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

TL;DR

This paper demonstrates that the complex admittance in disordered media with power-law distributed jump rates is not self-averaging, meaning sample-to-sample variations significantly affect the measured properties, especially in low-dimensional systems.

Contribution

It reveals the non-self-averaging nature of complex admittance in disordered media with power-law jump rate distributions and extends the analysis to higher dimensions.

Findings

Complex admittance is non-self-averaging in one-dimensional disordered media.

The disorder-averaged admittance differs from individual sample admittances.

Higher-dimensional media also exhibit non-self-averaging behavior.

Abstract

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.