Resistance fluctuation in a one-dimensional conductor with static disorder

Abstract

The non-self-averaging resistance of a one-dimensional conductor with static disorder is reexamined by the method of invariant imbedding, leading to a Fokker-Planck equation for its probability distribution ${W}_{\ensuremath{\rho}}(\ensuremath{\rho}, l)$, with varying sample length $l$. An exact two-point recursion relation for the moments $〈{\ensuremath{\rho}}^{n}〉$ is given along with a closed-form solution for ${W}_{\ensuremath{\rho}}(\ensuremath{\rho}, l)$ for the case of Gaussian white-noise disorder. The latter confirms $\mathrm{ln}\ensuremath{\rho}$ as the correct scale variable. The treatment admits generalization to the case of $N$ channels and to general disorder.