Generalized fractal dimensions

The three dimensions introduced above, Dp, Di and Dq, are actually three members of an (uncountably infinite) hierarchy of generalized fractal dimensions introduced by Heiitschel and Procaccia [hent83]. The hierarchy is defined by generalizing the information function /(e) (equation 4.88) used in defining Dj to the i/ -order Renyi information function, Io e) - 

It can be shown that Di, is a non-decreasing function of V [hent83] i.e. that 

The table below gives a few selected values of for the logistic and Henon map attractors  

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be 

It is easy to see that K = 0 for regular trajectories, while completely random motion yields K = 00. Deterministic chaotic motion, on the other hand, results in K being both finite and positive. 

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The adsorbed layer is thicker and bound less strongly for the real chain (since for weak adsorption 0 < < 1) because it pays a higher confinement penalty than the ideal chain. The excluded volume interaction of real chains make them more difficult to compress or adsorb than ideal chains. These scaling calculations can be generalized to adsorption of a polymer with general fractal dimension Ifir. 

The adsorbed layer thickness for a polymer with general fractal dimension Xfu is derived in Problem 3.18 ... 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Although it is not the only such measure, the fractal dimension docs quantify the intuitive belief that the Cantor set is somewhere in-between a point and a line. We will consider generalizations of fractal needed in later chapters,... 

Measure Entropy In the same way as the information dimension, Dp generalizes the fractal dimension. Dp, of an attractor. 4, by taking into account the relative frequency with which the individual e-boxes of a partition are visited by points on the attractor, so too the finite set entropy generalizes to a finite measure entropy,... 

Generalized Renyi Entropies and Dimensions A hierarchy of generalized entropies and dimensions, SQ B,t), s[ B,t), S B,t),. .. - analogous to the hierarchy of fractal dimensions, Dq, Di,. .., introduced earlier in equation 4.94 for continuous systems may also be defined ... 

However, more sophisticated methods may be used to attain a good feel for the shape factor, namely its fractal dimension. This is most conveniently carried out by use of imaging techniques. The general principle of this is shown in Figs. 11 and 12. 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

Van der Waals forces between solid/gas interactions and the liquid/gas surface tension forces represent the limiting cases, but in general both the forces competitively affect the adsorption process. Therefore, in determining the surface fractal dimension of the carbon specimen, it is very important to use appropriate relation between C and dFSF. According to Ismail and Pfeifer,111... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

Therefore, the estimation 0 im problem brings to the question of fractal dimension Df determination. At present two methods of indicated dimension determination one exist. First method consists of using of chemical reactions fractal kinetics general relationship [9] ... 

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. 

The power-law relation in Eq. 6.1 can be interpreted physically as indicative of a cluster fractal.12 The exponent D is then termed the cluster fractal dimension. Some basic concepts about cluster fractals are introduced in Special Topic 3 at the end of this chapter. Suffice it to say here that Eq. 6.1 can be pictured as a generalization of the geometric relation between the number of primary particles in a cluster that is d-dimensional (d = 1, 2, or 3) and the d-dimensional size of the cluster. For example, if a cluster is one-dimensional (d = 1), it can be portrayed as a straight chain of, say, circular primary particles of diameter L0. The number of particles in a chain of length L is... 

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

In general, in order to include more types of porous media the random fractal model can be considered [2,154,216]. Randomness can be introduced in the fractal model of a porous medium by the assumption that the ratio of the scaling parameters c X/A is random in the interval [c0,1 ], but the fractal dimension I) in this interval is a determined constant. Hence, after statistical averaging, (66) reads as follows ... 

Note that in our approximation, due to the randomized character of the fractal medium the average porosity of the disordered porous glasses determined by (70) depends only on the fractal dimension Dr and does not exhibit any scaling behavior. In general, the magnitude of the fractal dimension may also depend on the length scale of a measurement extending from X to over... 

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