Stretched exponential relaxation in systems with random free energies

A spin glass phase is characterized by a large number of quasi degenerate states (or valley bottoms). Their free energies Fa = F0 + fa, where fa is a non extensive fluctuation, have recently been shown to be random independent variables, as in Derrida's model. Relaxation to equilibrium of such systems is considered and a simple approximation to the transition probability (in the master equation) is considered that leads to a (random) relaxation time τ ∼ exp β ΔF with — β(F0 + ΔF) = In Σ exp( — β Fa). The ensuing relaxation to equilibrium is shown to be a stretched exponential a behaviour common to a wide class of materials. The dependence of the result on the particular choice of the transition probability is touched upon.