Surface effects in invasion percolation

Cafiero, Raffaele and Caldarelli, Guido and Gabrielli, Andrea Surface effects in invasion percolation. Physical Review E, 56 (2). R1291-R1294. ISSN 1539-3755 (1997)

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Abstract

Boundary effects for invasion percolation are introduced and discussed here. The presence of boundaries determines a set of critical exponents characteristic of the boundary. In this paper we present numerical simulations showing a remarkably different fractal dimension for the region of the percolating cluster near the boundary. In fact, near the surface we find a value of $D^{sur}=1.65\pm 0.02$,(for IP with trapping $D_{tr}^{sur}=1.59\pm 0.03$), compared with the bulk value of $D_{sur}=1.88\pm 0.02$ ($D_{tr}^{sur}=1.85\pm 0.02$). We ﬁnd a logarithmic crossover from surface to bulk fractal properties, as one would expect from the ﬁnite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organizes into an asymptotic shape characterized by a discontinuity at a value $x_c=0.5$, which coincides with the bulk critical threshold. The exponent $\tau^{sur}$ of the boundary avalanche distribution for IP without trapping is $\tau^{sur}=1.56\pm 0.05$; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the ﬁxed scale transformation and the run time statistics, which are particularly suited for the study of irreversible self-organized growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data.

Item Type: Article 10.1103/PhysRevE.56.R1291 PACS: 05.40.+j, 68.70.+w Q Science > QC Physics Economics and Institutional Change Ms T. Iannizzi 24 Feb 2012 13:19 27 Feb 2012 15:00 http://eprints.imtlucca.it/id/eprint/1177