ANALYSIS OF CERTAIN BINARY COLLISION APPROXIMATION CLOSURES OF THE BBGKY HIERARCHY

The closure of the BBGKY hierarchy to obtain the Boltzmann equation re quires, in particular, restricting particle interactions to include on ly isolated binary collisions, Boercker and Dufty accomplish this by a pproximating the three-particle reduced density operator in a particul ar manner that favours correlation between two of the particles, while ignoring the correlation with the third. The tradition of most other closures has more closely followed Boltzmann's original thinking to co mpletely neglect any reference to three-particle effects while assumin g a generalized form of molecular chaos for the pair density operator. The two closures are compared in two ways: (a) by finding iterated se ries solutions of the BBGKY hierarchy and of the Boltzmann equation; ( b) by computing an exact correction to the quantum Boltzmann equation. A consequence of the comparison of the iterated series shows that an important, but little emphasized, difference between the BBGKY and Bol tzmann hierarchies is the effective instantaneousness of binary collis ions in the latter. The form for the correction found is shown to vani sh for either closure provided the instantaneousness of the binary col lisions is imposed. It is shown moreover that the correction is closel y related to the three-body collision integral arising in the standard theory of the density corrections to the Boltzmann equation. We also comment on the related work of Klimontovich, who introduces an approxi mation analogous to that of Boercker and Dufty.