Sierpinski gasket

Fig. 6.4. The Sierpinski gasket on the 3rd stage. The open and filled circles give the positions of sites for type a and type b gaskets, respectively.

In this Section following [9], we analyse the A + B —> 0 reaction with immobile reactants on the so-called Sierpinski gasket described below. We will proceed to show that in this case equation (6.1.1) with a = d/2 transforms into... 

In simulations [9] Sierpinski gaskets on the 12th stage, containing 177147 or 265722 sites, were used respectively. The number No of randomly distributed A or B particles was 10 percent of the total number of sites. The random mutual annihilation of dissimilar particles was simulated through a minimal process method [10] from all AB pairs at each reaction step one pair was selected randomly, according to its reaction rate (3.1.2) the time... 

Figure 6.5 displays a typical distribution of A and B particles at t = 100 for a realization of the annihilation process on the Sierpinski gasket of the first kind at the 9th stage. The segregation of dissimilar particles, resulting from initial concentration fluctuations, is clearly visible at this reaction stage. 

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has... 

In order to determine the constant 7, we computed the radial distribution functions for the two types of the Sierpinski gaskets under consideration. In Fig. 6.6 these functions are plotted, as averaged over all sites of the finite gaskets at the 11th stage. Due to the finite size of the structures, deviations... 

Fig. 6.6. Radial distribution functions rdf(r) for the two types of the Sierpinski gaskets a and b (dots) and ideal rdfs (solid curves) with d = 1.58 and 7 = 3.65 (a) or 7 = 5.2 (b) (see equation (6.1.30)). Note, that both axes are logarithmic.

We performed numerical simulations of the annihilation process for the Sierpinski gaskets up to the 12th stage, starting with 10 percent of all sites randomly filled with A particles and another 10 percent with B particles. We chose the interaction range ro = 4.27 (in units of the nearest-neighbor distance ao, which leads to the dimensionless initial reactant concentration n(0) = 1 (in units of r d, d = 1.585). Note, that time is still measured in units of... 

Fig. 6.7. Decay of reactant concentration obtained from direct computer simulations of the reaction on the Sierpinski gaskets of type a and b (curve a and b, respectively), from the numerical evaluation of equations (5.1.14) to (5.1.16) (curve c) and from the lower-level approximations, neglecting correlations between similar particles (X(r,t) = 1, curve d) or neglecting all spatial correlations (X(r,t) = Y(r, t) = 1, curve e).

Sometimes, as in the case of particle segregation on fractals (e.g., the planar Sierpinski gasket discussed in Section 6.1) this effect indeed is self-evident [88-90]. Its analytical treatment for particle accumulation was presented in [91, 92] we reproduce here simple mesoscopic estimates following these papers. Particle concentrations obey the kinetic equations... 

From the scaling properties of G(x, t) one can derive that S = const(d, 0)T)ft/d with d = 2dj(2 + 9) the spectral dimension of the fractal. The growth of the cluster s sizes goes on until l L where L is the whole system s size. The further growth of clusters and accumulation of particles stop because the same quantity L is the characteristic scale of a pair of different particles created in the system according to [91] there is no accumulation effect when particles are created by pairs on fractals of the Sierpinski gasket type. 

Now we can turn to the numerical simulations carried out in [88, 89] for the Sierpinski gasket having d- 1.59 and d = 1.36. The value of 6 can be estimated under the assumption that the expression (r2) = 8t2Z2+e) holds up to the time of a single step. As the particle takes one step of unit length per unit time we can conlude that 5 = 1. The total concentration of particles is... 

Similarly, the first iteration in the generation of the Sierpinski gasket, (Figure 1.2 A) involves the reduction of the scale by a factor r = 2 and results in 3... 

Another well-known fract,al is the Sierpinsky gasket (Fig. 2.10). The Sierpinsky gasket consists of three congruent pieces. Magnified by 2 they are identical with the whole fractal. Therefore, the dimension of the Sierpinsky gasket is d = ln(3)/ln(2) 1.59. 

Another example of a regular fractal is a Sierpinski gasket shown in Fig. 1.13. Start with a filled equilateral triangle [Fig. 1.13(a)], draw the... 

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