Relationship between Classical Motion in Random Media and Quantum Localization

We establish the mapping of the master equation for classical motion in random media to a quantum system, describing the motion of a quantum particle in a random potential. Invoking dynamic scaling, the mapping connects the time-dependent mean square displacement of classical motion with the density of states in terms of the classical correlation length. These connections are illustrated for a model with exactly solvable diffusion properties, confirming dynamic scaling. The numerical results for the quantum counterpart reveal energy-dependent localization in terms of the real part of the Lyapunov exponent, which is proportional to the inverse localization length.