Resistance fluctuation in a one-dimensional conductor with static disorder

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 $W_{\rho }(\rho , l)$, with varying sample length $l$. An exact two-point recursion relation for the moments $〈{\rho }^n〉$ is given along with a closed-form solution for $W_{\rho }(\rho , l)$ for the case of Gaussian white-noise disorder. The latter confirms $\ln \rho$ as the correct scale variable. The treatment admits generalization to the case of $N$ channels and to general disorder.