Continuous time random walk model as a model of anomalous relaxation

The microscopic models of anomalous structural relaxation are proposed and studied within the framework of the continuous time random walk. The Cole-Cole type relaxation is observed in the model with the fractal nature where the waiting time distribution, psi(t), of the random walker is given by the power-law, psi(t) alpha h(-1-a), with a < 1. The relaxation also becomes of the Cole-Cole type even when we introduce the correlation between the direction of two successive jumps, suggesting the robustness of this property. On the other hand, the stretched-exponential type relaxation appears when the fractal nature of the model is removed by the upper cutoff of the waiting time. These results show that the type of the relaxation is strongly related to the (non-) fractal nature of the system.