Mayer cluster integral

The paper by Montroll and Yu makes use of a general formalism for the investigation of the properties of lattices with two types of component, type I and type II. The formalism is in the spirit of the Mayer cluster integral approach. Let us consider a pure type 1 component lattice with the effect of the type II components imposed upon it. Type II components are considered first as single units, then as pairs, triples, etc. For the DNA problem, the pure lattice is a one-dimensional string of A-T (or G-C) base pairs, witli the G-C (or A-T) base pairs acting as the type II components. 

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... 

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-... 

One of the newer theoretical treatments, based on the pioneering statistical thermodynamic work of McMillan and Mayer (6) y as mathematically formulated by Friedman W, does appear to hold significant promise as a theory of sufficient generality that it may eventually embody other working theories as demonstrated special cases. This theory, known as the cluster integral expansion theory (j ) or simply as cluster theory (9)y has been developed to the point where applications have been made to calculating... 

B(r3, rJBirJ+BMBirJBirJ and so on. Then it can easily be shown that B(rt,..., r,) is different fron zero only when rlt..r, are near together (i.e., in the range of the intermolecular forces) or form a "cluster." Mayer s17 cluster integrals are defined by... 

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.

Singh, J.K., Kofke, D.A. Mayer sampling calculation of cluster integrals using Ifee-energy perturbation methods. Phys. Rev. Lett. 92, 220601 (2004)... 

In the limit y 0, this expression reduces to Eq. (35). For nonzero y, all cluster integrals exist. Following Mayer then, we will perform a topological reduction for nonzero y and then take the limit y 0 at the end of the calculation. 

We also note that for a one component system Mayer s reducible cluster integral is given by... 

Note that (6.23) differs from the expression given in Section 1.8. The latter is obtained from the former if the total potential energy C/3(Xi, X2, X3) is pairwise additive. Equations (6.22) and (6.23) are special cases of a more general scheme which provides relations between virial coefficients and integrals involving interactions among a set of a small number of particles. This is known as the Mayer cluster theory [see, for example, Mayer and Mayer (1940), Hill (1956), and Munster (1969)]. 

If we apply Mayer s theory of condensation to AHS systems, we find that in the cluster expansions (7.13a) for the molar volume v and (7.13b) for the pressure p. The coefficients (cluster integrals) bi are constmcted by the special form... 

Mayer showed that the two-body reduced coordinate distribution function can be expressed as a series of powers and logarithms of the density, with a second type of cluster integrals as coefficients. ... 

In a general theory of solutions, McMillan and Mayer demonstrated the formal equivalence between the pressure of a gas and the osmotic pressure of a solution. Hence the ratio of the osmotic pressure O of a dilute solution to the concentration (number density) p of the solute can be expanded in a power series in p and the coefficients of the series can be expressed, as in the theory of a real gas, in terms of cluster integrals determined by intermolecular potential energy functions. The only difference is, as already mentioned, that in the solution these potentials are effective potentials of average force, which include implicitly the effects of the solvent, modelled as a continuum. 

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... 

The products in the integrand correspond to the ordered sequence of the bond vectors. The evaluation of the right hand side integral may be made following Mayer s cluster theory of imperfect gases. We arrive at a cluster series... 

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