Probability and Scaling in One-Dimensional Disordered Systems

Abstract

The single parameter scaling theory of localization [1] is central to much of our present understanding of transport in disordered systems predicting as it does the existence of a metal-insulator transition in dimension d = 3 and the localization of electron states for d = 2 and d = 1. The central assumption of the theory, embodied in the definition of the beta function [1], is a smooth one parameter scaling behavior of the conductance g with system size. However as is now widely appreciated this assumption is not correct since it ignores the existence of fluctuation phenomena such as the universal conductance fluctuations (UCF). The basic problem is that in the absence of inelastic effects the conductance of a disordered system does not self average as the system size increases but remains sensitive to the exact microscopic arrangement of impurities. In the localized regime the fluctuations in g diverge exponentially with increasing system size, and even in the metallic regime they are not zero (the UCF mentioned above). The question then arises of how to reconcile the scaling theory with the fluctuating behavior of g. One solution is to propose that the distribution p(g) of g obeys a single parameter scaling, for example

$${p_L}(\ln g) = p(\ln g,{\left\langle {\ln g} \right\rangle _L})$$

((1))

$$\frac{{\partial {{\left\langle {\ln g} \right\rangle }_L}}} {{\partial \ln L}} = \beta ({\left\langle {\ln g} \right\rangle _L}).$$

((2))