Abstract
We study diffusion in systems with static disorder, characterized by random transition rates {}, which may be assigned to the bonds [random-barrier model (RBM)] or to the sites [random-jump-rate model (RJM)]. We make an expansion in powers of the fluctuations around the exact diffusion coefficient in the low-frequency regime, using diagrammatic methods. For the one-dimensional models we obtain a systematic expansion in powers of of the response function (transport properties) and Green's function (spectral properties). The frequency-dependent diffusion coefficient in the RBM is found as , where includes up to fourth-order fluctuations and up to sixth order. In the RJM, .. Similarly, we obtain results (very different in RBM and RJM) for the frequency-dependent Burnett coefficient and the single-site Green's function [which determines the density of eigenstates and the inverse localization length of relaxational modes of the system]. The spectral properties of both models are identical and agree with exact results at low frequencies for the spectral properties of random harmonic chains. The long-time behavior of the velocity autocorrelation function in RBM is and for the Burnett correlation function , with coefficients that vanish on a uniform lattice. For the RJM, and . The long-time behavior of the moments of displacement and and the staying probability are calculated up to relative order . A comparison of our exact results with those of the effective-medium (or hypernetted-chain) approximation (EMA) shows that the coefficient in as given by EMA is incorrect, contrary to suggestions made in the literature. For the RJM all results can be trivially extended to higher-dimensional systems.
- Received 28 June 1983
DOI:https://doi.org/10.1103/PhysRevB.29.1755
©1984 American Physical Society