One-dimensional random walks on a lattice with energetic disorder

Monte Carlo results are obtained for random walks of excitation on a one-dimensional lattice with a Gaussian energy distribution of site energies. The distribution Ψ(t) of waiting times is studied for different degrees of energetic disorder. It is shown that at T=0, Ψ(t) is described by a biexponential dependence and at T≠0 the distribution Ψ(t) broadens due to the power-law ‘‘tail’’ ${\mathit{t}}^{\mathrm{-}1\mathrm{-}\gamma }$ that corresponds to the description of Ψ(t) in the framework of the continuous-time random walk model. The parameter γ depends linearly on T for strong (T→0) and moderate disorder. For the case of T=0 the number of new sites S(t) visited by a walker is calculated at t→∞. The results are in accordance with Monte Carlo data. The survival probability Φ(t) for strong disorder in the long-time limit is characterized by the power-law dependence Φ(t)∼${\mathit{t}}^{\mathrm{-}\mathrm{\beta }}$ with β=cγ, where c is the trap concentration and for moderate disorder the decay Φ(t) is faster than ${\mathit{t}}^{\mathrm{-}\gamma }$.