Fractals square lattice

From this relationship, we obtain A = 1/3 since the value of ds is 4/3 for A + A reactions taking place in random fractals in all embedded Euclidean dimensions [9, 19]. It is also interesting to note that A = 1/2 for an A + B reaction in a square lattice for very long times [12]. Thus, it is now clear from theory, computer simulation, and experiment that elementary chemical kinetics are quite different when reactions are diffusion limited, dimensionally restricted, or occur on fractal surfaces [9,11,20-22]. 

Figure 4.11 A percolation fractal embedded on a 2-dimensional square lattice of size 50 x 50. Cyclic boundary conditions were used. We observe, especially on the boundaries, that there are some small isolated clusters, but these are not isolated since they are actually part of the largest cluster because of the cyclic boundary conditions. Exits (release sites) are marked in dark gray, while all lighter grey areas are blocked areas. Reprinted from [87] with permission from American Institute of Physics.

By using the above-described algorithms to generate fractal sets, fractal sets constructed on square lattices have been obtained [24]. 

Stauffer and Aharony [1] have studied chaotic fractal ensembles on square lattices where all bonds were identically colored at the initial stage and later... 

Consider now fractal ensembles grown on rectangular subsets of the square lattice, lx x ly(lx / ly), further referred to as rectangular generating cells. Hence, the characteristic length of the system, Iq, can be chosen in various ways. For lx> 1(4 1 is trivial) the simplest and most natural choice is Iq = lx. As can... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

In the case of fractal substrates, one has to distinguish between two main subclasses of structures, namely deterministic and random fractals. Within the class of deterministic fractals, one additionally has a subdivision in finitely and infinitely ramified fractals. Here, (either finite or infinite) ramification refers to the number of cut operations which are required to disconnect any given subset of the structure, the upper limit of which is independent of the chosen subset [7,8]. An example of a finitely ramified structure is the Sierpinski triangular lattice, whereas the Sierpinski square lattice is an example of an infinitely ramified structure. See Figs. 2(a) and 6 in Section 4 for the respective sketches of these structures in d = 2. 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

Renormalisation group (RG) techniques have been applied to several finitely ramified structures, so that results are available for some deterministic fractals including Sierpinski triangular lattices [40-47] (for a comprehensive discussion see Ref. [48]). For infinitely ramified structures, there is no RG result available and one has to rely on numerically evaluating SAWs on these fractals (note, however, the study of Taguchi [49] of SAWs on Sierpinski square lattices). Nonetheless, even in the former case when RG results are available, it is instructive to apply munerical schemes as mentioned in the Introduction. 

SAWs on Sierpinski square lattices do depend on both n,k) and the lacunarity (for instance, the exponent s is known to increase for increasing lacunarity [49,55]). Also here, the discussion below follows the majority in the literature and is restricted to the most symmetric configuration with the smallest lacunarity, see Fig. 6. Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance r, so that there is only one mass fractal dimension ds, M r . ... 

Figure 1. Fractal dimension of clusters on square lattice. The uppermost arrow shows maximal dimension d = D = 2. The percolation threshold point is shown by the pair of arrows below the upper right corner of the diagram. The lattice animal limit is marked by the o symbol, the dashed line marks the (uncertain) course of the 0 dependent dimensionality at /3 < 0. The two arrows along the left vertical axis mark the vedues of self-avoiding random walk at d = 4/3 and the limiting value of the dimensionality (d = 1) of straight clusters in the limit /3 — —oo.

The behaviour of surface reaction is strongly influenced by structural variations of the surface on which the reaction takes place [23], Normally theoretical models and computer simulations for the study of surface reaction systems deal with perfect lattices such as the square or the triangular lattice. However, it has been shown that fractal-like structures give much better description of a real surface [24], In this Section we want to study the system (9.1.39) to (9.1.42). 

Thus, a knowledge of the function Y(Iq,Pq) is crucial for determining the properties of the fractal model. For small initial lattices this function can be calculated exactly. The results for square generating cells of Iq = 2, 3,4 as well for the more general case of rectangular generating cells are given in the Appendix. 

This generalized des Cloizeaux relation is in very good agreement with numerically ob-tmned values [74], The second term in Eq. (43) has its origin in the (self-similar) disordered nature of the backbone of critical percolation clusters and is expected to be absent on deterministic fractals such as the Sierpinski lattice. EE results support this conclusion as shown in Section 4 for both triangular and square Sierpinski lattices. 

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