Subject partition function

When we deal with the subject of reactions in solution, which has been our primary emphasis, we are not concerned with ab initio calculations of Ki, since in general the partition functions are unavailable for the participants. 

Return now to the assertions of the Introduction. The explanation of assertion (1) was pointed out previously. Assertion (2), that the PDT is a practical computational tool, was the subject of Sect. 9.2.3. See especially the discussion "the general computational tricks work for the PDT also, emphasizing the general statistical methods of stratification and importance weighting, and their correspondence to the natural theoretical analyses of the PDT partition function. 

The mesoscopic description is introduced by defining functions 4> (q) and 4>B(q) that have the meaning of averaged over some mesoscopic volume values of the microscopic concentration operators. The conditional partition function, Z(4>t) (y =A,B), is the partition function for the system subject to the constraint that the microscopic operators 4>T(q) are fixed at some prescribed... 

The incompressibihty constraint 5(A + < >B — 1) has been explicitly included in partition function (37), and the continuous chain model, Eq. (19), is being used. Q is the partition function of an independent single chain subjected to the fields Ua and Ub, and it can be evaluated exactly. One writes the partition function as Q = f drq(r, 1), where... 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

The last two equations identify 0A(r) and average densities of A and B monomers at r, as calculated in an ensemble of non-interacting polymers subject to the fields wA(r) and WofV), which act on A and B monomers, respectively. Once the partition function of this problem is known, eqns A.7-A.11 can be solved and the free energy obtained. 

Another kind of self-consistent theory used a continuum diffusion representation to describe the distribution of segments.21 23 The segments were assumed to be subjected to an external potential of a mean field. An analytical approximation of the latter self-consistent theory was suggested by Milner et al. (the MWC model)24 on the basis of the observation that at high stretching the partition function of the brush is dominated by the classical path as the most probable distribution. Under this assumption, it was found that the self-consistent field is parabolic and leads to a parabolic distribution of the monomer density. Similar theories for polyelectrolyte brushes25 27 also adopted the parabolic distribution approximation. 

We will introduce this subject through discussion of a central example. Suppose that we have a sample ooi, (02,..., coj = O, from which we will estimate a partition function Z. This means that we have a function Z(fi) to be evaluated for our sample to produce a numerical value that we take to be the estimate of Z. We assume here the common circumstance that Z(fi) takes the form... 

This notation [4>mf emphasizes that this is a functional of the enclosed material being subject to the molecular field Evaluation of the partition function Eq. (7.55) typically would not be a trivial task, but Chapter 5, p. 100, can be brought to bear because this is a standard form. 

The subject of statistical mechanics was met briefly in Chapter 16. The underlying quantity which appears throughout the subject is that of the partition function q, which is defined for a molecule as the exponential function... 

Approximation of the orientation factor Zoriem the partition function according to Eq. (9) of the text provides the key to the earlier version of the lattice theory of rodlike particles not subject to orientation-dependent interactions. Although approximate, this formulation offers advantages of simplicity that for most purposes outweigh the incident errors. The latter are generally small (see Table 1). 

Boltzmann constant, and g, is a statistical weight for the ith excited state. The summation over all possible states is the electronic partition function. If the flame temperature is constant throughout the analysis, the signal level will be subject only to the amount of sample in this region. Thus the intermediate zone is usually aligned with the optical path and is of most importance for analytical measurements. However, this alignment of the optical path should also be optimized for the particular element to be quantitated. 

We can also assume that these configurations are subject to a constraint. For instance, the constraint may be that a set of points 0 belonging to the molecules have definite space positions and then 0 represents the set of these positions. We thus define the restricted partition function Q (0, 0) by... 

Lifshitz represents a polymer containing N monomers as a chain with N independent links subjected to interactions. Let SI be the configuration of such a polymer, °Hr Sl) the weight associated with the free chain and t(Sl) the interaction energy (in thermal units 1 //f). The partition function Z can be written in the following symbolic form... 

Both of these characteristic properties of the nonideal chain, its partition function and its average squared end-to-end length, can be calculated using the self-similarity based RG techniques described in the previous sections but, because these two properties depend differently on their initial values, they are subject to two different renormalization groups. The situation just described closely resembles our previous treatments in Sections II.E and... 

It is important to digress briefly on the meaning of Bjerrum s approach, which has often been the subject of controversy. Onsager considered that one may say that Bjerrum applies different approximations to different regions of space in evaluating the partition function of the electrolytic solution. In no case may Bjerrum association be taken to imply necessarily a chemical association in the way protons and acetate ions associate to yield the new chemical species acetic acid. Bjerrum pointed out that for this chemical process intermediate species between the associated and dissociated states do not exist in measurable concentrations but between free and associated ions intermediate forms exist in finite concentrations. We shall illustrate this point below. This approach deals with that part of the interaction which is underestimated by the approximation of Debye and Huckel and assumes that ions in the inner region may be considered to form a neutral pair, not affecting the rest of the solution. 

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

Kosmas and Freed [41] presented another approach to scaling laws. Differing from the theory described above, it starts from the partition function for a solution of continuous chains which interact subject to the binary cluster approximation. For example, their theory derives for osmotic pressure (in three dimensions) a general scaling law which, in our notation, may be written... 

D24.3 The Eyring equation (eqn 24.53) results from activated complex theory, which is an attempt to account for the rate constants of bimolecular reactions of the form A + B iC -vPin terms of the formation of an activated complex. In the formulation of the theory, it is assumed that the activated complex and the reactants are in equilibrium, and the concentration of activated complex is calculated in terms of an equilibrium constant, which in turn is calculated from the partition functions of the reactants and a postulated form of the activated complex. It is further supposed that one normal mode of the activated complex, the one corresponding to displaconent along the reaction coordinate, has a very low force constant and displacement along this normal mode leads to products provided that the complex enters a certain configuration of its atoms, which is known as the transition stale. The derivation of the equilibrium constant from the partition functions leads to eqn 24.51 and in turn to eqn 24.53, the Eyring equation. See Section 24.4 for a more complete discussion of a complicated subject. 

The conformational partition function is subject to the same types of manipulations as are other partition functions encountered in statistical mechanics. Thus the average conformational energy of the chain is obtained from the temperature dependence of Z . 

---