Classical diffusion in a random environment and quantum mechanics with a random potential

The problems of classical diffusion in a random environment and of quantum mechanics with a random potential are related by the transformation of the Fokker-Planck equation into the imaginary time SchrɆdinger equation. The hypothesis of dynamic scaling allows to connect the long-time properties of diffusion to the low-energy spectrum of the quantum problem. For the specific model where the drift force is a white noise, the anomalous sublinear diffusion found by Sinai is related to logarithmic anomalies in the inverse localization length and in the density of states. © 1988 IOP Publishing Ltd.