We show that for all Euclidean dimensions d ζ̃= d ̅ w - d ̅ f ; where L R ∼ξ ζ̃ is the effective resistance between two points separated by a distance comparable with the correlation length ξ; d ̅ f is the fractal dimension of the backbone; and d ̅ w is the fractal dimension of a random walk on the same backbone. We also find a relation between the backbone and the full percolation cluster; d ̅ w - d ̅ f = d w - d f . Thus the Alexander-Orbach conjecture ( d f / d w =2 / 3 for d > \~2) fails numerically for the backbone.
%0 Journal Article
%1 Stanley1984Flow
%A Stanley, H. Eugene
%A Coniglio, Antonio
%D 1984
%I American Physical Society
%J Physical Review B
%K backbone, porous\_media percolation critical-phenomena
%N 1
%P 522--524
%R 10.1103/physrevb.29.522
%T Flow in porous media: The "backbone" fractal at the percolation threshold
%U http://dx.doi.org/10.1103/physrevb.29.522
%V 29
%X We show that for all Euclidean dimensions d ζ̃= d ̅ w - d ̅ f ; where L R ∼ξ ζ̃ is the effective resistance between two points separated by a distance comparable with the correlation length ξ; d ̅ f is the fractal dimension of the backbone; and d ̅ w is the fractal dimension of a random walk on the same backbone. We also find a relation between the backbone and the full percolation cluster; d ̅ w - d ̅ f = d w - d f . Thus the Alexander-Orbach conjecture ( d f / d w =2 / 3 for d > \~2) fails numerically for the backbone.
@article{Stanley1984Flow,
abstract = {{We show that for all Euclidean dimensions d ζ̃= d ̅ w - d ̅ f ; where L R ∼ξ ζ̃ is the effective resistance between two points separated by a distance comparable with the correlation length ξ; d ̅ f is the fractal dimension of the backbone; and d ̅ w is the fractal dimension of a random walk on the same backbone. We also find a relation between the backbone and the full percolation cluster; d ̅ w - d ̅ f = d w - d f . Thus the Alexander-Orbach conjecture ( d f / d w =2 / 3 for d > \~{}2) fails numerically for the backbone.}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Stanley, H. Eugene and Coniglio, Antonio},
biburl = {https://www.bibsonomy.org/bibtex/2ff22dca0dd590e1d26bf7201c4cf6cfc/nonancourt},
citeulike-article-id = {4631157},
citeulike-linkout-0 = {http://dx.doi.org/10.1103/physrevb.29.522},
citeulike-linkout-1 = {http://link.aps.org/abstract/PRB/v29/i1/p522},
citeulike-linkout-2 = {http://link.aps.org/pdf/PRB/v29/i1/p522},
doi = {10.1103/physrevb.29.522},
interhash = {1d2fd60c4bb4c05667cfd43badaf0453},
intrahash = {ff22dca0dd590e1d26bf7201c4cf6cfc},
journal = {Physical Review B},
keywords = {backbone, porous\_media percolation critical-phenomena},
month = jan,
number = 1,
pages = {522--524},
posted-at = {2009-05-25 15:39:10},
priority = {2},
publisher = {American Physical Society},
timestamp = {2019-07-31T12:26:23.000+0200},
title = {{Flow in porous media: The "backbone" fractal at the percolation threshold}},
url = {http://dx.doi.org/10.1103/physrevb.29.522},
volume = 29,
year = 1984
}