Exact results for diffusion on a disordered chain

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Abstract
The authors study diffusion on a chain with static disorder, characterised by random transition rates (wn) using an expansion in powers of the fluctuation delta n=(wn-1-(w-1))/(w-1) around the exact diffusion coefficient D=1/(w-1) (low-frequency behaviour), or in powers of the fluctuation delta n=(wn-(w))/(w) around the bare diffusion coefficient D0=(w) (high-frequency behaviour). The former method yields a systematic expansion of the frequency-dependent diffusion coefficient U0(z), the single-site Green function (p0(z)) (spectral properties) and related quantities in powers of square root z(z to 0); the latter method yields a systematic 1/z expansion (z to infinity ). The exact results for the transport properties show that the approximate results of effective medium or hypernetted chain approximations are incorrect at low frequencies beyond the square root z terms. The results for the spectral properties of the chain agree with exact results for low and high frequencies.