Response functions of the diffusion model of one-dimensional disordered systems

The configurational average response of discrete one-dimensional disordered systems modeled by the classical diffusion equation is investigated. Perturbation expansions of the system response functions based on the average deviation $〈\Delta W〉$ of the nearest-neighbor interaction constants $W_n$ are developed in the frequency domain. It is shown that for probability distributions $\rho \mathrm{}\left(W\right)\mathrm{}$ such that $〈W^{-1\mathrm{}}〉$ is finite, a frequently applied effective-medium approximation is exact to second order in $\Delta W$ for all frequencies. The frequency dependence of the hopping conductivity is a second-order effect.