Hopping conductivity of a one-dimensional bond-percolation model in a constant field: exact solution

Abstract

The recent work of Odagaki and Lax on the ac hopping conductivity in a one-dimensional bond-percolation model is generalized to include a constant applied field. The corresponding biased random-walk problem on a finite chain of arbitrary length and with properly terminated ends is solved analytically. The solution is extended to include all continuous-time random walks. (It is thus directly applicable, for instance, to spectral diffusion with asymmetric transfer rates and memory effects.) The diffusivity (and thence the conductivity and relative permittivity) on a chain of N sites is obtained exactly, in closed form. A configuration averaging over N yields the diffusivity and relative permittivity for the randomly interrupted infinite chain. Our analytic results are valid at all frequencies, and are in a form amenable to extensive numerical computation. This is done, and the noteworthy features of the results are displayed graphically. All the known results in the field-free case are recovered as special cases of the expressions presented here.