Classical Diffusion in One-Dimensional Disordered Lattice

Classical diffusion of localized excitations is investigated on a one-dimensional chain with (energy-independent) nearest-neighbor transfer rates $W_{n,n+1\mathrm{}}=W_{n+1,n}$ that are independently distributed according to a probability density $\rho \mathrm{}\left(W\right)\mathrm{}$. An exact formal solution is derived for $〈P_0\mathrm{}\left(t\right)\mathrm{}〉$, the time development of the initial excitation. The long-time decay of $〈P_0\mathrm{}\left(t\right)\mathrm{}〉$, determined by the behavior of $\rho \mathrm{}\left(W\right)\mathrm{}$ near $W=0\mathrm{}$, is analyzed in detail for arbitrary probability densities $\rho \mathrm{}\left(W\right)\mathrm{}$.