Partition function lattice

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

We now calculate the density of the phonon scattering states. Since we have effectively isolated the transition amplitude issue, the fact of equally strong coupling of all transitions to the lattice means that the scattering density should directly follow from the partition function of a domain via the... 

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

Combining all contributions to the partition function of the disordered state of a lattice polymer solution, we obtain... 

The remark just made suggests that a natural place to begin our discussion of equilibrium equations is with the occupation of different charge states. Let a hydrogen in charge state i(i = +, 0, or - ) have possible minimum-energy positions in each unit cell, of volume O0, of the silicon lattice. (O0 contains two Si atoms, so our equations below will be applicable also to zincblende-type semiconductors.) To account for spin degener-ancies, vibrational excitations, etc., let us define the partition function... 

In the derivation of the mean-field partition function, it is necessary to know the probability for inserting a chain molecule in a given conformation into the system. The classical way to compute this quantity is by approximating it by a product of the local volume fractions of an unoccupied site (averaged over lattice layers). It was realised that, besides the density information, information on the bond distributions is also available. The bond distribution gives information on the average local order. Using this information, it becomes possible to more accurately access the vacancy probability. 

The summation is here over all possible configurations of the defects on the lattice, each defect being able to occupy any site on its particular sublattice. Configurations in which more than one defect is assigned to a particular lattice site are now included in the summation, but such configurations do not contribute to the partition function because of the definition of the functions ha,... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

The treatments of Flory,93 Gibbs and Di Marzio,91 and Milchev94 differ in the way they calculate the second factor ftnter- This microcanonical partition function describes the number of ways in which the K chains can be put on the lattice,... 

Lattice QCD at finite density is described by a partition function... 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The temperature ranges in which these simple behaviours are approximated depend on the vibrational frequencies of the molecules involved in the reaction. For the calculation of a partition function ratio for a pair of isotopic molecules, the vibrational frequencies of each molecule must be known. When solid materials are considered, the evaluation of partition function ratios becomes even more complicated, because it is necessary to consider not only the independent internal vibrations of each molecule, but also the lattice vibrations. 

We first calculate the potential energy ((/, kJ/mol) and vibrational Helmholtz free energy (A vib, kJ/mol) for the unit cell at fixed temperature (T, K) and lattice constants (a, m) using the full quantum mechanical partition function. These two terms, in conjunction with a work term in the presence of an applied stress, provides the the Gibbs free energy (G, kJ/mol). 

We will now present some of the results of the recent Monte Carlo computations of the chain partition function and the related thermodynamic functions for some three-dimensional lattices performed recently by McCrackin and Mazur.2... 

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