Category Archives: Statistical mechanics

Electromagnetism, Plectics, Statistical mechanics, String Theory

Superconductors via Superstrings

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Blogging on Peer-Reviewed ResearchNature has an article about a nifty and relatively new application of ideas born out of string theory: to understand what happens in high-temperature superconductors! The story goes something like this.

Take a sample of some material which can conduct electricity, and apply two kinds of outside influence upon it. First, stick it in a magnetic field pointing in some direction, and second, apply a temperature gradient in a direction perpendicular to the magnetic field. In some substances, an electric field will appear, perpendicular to both the magnetic field and the temperature gradient. This is called the Nernst effect. It doesn’t happen very much with ordinary metals, but in semiconductors — like silicon or germanium — it can be quite noticeable. It also appears in some superconductors, like Y-Ba-Cu-O and CeCoIn5 to name but two.

Sean A. Hartnoll et al. have cooked up a theory to explain the Nernst effect and other behaviors seen in the cuprate superconductors, ceramic compounds containing copper. Looking at the situation near the phase transition, where a substance is “on the verge” of changing from insulator to superconductor, they developed a theory involving the magnetic field, call it [tex]B[/tex], and fluctuations in the material’s density, [tex]\rho[/tex]. Then they looked at this theory in the conceptual mirror known as the AdS/CFT correspondence. This connection between seemingly disparate ideas takes you from a “conformal field theory,” the sort of math involved with the superconductor problem (among other things), to a theory of gravity in a type of universe called anti-de Sitter space. In this mirror-world description, the perturbations in [tex]B[/tex] and [tex]\rho[/tex] become magnetic and electric charges of a black hole sitting in the AdS universe!
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Group Theory, Neuroscience, Statistical mechanics, Transhumanism

Psychedelic Bibliographies

Advances in the History of Psychology, a blog operated out of York University, has posted annotated bibliographies of psychedelic research, both on general psychological research and on studies focusing specifically on LSD.

(Hah! And you thought I was just trying to make a strange juxtaposition in my title.)

The AHP folks note something which I find interesting but not wholly unexpected: while plenty of papers have been written about LSD and marijuana, the academic literature doesn’t appear to have histories dedicated to the two-carbon phenethylamines like 2C-B or other significant drugs like DMT, DOM or mescaline. These remarkable little molecules sometimes get mentioned in general discussions or in studies of other drugs, but they don’t appear to have peer-reviewed literature of their own. PiHKAL (1991) and TiHKAL (1997) seem to be the end of the line.

One unfortunate consequence of this lack is our inability to judge the universality of neurological reactions to chemical stimuli. In this context, I’d like to bring up the paper by Bressloff, Cowan, Golubitsky and Thomas in Neural Computation (2002), “What geometric visual hallucinations tell us about the visual cortex.”
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Bad Math, Network theory, Plectics, Statistical mechanics, Transhumanism

Don’t Make Baby Gauss Cry

Cosma Shalizi writes of “Power-Law Distributions in Empirical Data“:

Because this is, of course, what everyone ought to do with a computational paper, we’ve put our code online, so you can check our calculations, or use these methods on your own data, without having to implement them from scratch. I trust that I will no longer have to referee papers where people use GnuPlot to draw lines on log-log graphs, as though that meant something, and that in five to ten years even science journalists and editors of Wired will begin to get the message.

Mark Liberman is not optimistic (we’ve got a long way to go).

Among several important take-home points, I found the following particularly amusing:
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Classical Mechanics, Popularization, Statistical mechanics, Video

Missing No More

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Richard Feynman’s second Messenger Lecture, on the relation between physics and mathematics, is missing no more:

[VIDEO REMOVED FROM GOOGLE’S ARCHIVES]

This is the lecture in which Feynman presents an example I have appropriated before, concerning the necessity of knowing math before being able to do science, and how popularizations of physics often fail because they leave out the mathematics.

Feynman’s example goes like this: I can say that when a planet travels in its orbit, a line from the planet to the Sun sweeps out equal areas in equal times. I can also say that the force pulling on the planet is always directed toward the Sun. Both of these statements require a little math — “equal areas,” “equal times” — but it’s not really math, not a kind to give the layman heebie-jeebies. Given some time for elaboration, one could translate both of these statements into “layman language.” However, one cannot explain in lay terminology why the two statements are equivalent.
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Bad Math, Network theory, Plectics, Software, Statistical mechanics, Transhumanism, Transparency

Power-law Distributions in Empirical Data

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Throughout many fields of science, one finds quantities which behave (or are claimed to behave) according to a power-law distribution. That is, one quantity of interest, y, scales as another number x raised to some exponent:

[tex] y \propto x^{-\alpha}.[/tex]

Power-law distributions made it big in complex systems when it was discovered (or rather re-discovered) that a simple procedure for growing a network, called “preferential attachment,” yields networks in which the probability of finding a node with exactly k other nodes connected to it falls off as k to some exponent:

[tex]p(k) \propto k^{-\gamma}.[/tex]

The constant γ is typically found to be between 2 and 3. Now, from my parenthetical remarks, the Gentle Reader may have gathered that the story is not quite a simple one. There are, indeed, many complications and subtleties, one of which is an issue which might sound straightforward: how do we know a power-law distribution when we see one? Can we just plot our data on a log-log graph and see if it falls on a straight line? Well, as Eric and I are fond of saying, “You can hide a multitude of sins on a log-log graph.”

Via Dave Bacon comes word of a review article on this very subject. Clauset, Shalizi and Newman offer us “Power-law distributions in empirical data” (7 June 2007), whose abstract reads as follows:
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Quantum mechanics, Seminar Announcements, Seminar Summaries, Statistical mechanics

Quantum Mechanics Homework #1

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Well, in the past two days I’ve linked to an Internet quiz and some anime videos, so in order to retain my street cred in the Faculty Lounge, it’s time to post a homework assignment. Don’t worry: if you haven’t met me in person, there’s no way I can grade you on it (unless our quantum states are somehow entangled). This problem set covers everything in our first two seminar sessions on QM, except for the kaon physics which we did as a lead-up to our next topic, Bell’s Inequality. I’ve chosen six problems, arranged in roughly increasing order of difficulty. The first two are on commutator relations, the third involves position- and momentum-space wavefunctions, the fourth brings on the harmonic oscillator (with some statistical mechanics), the fifth tests your knowledge about the Heisenberg picture, and the sixth gets into the time evolution of two-state systems.

Without extra ado, then, I give you Quantum Mechanics Homework #1.
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Chaos Theory, Classical Mechanics, Plectics, Statistical mechanics, Topology

Chaos, Phase Transitions and Topology

Ben has suggested the following paper as a target for our discussion. We should, he says, have the necessary background by the end of the summer.

As the abstract says, the paper is divided into two main parts:
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Classical Mechanics, Information Theory, Statistical mechanics

Entropy for Non-Majors

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Every once in a while (well, actually, pretty frequently) I see a post out there in the Blagopelago which makes me feel bad about ranting so much and discussing science so little. Today’s entry in this category is Jacques Distler’s treatment of Boltzmann entropy. He explains his motivation as follows:

This semester, I’ve been teaching a Physics for non-Science majors (mostly Business School students) class.

Towards the end of the semester, we turned to Thermodynamics and, in particular, the subject of Entropy. The textbook had a discussion of ideal gases and of heat engines and whatnot. But, somewhere along the line, they made a totally mysterious leap to Boltzmann’s definition of Entropy. As important as Boltzmann’s insight is, it was presented in a fashion totally disconnected from Thermodynamics, or anything else that came before.

So, equipped with the Ideal Gas Law, and a little baby kinetic theory, I decided to see if I could present the argument leading to Boltzmann’s definition.

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Seminar Announcements, Statistical mechanics

stat mech problems up.

Please find the stat mech problem set here:

http://web.mit.edu/edown/www/statmech_pset.pdf

This covers the microcanonical and canonical ensembles. You should definitely be able to finish all of I and II by Monday 30th. Part III becomes more and more difficult as one progresses. I would expect everyone to have no problems completing A and B. C is not that hard if you’re willing to play with it. D is a bit more involved but should be within reach.

Let me know if you have any questions. eric

Information Theory, Neuroscience, Quantum mechanics, Statistical mechanics, Wobosphere fun

Where Was I When They Were Passing Out the Wit?

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Scott Aaronson has a new comment policy:

If you reject an overwhelming consensus on some issue in the hard sciences — whether it’s evolution, or general relativity, or climate change, or anything else — this blog is an excellent place to share your concerns with the world. Indeed, you’re even welcome to derail discussion of completely unrelated topics by posting lengthy rants against the academic orthodoxy — the longer and angrier the better! However, if you wish to do this, I respectfully ask that you obey the following procedure:

1. Publish a paper in a peer-reviewed journal setting out the reasons for your radical departure from accepted science.
2. Reference the paper in your rant.

If you attempt to skip to the “rant” part without going through this procedure, your comments will be deleted without warning. Repeat offenders will be permanently banned from the blog. Life is short. I make no apologies.

It looks like Dave Bacon can now talk about time travel, but my own conspiracy theories will have to wait. But soon, I promise, the real meaning behind supersymmetric quantum mechanics will be made clear. They laughed at me when I suggested that the BPS interpretation of shape invariance may have a non-topological origin. The fools — I’ll show them all!
Continue reading Where Was I When They Were Passing Out the Wit? →

Neuroscience, Plectics, Statistical mechanics

Jack Cowan at MIT

Second in today’s “e-mails from Eric” department is this announcement of a talk by Jack Cowan, a mathematics professor at U. Chicago. The talk, scheduled for 16:00 on Tuesday, 3 April in room 46-3189, is abstracted as follows:

We have recently found a way to describe large-scale neural activity in terms of non-equilibrium statistical mechanics [Buice & Cowan, in preparation]. This allows us to calculate (perturbatively) the effects of fluctuations and correlations on neural activity. Major results of this formulation include a role for critical branching, and the demonstration that there exist non-equilibrium phase transitions in neocortical activity, which are in the same universality class as directed percolation. This result leads to explanations for the origin of many of the scaling laws found in LFP, EEG, fMRI, and in ISI distributions, and provides a possible explanation for the origin of alpha, beta, gamma, delta and theta waves. It also leads to ways of calculating how correlations can affect neocortical activity, and therefore provides a new tool for investigating the connections between neural dynamics, cognition and behavior.

I suspect I’ve already seen some of these results, in the video “Spontaneous pattern formation in large scale brain activity: what visual migraines and hallucinations tell us about the brain” (2006). The one review of that video is, depressingly enough, a muddled remark about “observers” in quantum mechanics and what they must mean for consciousness (and you’ll probably catch me ranting about that, anon). Fortunately, the video itself is much more informative.

For a quick primer, see “Hallucinatory neurophysics” at Sean Carroll‘s old blog, Preposterous Universe.

UPDATE (11 June 2007): Yes, I ranted more about the “quantum mind.” See here and here.