Percolation spectral dimension

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. 

As it has been shown in Ref. [22], at chemical reactions on ftactal objects study corrections on small clusters in system availability are necessary. Just such corrections require using in theoretieal estimations not generally accepted spectral dimension [23], but its effective value application. For percolation systems two eases are possible [24] ... 

Another kind of Flory-type formula is suggested in [30], where it was argued that the spectral dimension dg of the fractal percolation cluster must be an intrinsic property ... 

The reactions of deposition or crystal growth are surface reactions. The reactants are adsorbed, more or less mobile molecules, e.g., A and in the fictitious reaction A -h B 0. These adsorbates form the substrate surface and growth is the annihilation reaction between the adsorbed reactants. The reaction rate r is expressed as usual (Chapter 6) in the reactant concentrations as r = k[ A][ B]. This can be done if the surface (the reaction space) can be considered to be a well-stirred reactor. In other words, the mobilities of A and B are high compared to the rate of the growth reaction. If that is no longer true and there is diffusion limitation the reaction can still be fitted to the above rate equation except that the reaction rate coefficient k is replaced by kit in which h — i —jS (with S being the spectral dimension). A characteristic value for his for the case of a reaction on a percolation cluster with a spectral dimension of... 

Referring to the previous relations in the high-dimensional case, we conclude that = 2 is a critical spectral dimension beyond which S N) always increases linearly with t after all, we cannot visit more than N sites in N steps. For example, in the case of diffusion over the structure of a Brownian motion, itself contained in a plane d = 2) or a 3-dimensional space d = S), Df = 2 and Dyf = 4, and we find that S N) For diffusion over a percolating cluster in the plane d = 2) at the threshold, we find that D = 1.89, Dy, = 2.87, and S N) But for diffusion in a plane, with D( = d = 2 and Dy, = 2, S N) t and for diffusion in a 3-dimensional space, with D = d = S, the result is still true. 

The previous remarks imply that for static purposes, a polymer sol can always be thought of as a set of interacting quenched clusters with the appropriate C simple cases, the relevant interaction is excluded volume. Let us take the percolation model in d = 3 as an example, and try to calculate the fractal dimension D 2.5 of the interacting clusters, using only known information about C As well as the size distribution exponent r 2.2, much is known about the cluster connectivities in particular, the clusters have spectral dimension... 

In the percolation problem one has for the spectral dimension approximately, d = 4/3, independent of the space dimensions according to the Alexander and Orbach conjecture. It is also the mean field value for lattice animals (branched structures defined on lattices) or Cayley tree-like structures. Hence the Cayley tree corresponds to the mean field solution to percolation. ... 

Alexander and Orbach (1) have observed, from various simulations, that the spectral dimension is always close to 4/3 in the Ccise of percolation with a space dimension, d, between 1 and 6. For z = 1,94, the value ds = 1.35 was found. 

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] ... 

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