Averages Involving Functions in the Phase Space of Two Molecules Only

3 Averages Involving Functions in the Phase Space of Two Molecules Only 

Another relevant set of average quantities involves interactions between pairs of beads belonging to different molecules. Specifically we will be concerned with 

We start by replacing the Dirac delta function in Eq. (5.12) by the integral of the product of two delta functions  

Because of the delta functions, the b can be regarded as functions of the pj, and hence the index i on by is not needed. The similarly can be regarded as functions of r and rj, and the indices i and j on may now be removed. Next, we perform the integrations over all the x-variables and use the definition for the doublet distribution function given m Eq. (4.6) this gives  

In going from Eq. (5.12) to this expression, all the variables have been integrated out, except for those involving two molecules the integration over the momenta has been indicated symbolically by use of the double-bracket notation. Note that in Eq. (5.16), when a and P are the same, Cf describes interactions between beads of two different molecules of species a. 

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