An example the second virial coefficient

The reduction to cumulant s tremendously reduces the number of diagrams to be explicitly considered. Furthermore, each cumulant yields a single factor of the volume i of the system. The cunmlants therefore will play a prominent role in our construction of the thermodynamic limit Q — qg for a many-chain svstem, 

As a first application we calculate the second virial coefficient of the osmotic pressure for a mono disperse solution nm = n. According to Eq. (1,1) it is defined as 

The evaluation of this expression makes for an instructive example of the manipulation of diagrams. 

In the diagrams of the second line of Fig. 4,12 the vertex connecting the polymer lines does not carry momentum. Therefore the two propagators attached to the upper end of this vertex (distinguished point f) combine to yield a single propagator 

The index f survives only as part of the ordered summation. Adding the three diagrams we sum f over all those segments of its chain not occupied by the other interaction. This yields a factor of n — 3. The remaining contribution of the upper polymer line is identical to the first order contribution to 3 (0,0 n). The sum of these three diagrams therefore is found as 

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