After looking at some incommensurate and quasicrystalline models in two dimensions, Shelly Shumway and I began to focus on the dynamics of the simplest of frustrated atomic models: a one dimensional chain of N atoms in a periodic potential with M wells (the Frenkel--Kontorova model). The phase transition here occurs at zero temperature. Shelly showed that the transition is sluggish indeed if N/M approaches an irrational limit: if one cooled at a fixed rate R, the ordered regions would grow only as log |log R|. As Landau used to say, the logarithm is not a function for the same reason that a hen is not a bird. No matter how slowly one cooled, the system would get stuck in a glassy, disordered state. We then used the model to try out ideas for acceleration methods. Suppose, instead of molecular dynamics, we allowed the atoms to jump from well to well? Slight improvement: still the same log-log scaling. Suppose we allowed clusters of atoms to jump? If the cluster jumps are finely tuned, to move each atom just the right distance, we could get relaxation --- what we needed was a numerical enzyme, tuned to the chemical reaction needed. Finally, could we get the computer, without hints, to find this enzyme? Shumway used Darwinian evolution! Starting from a population of random single--ball moves, and allowing for reproduction between pairs of the fittest moves (including mutations and affirmative action), Shumway's ecology evolved a 14--atom Monte Carlo move, fine--tuned to seven digit accuracy. Incidentally, her glassy system found a state within machine accuracy of the true ground state.
James P. Sethna, email@example.com
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).