Computer-simulated hopping in a random one-dimensional system

Classical single-particle hopping has been investigated by Monte Carlo calculations for a one-dimensional (1D) lattice with random nearest-neighbor hopping rates $\Gamma$ distributed as ${\Gamma }^{-\alpha }$ between ${\Omega }_1$ and ${\Omega }_0$ in accordance with the recent theory of Bernasconi et al. We find agreement if the minimum rate ${\Omega }_1$ is finite, but not for ${\Omega }_1=0\mathrm{}$. We show that the predicted $t\to \infty (\omega \to 0)$ limit is not reached until extraordinarily long times, much longer than the $5\times {10}^3{\Omega }_0^{-1\mathrm{}}$ of the simulations; so this, rather than an incorrect theoretical $\omega \to 0$ limit, is the likely cause of the discrepancy. Implications for observance of the predicted conductivity transition in potassium hollandite are discussed.