Scale invariance in biased diffusion

We have studied hopping transport on a one-dimensional disordered chain in which the hopping rates are characterized by scale-invariant distributions. As a model for dispersive transport, the distribution contains a large weight of small hopping rates. Emphasis has been placed on the range dependence of the diffusion coefficient D(L)∼${\mathit{L}}^{\mathrm{-}\mathrm{\theta }}$ and the frequency dependence of the electrical conductivity σ(ω)∼${\mathrm{\omega }}^{\mathit{x}}$, with x=θ/(2+θ), so that comparison with conductivity experiments can be made. In the presence of a biased electric field, the frequency dependence of the conductivity σ(ω) is found to cross over from ${\mathrm{\omega }}^{\mathit{x}}$ (diffusive limit) to ${\mathrm{\omega }}^{\mathit{y}}$ with y=θ/(1+θ) (drift limit). The transport exponents are calculated explicitly in terms of the parameters of the distribution. Numerical simulations have also been carried out to confirm the above scaling results. Excellent agreement is found.