Exact enumeration

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. 

Historically, it had been believed that the finite lamellar thickness observed in experiments is completely due to kinetic control and that, if sufficient time is granted for the lamellae, the thickness would progressively grow to the extended chain value. The exact enumeration calculations and an exactly solvable model... 

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... 

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. 

However, for the enumerations considered in the present paper no such complications arise and it is reasonable to rely on the results of extrapolations. The suggestion made by Flory and Fisk31 that the exact enumerations in the range n = 1 to 14 are dominated by immediate reversals is incorrect. The discussion of Section IV-G indicates that the nth step in the enumeration eliminates the effect of polygons of order n, and the contribution of the nth stage is of order l/nc. Thus the exact enumeration procedure is equivalent to assessing the asymptotic behavior of l/nc from the first 14 terms. 

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... 

Krukowski et al. [24] studied the effect of molecular shape in details by performing exact enumerations on lattice models of different molecular shapes. They calculated the entropic component of the chemical potential, i.e.,... 

Exact enumeration ) and Monte Carlo ) computations Indicated that this... 

Most or all of these features have also been found (or extrapolated) from exact enumeration and Monte Carlo techniques. 

Figure 9.30. Kinetics of the photoinduced back ET reaction for (NH3)5Fe"(CN)(Ru) "(CN)5 in water at BOOK with F = 2500cm , — AG = 3900cm and , = 3800cm . Ohmic spectral density with exponential cutoff at 220 cm is assumed for the solvent. Exact enumeration method (dashed) and the transfer matrix path integral approach (solid) are compared with the Golden Rule prediction (dash-dotted). (Reproduced from [132] with permission. Copyright (1998) by the American Institute of Physics.)...

The first SCF result that we discuss here is shown in Fig. 7. In this case, bd has been varied from 0 to 1.5, with steps of 0.5, for the case that the lengths of the two chains is varied under the constraint that the sum of the two is fixed to 200. The most simple system is found for Nb=Nd = 100 and Xwi = 0. In this case, we have a spherical homodisperse, athermal brush. For this situation, the (dimensionless) segment potential is simply given by u r) = ln[l - (ps(r)] [73]. In short, within a freely jointed chain model, we generate all possible conformations of the polymer chains with the constraint that the first segment is positioned at r = 6 (next to the surface). Depending on the positions visited by, e.g. a conformation c, we can exactly enumerate the potential felt by this conformation Mc. The statistical weight of... 

What makes this problem so difficult—and indeed most copolymer problems difficult— is the necessity to take the average over all possible 2" distributions of monomers on the polymer. Nevertheless, some exact enumeration studies of this problem have been conducted. Martin [16] estimated the location of the transition using exact enumeration methods. An open question is the value of the crossover exponent, (j>, which describes the shape of the free-energy near the adsorption critical temperature. The conclusion from the above study, and others, is that the difference between this exponent and its homopolymer counterpart, if it exists, is too small to be detectable by any current numerical studies. 

Golding and Kantor [19] and Kantor and Kardar [20] stndied a version of this problem by both exact enumeration and Monte Carlo methods. In the former situation described above, they find a collapse transition much like the transition discnssed above in the... 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

Here, dy is the fractal dimension of the percolation cluster. This first simple generalization is in a good agreement with results of Monte Carlo simulation [16-21] and exact enumerations [22-28] (see the Table 1). 

The exponent Vp for a SAW on a percolation cluster in several dimensions. FL Flory-like theories, EE exact enumerations, US, RG real-space and field-theoretic RG. The first line shows i saw for SAW on the regular lattice (d = 2 [13], d = 3 [11]). 

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