Percolation exponents table

Percolation exponent values are given in Table 1, they are calculated from Monte Carlo simulations performed on a lattice at a space dimensionality d. Exponent values are very sensitive to d (the mean-field values are reached for d > 6) but insensitive to the exact geometry of the lattice (number of first neighbour) for a given dimensionality. 

Table 3. Qimparison of gelation (percolation) exponents (left part) with exponents in thermal phase transitions (right part) for both dassical and modem theories...

Comparing the results in Table 1, we see that the classical exponents typically differ from three-dimensional percolation exponents by a factor of about two. Even experiments of moderate accuracy may distinguish between two such drastically different pre-... 

Table III summarizes results for the critical exponents and critical coverages, obtained by percolation theory analysis of the protonic conduction processes for lysozyme and purple membrane, and compares these values with theory.

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. 

Table 1.2 Values of the percolation critical exponents in various dimensions...

These powers a, (3, 7, p and i/ are called the critical exponents. These exponents are observed to be universal in the sense that although Pc de-pends on the details of the models or lattice considered, these exponents depend the only on the lattice dimensionality (see Table 1.2). It is also observed that these exponent values converge to the mean field values (obtained for the loopless Bethe lattice) for lattice dimensions at and above six. This suggests the upper critical dimension for percolation to be six. 

Table 1.4 Percolation thresholds and exponents for central force elastic percolation...

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. 

In the classical theory, however, the neglect of loops significantly affects the size distribution and other properties of the clusters as one approaches the gel point. Some of the critical exponents that describe these properties in the classical theory and in percolation theory near p Pc are compiled in Table 5-1 (Martin and Adolf 1991). 

The power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

Note that whenever a new exponent is defined, there is also a scaling relation that calculates this exponent from t and a. There are only two independent exponents that describe the distribution of molar masses near the gelation transition, with the other exponents determined from scaling relations. Table 6.4 summarizes the exponents in different dimensions that have been determined numerically, along with the exact results for 1, d=2, and d>6. It turns out that t/=6 is the upper critical dimension for percolation, and the mean-field theory applies for all dimensions d>6. 

Table 2.3. Scaling exponents for classical and percolation theories of gelation ...

After having discussed the behavior of SAWs on deterministic fractals, we move on to the second major topic of this chapter, namely the numerical study of SAWs on random fractals, the latter modelled by percolation. As non-trivial changes in the exponents characterizing the structure of SAWs on the incipient percolation cluster (and as a consequence on its backbone) axe only expected at criticality [5], i.e. for probability p of available sites being p = Pc, the following discussion is restricted to this case. A summary of exponents and fractal dimensions characterizing critical percolation is given in Table 1. 

In these expressions, tg is the reaction time at gel point, s and t are the static scaling exponents which describe the divergence of the static viscosity, nO 6 , at trstatic elastic modulus. Go at tggelation mechanism has been discussed on the basis of several models based on the percolation theory (for review, see ref 16), that provide power laws for the divergence of the static viscosity and the elastic moduli. Characteristic values for the s, t and A exponents are predicted by each of these models (Table I). 

Vojta and Sknepnek also performed analogous calculations for the quantum percolation transition at p = pp, J < 0.16/ and the multicritical point 2itp=pp,J = 0.16/. A summary of the critical exponents for all three transitions is found in Table 3. The results for the percolation transition are in reasonable agreement with theoretical predictions of a recent general scaling theory of percolation quantum phase transitions P/v = 5/48, y/v = 59/16 and a dynamical exponent oi z = Df = (coinciding with the fractal dimension of the critical percolation cluster). 

Table 3 Critical Exponents of the Generic Transition, Percolation Transition, and Multicritical Point of the Dimer-Diluted Bilayer Quantum Heisenberg Antiferromagnet...

In contrast to critical exponents, values of percolation thresholds depend on the lattice topology and the type of the percolation problem considered. Percolation thresholds for a few known lattice types are listed in Table 3.2. In ID, it is trivially Pc = 1. In 2D, values forpc are known exactly for specific lattice types. In 3D, values of Pc can only be found with the help of computer simulations. 

Table 7.3 gives the predicted exponents n, for microbial invasion (j8 = 1) and enzyme diffusion P = V2) for two and three dimensions and fractal dimensions of percolation clusters at, above and below the percolation threshold, using equation 7.12. The fractal dimension in each case increases with p with > = at/7 = 1. The initial exponents are independent of p since the fractal pathways have not yet been developed and the starch is immediately accessible on the surface. Thus, the initial slope reflects the intrinsic behaviour A The major exponent is dominated by the fractal dimension existing near p. For the poly-disperse samples, a fractal dimension of Z) = 1.59 better describes the data for both invasion and diffusion compared to Z) 1.8 for the monodisperse samples. The difference in fractal dimension is due to the coarseness of the poly-disperse pathways (average diameter 10 im) compared to the monodisperse blend (diameter = 1 pm). In general, the theoretical exponents in Table 7.3 are in agreement with the simulation exponents. 

The results of the conductivity measurements are given in Figure 4.10 [/) along with the linear fittings of the equation given in Table 4.1. The determined percolation thresholds, ultimate conductivities, and critical exponent values are summarized in Table 4.3. 

Although this method removes the uncertainty about the location of the critical point, it does not necessarily help to distinguish between theories. For example, the exponent in Eq. 51 has the value y/v = 2 for both percolation and the classical theory. (See values in Table 3.) Fortunately, this is not the case for every pair of properties, so the general approach can be successfully applied, as we shall see. 

Table 1. Critical exponents in three-dimensional percolation and classical or...

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