Fractal distributions

The A1 AI2O3 composite grown at low temperatures (450-500 °C) and low pressure (10 -10 mbar) consists of aluminum particles (diameters ranging from 1-50 nm depending on reaction time), which are embedded in an almost amorphous AI2O3 matrix. The sizes of the particles seem to follow a fractal distribution with a fractal exponent of 2.4 [24] which we have already found for other metal/metal-oxide composites grown by similar CVD processes [22,29]. The amorphous aliuninum oxide is transformed to the crystalhne 7-AI2O3 at temperatures aroimd 550-600 °C. 

Since gravitational phenomena are conventionally considered to arise as mass-driven perturbations of a flat inertial background, the main result of the analysis—that the flat inertial background is irreducibly associated with a nontrivial fractal distribution of material—must necessarily lead to novel insights into the nature and causes of gravitational phenomena. 

We mentioned the possibility of permeability being distributed as a fractal in conjunction with Figure 10. A fractal distribution would show C decreasing moiiotonically at a rate prescribed by its factal dimension. This is not the case with the eolian outcrop which clearly shows two scales of heterogeneity. Given its correspondence with the geologic features, the two-scale interpretation is the only possible consistent interpretation. 

Mandelbrot [23] has shown that the most random type of height distribution to be expected on earth is of a fractal type. The same should be true for a value distribution in the v-dimensional sequence space. Such a fractal distribution is highly connective, that is, anything but uncorrelated. Moreover, we know that functional efficiency is clustered around certain sequences. The functional efficiency of an enzyme depends on the correct spatial arrangement of certain amino acid residues that comprise the active center this is achieved by three-dimensional folding of the polypeptide chain [24]. Hence there exists a correlation ... 

Figure 11 (a) Fractally distributed martensite phase a (dark) and... 

Two models of aggregation result in fractal distributions of uniform particles. In the Diffusion-Limited Aggregatioij ) model, particles are launched one at a time from random positions on an infinitely distant horizon and travel by random walks... 

Turcotte, D.L. and Huang, J. (1995). Fractal distributions in geology, scale invariance, and deterministic chaos. In Fractals in the Earth Sciences, Barton, C.C. and La Pointe, P.R. (eds). Plenum Press, New York, pp. 1-40. 

Similar equations arise for the transverse relaxation rate l/7 2 . The problem now is to extract the pore distribution, W , from the observed RJ,t). From a mathematical point of view, this could be done by a Laplace inversion of Eqn (28.3). Examples of this method can be found in the review paper of W. P. Halperin et al. Other methods have used a prerequisite distribution that must be verified a posteriori. This has been carried out recently for the nuclear relaxation of and of methanol and nitromethane adsorbed on an organic polymeric resin crosslinked by paramagnetic divalent metal ions (Fig. 28.2). The results have been interpreted with a fractal distribution of categories of quasi-disconnected spherical pores, each being composed of N" spherical pores of radius R = RJ 1), with = log(/ o/ n.in)/log Introducing iht fractal dimension Df through the relation N a , leads to... 

If we consider the fractal distribution fV of categories of pores given in Eqn (28.5), the attenuation of the echo may be written as the following weighted sum over the restricted and free diffusion contributions... 

The differential equation of Laplace (V c = 0) describes the formation of fractal distribution of solid matter that results from very different processes (Figure 7.5). Diffusion-limited aggregation (c is the concentration), electrogalvanic deposition (c is the electric potential), and viscous invasion (c is the local pressure) are three Laplacian processes that produce similar fractal distributions. They all imply a strong positive feedback and have the same mathematics. The first two are of significance for materials synthesis. 

Another example of a fractal surface is anodically etched p-type silicon. It has a fractal porosity and shows photoluminescence and even electroluminescence, which would be impossible in compact bulk silicon. Electrons behave differently in fractally distributed solids than in three-dimensional extended lattices they are supposed to be more localized. The density of states p in a three-dimensional box is... 

We interpret that the ribbons in the swollen pellicles consist of MFs oriented parallel to the ribbon axis as illustrated in Figure 12 (b-3). The power-law exponent si=2.35 (Figures 11 and 12(a)) reveals that mass centers of the MFs are fluctuating in the lateral direction perpendicular to the ribbon axis with the lateral mass fractal dimension Om l = si - 1 = 135. The mass fractal distribution of the MFs in the lateral direction has the upper cutoff length Ls2=2n/ ( s2 600nm and the low cutoff length Lsi = 2n/[Pg.393]

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