Distribution function virial expansion

In this section we shall explain somewhat the results which we have just presented. We are interested this time in the evolution equation for the one-particle distribution function. We write down the virial series expansion of the transport equation and we recall that every contribution to this equation is proportional to V n+d, where n is the number of particles which are involved... 

Equation (95) is obtained from the virial expansion of the equation of state for rigid spheres for higher densities the rigid-sphere equation of state obtained from the radial distribution function by Kirkwood, Maun, and Alder has to be used (K10, Hll, p. 649). When Eq. (95) is substituted in Eqs. (92), (93), and (94) one then obtains the rigorous expressions for the coefficients of viscosity, thermal conductivity, and selfdiffusion of a gas composed of rigid spheres. 

The virial expansions of these distribution functions [135] are given by... 

Time-dependent correlation functions. Similar pair and triplet distributions, which describe the time evolution of a system, are also known [318]. These have found interesting uses for the theory of virial expansions of spectral line shapes, pp. 225 ff. below [297, 298],... 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

Section 2 brings the cluster development for the osmotic pressure. Section 3 generalizes the approach of Section 2 to distribution functions, including a new and simple derivation of the cluster expansion of the pair distribution function. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 contains an application of our general solution theory to compact macromolecular molecules. Section 6 contains the second osmotic virial coefficient of flexible macromokcules, followed in Section 7 by concluding remarks. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

We round out this introduction to the virial equation of state by reference to its theoretical foundation. Thus statistical mechanics permits deduction of an expression for pVin terms of either the grand partition function or the radial distribution function. The leading term in the expansion of the latter function corresponds to pairwise interaction between molecules, and indicates the following relation between the second virial coefficient and the potential energy (r) of the interacting pair, when this depends only on the distance r between molecular centres ... 

As attractive and convenient as virial expansions are, the problem is that they do not converge at liquid densities. It turns out that a many-body approach is required for a theoretical study of liquids, and we shall develop the standard statistical mechanical n-body distribution function formalism in the next section. 

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... 

The procedure adapted by Mayer to derive the free energies of ionic solutions from the virial expansion has been applied to the pair-correlation functions by Meeron (1957). The pair-distribution function can then be written as... 

As the density of the fluid is increased above values for which a linear term in the expansion of equation (5.1) is adequate (crudely above values for which a third virial coefficient is adequate to describe the compression factor of a gas), the basis of even the formal kinetic theory is in doubt. In essence, the difficulty arises because it becomes necessary, at higher densities, to consider the distribution function, in configuration and momentum space, of pairs, triplets etc. of molecules in order to formulate an equation for the evolution of the single-particle distribution function. Such an equation would be the generalization to higher densities of the Boltzmann equation, discussed in Chapter 4 (Ferziger Kaper 1972 Dorfman van Beijeren 1977). 

A chain conformational renormalization group method has been developed by Miyaki and Freed " for star polymers. Explicit expressions for the distribution functions for intersegment distance vectors and their moments, b, the osmotic second virial coefficient and the related functions, etc., have been reported based on the 8-expansion approximation. A detailed comparison of these results with those of linear chains, scaling predictions and the experimental data is also reported in the literature. 

Unfortunately, the radial distribution function cannot be determined with sufficient accuracy at all densities. For low density fluids, however, solution is possible through a Taylor s series expansion around zero density, which will lead to the virial equation. 

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