Percolating Rules

If the sites of a lattice are randomly colored either black (with probability Pbiack or white (with probability Pwhite = 1 — Pbiack) then there exists a percolation threshold Pc, such that if, say, Pbiack Pc, the lattice consists of isolated clusters of black sites immersed in a white sea , and if pbiack Pc a connected black structure percolates (i.e. spans) the entire lattice. 

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. 

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Of particular interest is the long-term behavior of voting-rule systems, which turns out to very strongly depend on the initial density of sites with value cr = 1 (= p). While all such systems eventually become either stable or oscillate with period-two, they approach this final state via one of two different mechanisms either through a percolation or nucleation process. Figure 3.60 shows a few snapshots of a Moore-neighborhood voting rule > 4 for p = 0.1, 0.15, 0.25 and 0.3. 

Fig. 3.61 Some snapshots of the evolution of a 5-neighbor von Neumann neighborhood percolating voting rule V3 the initial densities are (a) p = 0.35 < pc and (b) p = 0.50 = pc-...

Figure 3.61 shows a few steps in the contrasting evolutions of the percolating voting-rule system V3 when p < pc and p Pc-... 

Many more such relationships can be derived in a similar manner (see [ma85] or [stan71]). For our purposes here, it will suffice to merely take note of the fact that certain relationships among the critical exponents do exist and are in fact commonly exploited. Indeed, we shall soon sec that certain estimates of critical behavior in probabilistic CA system are predicated on the assumptions that (1) certain rules fall into in the same universality class as directed percolation, and (2) the same relationships known to exist among critical exponents in directed percolation must also hold true for PC A (see section 7.2). 

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

Thus, one could expect to find a droplet morphology at those quench conditions at which the equilibrium minority phase volume fraction (determined by the lever rule from the phase diagram) is lower than the percolation threshold. However, the time interval after which a disperse coarsening occurs would depend strongly on the quench conditions (Fig. 40), because the volume fraction of the minority phase approaches the equilibrium value very slowly at the late times. 

In all of these cases, it does seem that the law that you do not get something for nothing applies. The conductivity decreases as would be expected from percolation theory, if the conducting polymer is dispersed, and according to the volume fraction if it forms a network. The mechanical properties improve as one might expect from an additivity rule applied to the two components. 

The bed scale corresponds to the whole bed or to a volume containing a large number of particles. That is the level at which we want to derive models for the investigated transport processes. However these processes are generally ruled by gas-liquid-solid interactions occurring at the particle scale. That is the reason why it is necessary to model these processes at the particle scale. The change of scale or volume averaging between both levels is ruled by the percolation process, i.e., by the velocity distribu-... 

Both problems share the common property that P(p) is of measure zero for p < Pc, with the critical threshold value pc as a function of the type of lattice considered. Few exact formulas exist for P(p) or even pc. There are, however, a number of empirical rules. For instance, for the bond percolation problem,... 

The minimally varying U and Th systematics over this record suggest stable drip routes changes in percolation pathways can thus be ruled out as cause for the shifts in isotopic values. 

Fulvic acid plays a major role in the transport and deposition of Fe, AI, and other metals in soils. The acid is produced by organic decay in the top of the soil s A horizon. Fulvic acid ligands can form soluble complexes with Fe + and AP+ and other metals, which facilitates metal movement downward through the soil. As a rule of thumb, if the molar ratio of metals/fulvic acid is less than 1/1, the metals are water soluble and mobile (Schnitzer 1971). If that ratio exceeds 1/1, the metals become insoluble and immobile. Thus, as fulvic acids are destroyed by aerobic decay or other processes during downward percolation, the metals precipitate, typically in the soil s B horizon. Precipitation of Fe and Al (and also Mn) oxyhydroxides, in turn, leads to coprecipitation and concentration of trace metals such as Cu, Cd, Zn, Co, Ni, and Pb in the soil (cf. Suarez and Langmuir 1975). 

The presence of adsorption hysteresis cannot be reconciled with the laws of classical thermodynamics. It is evident that there are various forms of adsorption hysteresis [7], which require different explanations [7, 39, 40]. In the capillary condensation range, well-defined hysteresis loops are generally associated with delayed condensation or percolation [11] through pore networks or ink bottles [38, 39]. In the case of activated carbons, delayed condensation is likely to be the most important mechanism [11], but we cannot rule out the other effects. 

Equations 7.3 and 7.4 are known in materials science as the rule of mixtures. In practice, the ice phase in most water ices is quite connected, but not totally continuous in any direction. This is reflected in the fact that the moduli of water ices that are above the percolation threshold in Figure 7.17 lie closer to the upper line than to the lower. 

The phenomenon of surface-enhanced infrared absorption (SEIRA) spectroscopy involves the intensity enhancement of vibrational bands of adsorbates that usually bond through contain carboxylic acid or thiol groups onto thin nanoparticulate metallic films that have been deposited on an appropriate substrate. SEIRA spectra obey the surface selection rule in the same way as reflection-absorption spectra of thin films on smooth metal substrates. When the metal nanoparticles become in close contact, i.e., start to exceed the percolation limit, the bands in the adsorbate spectra start to assume a dispersive shape. Unlike surface-enhanced Raman scattering, which is usually only observed with silver, gold and, albeit less frequently, copper, SEIRA is observed with most metals, including platinum and even zinc. The mechanism of SEIRA is still being discussed but the enhancement and shape of the bands is best modeled by the Bruggeman representation of effective medium theory with plasmonic mechanism pla dng a relatively minor role. At the end of this report, three applications of SEIRA, namely spectroelectrochemical measurements, the fabrication of sensors, and biochemical applications, are discussed. 

Thanks to the percolation theory, we know the laws which rule the electric conductivity of a mixture of insulating and conducting powders. Thrrs, if n is the conducting powder proportion, the conductivity ct of the mixture is calculated using the following expression ... 

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