Site percolation

Voting rule systems approaching their final state through percolation display much of this same behavior. There is a critical initial density, pc, such that for p > Pc, a connected network of cr = 1 valued sites percolates through the lattice. If p < pc, on the other hand, a similar a 0 valued lattice-spanning structure percolates through the lattice. In either case, the set of sites with the minority value consists of a disconnected sea of isolated islands, and a finite number of islands persist to the system s final state as long as the initial density p > 0. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

The spatial temperature distribution established under steady-state conditions is the result both of thermal conduction in the fluid and in the matrix material and of convective flow. Figure 2. 9.10, top row, shows temperature maps representing this combined effect in a random-site percolation cluster. The convection rolls distorted by the flow obstacles in the model object are represented by the velocity maps in Figure 2.9.10. All experimental data (left column) were recorded with the NMR methods described above, and compare well with the simulated data obtained with the aid of the FLUENT 5.5.1 [40] software package (right-hand column). Details both of the experimental set-up and the numerical simulations can be found in Ref. [8], The spatial resolution is limited by the same restrictions associated with spin... 

Fig. 2.9.10 Maps of the temperature and of the experimental data. The right-hand column convection flow velocity in a convection cell in refers to numerical simulations and is marked Rayleigh-Benard configuration (compare with with an index 2. The plots in the first row, (al) Figure 2.9.9). The medium consisted of a and (a2), are temperature maps. All other random-site percolation object of porosity maps refer to flow velocities induced by p = 0.7 filled with ethylene glycol (temperature thermal convection velocity components vx maps) or silicon oil (velocity maps). The left- (bl) and (b2) and vy (cl) and (c2), and the hand column marked with an index 1 represents velocity magnitude (dl) and (d2).

Resulting maps of the current density in a random-site percolation cluster both of the experiments and simulations are represented by Figure 2.9.13(b2) and (bl), respectively. The transport patterns compare very well. It is also possible to study hydrodynamic flow patterns in the same model objects. Corresponding velocity maps are shown in Figure 2.9.13(d) and (c2). In spite of the similarity of the... 

Fig. 2.9.13 Qu asi two-dimensional random ofthe percolation model object, (bl) Simulated site percolation cluster with a nominal porosity map of the current density magnitude relative p = 0.65. The left-hand column refers to simu- to the maximum value, j/jmaK. (b2) Expedited data and the right-hand column shows mental current density map. (cl) Simulated NMR experiments in this sample-spanning map of the velocity magnitude relative to the cluster (6x6 cm2), (al) Computer model maximum value, v/vmax. (c2) Experimental (template) for the fabrication ofthe percolation velocity map. The potential and pressure object. (a2) Proton spin density map of an gradients are aligned along the y axis, electrolyte (water + salt) filling the pore space...

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

If S is smaller than the site percolation threshold for the square lattice Sc = 0.59275 we obtain a system which consists only of finite clusters. In principle these clusters can be completely occupied by one-kind species. For the case that no desorption is allowed this represents a poisoned state for which the production rate Rco2 goes to zero as t —> oo. Then the whole system consists of finite clusters poisoned by particles A or B. For this state the condition Ca + Cb = S holds where C is the density of particles of type A (C a + Cb + Co = S). 

Carlo Tests of the Universality in a Correlated-Site Percolation Problem. 

Richard Zallen, The Physics of Amorphous Materials (New York Wiley-Interscience, 1998), chap. 4. The percolation model distinguishes between bond percolation, in which a pairwise connection exists between sites with bonds either connected or unconnected (percolation threshold is reached at 18 percent), and site percolation, in which all sites are connected and the sites, rather than the bonds between sites, establish the character of connectivity (percolation threshold is reached at 15 percent). If the analogy holds, and the number of connected bonds is the critical component for women in an academic department, then the critical mass threshold is reached at 18 percent. 

The transition to a model of random bonds neglects the correlations imposed by construction between passing bonds around an A site, as is shown distinctly in Fig. 4.15, where two lattices satisfying the equivalence (4.83) are compared one lattice with random bond distribution, and one lattice with random site suppressions. Aggregation of bonds is easily discernable in the case of site percolation. However, as a matter of fact, these correlations have no importance in the case of conductivity, so that we may obtain a good approximation when leaving them out.178 It will be shown below that this approximation is questionable for more local properties where the microscopic arrangements of bonds may be crucial. 

From now on, we wish, in the spirit of the site percolation in electrokinetics (Section IV.C.4.a), to neglect the bond correlations. Thus, we consider an effective medium around the energy vA where the excitation will propagate it is clear that this medium correctly describes the propagation, but that it will not correctly describe, for example, the density-of-states distribution, since it contains also fictitious B sites at the energy vA. Therefore, by means of this restriction, the HCPA method is then directly transferable to the naphthalene triplet lattice, with probability cL = cA of having a passing bond (4.83). The curves of Fig. 4.18 are likewise transferable, but, because of the fictitious B sites, the density of states around vA is not normalized at the real concentration of the A sites (as was possible for the CPA cases cf. Fig. 4.11). 

A description of the percolation phenomenon in ionic microemulsions in terms of the macroscopic DCF will be carried out based on the static lattice site percolation (SLSP) model [152]. In this model the statistical ensemble of various... 

Here the dimensionless time z=t/t is normalized by the characteristic relaxation time t, the time required for a charge carrier to move the distance equal to the size of one droplet, which is associated with the size of the unit cell in the lattice of the static site-percolation model. Similarly, we introduce the dimensionless time zs = ts/t where ts is the effective correlation time of the s-cluster, and the dimensionless time z = tm/t. The maximum correlation time t, is the effective correlation time corresponding to the maximal cluster sm. In terms of the random walker problem, it is the time required for a charge carrier to visit all the droplets of the maximum cluster sm. Thus, the macroscopic DCF may be obtained by the averaging procedure... 

Figure 34 shows the temperature dependencies of the static fractal dimensions of the maximal cluster. Note that at percolation temperature the value of the static fractal dimension Ds is extremely close to the classical value 2.53 for a three-dimensional lattice in the static site percolation model [152]. Moreover, the temperature dependence of the stretch parameter v (see Fig. 34) confirms the validity of our previous result [see (62)] Ds = 3v obtained for the regular fractal model of the percolation cluster [47]. 

Nevertheless, in a previous study dealing with inert matrices of naltrexone-HCl [74], two different excipient percolation thresholds pc2 were found for the matrixforming excipient Eudragit RS-PM the site percolation threshold related to a change in the release kinetics and the site-bond percolation threshold derived from the mechanical properties of the tablet, where the excipient failed to maintain tablet integrity after the release assay. 

An evident change in the release kinetics between tablets containing 80 and 90% w/w of drug can be observed in Table 23 (from k 0.57 to k 0.7 in the Peppas equation). Therefore, the site percolation threshold of the excipient can be estimated between the matrices containing 80 and 90% w/w of dextromethorphan hydrobromide (10-20% v/v of excipient). Above this threshold, a percolating cluster of excipient particles exists. These particles are able to control the drug release kinetics, but their cohesion forces can be insufficient to maintain tablet integrity after the release assay. 

Pc, at which a spanning cluster occurs is called the percolation threshold. There are two versions of percolation, site and bond. With bond percolation, the sites are initially filled and the bonds are added to connect the sites. With site percolation, a grid placed over a region is gradually filled with spheres. The percolation threshold is lower for bond percolation than for site percolation because a bond is attached to two sites while a site is connected to a maximum of z bonds. Taking the coordination number, z, around the sites, the threshold for bond percolation is seen to be close to that of the classical theory [21]... 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

Structure Coordination z Probaldlify of percolation 1 z - 1 Probability of site percolation P - Probability of bond percolation Space fUUng factor V I rcolation volume fraction 0 = opf"... 

Site and bond percolation can be shown to be equivalent, and for simplicity from this point we discuss only site percolation. 

Fig. 41. Percolation probability for finite-sized lattices. Computer calculations of the percolation probability, P(p), as a function of the site-filling probability, p, for two-dimensional square lattices of varied dimension O, 10 x 10 , 20 x 20 , 40 x 40. Each curve is an average over a set of site percolation simulations for a lattice size. The site percolation threshold for an infinite two-dimensional square lattice is 0.593. Nonzero values of P p) below the infinite lattice threshold reflect the variance of the threshold value for finite lattices (unpublished results).

Figure 41 shows the percolation probability P(p), determined by averaging Monte Carlo simulations for site percolation on a two-dimensional square lattice, for finite lattices of varied size. For an infinite lattice of this type, pc = 0.593. The nonzero values of P p) below p = 0.593 reflect the dispersion in pc found for finite lattices. A protein, with several hundred water sites on its surface, would fall in the range of lattice sizes modeled in Fig. 41. The shape of the P(p) function is not strongly affected by lattice size.

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