Absence of self-averaging in the complex admittance for transport through disordered media

Random walk models in one-dimensional disordered media with an oscillatory input current are investigated theoretically as generic models of the boundary perturbation experiment. It is shown that the complex admittance obtained in the experiment is not self-averaging when the jump rates $w_i$ are random variables with the power-law distribution $\rho(w_i)\sim {w_i}^{\alpha-1} (0 < \alpha \leq 1)$. More precisely, the frequency-dependence of the disorder-averaged admittance $<\chi>$ disagrees with that of the admittance $\chi$ of any sample. It implies that the Cole-Cole plot of $<\chi>$ shows a different shape from that of the Cole-Cole plots of $\chi$ of each sample. The condition for absence of self-averaging is investigated with a toy model in terms of the extended central limit theorem. Higher dimensional media are also investigated and it is shown that the complex admittance for two-dimensional or three-dimensional media is also non-self-averaging.