Analysis of certain binary collision approximation closures of the BBGKY hierarchy

The closure of the BBGKY hierarchy to obtain the Boltzmann equation requires, in particular, restricting particle interactions to include only isolated binary collisions. Boercker and Dufty accomplish this by approximating the three-particle reduced density operator in a particular 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 completely neglect any reference to three-particle effects while assuming a generalized form of molecular chaos for the pair density operator. The two closures are compared in two ways: (a) by finding iterated series 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 Boltzmann hierarchies is the effective instantaneousness of binary collisions in the latter. The form for the correction found is shown to vanish for either closure provided the instantaneousness of the binary collisions is imposed. It is shown moreover that the correction is closely 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 approximation analogous to that of Boercker and Dufty.