Graphical Representation of the Grand Partition Function

Using the above definitions, the grand partition function for a one-component system may be expressed as a sum of simple graphs on black z circles connected by bonds, which are defined by 

In the special case of strictly pairwise interactions Eq. (77) becomes [cf. Eq. (49)] 

Qiister expansion of S. Wf bonds connect N black z circles. 

For example, in the case of three components of species a, 6, and c the second graph in Fig. 5 is given by 

Since the graphical representations have the same form for single- or multicomponent systems, it is sufficient to derive the rules for representing various statistical-mechanical functions in terms of graphs for just the simple single-component case. The methods used in this section all apply to many-component systems even when the many-component character is represented explicitly in the analytical expressions, but the notation becomes quite complicated. 

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