Topological reduction

The task of finding the one diagram in P that corresponds to a given diagram in M is somewhat more complicated and is essentially the reverse of the above procedure. Let M be a particular diagram in M We look for a pair of reducible points whose residual is a member of N. (See Section 2 for 

Suppose we find two residuals (and the corresponding pairs of reducible points) each of which is a graph in Ny sudi that in M the first residual contains all the points and bonds in the second residual. If we were to remove the second residual and replace it by a G bond, we could no longer remove the first, because the first residual would now contain a G bond and hence could no longer be a member of N. If, instead, we started by removing the first residual and repladng it by a G bond, the second residual would no lon r be there to remove in a subsequent step. The appropriate procedure to follow is the second one, i.e., if one residual in M contains another, replace the larger one by a G bond. (Note that sometimes the two residuals correspond to the same pair of reducible points.) 

These considerations may be summarized in the following way. For a diagram M, we want to find a set of residuals (and their associated pairs of reducible points) such that each residual is a member of V, no residual in the set has any bonds of M in common with another residual in the set, and each residual not in the set that is a member of N must be wholly contained within some member of the set. If there is a unique set of such residuals for M, then these residuals are replaced by G bonds between the pairs of reducible points. The unique diagram obtained in this way is the diagram in P to which M corresponds. This answers the third question above. (If no such unique set exists, the topological reduction cannot be performed.) 

Now that we know how to find the diagram in P corresponding to a particular A/, we can characterize P as the set of all topologically different 

---

The classification and topological reduction of the diagrams for pair correlation functions leads to the Wertheim modification of the Ornstein-Zernike (WOZ) equations [9],... 

The next step in the theory is to make a topological reduction so that the dependence upon the f bonds is replaced by a dependence upon bonds. Lupkowski and Monson show that this topological reduction cannot be carried out unless the reference total correlation functions are decomposed in the manner described in Section III.B in connection with the development of the CSL integral equation. This result is analogous to the derivation of the CSL equation as a topological reduction of the interaction site cluster series for h. They do, however, show that there is subset of diagrams in Eq. 

The standard application of the usual statistical mechanical graph techniques of topological reduction lead to the definition of two correlation functions, Cgfl) and Ci(l), and eventually to an analogue of the Ornstein-Zernike equation. 

Of course the concept of topologically reduction can be also appHed to supercyclic structures these are reducible to simple cycUc structures, as illus-... 

Fig. 15 Cartoons illustrating the topological reduction of a bicyclic assembly to a closed acyclic assembly (a), and of a multicyclic polymeric assembly to an acyclic polymer (b)...

Fig. 16 Cartoons illustrating the topological reduction of a supercyclic assembly to a cyclic architecture...

This concept of a pair of reducible points plays an important role in the process of topological reduction, which will be discussed in Section 4. It should be noted that a pair of reducible points can have more than one residual. Each residual has the property that a path from any point in the residual to a point not in the residual must pass through one of the pair of reducible points. 

We can now summarize conditions needed for the topological reduction described above to be possible ... 

In Section 5, we will describe some strategies that use topological reduction as a tactic, and Section 6 discusses some particular examples of the use of topological reduction. It should also be noted that some of the results stated without proof in Section 3 can be easily proven using this theorem. In particular, Eqs. (21), (26), and (27) can be derived from Eqs. (16) and (17) by considering the topological reductions of (21) to (17), (26) to (17), and (27) to (16). 

If the individual terms in a cluster series are divergent, perform a topological reduction to eliminate the divergence. Then it is usually necessary to resort to strategy 1 or 2 for the new series. 

Even if the individual terms are merely large and not divergent, perform some topological reduction to exploit any systematic cancellation that exists among the various terms. Then use strategy 1 or 2. 

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. 

Figure 6 illustrates some of the features of this topological reduction.) Here V RiNG is the sum of all the ring diagrams. By Fourier transform techniques, it can be shown that, in the limit y 0,... 

The last restriction in this equation is needed to satisfy the conditions of the theorem which allows the topological reduction. In this series, there is one term with no yo 8f bonds and only one term with one yo 8f bond. This latter term has only two field points and just one bond. There are an infinite number of graphs with two yo 8f bonds, and even the series for these terms looks like a low-density virial series which might not be expected to be convergent or meaningful at high densities. 

To obtain a result that looks more respectable at high density, we can now eliminate the /o bonds and introduce Hq bonds by a topological reduction. We obtain ... 

In Eqs. (23)-(25) we see how the separation of the potential leads naturally to the definition of fo and rp bonds. Equations (26) and (27) then express si and g in terms of these new bonds. Our object is to collect together all terms in these series that are of zeroth order in all of first order, etc. We also wish to transform them so that they no longer look like expansions in powers of the density. One step that helps to accomplish these goals is to perform a topological reduction using an ho bond to eliminate the fo bond, as we did in Section 5.3. 

Performing the topological reduction on the series for si is somewhat less straightforward, since graphs that are rings of

[Pg.35]

We shall now use topological reductions to manipulate the cluster expansion for the pair correlation function of such a model fluid. 

There is an extensive cancellation of terms in this series, because of the directionality properties of the hydrogen bond. To specify this cancellation precisely, let us define an overlapping pair of points as a pair of points that are both directly bonded to the same third point by /hb bonds. We now perform a topological reduction to introduce an Cq bond, defined by... 

This topological reduction satisfies the conditions of a slight generalization of the theorem in Section 4, and we then have... 

---