Self-consistent transport dynamics for localized waves

We find that the Vollhardt and Wolfle self-consistent theory of Anderson localization makes simple predictions for transport dynamics in unbounded one- and two-dimensional media. These predictions are derived and explored and compared with direct numerical simulations.

Figure 1

The solution (12) to the dynamic self-consistent transport equations in 1D, expressed as a function of dimensionless time and for several distances $x$ from the source. Length units are such that localization length $\xi =1$. Time units are such that $D_0=1$. The behavior is similar to that observed in direct numerical simulations [8]. Logarithms, log, are in base $e$ for all figures.

Figure 2

As in Fig. 1, but expressed as a function of distance for several times. The profile may be observed to approach $\exp (-x)∕2$ at late time.

Figure 3

The solution of the dynamic self-consistent transport equation in 2D as evaluated by numerical integration of integral (18). Length units are such that $\xi =1$; time units are such, as in Figs. 1, 2, that the branch point ${\Omega }_{bp}=i∕4$—i.e., $\tau =1$. One of the curves from Fig. 1 is overlaid for comparison.

Figure 4

The integral (18) expressed as a function of distance $x$ at several times.

Figure 5

The localization length ${\xi }_L$ in a strip of width $L$ is compared with that of an unbounded 2D system with the same microstructure. The solid line is the prediction of the self-consistent theory, Eq. (23). The isolated points are taken from the direct numerical simulations of MacKinnon and Kramer [10].

Figure 6

Comparison of transport dynamics at $x=2$ in strips of various widths $L$. Microstructural parameters are identical to those of Figs. 3, 4.

Figure 7

Comparison of transport dynamics at $t=1$ in strips of various widths $L$.

Figure 8

The integration contours $C_p$ are mapped from the original contours in the $\Omega$ plane $(-\infty -i\delta ,\infty -i\delta )$.

Figure 9

Comparison of the exact (solid lines) and asymptotic (dashed lines) spatial dependence of transport dynamics in 2D.

Figure 10

Comparison of the exact (solid lines) and asymptotic (dashed lines) temporal dependence of transport dynamics in 2D.

Figure 11

The predictions for transport dynamics in 1D are plotted versus $x^2∕t$. A phenomenological form $\exp (-x-x^2∕4D_{\mathit{eff}}t)$ approximately fits the data.

Figure 12

Transport dynamics in 2D approximately fits the phenomenological form $\exp [-x-{(x^{2+n}∕D_{\mathit{eff}}{\xi }^{n}t)}^{\gamma }]$ with $\gamma =0.8$ and $n=0.25$.

Figure 13

The slope of the exponent $S^D\left(p_s\right)$ versus $\log t$ for 1D and 2D and strips of different widths $L$ is close to $-1$, except in 2D where it varies but has a value near $-0.8$.

Figure 14

The slope of the exponent $S^D\left(p_s\right)$ versus $\log x$ for 1D and 2D and strips of different widths $L$ is close to $+2$ except in 2D where it varies but has a value close to 1.8.