Diffusion in systems with static disorder (original) (raw)

%'e study diffusion in systems with static disorder, characterized by random transition rates I w"), which may be assigned to the bonds [random-barrier model (RBM)] or to the sites [randomjump-rate model (RIM)]. We make an expansion in powers of the fluctuations 5"=(to"'-(w '))/ (w ') around the exact diffusion coefficient D=1/(to ') in the low frequency regime, using diagrammatic methods. For the one-dimensional models we obtain a systematic expansion in powers of Vz of the response function (transport properties) and Green's function (spectral properties}. The frequency-dependent diffusion coefficient in the RBM is found as Uo(z)=D-, a2V-Dz +cxoz+a)z + ' ', where K2= (5),cxo mcludes lip to fourth-order flllctuatlons and cx) lip to sixth order. In the RJM, Uo(z) =B. Similarly, we obtain results (very different in RBM and RJM) for the frequency-dependent Burnett coefficient Uq(z} and the single-site Green s function Go(z) [which determines the density of eigenstates M(e} and the inverse locaHzation length y(e} of relaxational modes of tlM system]. The spectral properties of both models are ideIltlcal Slid agree with exact results at low frequencies for the spectral properties of random harmonic chains. The long-time behauior of the velocity autocorrelation function in RBM is q& (t)=()t 'r~+(.)t '~' and for the Burnett correlation function p4(t)=(.~~)t ', with coefficients that vanish on a uniform lattice. For the RJM, g2(t)=&6+(t) and y4(t)=()t '~. The long-time behavior of the moments of displacement (n), and (n4), and the staying probability Po(t) are calculated up to relative order t~. A comparison of our exact results with those of the effective-medium (or hypernettedchain) approximation (EMA) shows that the coefficient ao in Uo(z) as given by EMA is incorrect, contlary to suggestions made ln the literature. For the RJM all results can be tlivially extended to higher-dimensional systems.