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- W1993670115 abstract "In this paper we investigate the reaction efficiency of diffusion-controlled processes on finite, planar arrays having physical or chemical receptors. This problem translates into the statistical-mechanical one of examining the geometrical factors affecting the trapping of a random walker on small lattices of dimension d=2, having N sites and average valency ensuremath{nu}ifmmodebarelsetextasciimacronfi{}. Extensive calculations of the site-specific average walk length 〈n〉 before trapping, a measure of the efficiency of the underlying diffusion-reaction process, have been carried out on triangular, square-planar, hexagonal, and Penrose platelets for N=16 and N=48. From the variety of distinct lattices considered, and the data generated, three general conclusions can be drawn. First, for fixed N, the smaller the number ${mathit{N}}_{mathit{b}}$ of vertices defining the boundary of the finite lattice under consideration, the smaller the value of the (overall) average walk length 〈nifmmodebarelsetextasciimacronfi{}〉 of the random walker before trapping. Second, for fixed N and fixed ${mathit{N}}_{mathit{b}}$, the smaller the value of the (overall) root-mean-square distance (r${mathrm{ifmmodebarelsetextasciimacronfi{}}}^{2}$${)}^{1/2}$ of the N lattice sites relative to the center of the array, the smaller the value of 〈nifmmodebarelsetextasciimacronfi{}〉. Third, for fixed {N,${mathit{N}}_{mathit{b}}$,(r${mathrm{ifmmodebarelsetextasciimacronfi{}}}^{2}$${)}^{1/2}$}, 〈nifmmodebarelsetextasciimacronfi{}〉 decreases with an increase in the (overall) average valency ensuremath{nu}ifmmodebarelsetextasciimacronfi{} of lattice sites comprising the array. Thus there are similarities but also real and significant differences in the conclusions drawn here in studying stochastic processes taking place on small, finite lattices of arbitrary shape and those found in studying nearest-neighbor random walks on infinite, periodic lattices of unit cells characterized by a given (N,d,ensuremath{nu}). We comment on these and on the possible relevance of this work to one aspect of morphogenesis, viz., predicting the morphologies assumed by small platelets when growth is optimized with respect to (chemical or physical) signal processing at receptor sites." @default.
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- W1993670115 date "1994-02-01" @default.
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- W1993670115 title "Reaction efficiency of diffusion-controlled processes on finite, planar arrays" @default.
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- W1993670115 doi "https://doi.org/10.1103/physreve.49.1049" @default.
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