Lattice enumeration

The properties of the primary valence lattice enumerated above— in fusibility, insolubility, strength and density—make it difficult to... 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

Certain measures re.spond particularly strongly to the intrinsic structural symmetry of r lattices cycle number (= C ) enumeration, for example, does not identify two cycles all of whose states are related by a spatial translation. Specific profiles may, therefore, be interrupted by a series of pronounced peaks at gj = (see figures (3.47-a,d), (3.48-d) and (3.49-a), for example). 

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

The dimer problem effectively consists of exactly enumerating the number of ways an arbitrary lattice can be decomposed into non-intersecting edges, without any leftover links covering an n X n chessboard with n /2 dominoes, for example, so that the entire board is covered without overlap or gaps. 

Arrangements in which the sets of x contiguous lattice cells chosen for occupation by polymer molecules are identical but which differ only in the permutation of the polymer molecules over these sets would be counted as different in this enumeration scheme. Since the polymer molecules actually are identical, it is appropriate to eliminate this redundancy. We require merely the number Q. of ways in which ri2 sets of X consecutively adjacent cells may be chosen from the lattice the order in which the sets are filled by polymer molecules is immaterial. Since U2 differentiable polymer molecules (labeled, for example, in the order of their insertion into the lattice) could be assigned to a given arrangement of U2 sets of sites in ri2 different ways, it follows that... 

In the simplest version of the transformation, the state of occupancy of a given bond lattice point is independent of the states of occupancy of other bond lattice points this corresponds to neglecting interaction between hydrogen bonds. The calculation of the distribution of possible occupation states of the bond lattice replaces the enumeration of occupancy states of the basic lattice section in the cell model, but in the simplest model the bond lattice occupancy distribution only accounts for a subset of possible basic lattice section occupancies. 

The collection of all symmetry operations that leave a crystalline lattice invariant forms a space group. Each type of crystal lattice has its specific space group. The problem of enumerating and describing all possible space groups, both two dimensional three dimensional, is a pure mathematical problem. It was completely resolved in the mid-nineteenth century. A contemporary tabulation of the properties of all space groups can be found in Hahn (1987). Bums and Glazer (1990) wrote an introductory book to that colossal table. 

For sufficiently short chains it is possible to calculate C , u , and all other features of interest exactly. Such enumerations were initiated independently because of their application to the statistical mechanics of interacting systems on crystal lattices,10 and a variety of analytical and computational methods (including the use of digital computers) has been employed to extend the enumerations to as large a value of n as practicable. These exact results are then used to conjecture the pattern of asymptotic behavior,... 

Table I (taken from Martin, Sykes, and Hioe16) contains the most recent exact enumerations of C for the triangular and fee lattices. Similar enumerations for other lattices have been given elsewhere ° 11 numerical analysis indicates that the close packed lattices lead to most rapid convergence, and these were therefore selected for an extensive enumeration project. It should be noted that C12 for the fee lattice is of order 1.8 x 1012. Using a direct enumeration procedure on a digital computer, the machine time required would be quite prohibitive. It is only by the way of sophisticated counting theorems17 and skilled programming that these numbers could be obtained.

Exact enumerations were subsequently undertaken for a number of lattices in two and three dimensions.13 Since random walk, and n2 for a completely stiff walk the form of Eq. (17) provides a reasonable interpolation between these extreme bounds. We should then expect... 

The results are plotted graphically as a function of 1 jn in Figure 3, and they suggest that 0 = f for all lattices in three dimensions. Corresponding enumerations in two dimensions suggest 0 =... 

If these conjectures are accepted the quantity /ne should tend to a limit as n - oo, and an estimate of this limit can be derived from the exact enumerations. Consequently an asymptotic formula can be put forward for the behavior of as a function of n, and this can be compared with Monte Carlo values for much longer walks. Such a comparison with walks of up to 600 steps on the tetrahedral and square lattices is reproduced in Table II, and the percentage deviations are recorded. An error of order 2% or 3% seems reasonable for a sample of about 1000 walks and the constanty of sign of eror may well be due to the enrichment 9 procedure introduced by Wall and Erpenbeck so as to overcome attrition. 

Comparison between asymptotic formulas based on direct enumeration and Monte Carlo estimates. Tetrahedral and square lattices. (Wall and Erpenbeck9). 

Applying this criterion to. a lattice model, v corresponds roughly to the volume of a unit cell, and / to a lattice spacing. Hence the values of n 10 for exact enumeration quoted in Section IV seem quite reasonable. Certainly there is no support for the claim by Flory and Fisk31 that the 6/5 power law is attained only for n > 10.6... 

Now that we have enumerated all of the 3D lattices, the 14 Bravais lattices, we can look in more detail at their symmetries. First of all, it must be recognized that every lattice point is a center of symmetry. The translation vectors tx, t2, and t3 are entirely equivalent to tj, -t2, and -t3, respectively. Therefore, in determining the point symmetry at each lattice point (which is what symmetry of the lattice means) we must include the inversion operation and all its products with the other operations. 

From a formal point of view, (2.2.53) describes random walks on a onedimensional lattice of enumerated sites. Unlike standard problems with constant transition probabilities between sites, in (2.2.53) these probabilities depend on a site number and are essentially non-linear. Figure 2.11 shows possible transitions in the model under consideration and the relevant transition rates. 

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