, this effect has been verified in numerous measurements since. The earliest example that I am aware of is this elegant experiment by Chambers in 1960, using an electrostatic “biprism” consisting of an aluminized quartz fiber flanked by two grounded metal plates (schematic here) to interfere two beams of electrons.  This was followed up by further electron holography measurements using toroidal ferromagnets, as well as by work studying oscillations in the resistance of tiny metal rings as a function of the magnetic field being applied through their core. More recently, these magnetoresistance oscillations have been observed in individual carbon nanotubes with the field applied parallel to the tube axis, which I think is pretty cool. I remember when I first learned about this effect: it was one of the first times I was truly, genuinely, acutely thrown by quantum mechanics. And it has profound consequences — namely, it suggests that the electromagnetic vector potential is in some sense more “real” than the electric or magnetic fields on their own.

in 1984, Berry went one step further, pointing out that the Aharonov-Bohm effect is a particular example of geometric phase, and that a geometric phase often arises in many quantum situations. In particular, if a quantum system is changed very slowly (that is, adiabatically) such that it is eventually brought back to its initial conditions in parameter space, it turns out that it remembers the path it took: it picks up a phase factor that depends on the geometry of the path it took through parameter space. For example, if you subject a fixed electron to a constant magnetic field that changes in direction — say the magnetic field vector sweeps out an arbitrary closed loop on the surface of a sphere centered on the electron — it turns out that the electron state picks up a Berry’s phase proportional to the solid angle subtended by the path relative to the origin. That’s it. Isn’t that crazy?

The idea of a Berry phase (and the way in which it links physical effects to topological quantities) is quite general, and has found applications in many physical systems. For example, the quantum Hall effect can be understood as an example of Berry’s phase applied to 2D electronic systems, while the anomalous Hall effect for dilute magnetic semiconductors has recently been linked to Berry’s phase, as well. Graphene is a nice recent experimental system for studying Berry’s phase for electrons in two dimensions: electrons in graphene can be understood using the Dirac equation for spin-1/2 particles, and are characterized by “pseudospin”. Just as in the Berry phase example I gave earlier, an electron in graphene that completes a cyclotron orbit in an applied magnetic field has its pseudospin rotated by 360 degrees, and thus picks up a phase shift of pi in its wavefunction. The consequences of this have recently been observed in quantum Hall measurements of monolayer and bilayer graphene.  In related work, topological insulators and the quantum spin Hall effect have recently begun receiving a huge amount of attention from the physics community, because of their unusual properties — while they are insulating in the bulk, they can support unique “surface states”. I don’t fully understand the theory of these, but the main framework within which they appear to be studied is by describing them as topologically ordered states, characterized by topological invariants such as Chern numbers and a non-trivial Berry phase.

An interesting side note: in all of these Aharonov-Bohm/Berry phase experiments, the quantum phase is measured through some kind of interference process. Recently, Manoharan’s group at Stanford has done some pretty cool STM experiments to directly measure quantum phase information, by comparing the STM signal of physically different but electronically identical quantum corral-type nanostructures.

One potential problem: the Aharonov-Bohm effect was apparently previously predicted by Ehrenberg and Siday ten years earlier, and Berry phase was apparently discussed by Pancharatnam some 28 years before Berry’s paper! On the other hand, history suggests that this may not be enough to deter the prize committee.

Update: apparently Thomson Reuters agrees with my pick for Physics…

Kepler’s Year

As some readers may already know, 2009 is the International Year of Astronomy, commemorating 400 years since Galileo raised his telescope to the stars and Kepler put forward his laws of planetary motion in his treatise Astronomia Nova. This put the Copernican idea of heliocentrism on a more rigorous scientific footing, and would eventually pave the way for Newton to formulate his theory of gravity.

Even aside from these achievements, Kepler is particularly fascinating character in the history of science. His work always seemed to be strongly guided by a Platonic view of a perfectly structured, geometrically harmonic universe. For example, in his Mysterium Cosmographicum – a lesser-known precursor to Astronomia Nova – Kepler devised an elegant, if incorrect, model of the six known planets as following orbits along spheres inscribed within and circumscribed around five platonic solids. Another example is his extensive exploration of tesselations and sphere-packings, which he often used to explain the structure of various materials, such as snowflakes.

A well-known result of this is Kepler’s conjecture on sphere packings – namely, that the maximum possible volume fraction of a container that can be filled by equally-sized spheres is that of face-centered cubic packing, slightly more than 74%. This was only proved by Thomas Hales recently, using a good deal of computational machinery.

A related problem – part of Hilbert’s list of some of the most important problems in mathematics – involves finding the densest packings of regular polyhedra. While cubes and truncated octahedra can tile space (so their densest packing has a volume fraction of 100%), it turns out that none of the other five Platonic solids or thirteen Archimedean solids can. A good deal of recent work has focused on trying to find the densest possible packing of regular tetrahedra, mainly by construction. In two very nice recent papers, however, Sal Torquato and his student Yang Jiao have come up with a computational scheme for generating very, very dense packings of polyhedra, calculating the densest known packings of the Platonic and Archimedean solids. I won’t go into the details of the algorithm – the follow-up paper does a very nice job of explaining it. The basic idea is to start with a randomized ‘dilute’ configuration of the polyhedra in a box of some shape and either randomly move a randomly chosen polyhedron by a small amount, or deform the box by some amount, only allowing changes that increase the volume fraction while still preventing polyhedra from overlapping. One can imagine that iterating this many times would result in very, very dense packings — the hope, of course, is that these are the densest possible polyhedral packings. This remains to be proven.

The cool thing that Torquato and Jiao found is that for all the Platonic and Archimedean solids possessing central symmetry, the densest possible packings they found had volume fractions equal to the volume fractions of Bravais lattice packings of the same solids, to within a few hundredths of a percent. A nice Kepler-style argument using inscribed spheres gives upper bounds on the volume fractions of these densest possible packings – these are larger by only ~3-13%. Taken together, these results hint at a possible “Kepler Conjecture” for polyhedral packings: namely, that the densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings. Very cool — this suggests that quite complicated polyhedral packings might be able to be understood using some very simple rules.

On the other hand, Torquato and Jiao found that for the two polyhedra not possessing central symmetry — the tetrahedron and the truncated tetrahedron — the volume fractions of the densest lattice packings grossly underestimated the volume fractions of the densest packings they found using simulations. This leads to a converse conjecture: in particular, that the densest possible packing of any convex, congruent polyhedron without central symmetry is not a Bravais lattice packing — rather, it is significantly more complicated. (A nice side result is that the densest packing they found possesses no long-range order — more on this later.) It is not clear at this point what rules, if any, would dictate what the densest possible packing of these polyhedra look like.

One word of caution: as I mentioned before, because this approach is computational, none of these densest-known packings has been proved to be the densest-possible. Because the algorithm requires some choice of the starting ‘dilute’ configuration, it is possible that this choice will influence the final structure the algorithm settles at. In fact, Torquato and Jiao already found this to be the case in their search for the densest possible tetrahedral packing, as they note in their second paper — using a different initial condition, they found a densest packing ~4% more dense than the one they initially reported in the first paper.

While they don’t mention it, I think these results are particularly interesting in the context of amorphous systems, such as glasses. For example,why can a simple liquid metal be supercooled below its freezing point without crystallizing, potentially forming a glass? F. C. Frank put forward a very nice explanation for this. Considering the atoms of the liquid metal as spheres interacting via a non-directional Lennard-Jones potential, it turns out that the local energy density can be minimized by forming “locally-preferred” tetrahedral clusters. These then come together to form polytetrahedral Frank-Kasper phases, because forming these clusters requires less energy than forming crystalline clusters. The only problem with these phases is that they cannot tile space and are geometrically ‘frustrated‘ — the system is not in the crystalline state of lowest possible free energy, but is rather trapped a local free energy minimum in phase space. A significant amount of work has focused on trying to understand these kinds of structures, and connecting the geometric frustration inherent in these phases to the physical properties of supercooled liquids and glasses. It is not too surprising, then, that definitively finding the densest possible packing of tetrahedra (and analyzing the physical properties of such a structure) could help flesh out these connections — and Torquato and Jiao’s work seems to point the way.

Managing Information on the Web, Wolfram-Style

While Steven Wolfram may not the most, um, orthodox figure in the scientific community (see, for example Steven Levy’s bio, or Cosma Shalizi’s review of the modestly-titled A New Kind of Science), I don’t think anyone doubts the usefulness of Mathematica and the various things associated with it (e.g. MathWorld and the Demonstrations Project). And now apparently his latest production – WolframAlpha, Wolfram’s new Mathematica-based search engine – will be released to the public this Monday. It looks quite interesting.

Finding useful information on the internet can be difficult and incredibly annoying, particularly for scientists or anyone in search of statistics of some sort. Google and Wikipedia, while useful, can often be inefficient or yield inadequate results. Many new search engines tailored to various interests seem to have emerged recently, but I am not aware of any current tools that satisfactorily tackle this particular (non-trivial) problem. One solution for anyone interested in biology is bionumbers, a searchable database of useful biological facts and data taken straight from the literature — but I think it’s quite clear that a more general and comprehensive solution (which WolframAlpha purports to be) would be very cool.

Judging from Wolfram’s promo video and reviews on pcworld, techreview and semantic universe, Alpha seems to be bionumbers made significantly more powerful and comprehensive. You probably won’t want to use it over google to find movie times or track your favorite celebrities’ lovelives; but you will want to use it to find various kinds of quantitative information: various metrics of the weather in Springfield, MA on the day David Ortiz was born, the location and sequence of some gene, the flowfield over a particular airfoil, the current position of the International Space Station, or data on blood cholesterol and potassium levels of middle-aged male smokers, for example. I look forward to pushing the limits of this tool, but it looks very useful.

Not be outmatched, Google recently announced plans to implement a similar kind of service using publicly-available data. I’m not sure when they will be releasing it, though, or how it will compare to WolframAlpha.

Determining Linguistic Structures Using Entropy?

Here’s an interesting recent controversy. While I know absolutely nothing about the subject, the ideas and questions raised are interesting, so here’s a quick summary of the different opinions.

In one corner – Rao et al.
In a recent high-profile Brevia published in Science a week ago, Rao et al. suggest that

1. the degree of randomness in linguistic systems is significantly different from that of nonlinguistic systems, and
2. the degree of randomness of the script of the Indus civilization is similar to that of other linguistic systems – in particular, Sumerian and Old Tamil. (The similarity to Old Tamil is particularly striking because it seems to support the somewhat controversial opinion of some “that the Indus peoples spoke and wrote a Dravidian language” – here I’m quoting from Farmer et al.’s rebuttal.)

Based on this, their claim is that the Indus script encodes some kind of linguistic structure, in stark contradiction to some well-known work by Farmer et al. arguing that the Indus script is “a simple nonlinguistic sign system common in the ancient world”. Unsurprisingly, this has set off a number of critical responses, and it’s always fun to see discussion and debate of this sort go on.

The way Rao et al. quantify the degree of randomness of any given sequence of units, or “tokens” (for example, words or characters in English) is by computing the conditional entropy, a standard measure of randomness in information theory. Simplistically, this quantity is a measure of how flexibly different tokens can be ordered: in a nonlinguistic system where different tokens are ordered at random – what Rao et al. call a “Type 1 nonlinguistic system” – the conditional entropy is high, while in a nonlinguistic system where a given token must be followed by another specific token – a “Type 2 nonlinguistic system” – the conditional entropy is low. Intuitively, it is perhaps not surprising that linguistic systems fall somewhere in between: Rao et al. verify this by computing the conditional entropy for a few different linguistic systems, as well as two synthetic nonlinguistic systems (type 1 and type 2). They use this to support their first claim. Furthermore, they compute the conditional entropy for sequences of signs from the Indus script and – surprise, surprise – find that it falls somewhere in between the type 1 and type 2 nonlinguistic systems, just like the other linguistic systems they studied. They use this to support their second claim.

In the other corner – Farmer et al., Liberman, Pereira, Shalizi, Sproat, and others.
Farmer et al. – whose work Rao et al.’s contradicts – have written a pretty strong response to Rao et al.’s paper. Among other things, Farmer et al. claim that their original work from 2004 “awakened resistance from Indian nationalists and researchers whose entire careers have been linked to the Indus-script thesis, one of whom is listed as coauthor of [Rao et al.'s] study”; and, “if [Rao et al.'s] paper had been properly peer reviewed it would not have been published.” Ouch. Here are their main critiques of this work:

• Rao et al. used “synthetic” type 1 and type 2 nonlinguistic data in their calculations – that is, they created it according to certain rules. In a sense, these are designed to represent two different extremes on the “conditional entropy spectrum”, and as such it is not surprising that linguistic systems fall somewhere in between. Other nonlinguistic systems might, as well – so, claim #1 is unsubstantiated.
• The idea that the Indus signs are in some linguistic way related to Old Tamil does not make sense historically: for example, “the first attestation
  of Old Tamil came nearly two thousand years after the Indus civilization disappeared”.

Others have weighed in on this as well, including Mark Liberman, Fernando Pereira, Cosma Shalizi, and Richard Sproat. In particular, Liberman, Shalizi and Sproat have come up with simple counter-examples to Rao et al.’s data, showing instances of nonlinguistic datasets that show at least qualitatively similar behavior to Rao et al.’s linguistic datasets. It appears that at least for now, Pereira’s comment that language is “a system… carrying lots of specific information that cannot be captured by a single statistic” seems to hold.

Type-1.5 superconductors

Superconductors are generally classified as being type-I or type-II; Doug Natelson touched on this as part of his recent series of pedagogical posts explaining solid-state physics concepts. Type-I superconductors usually do not admit an external magnetic field in the superconducting state: they turn “normal” above a critical value of the field. Type-II superconductors do admit a magnetic field for some field strengths above the critical value, while still being able to superconduct: this is known as the “mixed” state.

In the presence of an externally-applied magnetic field, “vortices” form in a superconductor, with a normal nonsuperconducting core of size ~ (the “coherence length” over which Cooper pairs – quasiparticles consisting of pairs of ‘bound’ electrons – are extended). This is surrounded by a region of size ~ in which a supercurrent circulates (where is known as the penetration depth), and hence produces its own opposing magnetic field. Forming such a vortex requires some energy; straightforward calculations show that the interfacial energy per unit length associated with a vortex is proportional to . If , forming such vortices increases the free energy of the system, and vortices tend to attract and annihilate – as in the case of type-I superconductors. If , on the other hand, a “lattice” of repulsive vortices is formed – as in the case of the mixed state of type-II superconductors.

In some materials, electrons can exist in two different bands ( and ), reflecting different kinds of bonding. A classic example of this is graphite. The electrons in the highest occupied states in a structurally similar superconductor, MgB₂, are similarly - or -bonded. This can be thought of as resulting in two different kinds of Cooper pairs with two different values of and . The interesting thing is that in MgB₂, the quasiparticles associated with the electrons have (type-I), while the quasiparticles associated with the electrons have (type-II).

The coupling between these two different states is so weak that MgB₂ was predicted - and has now been found – to be a so-called “type-1.5″ superconductor — that is, one with behavior combining aspects of type-I and type-II superconductivity. In this case, the vortices repel each other (as in type-II superconductors) over short distances while they attract each other (as in type-I superconductors) over long distances. In a previous post, I noted that competition between long-range repulsive and short-range attractive forces often leads to spatially inhomogeneous and anisotropic phases in various systems: examples include “stripe” or “bubble” phases in blockcopolymers, “pasta phases” of the crusts of neutron stars or DNA-intercalated lipid bilayers, stripe formation in ferrofluids, and anisotropic phases in two-dimensional electron gases in the presence of moderately large magnetic fields. Similarly, one might expect the competition between short-range repulsive and long-range attractive forces between vortices to give rise to interesting pattern formation in MgB₂ at low applied fields.

This is what Moshchalkov et al. set out to explore. One way to visualize the flux vortices of a superconductor is using so-called ‘magnetic decoration’: that is, by sprinkling ferromagnetic powder onto the surface of the superconductor. The powder is then attracted by the vortex flux lines and forms a pattern representative of the flux vortices. Using this technique, Moshchalkov et al. found indeed that the vortices in MgB₂ were inhomogeneously distributed, often forming stripes separated by regions of ‘normal’ phase – thus confirming that MgB₂ is a type-1.5 superconductor.