Finite ensemble averages of the zero-temperature resistance and conductance of disordered one-dimensional systems

Abstract

Because of the unusual width of the probability distributions of the zero-temperature response functions for disordered systems their mathematical expectation value will not represent the results of a numerical or physical experiment. We show that it is possible to derive analytically a scaling law for the average resistance $\overline{\rho }$ and the average conductance $\overline{g}$ taken over a finite ensemble of $m$ systems, which is of a different analytic form than $〈\rho 〉$ and $〈g〉$, and which gives good qualitative and quantitative agreement with numerical results. The conditions under which this new scaling behavior might be observed experimentally are discussed. Our result also rigorously proves that $\frac{\ln \overline{\rho }}{〈\ln \rho 〉}=-\ln \overline{g}〈\ln \rho 〉\to 1$ as $L\to \infty$; thus, as expected, it is the geometric means of the zero-temperature response functions which are the relevant quantities in this limit.