Anomalous diffusion on a random comblike structure

Shlomo Havlin, James E. Kiefer, George H. Weiss

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We have recently studied a random walk on a comblike structure as an analog of diffusion on a fractal structure. In our earlier work, the comb was assumed to have a deterministic structure, the comb having teeth of infinite length. In the present paper we study diffusion on a one-dimensional random comb, the length of whose teeth are random variables with an asymptotic stable law distribution (L)1/4 L-(1+) where 0<1. Two mean-field methods are used for the analysis, one based on the continuous-time random walk, and the second a self-consistent scaling theory. Both lead to the same conclusions. We find that the diffusion exponent characterizing the mean-square displacement along the backbone of the comb is dw=4/(1+) for <1 and dw=2 for 1. The probability of being at the origin at time t is P0(t) 1/4 t-ds/2 for large t with ds=(3-)/2 for <1 and ds=1 for >1. When a field is applied along the backbone of the comb the diffusion exponent is dw=2/(1+) for <1 and dw=1 for 1. The theoretical results are confirmed using the exact enumeration method.

Original languageEnglish
Pages (from-to)1403-1408
Number of pages6
JournalPhysical Review A
Issue number3
StatePublished - 1987


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