Dynamics of complex systems above the glass temperature

We propose a phase space approach to understanding non-exponential relaxation above the glass temperature in complex physical systems. Relaxation of a system consisting of many interacting elements (atoms, molecules, spins, etc.) at a given temperature can be considered as the random walk of the point representing the whole system among those configurations which are thermodynamically permitted at that temperature. The phase space is a closed (sphere-like) high dimensional space. It can be demonstrated numerically that random walks on a fractal inscribed in a sphere-like space lead to stretched exponential relaxation of the memory function. As stretched exponential decay is observed experimentally and numerically in very many glassy systems as the freezing temperature is approached, we suggest that this is the signature of a ubiquitous fractal phase space structure necessarily preceding a glass transition. Relaxation that resembles stretched exponential decay but which is not strictly stretched exponential could indicate a subtly different physics.