Diffusion in systems with static disorder

We study diffusion in systems with static disorder, characterized by random transition rates {$w_n$}, 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 ${\delta }_n=\frac{(w_n^{-1\mathrm{}}-〈w^{-1\mathrm{}}〉)}{〈w^{-1\mathrm{}}〉}$ around the exact diffusion coefficient $D=\frac{1}{〈w^{-1\mathrm{}}〉}$ in the low-frequency regime, using diagrammatic methods. For the one-dimensional models we obtain a systematic expansion in powers of $\sqrt{z}$ of the response function (transport properties) and Green's function (spectral properties). The frequency-dependent diffusion coefficient in the RBM is found as $U_0\mathrm{}\left(z\right)=D-\frac{1}{2} {\kappa }_2\sqrt{\mathrm{Dz}}+{\alpha }_{0}z+{\alpha }_1{z}^{\frac{3}{2}}+\cdots$, where ${\kappa }_2=〈{\delta }^2〉,{\alpha }_0$ includes up to fourth-order fluctuations and ${\alpha }_1$ up to sixth order. In the RJM, $U_0\mathrm{}\left(z\right)=D$.. Similarly, we obtain results (very different in RBM and RJM) for the frequency-dependent Burnett coefficient $U_2\mathrm{}\left(z\right)\mathrm{}$ and the single-site Green's function ${\hat{G}}_0\mathrm{}\left(z\right)\mathrm{}$ [which determines the density of eigenstates $\mathcal{N}\left(\epsilon \right)$ and the inverse localization length $\gamma \left(\epsilon \right)$ 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 ${\varphi }_2\mathrm{}\left(t\right)\mathrm{}\simeq \left(\cdots \right)t^{\frac{-3\mathrm{}}{2}}+\left(\cdots \right)t^{\frac{-5\mathrm{}}{2}}$ and for the Burnett correlation function ${\varphi }_4\mathrm{}\left(t\right)\mathrm{}\simeq \left(\cdots \right)t^{\frac{-3\mathrm{}}{2}}$, with coefficients that vanish on a uniform lattice. For the RJM, ${\varphi }_2\mathrm{}\left(t\right)=D{\delta }_{+}\mathrm{}\left(t\right)\mathrm{}$ and