Cayley-tree lattice

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. 

Cayley tree or Bethe lattice with functionality/ = 3. 

The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. In a one-dimensional lattice, the percolation threshold is obviously equal to one. For higher dimensions, pc will be smaller than one. To illustrate this central result, we consider the Bethe lattice (also called the Cayley tree). 

The Cayley tree is formed of nodes from each of the z links that join the next nearest neighbours (Fig. 3.14). If one starts at an arbitrary node and progressively builds the tree, one gets a lattice of size l + z+ + — + = zR+1 — 1 if the generation of R nodes is considered. 

Figure 3.14 Cayley tree, or Bethe lattice, with z = 3 branches. The generation of three branches axe shown.

Percolation theory (Stauffer and Aharony 1995) describes the random growth of molecular clusters on a d-dimensional lattice. It is intended to describe gelation in a better way then classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Cayley tree) because it overcomes limitations regarding mean-field assumptions like unlimited mobility and accessibility of all groups (Stauffer et al. 1982 De Gennes 1979). 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

Dendrimeric molecules with branched tree-like structures are an interesting novel class of polymers with well controlled structure and size (Figure 11). Theoretical interest in these Cayley trees (also known as Bethe lattices) arises from their peculiar dimensionality the connectivity between different... 

The role of positional fluctuations in polymer networks is central to some theories of elasticity, and has been investigated with an MC method based on a modified bond-fluctuation model (265). The simple model used in the simulations gave results close to those calculated from theory for a Bethe lattice (also known as a Cayley tree). More extensive results bearing on the role of fluctuations in polymer networks have been reported by Grest and co-workers (225). They find that entanglements limit fluctuations, giving behavior similar to the description provided by the tube model. 

Linke et al. (1983) and Franz (1984, 1986) developed a different version of the lattice-gas concept. These models emphasize the experimental observation (Secs. 3.4 and 4.4) that thermal expansion of liquids is mainly achieved by a reduction of the average near-neighbor coordination number. A given structure such as the artificial, but mathematically convenient Cayley Tree or Bethe lattices can, when partially populated, be viewed as a crystalline alloy of atoms and vacancies. Tight-binding methods then permit calculation of the electronic structure, in particular the density of electronic states. Franz made use of quantum percolation theory to model the DC conductivity. A more recent model (Tara-zona et al., 1996) employs a body-centered-cubic lattice which, when fully occupied, provides a reasonable approximation to the local structure of liquid metals near the melting point. 

Rather remarkably for a network of a given chain density one can actually calculate Vj since when a chain moves in a lattice of other chains the paths available to it are equivalent to a Cayley tree i.e. to a random walk which always retraces its steps. Details are given in refs 4 and 5 (see also ref 6). Since it appears the only tractable representation available to us, we adopt (3.7) and take entanglements to be equivalent to slip links i.e. are characterized by an Y and are quadrifunctional. 

Disaster approaches for cluster (a) with e = 3 growing on a Cayley tree (or Bethe lattice), because lattice contains no loops, so its density increases without limit (b) as the radius grows. Additional bonds forming at the periphery indicated by thick lines. 

In the percolation problem one has for the spectral dimension approximately, d = 4/3, independent of the space dimensions according to the Alexander and Orbach conjecture. It is also the mean field value for lattice animals (branched structures defined on lattices) or Cayley tree-like structures. Hence the Cayley tree corresponds to the mean field solution to percolation. ... 

It is obvious that Fig. 2 represents a drastic simplification of reality for complicated molecules The merit of this theory is that it gives a good qualitative picture of real gelation, the first indication for the universality principle that complicated molecular details are not very relevant for the main results. Physicists call Fig. 2 a Bethe lattice, and this gelation process is then called percolation on a Bethe lattice . Hie macromolecules are also designated as Cayley trees since, like trees in a forest, they have no cyclic links between their branches. Many other problems of theoretical physics, besides percolation, have been studied on Bethe lattices. When critical exponents were found they usually agreed with those obtained by using simple approximations for real lattices like mean field (or molecular field) approximation, Landau ansatz for phase transitions, van der Waals equation, etc. We will thus also denote them as mean field approximations. 

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