Many physicists study “disordered systems,” such as materials like glasses where the molecules making up the material are arranged randomly in space, in contrast to crystals, where all the particles are arranged in beautiful repeating patterns.

The symmetries of crystals make them much easier to analyze than glasses, and new theoretical methods had to be invented before physicists could make any headway in computing the properties of disordered systems. Those methods have turned out to be closely connected to approaches, such as the “belief propagation” algorithm, that are widely used in computer science, artificial intelligence, and communications theory, with the result that physicists and computer scientists today regularly exchange new ideas and results across their disciplines.

Returning to the physics of disordered systems, physicists began working on the problem in the 1970’s by considering the problem of disordered *magnets* (also called “spin glasses”). My Ph.D. thesis advisor, Philip W. Anderson summarized the history as follows:

“In 1975 S.F. (now Sir Sam) Edwards and I wrote down the “replica” theory of the phenomenon I had earlier named “spin glass”, followed up in ’77 by a paper of D.J. Thouless, my student Richard Palmer, and myself. A brilliant further breakthrough by G. Toulouse and G. Parisi led to a full solution of the problem, which turned out to entail a new form of statistical mechanics of wide applicability in fields as far apart as computer science, protein folding, neural networks, and evolutionary modelling, to all of which directions my students and/or I contributed.”

In 1992, I presented five lectures on “Quenched Disorder: Understanding Glasses Using a Variational Principle and the Replica Method” at a Santa Fe Institute summer school on complex systems. The lectures were published in a book edited by Lynn Nadel and Daniel Stein, but that book is very hard to find, and I think that these lectures are still relevant, so I’m posting them here. As I say in the introduction, “I will discuss technical subjects, but I will try my best to introduce all the technical material in as gentle and comprehensible a way as possible, assuming no previous exposure to the subject of these lectures at all.”

The first lecture is an introduction to the basics of statistical mechanics. It introduces magnetic systems and particle systems, and describes how to exactly solve non-interacting magnetic systems and particle systems where the particles are connected by springs.

The second lecture introduces the idea of variational approaches. Roughly speaking, the idea of a variational approach is to construct an approximate but exactly soluble system that is as close as possible to the system you are interested in. The grandly titled “Gaussian variational method” is the variational method that tries to find the set of particles and springs that best approximates an interacting particle system. I describe in this second lecture how the Gaussian variational method can be applied to heteropolymers like proteins.

The next three lectures cover the replica method, and combine it with the variational approach. The replica method is *highly* intricate mathematically. I learned it at the feet of the masters during my two years at the Ecole Normale Superieure (ENS) in Paris. In particular, I was lucky to work with Jean-Philippe Bouchaud, Antoine Georges, and Marc Mezard, who taught me what I knew. I thought it unfortunate that there wasn’t a written tutorial on the replica method, so the result were these lectures. Marc told me that for years afterwards they were given to new students of the replica method at the ENS.

Nowadays, the replica method is a little less popular than it used to be, mostly because it is all about computing averages of quantities over many samples of systems that are disordered according to some probability distribution. While those averages are very useful in physics, they are somewhat less important in computer science, where you usually just want an algorithm to deal with the one disordered system in front of you, rather than an average over all the possible disordered systems.

Tags: disordered systems, replica theory, spin glasses, statistical mechanics, theory of glasses, variational methods

September 7, 2007 at 12:01 am |

[…] subject too seriously, focusing instead on neural networks and spin glass theory. (As I mentioned in this post, Anderson also made seminal contributions to those fields, and he let me work on whatever I was […]

October 3, 2007 at 10:22 pm |

[…] If you’re interested in quantum systems like e.g. the Hubbard model, the expansion approach has the advantage that it can also be applied to them; see my paper with Georges, or the lectures by Georges on his related “Dynamical Mean Field Theory,” or this recent paper by Plefka, who recently returned to the subject. Also, if you’re interested in learning more about spin glasses or other disordered systems, or about other variational derivations for mean-field theory, please see this post. […]