# Numerical Enzymes

## Evolving Enzymes: Accelerating Relaxation in Glassy Systems.
(Shumway;
51, 54)

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.

## Other Coarsening

Last modified: December 12, 1994

James P. Sethna, sethna@lassp.cornell.edu
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).