Classical Diffusion in One-Dimensional Disordered Lattice

Abstract

Classical diffusion of localized excitations is investigated on a one-dimensional chain with (energy-independent) nearest-neighbor transfer rates ${W}_{n,n+1}={W}_{n+1,n}$ that are independently distributed according to a probability density $\ensuremath{\rho}(W)$. An exact formal solution is derived for $〈{P}_{0}(t)〉$, the time development of the initial excitation. The long-time decay of $〈{P}_{0}(t)〉$, determined by the behavior of $\ensuremath{\rho}(W)$ near $W=0$, is analyzed in detail for arbitrary probability densities $\ensuremath{\rho}(W)$.