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

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 $ln\mathcal{W}\left(x\right)$ 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 $ln\mathcal{W}\left(x\right)$ is independent of $L$ and equal to $-x/\ell$, with $\ell$ the mean free path. At the beginning of the sample, $\mathrm{var}[ln\mathcal{W}(x\left)\right]$ 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.

Figure 1

Top panel: The scattering problem for the 1D disordered waveguide of length $L$ described in the text. Lower panels: Theoretical results (solid lines) and computer simulations (various symbols and dotted lines) for the profiles (a) ${〈ln\mathcal{W}\left(x\right)〉}_s$ and (b) $\mathrm{var}{[ln\mathcal{W}\left(x\right)]}_s$ as functions of $x/\ell$, for three values of $s=L/\ell$. For the variance, Eq. (12) was complemented with Eq. (13) when $x=L$ (see the arrows): Their combination accounts for the “bending” shown by the simulation. Agreement is excellent. Simulations consist of ${10}^5$ realizations, with $kd=0.1$ and $k\ell =178$.

Figure 2

Evolution of the statistical distribution of $ln\mathcal{W}\left(x\right)$ for $x/L=1.0$, $3/4$, $1/2$, and $1/16$, $x=L$ corresponding to $lnT$. The histograms are the results of computer simulations with ${10}^5$ realizations each, and $kd=0.1$, $k\ell =178$. All cases are in the localized regime, as $s=L/\ell =22.4$. The ordinate gives the number of events $N_i$ falling in box $i$ of the histogram. The continuous curves are Gaussians with the parameters discussed in the text. For $x$ not too close to zero, the agreement between the theoretically constructed Gaussians and the computed generated histograms is excellent.

Figure 3

Results from microwave experiments for (a) $\langle ln\mathcal{W}\left(x\right)\rangle$, (b) $\mathrm{var}[ln\mathcal{W}(x\left)\right]$. The solid line in (a) shows a comparison to the theoretical prediction of a linear fall-off as in Eq. (9). (c) Experimental results for the PDF of $ln\mathcal{W}\left(x\right)$ at different locations for $s=5$. Lines are Gaussian fit curves.