Model of anomalous relaxation in supercooled liquids: Random walk in fractal space and time

Anomalous structural relaxation observed in the supercooled liquids has been one of the most exciting problems for the last decade. Toward the full understanding of this phenomenon from the microscopic viewpoints, we report in this article the results of our studies on the several random walk models, i.e., random walk on fractal and non-fractal structures and the fractal time random walk model. It is found that the relaxation becomes of the Cole-Cole type, a famous empirical law, when the models have the fractal nature while the stretched-exponential type relaxation is observed in non-fractal structures. This work indicates that the concept of `fractal' introduced with mathematical mind has important relevance to realistic physical systems.