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Diffusion on random systems above, below, and at their percolation threshold in two and three dimensions

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Abstract

A detailed Monte Carlo study is presented for classical diffusion (random walks) on randomL * L triangular andL * L* L simple cubic lattices, withL up to 4096 and 256, respectively. The speed of a Cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual CDC Cyber 76 computer. To reach the asymptotic scaling regime, walks with up to 10 million steps were simulated, with about 1011 steps in total forL=256 at the percolation threshold. We review and extend the dynamical scaling description for the distance traveled as function of time, the diffusivity above the threshold, and the cluster radius below. Earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient Monte Carlo data. The conductivity exponent μ is found to be 2.0 ± 0.2 in three and 1.28 ± 0.02 in two dimensions. Our data in three dimensions follow well the finite-size scaling theory. Below the threshold, the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2/5 independent of dimensionality. The correction-to-scaling exponent atp c seems to be larger in two than in three dimensions.

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Pandey, R.B., Stauffer, D., Margolina, A. et al. Diffusion on random systems above, below, and at their percolation threshold in two and three dimensions. J Stat Phys 34, 427–450 (1984). https://doi.org/10.1007/BF01018553

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