Percolation fracton dimensions

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

Ardyralds [12] showed that at the study of ohemieal reactions on fractal objects the cor-rections on small clusters availability in the system were necessary. Just such corrections require the usage in theoretical estimations not generally accepted spectral (fracton) dimension ds [13], but its effective value. For percolation system two cases are possible [12] ... 

FRACTON DIMENSIONS FOR ELASTIC AND ANTIFERROMAGNETIC PERCOLATING NETWORKS... 

This article presents the evaluation of the fracton dimensions via the calculations of the densities of states (DOS) for both percolating elastic and antiferromagnetic networks d = 2-4). The latter belongs to a different universality class that for scalar elasticity. We claim the fracton dimension Saf for antiferromagnetic fractons to be very close to unity independent of the Euclidean dimension d,... 

DfIzAF and Df the fractal dimension of the percolating network, respectively. Prom eq. (2), the fracton dimension dAF of antiferromagnetic fractons is given by... 

Anomalous subdiffusion occurs on percolation clusters or on objects that in a statistical sense can be described as fractal, by which we mean that selfsimilarity describes simply the scaling of mass with length. Connections between v, the fractal dimension of the cluster, D, and the spectral dimension, d, have been established, relations that were originally derived by Alexander and Orbach [35], who developed a theory of vibrational excitations on fractal objects which they called fractons. An elegant scaling argument by Rammal and Toulouse [140] also leads to these relations, and we summarize their results. 

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