Relationship between classical diffusion, intermittency, and the motion of a quantum particle in modulated or random media

Using the mapping of classical and spatially continuous diffusion onto a quantum system, we establish the connection between long-time diffusive behavior and the leading low-frequency properties in the density of states. The mapping also relates the hitherto independent fields of classical diffusion in random media and localization. Moreover, the approach is extended to spatially discrete diffusion. Here, the low-frequency scaling behavior is connected to intermittency in one-dimensional maps and confirmed by numerical results for a model exhibiting sublinear diffusion. © 1986 The American Physical Society.