Fractal conception medium

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

Use of the concept of the fractal set allows one to examine the dependence of physical properties on the behavior of hierarchical structures. Such structures appear in stochastic inhomogeneous medium. 

Equation (6.12) faces one main difficulty, namely the determination of the explicit functional form of D(s). It is important to stress that D s) does not have exactly the same properties as the classical diffusion coefficient, and we refer to it here as the conductivity. Likewise, we define the resistivity of the medium as R = l/D. We expect the resistivity to be proportional to the number of steps of the particle, and arguments from random walks on fractals should be useful to determine R. Walks on fractals are characterized by the existence of two scales. Divide the medium into small blocks of size, such that the diffusion is normal within the small blocks, f. At scales larger than f, the effect of the heterogeneities becomes important, and motion depends on the fractal parameters. The self-invariance properties of the fractal are not valid at short distances. Similarly, the idealized concept of self-similarity at all scales does not hold for fractals in practice. 

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. 

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