Single-parameter scaling and maximum entropy inside disordered one-dimensional systems: Theory and experiment

Abstract

The single-parameter scaling hypothesis relating the average and variance of the logarithm of the conductance is a pillar of the theory of electronic transport. We use a maximum-entropy ansatz to explore the logarithm of the particle, or energy density $ln\mathcal{W}(x)$ at a depth $x$ into a random one-dimensional system. Single-parameter scaling would be the special case in which $x=L$ (the system length). We find the result, confirmed in microwave measurements and computer simulations, that the average of $ln\mathcal{W}(x)$ is independent of $L$ and equal to $\ensuremath{-}x/\ensuremath{\ell}$, with $\ensuremath{\ell}$ the mean free path. At the beginning of the sample, $\mathrm{var}[ln\mathcal{W}(x)]$ rises linearly with $x$ and is also independent of $L$, with a sublinear increase and then a drop near the sample output. At $x=L$ we find a correction to the value of $\mathrm{var}[lnT]$ predicted by single-parameter scaling.