Partition sum

The logarithm of the partition sum of the annealed species under the sign of integration in Eq. (1) makes the analytical treatment difficult. Therefore, the replica method, see, e.g.. Ref. 1, is used, or, in other words the following identity is exploited... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

Dimer concentrations. With the pair partition sums thus defined, we may compute the bound dimer concentrations from the mass action law,... 

Free single particle partition sum. The one-particle sum over states of the ideal gas is easily evaluated if the free-particle Hamiltonian (with V (R) = 0) is used,... 

Larger clusters. With the assumptions made, the triplet partition sum is given by [184]... 

The Zr normalizes the expression it is chosen so that the sum of P over all free and bound states equals unity. In other words, Zr equals the pair partition sum of relative motion, that is Eq. 2.16 or the sum of Eqs. 2.21 and 2.22, with the center of mass contribution v/X M suppressed. (The for like pairs characteristic factor of 1/2 is also commonly suppressed at this point.)... 

The computation of the fractions r is straightforward. For equilibrium, the para-H2 and ortho-H2 densities, np, n0, are obtained from the partition sum of rotational states, but if an excess of para-H2 is to be accounted for (as is often the case in astrophysics), then the np, n0 are input in some form. The number of para-H2 pairs in the volume V is n2V2 which enter the expression for reven with a multiplicity of unity. The number of ortho-H2 pairs is jn2V2 of which the fraction 5/9 enters reven, and the fraction 4/9 enters r0dd- The number of para-H2-ortho-H2 pairs without exchange symmetry is npn0V2, and the total number of H2-H2 pairs equals (np + n0)2V2. Hence, we have... 

At high temperatures, vibrational states must also be included in the partition sum above. The nuclear weights are gj for hydrogen we have, for example, gj = 1 for even j, and gj = 3 for odd j. However, we mention that in low-temperature laboratory measurements as well as in astrophysical applications, para-H2 and ortho-H2 abundances may actually differ from the proportions characteristic of thermal equilibrium (Eq. 6.53). In such a case, at any fixed temperature T, one may account for non-equilibrium proportions by assuming gj values so that the ratio go/gi reflects the actual para to ortho abundance ratio. Positive frequencies correspond to absorption, but the spectral function g(co T) is also defined for negative frequencies which correspond to emission. We note that the product V g a> T) actually does not depend on V because of the reciprocal F-dependence of Pt, Eq. 6.52. 

In the following discussion we will consider an ensemble of molecules that represents a liquid system. Assuming there is no correlation between molecular interactions and geometrical restraints, the partition sum Z of an ensemble can be factorized into three contributions ... 

The second factor on the right-hand side of Eq. (5.1) is called the combinatorial factor. It is the partition sum of the equivalent ensemble of molecules that interact only through steric restraints. The combinatorial factor takes into account all size and shape effects of the molecules. There is no exact expression for Zc but, by fitting the simplified theoretical models to thermodynamic data of... 

The third factor, ZR, in Eq. (5.1) is called the residual contribution in the chemical engineering notation and it arises from all kinds of non-steric interactions between molecules, i.e., usually from vdW, electrostatic, and hydrogen bond interactions. Despite its name, it is the most important contribution in most liquids. The basic assumption of surface-pair interaction models is that residual—i.e., non-steric—interactions can be described as local pairwise interactions of surface segments. The residual contribution is just the partition sum of an ensemble of pairwise interacting surface segments. 

In view of our earlier assumption that the residual partition sum ZR depends only on the segment composition, we have... 

The simplest model for the residual part of the partition sum uses the assumption of random mixing (RM). This corresponds to... 

Consider an ensemble S of N objects, belonging to m classes (surface segment types in COSMOSPACE) of identical objects (surface segments in COSMOSPACE). Let N, be the number of objects of class i. Then x, =Ni/N is the relative concentration. Let there be N sites that can be occupied by the N objects. Each two of these sites form pairs. Hence, all objects occupying the N sites are paired. The interaction energy of the pairs will be described by a symmetric matrix, Ey, where i and j denote two different classes of objects. Let Z be the partition sum of the ensemble. From statistical thermodynamics, the chemical potential of objects of class i is given by... 

In the next step we derive a relationship for the chemical potentials in ensemble S. For that, we first choose any object a. We can sort all states of the ensemble with respect to the partner P of a. The partition sum of the system can be written as a sum over all partners ft, taking into account the Boltzmann weight exp(—E xp/kT) of the interaction of a with / multiplied by the partition sum Z( a> / ) of a system missing objects a and / . A factor N/2 arises from the fact that a pair a) can be taken away from any of the N/2 pair positions of the ensemble, and a factor of 2 arises from the two possible orientations of that pair on each pair position. Hence we have... 

All the statistical properties of the polymer will depend on the partition sum. For instance, the average end-to-end distance of a polymer chain that is free to move is given by... 

Before ending this section we would like to point out the relationship between the statistical properties of a polymer chain and a variety of other physical problems. The partition sum, as written in Eq. (5), is in the form of a path integral over all possible chain conformations. Since many physical problems can be formulated in terms of path integrals, we can map the problem of a Gaussian polymer in a random media to a wide... 

In order understand the conformational statistics of a Gaussian chain in a random potential, we map the partition sum to an imaginary time Schrodinger eqnation. This mapping (see Ref. [7] Eqs. (3.12)-(3.18)) is given by... 

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