Property of a fractal

In the real world, however, the objects we see in nature and the traditional geometric shapes do not bear much resemblance to one another. Mandelbrot [2] was the first to model this irregularity mathematically clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Mandelbrot coined the word fractal for structures in space and processes in time that cannot be characterized by a single spatial or temporal scale. In fact, the fractal objects and processes in time have multiscale properties, i.e., they continue to exhibit detailed structure over a large range of scales. Consequently, the value of a property of a fractal object or process depends on the spatial or temporal characteristic scale measurement (ruler size) used. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

Fractals can be considered as disordered systems with a non-integral dimension, called the fractal dimension. An important property of fractal objects is that they are self-similar, independent of scale. This means that if part of them is cutout, and then this part is magnified, the resulting object will look exactly the same as the original one. The other distinct property of a fractal is the power law or scaling behavior, where the property and variable of a system are related in the following manner ... 

On scales n < 5 (Fig. 43) the composite exhibits the properties of a fractal object with the characteristic power law dependence of such properties (resistivity p and Hall coefficient R) on the scale ... 

We start again with the basic property of a fractal surface, as given in eq. [3a]... 

The spectral dimension d is a true property of a fractal and is determined only by its connectivity. It differs from the mass scaling index [see Equation (11.1)] or fractal dimension df and from the scaling index of the diffusion constant 8 by the fact that it does not depend on the way in which a fractal has been inserted into the Euclidean space with dimension d. The dependence of d and 8 on df is described by Equation (11.3). 

As far as transport properties of a fractal structure are concerned, the mean square displacement (MSD) of a particle follows a power law, (r ) where r is the distance from the origin of the random walk and is known as the random walk dimension. In other words, diffusion on fractals is anomalous, see Sect. 2.3. Recall that for normal diffusion in three-dimensional space the MSD is given by (r ) = 6Dt. For fractals, dy, > 2, and the exponent of t in the MSD is smaller than 1. We introduce a dimensionless distance by dividing r by the typical diffusive... 

This section attempts to examine macromolecular geometry, and in particular dendritic surface characteristics, from the perspectives of self-similarity and surface irregularity, or complexity, which are fundamental properties of basic fractal objects. It is further suggested that analyses of dendritic surface fractality can lead to a greater understanding of molecule/solvent/dendrimer interactions based on analogous examinations of other materials (e.g., porous silica and chemically reactive surfaces such as found in heterogeneous catalysts). 52 ... 

Panella and Krim suggested that the high value n = 4.7 was more consistent with the properties of a self-affine surface rather than a self-similar one. Self-affine fractals are associated with asymmetric scaling, that is different scaling relations in different directions (Avnir, 1997). 

The properties of fractal ceramic powders depend on their fractal dimensions. The density, p, of a fractal particle depends on its radius, R, and the fractal dimension ... 

The textural properties of a fat are influenced by all levels of structure, particularly microstructure. The microstmcture includes the spatial distribution of mass, particle size, interparticle separation distance, particle shape, and interparticle interaction forces (49-51). Methods that can be used for the characterization of microstructure in fat systems include, among others, small deformation rheology and polarized light microscopy, employing a fractal approach (49-51). 

The power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

There are a number of different fractal dimensions commonly used to describe a specific property of a system. These fractal dimensions and methods to obtain them are explained in detail in Boundary and Surface Fractal Dimensions and Mass Fractal Dimension. Fractal dimensions used in specific applications will be shown also in the related section. 

The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin-Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefficient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin-Strikman medium. 

The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. 

Thus, a knowledge of the function Y(Iq,Pq) is crucial for determining the properties of the fractal model. For small initial lattices this function can be calculated exactly. The results for square generating cells of Iq = 2, 3,4 as well for the more general case of rectangular generating cells are given in the Appendix. 

In the following section, the detailed analysis of effective Hall properties of a 3D two-component composite will be carried out based on the fractal structure model and the iterative averaging method. 

Fractal Properties. Figure 43 shows the dependence of the effective Hall properties of a composite on the scale (the number of iterations which according to Eq. (196) is equal to n = — 1) near the percolation thresholdpc. Note that... 

Apparently from the plots of log10 K (Fig. 47a) and log10 p (Fig. 47b) of the fractal ensemble versus the iteration step, number n, all these elastic properties behave like fractals before an eventual levelling off. The latter is obviously associated with the upper limit of fractal-like asymptotics, above which the elastic properties of a system are no longer p dependent on the scale—that is, on the iteration number (the loss of the self-similarity property occurs at iteration step n q = logc/log/o which defines the correlation length c at the given concentration, p). 

We have also studied the elastic properties of a nonuniform medium with chaotic structure in which one phase has a negative shear modulus. The analysis may be made using the fractal hierarchical structure model. 

The surface property of a solid is characterized by the nature of the surface boundary. The surface boimdary is expected to be related to the underlying geometric nature of the surface, hence its fractal dimension. Many properties of the solid depend on the scaling behavior of the entire solid and of the pore space. The distribution of mass in the porous solid and the distribution of pore space may also reflect the fractal nature of the surface. If the mass and the surface scale are alike, that is, have the same power-law relationship between the radius of a particle and its mass, then the system is referred to as a mass fractal. In a similar manner if the pore volume of porous material has the same power-law relationship between the pore volume and radius as that of the surface, then it is described as a pore fractal. 

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