Effect of periodic boundary conditions in the calculation of ac conductivity for a one-dimensional percolation model

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Abstract
The calculation of the ac hopping conductivity for a linear chain with random interruptions requires the solution of the hopping problem for an ordered segment of arbitrary finite length. This problem was treated recently by the author using periodic boundary conditions and, exactly, by Odagaki and Lax (OL) who considered properly terminated segments. It is shown that the results obtained using the OL definition of the averaged complex diffusion constant in the author's treatment for periodic boundary conditions coincide with the exact results to leading order for strong disorder, both at low and high frequencies. These new results are compared in detail with those of OL for the whole range of disorder. An important normalization correction is included in the author's earlier results, which were based on a different definition of the averaged diffusion constant. It is shown that the two definitions of the diffusion constant may lead to substantial quantitative differences.