Fractals, self-similar random

Since diffusing species move randomly in all directions, the diffusing species may sense the self-affine fractal surface and the self-similar fractal surface in quite different ways. Nevertheless a little attention has been paid to diffusion towards self-affine fractal electrodes. Only a few researchers have realized this problem Borosy et al.148 reported that diffusion towards self-affine fractal surface leads to the conventional Cottrell relation rather than the generalized Cottrell relation, and Kant149,150 discussed the anomalous current transient behavior of the self-affine fractal surface in terms of power spectral density of the surface. 

It should be stressed here that the specific power dependences from the above self-affine fractal interfaces are maintained even during the relatively long time (or number of random jumps) interval. This implies that the morphology of the self-affine fractal interfaces tested is possibly characterized by the self-similar fractal dimension within a relatively wide spatial cutoff range. 

Bearing in mind that diffusing ions move randomly in all directions, it is reasonable to say that the diffusing ions sense selfsimilar scaling property of the electrode surface irrespective of whether the fractal surface has self-similar scaling property or self-affine scaling property. Therefore, it is experimentally justified that the fractal dimension of the self-affine fractal surface determined by using the diffusion-limited electrochemical technique represents the apparent self-similar fractal dimension.43... 

In media of fractal structure, non-integer d values have been found (Dewey, 1992). However, it should be emphasized that a good fit of donor fluorescence decay curves with a stretched exponential leading to non-integer d values have been in some cases improperly interpreted in terms of fractal structure. An apparent fractal dimension may not be due to an actual self-similar structure, but to the effect of restricted geometries (see Section 9.3.3). Another cause of non-integer values is a non-random distribution of acceptors. 

Simulation of structure formation on a lattice [7,100] demonstrated that randomly formed branched clusters also fulfill self-similarity conditions and gave fractal dimensions of [7,104,105] ... 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

Pouzot et al. (2004) reported applicability of the fractal model to )6-lactoglobulin gels prepared by heating at 80 C and pH 7 and O.IM NaCl. They suggested that the gels may be considered as collections of randomly close packed blobs with a self-similar structure characterized by a fractal dimension Z)f 2.0 0.1. 

A fractal that is not self-similar is constructed as follows. A square region is divided into nine equal squares, and then one of the small squares is selected at random and discarded. Then the process is repeated on each of the eight remaining small squares, and so on. What is the box dimension of the limiting set ... 

In general, self-similar fractal surfaces do not exist in the real world. The fractal models may only approximate random surfaces. In addition, Eq. (183) for ( ) = 0.5 is formally identical with the semi-infinite porous model presented in the next section. The fractal model in the presence of diffusion is discussed in Refs. Ill and 118. Experimental verifications of the fractal model were also carried out for some electrodes. It was... 

Note It is interesting to compare the discussion in this section with that of Section 6.2.1 on the conformation of random-coil linear polymers. Also in that case a larger molecule, i.e., one consisting of a higher effective number of chain elements is more tenuous. Equation (6.4) reads rm = b(n )v, where rm may be considered proportional to the parameter R in Eq. (13.12) b then would correspond to a in (13.12), and to Np. For a polymer molecule conformation that follows a self-avoiding random walk, the exponent v is equal to 0.6. Rewriting of Eq. (6.4) then leads to n = (rrn/b)1 67, which is very similar to Eq. (13.12) with a fractal dimensionality of 1.67. Depending on conditions, the exponent can vary between about 1.6 and 2.1. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

The authors [4, 5] considered random walks without self-intersections or self-avoiding walks (SAW) as the critical phenomenon. Monte-Carlo s method within the framework of computer simulation confirmed SAW or polymer chain self-similarity, which is an obligatory condition of its fac-tuality. Besides, in Ref [4], the main relationship for polymeric fractals at their treatment within the framework of Flory conception was obtained ... 

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. 

Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are quasi or statistically self-similar over a finite length scale that is most often determined by the characterization technique that is employed. An object or process can be classified as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data ... 

An ideal or regular fractal structure exhibits self-similarity over all characterization length scales, i.e. the structure can be decomposed into smaller copies of itself, so that when any portion of the structure is magnified it will appear identical to a larger part. Since natural structures tend to be self-similar over only a finite range of length scales, they are most often referred to as random fractals. 

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