Diffusion in systems with static disorder

Abstract

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 ${\ensuremath{\delta}}_{n}=\frac{({w}_{n}^{\ensuremath{-}1}\ensuremath{-}〈{w}^{\ensuremath{-}1}〉)}{〈{w}^{\ensuremath{-}1}〉}$ around the exact diffusion coefficient $D=\frac{1}{〈{w}^{\ensuremath{-}1}〉}$ 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 ${U}_{0}(z)=D\ensuremath{-}\frac{1}{2} {\ensuremath{\kappa}}_{2}\sqrt{\mathrm{Dz}}+{\ensuremath{\alpha}}_{0}z+{\ensuremath{\alpha}}_{1}{z}^{\frac{3}{2}}+\ensuremath{\cdots}$, where ${\ensuremath{\kappa}}_{2}=〈{\ensuremath{\delta}}^{2}〉,{\ensuremath{\alpha}}_{0}$ includes up to fourth-order fluctuations and ${\ensuremath{\alpha}}_{1}$ up to sixth order. In the RJM, ${U}_{0}(z)=D$.. Similarly, we obtain results (very different in RBM and RJM) for the frequency-dependent Burnett coefficient ${U}_{2}(z)$ and the single-site Green's function ${\stackrel{^}{G}}_{0}(z)$ [which determines the density of eigenstates $\mathcal{N}(\ensuremath{\epsilon})$ and the inverse localization length $\ensuremath{\gamma}(\ensuremath{\epsilon})$ 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 ${\ensuremath{\phi}}_{2}(t)\ensuremath{\simeq}(\ensuremath{\cdots}){t}^{\frac{\ensuremath{-}3}{2}}+(\ensuremath{\cdots}){t}^{\frac{\ensuremath{-}5}{2}}$ and for the Burnett correlation function ${\ensuremath{\phi}}_{4}(t)\ensuremath{\simeq}(\ensuremath{\cdots}){t}^{\frac{\ensuremath{-}3}{2}}$, with coefficients that vanish on a uniform lattice. For the RJM, ${\ensuremath{\phi}}_{2}(t)=D{\ensuremath{\delta}}_{+}(t)$ and ${\ensuremath{\phi}}_{4}(t)\ensuremath{\simeq}(\ensuremath{\cdots}){t}^{\frac{\ensuremath{-}1}{2}}$. The long-time behavior of the moments of displacement ${〈{n}^{2}〉}_{t}$ and ${〈{n}^{4}〉}_{t}$ and the staying probability ${P}_{0}(t)$ are calculated up to relative order ${t}^{\frac{\ensuremath{-}3}{2}}$. A comparison of our exact results with those of the effective-medium (or hypernetted-chain) approximation (EMA) shows that the coefficient ${\ensuremath{\alpha}}_{0}$ in ${U}_{0}(z)$ 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.