Mayer expansions

Table 2. Combinatorial factors [10] and integrals [13-16] for Mayer expansions of the virial coefficients B3, B4, and Bs at low D.

Figure 2. Dimension-dependence of the cluster integrals contributing to the Mayer expansions for the virial coefficients B3, B4, and B5 [13,14,17]. Dashed lines indicate negative values.

Two of the problems mentioned in connection with the Mayer expansion are at least partially removed by means of a reformulation due to Ree and Hoover [20]. This expansion makes use of the modified (or Ree-Hoover) integrals, defined by... 

From a numerical standpoint the Ree-Hoover expansion is far better to work with than the Mayer expansion. There are two reasons for this. First, the inserted / functions act as additional constraints which render each integrand nonzero only over a much more limited region of configuration space. (It isn t hard to see that each Ree-Hoover integral corresponds to a different region of configuration space, and that some of those regions will be very small. For hard rods, disks. 

It should be noted, however, that the resulting expansions in terms of Mayer integrals are not truncations of the Mayer expansions.)... 

The convergence of the Mayer expansion and the Stell-Lebowitz expansions for the free energy is slow, and accurate estimates of the thermodynamic properties for a model electrolyte at concentrations near 1 M are difficult to obtain. A way out of this difficulty is to consider approximations for the radial distribution functions which correspond to the summation of a certain class of terms which contribute to all of the virial coefficients. The integral-equation approximations, such as the HNC, PY, and MS approximations, attempt to do just this. They also provide information on the structure of the solutions to varying degrees... 

An alternative approach may be made without using a differential equation, but treating the problem completely statically. Such an approach was first made by Yamamoto and Teramoto as early as 1952 3). These authors used the Ursell-Mayer expansion method and evaluated the first order coefficient which agreed with Grimley s result. This method has been pursued by Yamakawa and Kurata and also by Fixman. 

We shall follow the same Ursell-Mayer expansion method. However, different from the previous theories, we shall not use the Gausdan chain approximation. In addition, we shall develop the theory for arbitrary inter-s mental intoactions. Althou a pearl-necklace model wiU be adopted for final results and especially for comparison with other results, our theory is rigorous and is applicable to chains of arbitrary lengths and interactions. Thus, even apart from applications the theory has its own merits. Hi tact, it is clear, as go higher orders, the Gansaan chain approximation becomes not applicable. 

Coulomb potential multiplied by -p. The graphical representation of the virial coefficients in temis of Mayer/ -bonds can now be replaced by an expansion in temis ofy bonds and Coulomb bonds ). 

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

Theoretical investigations of quenched-annealed systems have been initiated with success by Madden and Glandt [15,16] these authors have presented exact Mayer cluster expansions of correlation functions for the case when the matrix subsystem is generated by quenching from an equihbrium distribution, as well as for the case of arbitrary distribution of obstacles. However, their integral equations for the correlation functions... 

J. R. Mayer (1842) made the first calculation of the mechanical equivalent of heat by comparing the work done on expansion of air with the heat absorbed. 

For ionic defects the individual terms in the formal virial expansions diverge just as they do in ionic solution theory. The essence of the Mayer theory is a formal diagram classification followed by summation to yield new expansions in which individual terms are finite. The recent book by Friedman25 contains excellent discussions of the solution theory. We give here only an outline emphasizing the points at which defect and solution theories diverge. Fuller treatment can be found in Ref. 4. 

In the preceding paragraphs of this section we have summed the terms arising from the partial expansion of the exponentials occurring in the coefficients of the powers of particle concentrations to obtain a series of multiple infinite sums, the terms of which are convergent. The terms in S(R) are of the same form as those in the Mayer solution theory, apart from replacement of integration by summation and the fact that mu differs from the solution value because of the discreteness of the lattice. The evaluation of wi - is outlined in the next section. It is found that the asymptotic form is... 

The establishment of a formula for the a functions essentially involves solving the order-disorder problem in a suitable notation. Mayer s method is similar to that discussed by Domb and Hiley201 following earlier work of Rushbrooke and Scoins78 and Fournet20 . We shall not discuss it in detail, but it may help to clarify the difference between the expansions of Section IV and that above by considering the evaluation of (cf. Eq. (78))... 

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

Note added in -proof. The application of the usual integral equation theories of the liquid state 2> to water has not been successful.1) A recent study by H. C. Andersen [J. Chem. Phys. 61, 4985 (1974)] promises to change this situation. Briefly, Andersen reformulates the well known Mayer cluster expansion of the distribution function 2> by consistently taking into account the saturation of interaction characteristic of hydrogen bonding. Approximations are selected which satisfy this saturation condition at each step of the analysis. Preliminary calculations (H. C. Andersen, private communication) indicate that even low order approximations that preserve the saturation condition lead to qualitative be-... 

Thermodynamic perturbation theory is used to expand the Boltzmann distribution in the dipolar interaction, keeping it exact in the magnetic anisotropy (see Section II.B.l). A convenient way of performing the expansion in powers of is to introduce the Mayer functions fj defined by 1 +fj = exp( cOy), which permits us to write the exponential in the Boltzmann factor as... 

Mulliken s formula for Nk implies the half-and-half (50/50) partitioning of all overlap populations among the centers k,l,... involved. On one hand, this distribution is perhaps arbitrary, which invites alternative modes of handling overlap populations. On the other hand, Mayer s analysis [172,173] vindicates Mulliken s procedure. So we may suggest a nuance in the interpretation [44] departures from the usual halving of overlap terms could be regarded as ad hoc corrections for an imbalance of the basis sets used for different atoms. But one way or another, the outcome is the same. It is clear that the partitioning problem should not be discussed without explicit reference to the bases that are used in the LCAO expansions. 

A more practical discussion is given in Section 2.3. At this point, let us mention the Mayer cluster expansion technique originally applied to the imperfect gas [J.E. Mayer, M. Mayer (1940)] but to which Allnatt and Lidiard [A. R. Allnatt, A.B. Lidiard (1993)] have drawn attention in the present context. In this approach, In 2... 

Note that the introduction of structural conditions and site exclusions suffices to obtain (apparent) interaction parameters, which differs from the concept of the Mayer cluster expansion approach. 

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