Projection onto Pair Configuration Space

The general pair phase-space kinetic equation is given in (7.32). Letting 

An equation with the form of the sink Smoluchowski equation may now be derived by introducing an operator that projects out the velocities of the solute molecules. This is just the pair version of the derivation presented in the previous section. 

An equation for P follows directly upon integration of (9.2) over primed velocities and application, of projection operator techniques. We find 

This equation may be reduced further. First, explicitly evaluating L (12 z) o, we find 

The second term in the brackets in (9.5) may also be reduced further. Using number conservation in elastic collisions, one may show that 

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