Stretched Exponential Relaxation on the Hypercube and the Glass Transition

R. M. C. de Almeida, N. Lemke, I. A. Campbell

Published 2000-03-21Version 1

We study random walks on the dilute hypercube using an exact enumeration Master equation technique, which is much more efficient than Monte Carlo methods for this problem. For each dilution $p$ the form of the relaxation of the memory function $q(t)$ can be accurately parametrized by a stretched exponential $q(t)=\exp(-(t/\tau)^\beta)$ over several orders of magnitude in $q(t)$. As the critical dilution for percolation $p_c$ is approached, the time constant $\tau(p)$ tends to diverge and the stretching exponent $\beta(p)$ drops towards 1/3. As the same pattern of relaxation is observed in wide class of glass formers, the fractal like morphology of the giant cluster in the dilute hypercube is a good representation of the coarse grained phase space in these systems. For these glass formers the glass transition can be pictured as a percolation transition in phase space.