Percolation probability

We have already commented on the equivalence between peripheral PCA and the problem of directed percolation (see footnote on page 343). It is easy to show that the pi and P2 of the isotropic system are given in terms of site- and bond- directed percolation probabilities - ps and pb, respectively - by the expressions pi = PaPb and P2 = PsPfe(2 — p ) see [domany84] and [kinzel85]. 

Figure 9.39 The plot of the dependence of percolation probability Q(Zq) on Zq according to Equation 9.78.

Solving the problem of sites is similar to that of bonds. One can also calculate the percolation probability, Q q), of formation of an infinite cluster of sites of one type, for example, B, depending on their numerical part q = B/(A+B), which is proportional to the part of the 3D volume occupied with phase B. The invariants of the problem of sites for 3D lattices of contiguous monospheres are [3,238]... 

Generally, percolation media can be characterized not only by the percolation probability, but also by several important quantities [8,121,234], Near the percolation threshold, r/lh, a number of... 

Let us consider some of these quantities. Let X=(J(q), that is, the percolation probability. The values of //i )j. for various lattices are shown in Table 9.9. 

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. 

For a system where the donors are distributed at random it can be shown mathematically that there exists a critical concentration C (also called "percolation concentration") below which the percofation (edge-to-edge connectivity) has a probability of zero and above which the percolation probability (P ) rises sharply with donor concentration (C). A mathematical relation ( ), for a substitutionally disordered binary lattice, is ... 

The percolation threshold pc is defined as the site-filling probability that marks the appearance of the lattice-spanning percolation cluster and the establishment of long-range connectivity. One can introduce the function P(p), called the percolation probability, which has the following significance When the fraction of filled sites is p, P p) is the chance of a randomly chosen site being both filled and part of the infinite cluster, or, in other words, P(,p) is the fraction of the entire system that is taken up by the infinite cluster. 

The values of the exponents quoted in Table XII have been estimated numericcilly by renormalization group techniques. Intuitively, there should be a close relationship between conductivity and percolation probability, and one would guess that their critical exponents should be identical. This is not true. Dead ends contribute to the mass of the infinite network described by the percolation probability, but not to the electric current it carries. Figure 39 shows the different growth of the percolation probability and the conductivity. It is convenient to set the conductivity equal to unity at = 1, as in Fig. 39. We note, in passing, that diffusivity is proportional to conductivity, in agreement with Einstein s result in statistical physics that diffusivity is proportional to mobility. 

Fig. 39. Behavior of the percolation probability, P, and the conductivity,

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.

The procedure for the calculation of connectivities c, following closely Seaton [8], can be summarized as follows The bond occupation probability / was obtained as a function of percolation probability F from the adsorption isotherms (Figure 1) using the pore size distribution as follows ... 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

The percolation probabilities obtained by Monte Carlo simulations on common lattices are shown in Fig. 7. Taking into account these results and the invariant [Eq. (3)], one can consider in applications that the percolation probability for the bond problems is a universal function dependent only on two parameters, i.e., d and zq. In particular, for three-dimensional lattices this probability can be represented as... 

The universal percolation probabilities for regular lattices are expected to be applicable also to irregular lattices. It is useful to consider two examples supporting this important statement. 

Finally, it is reasonable to present the equations describing percolation on the Bethe lattices (Fig. 6). The percolation probability can be calculated exactly for this model and is the same for bond and site problems 10). For example, the probability that all open walks from a chosen site are of the finite length, 1 — 9 b, can be represented as... 

Comparing Eqs. (8) and (3), one can conclude that the Bethe model is appropriate for describing the three-dimensional lattices only at z = 3 and 4. If z 3= 5, the percolation probability for the Bethe lattice differs considerably from that for regular lattices. 

Percolation media can be characterized not only by the percolation probability but also by other quantities (Table II)—for example, by the correlation length, which is defined as the average distance between two sites belonging to the same cluster. Near the percolation threshold, all these quantities are usually assumed to be described by the power-law equations (Table II). All current available evidence strongly suggests that the critical exponents in these equations depend only on the dimensionality of the lattice rather than on the lattice structure (72). Also, bond and site percolations have the same exponents. 

Finally, it is of interest to discuss briefly percolation on finite lattices. In the latter case, the percolation probability can be defined as the probability that an arbitrarily selected site (or bond) belongs to the largest cluster of the system. The scaling expression for 9 is usually assumed to be (72)... 

As pointed out above, the desorption process is dependent both on the void- and neck-size distributions,/(r) and C Fig. 13a) and the void and neck arrangements are random (the latter term means that the probability for an arbitrary void or neck to have a given value of the radius does not depend on the sizes of the neighboring voids and necks), the desorption process is mathematically equivalent to the bond problem in percolation theory. In particular, the probability that an arbitrary void is empty at a given value of the Kelvin radius during desorption is equal to the percolation probability 9 b(zo ) for the bond problem. Thus, the volume fraction of emptied voids under desorption [1 — Udes(rp)] can be represented as the product of the fraction of pore volume that may be emptied in principle at a given value of rp [1 - Uad(rp)] by the percolation probability b(zo ), i.e.,... 

Employing Eq. (24), we may assume that the percolation probability for the sublattice of voids with r > rp is the same as the universal percolation probability for the bond problem (Section II). The latter probability was originally calculated only for regular lattices. The sublattice of voids with r > rp is not regular. In addition, some of the voids with r > rp are not connected with the other voids having r > rp. However, the numerical results obtained by Yanuka (33) for a randomized cubic lattice (see also the discussion in Section II) support the hypothesis on the universality of the percolation probability for both regular and irregular lattices. 

Equations (24) and (30) solve the problem they relate the fractions of pore volume emptied from the condensate during adsorption and desorption with the percolation probability and the radius distributions of voids and necks. If these distributions do not overlap, i.e., < C (Fig. 13a), one can... 

To calculate the percolation probability in Eq. (34), Mason (18-20) and Palar and Yortsos (26,27) have employed the Bethe tree model. This model is known (see Section II) to yield quantitative results only at 3 z 5. In general, however, one can use in Eq. (32) the universal percolation probability calculated for regular lattices (see Section II). 

From the data presented in Fig. 27, we can conclude that the approach based on percolation theory permits one to obtain the neck-size distribution only in a relatively narrow range of radii. This is due to the fact that the percolation probability 9 b(z ) has a threshold (3>b(zq) = 0 at zq < 1.5) and increases from 0 to 1 in a relatively narrow range of 1.5 < z < 2.7. 

From Eq. (47) we can conclude that the effect of interconnection of various pores on mercury intrusion is very strong for laige necks at relatively low pressures (when Zoq < 2.5). Mercury penetration into large necks is blocked by small necks. For small necks at relatively high pressures (when Zoq > 2.5), the percolation probability 9 b2(zo ) is close to q (see, e.g., Fig. 7b), and Eq. (47) yields the same result as the model of independent cylinders. 

The conventional symbols for the bond occupation probability and the percolation probability are p and P, respectively, rather than q and 9 these symbols are not used in the present review to avoid confusion with the symbol for the pressure. 

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