Fractional dimension

The chemical reactivity of a self-similar surface should vary with its fractional dimension. Consider a reactive molecule that is approaching a surface to make a hit. Taking Fig. VII-6d as an illustration, it is evident that such a molecule can see only a fraction of the surface. The rate of dissolving of quartz in HF, for example, is proportional to where Dr, the reactive... 

The FFPE (19) contains the generalized friction constant r a and the generalized diffusion constant Ka, of dimensions [r]a] = s -2 and [i a] = cm2 s . The physical origin of these fractional dimensions will be explained in the next section. In what follows, we assume natural boundary conditions, that is, lim i-nx) W(x, t) = 0. The FFPE (19) describes a physical problem, where the system is prepared at to = 0 in the state W(x, 0). 

The Space Fractional Dimension the Cutoff Time of the Scaling in the Time Domain t0, the Characteristic Frequency co5, and Estimated Self-Diffusion Coefficient for the Polymer-Water Mixtures ... 

The average length of a nylon-6,6 polymer chain is about 50-100 pm (each polymer chain contains about 105 groups while the length of a polymer group rg is about 10 A). This length is comparable to the thickness of a sample 120-140 pm [275]. Thus, the movement of the chains is most likely occurring in the plane of the sample. This fact correlates with the values of the space fractional dimension da- For all of the samples d( C (1,2) (see Table IV). Thereby, the Euclidean dimension of the space in which chain movement occurs is dE = 2. 

Formation of platinum fractal-like structures is possible by PA-CVD (19% Ar, 80% O2, 1% SnMe4) on tin oxide thin films using Pt(acac)2 as starting material . The platinum aggregates show a dendritic structure of fractional dimension D ca 1.1-1.6 (Figure 10). The occurrence of such aggregates has been correlated to the concentration of the platinum precursor and to the radio-frequency power applied to the substrate electrode. Fabrication of microsensors integrated on silicon wafers with the help of photoresistors is possible . 

Mandelbrot introduced a new geometry in a book, which was first published in French in 1975, with a revised English edition [64] in 1977. In 1983 he published an extended and revised edition that he considered to be the definitive text [65]. Essentially he stated that there are regions between a straight line that has a dimension of 1, a surface that has a dimension of 2 and a volume that has a dimension of 3, and these regions have fractional dimensions between these integer limits. Kaye has presented an excellent review of the importance of fractal geometry in particle characterization [66]. 

For an ordinary (Euclidian) object, the exponent, Df, is equal to the dimension of the space in which the object exists. For example, in three dimensions, the mass of a sphere scales as its radius to the third power. In contrast, for a fractal object, the exponent Df is not equal to the dimension of space and, in general, is not an integer. However, Df is analogous to the dimension in the equation for Euclidian objects and is called the fractal (from fractional) dimension. 

Surface fractional dimensions of the adsorbents from industrial sludge... 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

On the other hand, it follows from the Poincare-Bendixon theorem (see Appendix A2.2) that an attractor cannot have a dimension equal to two (plane), one (line) or zero (point), since a chaotic nature of the limit set is then impossible (only stationary points and limit sets are then possible). Hence, a conclusion follows, confirmed by other methods, that the attractor of the system of equations (5.14) has a fractional dimension (more exactly, 2 + D, where D is a small positive number), see Appendix A2.7. 

As we have concluded in Chapter 5, in the case of the Lorenz system a trajectory always remains within a confined region of the phase space, being non-periodical. For t - oo, the trajectory approaches a certain limit set the Lorenz attractor. It follows from the Liouville theorem that the Lorenz attractor has a zero volume (since divF < 0). This implies that, apparently, the Lorenz attractor is a point (dimension zero), a line (dimension one), or a plane (dimension two). Then, however, the trajectory for t -> oo would have remained within a confined region on the plane and, by virtue of the Poincare-Bendixon theorem. Hence, a conclusion follows that the Lorenz attractor has a fractional dimension, larger than two. 

From the above analysis follows the necessity of defining sets of fractional dimension. Commonly, such sets are the fractal sets (Greek fractus — broken) introduced to physics by Mandelbrot. We will confine ourselves... 

The proportion of fluid elements experiencing a particular anomalous value of the Lyapunov exponent A / A°° decreases in time as exp(—G(X)t). In the infinite-time limit, in agreement with the Os-eledec theorem, they are limited to regions of zero measure that occupy zero volume (or area in two dimensions), but with a complicated geometrical structure of fractal character, to which one can associate a non-integer fractional dimension. Despite their rarity, we will see that the presence of these sets of untypical Lyapunov exponents may have consequences on measurable quantities. Thus we proceed to provide some characterization for their geometry. 

C curve, the topological dimension is equal to one and the Hausdorff-Besicovic dimension is equal to two and that is why it is considered fractal, despite the fact that neither of these dimensions is itself fractional. In most cases, however, a fractal has a fractional dimension. 

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

Kopelman and co-workers [10] also measured as a function of temperature. At low temperatures, all the polymers studied exhibit properties similar to the properties of fractals. As the temperature increases, h decreases, i.e., the d value increases. In some specimens, h 0 as the temperature is raised. This implies that all the effects described by fractional dimensions are associated with disorder [85]. A number of specimens also behave as fractals at room temperature. It is noteworthy that the d values for the polymers studied vary over wide limits, from 0.8 to 1.8. In the case of PMMA, d exactly corresponds to the spectral dimension determined by Raman scattering measurements [22, 35] it is 1.8 in both cases. 

As it is known through Ref [44], for the fiactal objects characterization as distinct from Euclidean (compact) ones the usage of three dimension, as minimum, is required—d, Z),and spectral (fraction) dimension d, which char-... 

S. Exactly Remember Goethe s comment Mathematicians are like the French... that we put at the start of Chapter 6 But, to be serious, what I really want to emphasize is this. First, it is not just interesting — it is often useful to master different ways of describing the same thing. (Never mind that they are mathematically identical ) This is exactly what Richard Feynmann illustrated for the law of gravity, in his wonderful book. Character of Physical Law [38]. Second, there are loads of examples in Physics where new achievements (and sometimes rather exciting ones ) were not expressed in the language of fractals, yet were very closely connected with them, due to the use of power laws and fractional dimensions. 

At describing the viscosity properties of diluted solution one usually proceeds from the linear dependence of an increment in viscosity on the polymer solution concentration. However, in the case of polar polymers to which CHT belongs there is a possibility of the occurrence of reversible agglomeration process which can take place not only in the area of semi-diluted solutions but even in the area of diluted ones. In this case the contribution to viscosity is made not by separate particles with V volume but by their aggregates whose volume V(n) depends not only on the number of particles constituting it, but also on their density characterized by fraction dimensions D [3] ... 

One quantitative measure of the structure of such objects is their fractal dimension D. Mathematicians calculate the dimension of fractal to quantify how it fills space. The familiar concept of dimensions applies to the object of classical or Euclidian geometry. Fractals have non-integer (fractional) dimensions whereas a smooth Euclidean line precisely fills a one-dimensional space. A fractal line spills over a two-dimensional space. Figure 13.2 shows subjects with increasing fractal dimension. 

In 1919, German mathematician Felix Hausdorff developed the concept of fractional dimension, a measure theory originated by Greek mathematician Constantin Caratheodory in 1914. Russian mathematician Abram Samoilovitch Besicovitch developed the idea of fractional dimension between 1929 and 1934. Taken together, the concepts dehned by Hausdorff and Besicovitch were used by Mandelbrot to... 

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