Bethe lattices

A natural question is just how big does Mq have to be to see this ordered phase for M > Mq. It was shown in Ref 189 that Mq <27, a very large upper bound. A direct computation on the Bethe lattice (see Fig. 2) with q neighbors [190,191] gives Mq = [q/ q — 2)f, which would suggest Mq 4 for the square lattice. By transfer matrix methods and by Pirogov-Sinai theory asymptotically (M 1) exact formulas were derived [190,191] for the transition lines between the gas and the crystal phase (M 3.1962/z)... 

FIG. 2 Phase diagram in the M-z plane for a square lattice (MC) and for a Bethe lattice q = A). Dashed lines Exact results for the Bethe lattice for the transition lines from the gas phase to the crystal phase, from the gas to the demixed phase and from the crystal to the demixed phase full lines asymptotic expansions. Symbols for MC transition points from the gas phase to the crystal phase (circles), from the gas to the demixed phase (triangles) and from the crystal to the demixed phase (squares). (Reprinted with permission from Ref. 190, Fig. 7. 1995, American Physical Society.)... 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

Fig. 7. Comparison of a the structure of the Bethe lattice with a functionality of 3 (only part of the system is shown) and b a two-dimensional square lattice [16]. For the Bethe lattice, each possible bond is shown as a line connecting two monomers. In FS theory an actual bond of these possible bonds is formed with probability p. For the square lattice, each bond that has been formed is shown as a short line connecting two monomers, while the monomers are not shown...

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

One example of a tree-based separator system is shown below in Fig. 2.8 where the Bethe lattice or Cayley tree is shown (Wilson, 1996). This graph can be expanded to any number of levels and can function with dilferent types of columns and electrophoretic elements. This is not the only graph that can function as a complex multidimensional separator system. But it is an example of something with multiple... 

FIGURE 2.8 Separator systems cascaded to form a Bethe lattice or Cayley tree where the point of introduction is the graph vertex 0 and solute can be sampled from any of the outward nodes at position 1, 2, 3, 4, and so on. The sample loops and valves are not shown. 

Figure 9.38 The simplest types of regular 2D lattices (A) the Bethe lattice (Z=3) (B) the honeycomb lattice (Z=3), The simple square lattice (Z=4), (C) the simple square lattice (Z=4), (D) Kagome lattice (Z=4), (E) the triangular lattice (Z=6).

The percolation processes were first developed by Flory [235] and Stockmayer [236] to describe polymerization process, which result in gelation, that is, the formation of very large networks of molecules connected by chemical bonds. But, their theory was developed only for a special kind of network, namely, the Bethe lattice, an infinite branching structure without any closed loops. Broadbent and Hammersley have developed a more general theory and have introduced it into the... 

Bethe lattices, percolation, 39 10 Bethe tree model, 39 26 BET method, in heterogeneous catalysis, 17 15-17... 

Fig, 8.6 The number of different contributions to the fourth moment about a given atom on a Bethe lattice with local coordination,. ... 

Fig. 8.7 Plotting (5+ 1 )/(s + 1) =1 versus local coordination for the pure s and sp cases, where s is the dimensionless shape parameter. The dashed curve gives the Bethe lattice result, eqn (8.23), in which there are no ring contributions. (After Cressoni and Pettifor (1991).)...

The model in Fig. 3.2 is sufficient to predict the general features of N E), but much more detailed calculations are needed to obtain an accurate density of states distribution. Present theories are not yet as accurate as the corresponding results for the crystalline band structure. The lack of structural periodicity complicates the calculations, which are instead based on specific structural models containing a cluster of atoms. A small cluster gives a tractable numerical computation, but a large fraction of the atoms are at the edge of the cluster and so are not properly representative of the real structure. Large clusters reduce the problem of surface atoms, but rapidly become intractable to calculate. There are various ways to terminate a cluster which ease the problem. For example, a periodic array of clusters can be constructed or a cluster can be terminated with a Bethe lattice. Both approaches are chosen for their ease of calculation, but correspond to structures which deviate from the actual a-Si H network. 

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. 

The Bethe lattice (or tree) is a lattice containing no closed loops (see, e.g., Fig. 6). The latter results in a simple analytical solution of the bond and site problems for these lattices. In addition, the general percolation properties for Bethe and regular lattices are often close. For these reasons, the Bethe lattices are rather popular in applied science publications, although these lattices have no physical sense. 

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. 

This is the so-called ideal value on the Bethe lattice, which, when ps = 1, recovers the Flory formula. 

At a glance, it can be understood that the intermolecular reaction on lattices has no dimension-dependence it is a function of functionality/alone, while the cyclization rate, Eq. (68), depends strongly on d according to the decreasing function of d. Thus, as the dimensionality increases, the cyclization rate alone naively declines, resulting in the known behavior of the percolation model p rmgj—>0 (Bethe lattice) as d—... 

The percolation model differs from real branching reactions in two points (1) intermolecular reaction of the percolation model has no dimension dependence (2) concentration in the ordinary chemical sense is absent in the percolation model. These differences arise from only the one fact that molecules are fixed on lattices, and give rise to the opposing dimensionality as d— °°, in real systems cyclization becomes predominant, whereas in the percolation model it is suppressed entirely (the Bethe lattice). In these respects, the percolation model is not commensurate with the general features of the branching processes as chemical reactions. 

Cayley tree or Bethe lattice with functionality/ = 3. 

The regular lattice constructed in this way is called a Bethe lattice (see Fig. 6.13). The mean-field model of gelation corresponds to percolation on a Bethe lattice (see Section 6.4). The infinite Bethe lattice does not fit into the space of any finite dimension. Construction of progressively larger randomly branched polymers on such a lattice would eventually lead to a congestion crisis in three-dimensional space similar to the one encountered here for dendrimers. 

A convenient way of presenting the mean-field model (though it is not the way it was originally defined) is by placing monomers at the sites of an infinite Bethe lattice, a small part of which is shown in Fig. 6.13. The Bethe lattice has the advantage of directly taking into account the functionality of the monomers/by adopting this functionality for the lattice. For trifunc-... 

In a bond percolation model on a Bethe lattice, we assume that all lattice sites are occupied by monomers and the possible bonds between neighbouring monomers are either formed with probability p or left unreacted with probability 1 —/ . In the simplest version, called the random... 

Fig. 6.16 Gel point calculation on a Bethe lattice. Each site that is already connected to the gel from its grandparent has/ - 1 possible additional connections (potential children). site is chosen as the starting point of this procedure makes no difference because they are all statistically identical for an infinite Bethe lattice. Let us assume that our starting ( parent ) site has already formed a bond with one of its neighbours (the grandparent site) as sketched in Fig. 6.16. We would like to calculate the average number of additional bonds the parent site forms with its/- 1 remaining neighbours (potential children ). The probability of each of these bonds being formed is p and is...

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