Fractal scale-invariance

The mechanism of Self-organized criticality, a concept first introduced by Bak, Tang and Wiesenfeld [bak87a], may possibly provide a fundamental link between such temporal scale invariant phenomena and phenomena exhibiting a spatial scale invariance - familiar examples of which are given by fractal coastlines, mountain landscapes and cloud formations [mandel82],... 

Evidence of the fraction of free monomer micelles at the clotting time would help to determine the appropriate form of the reaction kernel and whether growth is limited by diffusion or by the reaction itself. The growth of polymers from polyfunctional monomers, the formation of diffusion-limiting aggregates, and many other natural phenomena can all be scale invariant fractals with a similar fractal... 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

Three-dimensional objects occupy space that may similarly be characterized by an equivalency of length times breadth times width. Fractals are such objects they are irregularly shaped and built upon a constant repeating, microscopic fine structure. A polysaccharide gel, for example, is generated from an almost infinite number of scale-invariant nuclei (Birdi, 1993) multiplied many times into tertiary and quaternary structures. Assuming Rg to be the constant dimension of the fractal nucleus, floes conform to... 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

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. 

Another method used for data analysis of non-Euclidean objects is fractals. A true fractal object is scale invariant (i.e., exhibits seU-similarity) thus a fractal dimension is obtained ITom the outline of an object by varying the scale of analysis. The ITactal dimension (FD) of an irregular geometry is a measure of the space-filling... 

Applying fractal geometry to description of disordered media allows one to use the properties of scaling invariance—that is, to introduce macroscopic... 

Agglomerates are fractal-like in a statistical sense. Equation (8.1) describes the average radius of many agglomerates with the same Np and primary particle size. Agglomerates are not true fractals, because they are not infinitely scale-invariant. The lower limit on the size of an agglomerate is the primary particle (Np = 1). Fractal concepts break down for... 

A fractal is an object that displays scale invariant symmetry that is, it looks the same when viewed at different scales. Any real fractal object will have this scale invariance over only a finite range of scales. One important consequence of this symmetry is that the density autocorrelation function will have a power law dependence, which can be written as... 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

The main conclusion is that the fractal dimension of the distribution profile of acceptors around a donor is inversely dependent on the pore size. It is also important to notice that the same D values are obtained with all three donor/acceptor pairs. We interpret these D values as reflecting the geometry of the support as seen by an adsorbed molecule, and in particular that these D values are the surface fractal dimensions for adsorption, for the following reasons (a) The fact that the D values were found to be insensitive to the different Ro values of the three pairs and to the concentrations employed, is in keeping with the scale-invariance of the fractal model, (b) In a number of studies (21,43) it has been shown that for the same material, higher... 

Fractal models for soil structure and rock fractures are becoming increasingly popular (e.g., Sahimi, 1993 Baveye et al., 1998). The primary appeal of these models is their ability to parsimoniously parameterize complex structures. Scale symmetry or scale invariance, in which an object is at least statistically the same after magnification, is a fundamental property of fractals and can also be observed in numerous natural phenomena. Thus, it is logical that some investigators have examined theoretical transport in known prefractals. 

At t — 0, w(L, 0) oc fi where the superscript (3 is the roughness growth kinetics exponent along the x direction. After a certain time, a steady-state contour is reached, w(L, oo) oc L , or in other words, a scale invariant fractal is obtained (self-affined fractal). The exponent a is the surface roughness (only when it is measured at scale lengths shorter than the interface). 

There is, however, one basic principle, which derives from the fact that fractal clusters are scale invariant. When a gel is formed, clusters of size Rg make bonds with each other via strands at the periphery of the cluster, and the average number of the strands involved per cluster will not depend on cluster size. Since the apparent surface area of a cluster scales with R2 the number of junctions between clusters per unit area of cross section of the gel will scale with R 2. By using the equation for Rg we arrive at the following equation for the shear modulus of a fractal particle gel ... 

For t > tx the surface becomes a scale-invariant self-affine fractal. The value of a is related to Dfrac, the local fractal dimension of the surface [11], as Dfrac = 3 - or. It should be noted that due to the fact that hg, is proportional to t, Eqs. (3) and (4) can be formulated indistinctly in terms of these variables depending on the available experimental data. 

The idea that disorderly growth can lead to scale invariance was first pointed out by Witten and Sander (33) who were attempting to explain earlier observations of Forrest and Witten (34) of smoke aggregates. Computer simulations (33, -M) have also suggested that the resultant structures exhibit scale invariance and can be described as fractals. Two general classes of irreversible aggregation have emerged from the simulations. 

In the models of the irreversible aggregation, general concepts for the physical processes such as scaling and classes of the universality are widely used [10]. The sense of scaling (scale invariance) consists in abstracting from the details of structure and allocation of simple universal features that are the characteristic for a wide class of systems. Frequently used scaling parameters (indices) are the fractal dimensions. The hypothesis of the universality is closely connected to the hypothesis of scaling whose essence comprises... 

According to Eq. (2), Flory coil is a fractal, that is, an object, possessing the property of the scale invariance in dimensionality space df = (d+2)i3 ... 

From the facts presented above, it is evident that the copolymer sequences discussed here are correlated throughout their whole length. Also, it was found that any sufficiently large part of the averaged sequence has practically the same correlation properties as the entire sequence. This means that the generated sequences show scale invariance, a feature typical of fractal structures. 

S. Why, they ll learn that such things as a Gaussian coil, a swollen coil, and a randomly branched polymer are all fractals. This is interesting in its own right. Mind you, there are more things in life than polymers. It might be interesting to hear about other fractals, the scale invariance of different objects, and the mathematical idea of fractional dimensionality. 

Self-affine curves resemble self-similar curves, but have weaker scale-invariant properties. Whereas self-similarity expresses the fact that the shapes would be identical under magnification, self-affinity expresses that the two dimensions of the curve may have to be scaled by different amounts for the two views to become identical (Bassingthwaighte et al., 1994). A self-affine curve may hence also be self-similar if it is not, the curve will have a local fractal dimension D when magnified a certain amount, but this fractal dimension will approach 1 as an increasing part of the curve is included. Brownian motion plotted as particle position as a function of time gives a typical example of a self-affine curve. 

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