New method for a scaling theory of localization

Abstract

We base a scaling theory of localization on an expression for conductivity of a system of random elastic scatterers in terms of its scattering properties at a fixed energy. This expression, proposed by Landauer, is first derived and generalized to a system of indefinite size and number of scattering channels (a "wire"), and then an exact scaling theory for the one-dimensional chain is given. It is shown that the appropriate scaling variable is $f(\ensuremath{\rho})=\mathrm{ln}(1+\ensuremath{\rho})$ where $\ensuremath{\rho}$ is the dimensionless resistance, which has the property of "additive mean," and that scaling leads to a well-behaved probability distribution of this variable and to a very simple scaling law not previously given in the literature.