Approximate solutions of the Liouville equation. IV. The two-body additive approximation

The two-body additive approximation on the time-dependent Liouville distribution, first introduced in part I of this series, is put into the conventional form of a self-contained kinetic equation for the doublet distribution. From this point of view the approximation consists in truncating the BBGKY chain by expressing the triplet distribution as a functional of lower distributions at the same value of the time variable. To accomplish this, it is necessary to study two associated purely spatial integral equations. The doublet kinetic equation can then be written in terms of solutions of these integral equations and comparison with conventional methods of truncating the BBGKY chain can then be made. For the purpose of comparison a method of truncating the chain based on the Kirkwood superposition approximation is introduced and discussed briefly. The momentum structure of the resulting doublet kinetic equation is similar, but the nonlocality in space of our truncation introduces distinct differences in the spatial structure. The inconsistency between conventional truncations and the exact initial conditions used for the calculation of time-dependent correlation functions is pointed out. This inconsistency is not shared by the two-body additive approximation.