Hopping conduction in positionally disordered chains

Hopping conduction is studied in one-dimensional chains where carriers can move by hopping among randomly distributed localized centers. The distribution of a jump rate $w$ of the carrier between nearest-neighbor sites is shown to have a form $\frac{{\rho w}^{\rho -1\mathrm{}}}{w_0^{\rho }}$ ($0\le w\le w_0$), $\rho$ being a dimensionless density of the localized centers. With the use of the coherent medium approximation it is found that (i) an insulator-to-metal transition takes place at $\rho =1\mathrm{}$, (ii) there are five regimes for the behavior of the ac conductivity in the vicinity of the static limit, (iii) the ac part of the conductivity for densities slightly above and below $\rho =1\mathrm{}$ show a similar frequency dependence, and (iv) when $0<\mathrm{}\rho \le 1$ a carrier vanishes from its initial position after an infinite time even though the dc conductivity is zero.