Anderson Transition for Classical Transport in Composite Materials

The Anderson transition in solids and optics is a wave phenomenon where disorder induces localization of the wave functions. We find here that the hallmarks of the Anderson transition are exhibited by classical transport at a percolation threshold—without wave interference or scattering effects. As long range order or connectedness develops, the eigenvalue statistics of a key random matrix governing transport cross over toward universal statistics of the Gaussian orthogonal ensemble, and the field eigenvectors delocalize. The transition is examined in resistor networks, human bone, and sea ice structures.

Figure 1

Connectedness transition in composite structures. Images of the Arctic sea ice pack (photo credit D. K. Perovich) with increasingly connected ocean phase from left to right.

Figure 2

Short-range eigenvalue correlations. The ESDs for Poisson (blue dash dot) and WD (green dashed) spectra are shown in (a)–(c), along with ESDs for (a) the Arctic sea ice pack, (b) Arctic melt ponds, and (c) the 2D RRN.

Figure 3

Long-range eigenvalue correlations. (a)–(c) The eigenvalue number variance ${\mathrm{\Sigma }}^2(L)$ and (d)–(f) the spectral rigidity ${\mathrm{\Delta }}_3(L)$ for Poisson (blue dash dot) and WD (green dash) spectra are shown along with those of (a) Arctic pack ice, (b) Arctic melt ponds, (c) sea ice brine inclusions, (d) human bone microstructure, (e) the 2D RRN, and (f) the 3D RRN.

Figure 4

Delocalization of eigenvectors. (a) The electric field ${\chi }_1\mathbf{E}$ (in log scale) for a realization of the 2D RRN, with a system size $\mathit{L}=50$, volume fraction $p=p_c=1/2$, and ${\sigma }_1/{\sigma }_2=4.0\times 10^{10}$. (b) The IPR $I_j$ plotted versus the index $j=1,\dots ,N_1$ for a realization of the 3D RRN with $\mathit{L}=12$ and $p=1-p_c\approx 0.7512$. The vertical lines define the $\delta$ components of the spectral measure $\mu$ at $\lambda =0$, 1, while the horizontal line marks the GOE IPR value $I_{\mathrm{GOE}}=3/N_1$. (c) The $p$ dependence of the average IPR $⟨I⟩$ for the 2D and 3D RRN.