I’ve been poking around Cosma Shalizi’s website recently. He has a little bit of everything: a weblog, a set of book reviews he’s written, and a large collection of mini-essays (which he calls “notebooks”). The bulk of the material revolves around the related subjects of probability, machine learning, and dynamical systems (all with a strong physics flavor), but he touches on many other topics.
What triggered my current plunge into thermodynamics was this miniature book review by Cosma Shalizi of Richard S. Ellis’ Entropy, Large Deviations, and Statistical Mechanics:
In addition to being an excellent exposition of the rigorous theory of large deviations (especially for physicists, naturally!), this is also one of the most conceptually satisfying approaches to the foundations of statistical mechanics. In particular, it makes good probabilistic sense of the method of maximum entropy, without invoking weird sub-Bayesian ideas about statistical inference. (Namely, maximum Gibbs-Shannon entropy drops out as an approximate consequence of large deviations theory, when considering a small part of a large system, becoming exact only in the thermodynamic limit. As Ellis says, the core of this idea goes back to Boltzmann.)
I find the idea of statistical mechanics fascinating: that to describe the behavior of truly gigantic numbers of particles, all we need are a few bulk properties such as temperature and pressure. And to find out that it has a simple mathematical description in terms of probability theory, that’s the kind of thing that makes me want to know more.
Tragically, my library doesn’t have Ellis’ book, but I was able to track down home page, which has an extensive list of publications, many of which are available on-line. Two in particular give an overview of the relationship between statistical mechanics and large deviations:
Cosma also has a quick intro to large deviations. (In a rare lapse, Wikipedia has almost nothing. All that’s there is a pathetic little stub that I just created to fix what was there before, which was an incorrect redirect to extreme value theory.)
Something I’d always meant to do is learn something about thermodynamics. I’ve tried several times, but each time I get bogged down in the details of heat engines. This time, I found some very nice lecture notes by Nino Boccara. They provide an overview abstracted enough from the physical details to be informative to a mathematical audience. They take entropy as a primitive concept rather than deriving it from properties of heat engines.
(I’ve been trying to find other introductions from the same point of view of Boccara, but with more detail. I haven’t had any luck. Rather than sticking closely to ideal gases, Boccara takes a general view of the formalism, one that does not (in principle) even require energy as one of the state variables. His approach to statistical mechanics is related to the maximum entropy approach of E. T. Jaynes.)
Could someone with accurate knowledge of the state of the verification of Perelman’s proof of the Poincare conjecture comment on this article?
I would like to know if it is complete crap or not.
The latest Bulletin of the AMS is out. We previously linked to Frank Calegari’s introduction to modular forms. Medical image processing has become a major area of applied mathematics, one featured in a survey by Angenent, Pichon, and Tannenbaum. Golubitsky and Stewart discuss network dynamics from the point of view of groupoids.