Fracton dimension

Another important parameter which appears in connection with dynamical properties of fractals (such as diffusion) is the spectral (fracton) dimension d. Thus, in the diffusion-limited reactions, one has to replace d in (2.1.78) by d, i.e.,... 

The above-defined df and dt are structural parameters characterizing only the geometry of a given medium. However, when we are interested in processes like diffusion or reactions in disordered media, we need functional parameters, which are associated with the notion of time in order to characterize the dynamic behavior of the species in these media. The spectral or fracton dimension ds and random-walk dimension dw are two such parameters, and they will be defined in Section 2.2. 

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. 

Relaxation rate, which looks into the low-frequency vibrational modes of protein molecules, is considered to play a significant role in biological process. A fractal-related parameter, called the fracton dimension, d-p, has been used to describe the relaxation rate at low temperature.f ... 

The overall value of fracton dimension depends on the strength of interaction between molecules in the main chain and in the cross-links. Fracton dimension would be 2 if the interaction is strong as found in the main chain. However, as the interactions in the cross-links are usually very weak, their fracton dimension would be between 1 (no connectivity) and 2 (strong connectivity). ... 

Fracton dimension was used to fit a photolysis kinetic of denatured HbNO shown below ... 

To begin, recall that, in general, spaces can be characterized by three quantities dg, the dimension of the embedding Euclidean space, df the Hausdorff or fractal dimension, and ds the spectral or fracton dimension. A key to what follows is that for Euclidean spaces, these three dimensions are equal [57,58]. 

The complexity of the polymer structure is reflected in the large number of dimensions needed to describe it. Alexander and Orbach [28] proposed the use of spectral or fracton dimension for the description of the density of states on a fractal. The necessity of introducing is due to the fact that the fractal dimension defined by Equation (11.1) does not reflect this parameter. The investigators made use of the fact that anomalous diffusion of particles is expected on a fractal and, hence ... 

The first stage of the considered treatment is the formulation of polymeric fractal dimension ZT and its spectral (fracton) dimension d, which characterizes fractal object coimectivity degree [41], intercommunication. In this case Ihe value linear polymer chain and <7 =1.33 for very branched (cross-linked) chain. Using Flory-de Gennes mean field approximation, Vilgis obtained the following equation for/recalculation [39] ... 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

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] ... 

Anglaret E., Hasmy A., Courtens E., Pelous J., Vacher R. Fracton dimension of mutually self-similar series of base-catalyzed aerogels. J. Non-Cryst. Solids 1995 186 131-136... 

Here rf, is the fracton dimension defined by d, = 2dfld in which d/ is the fractal dimension and d the diffusion exponent [34]. For the case of constant injection rate on a fractal we do not have an analytical derivation. However, we recently calculated [35] the number of distinct sites visited on a fractal by N random walkers starting from the origin, 5at(0 with 8 = d / d - 1). This result can be shown to be... 

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... 

Here, / is the intensity and n is the Bose-Einstein factor. The values d D = 0.53, d = 1.3, and a = 1 lead to the observed exponent - 0.36.25 That the bending fracton dimension is the proper one to use in (4) is consistent with the fact that it is the bending modes that depolarize the scattered light. 

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