Three-Body Scattering Operator in Nonequilibrium Statistical Mechanics

A method is presented for calculating the time-dependent irreducible clusters, ${\beta }_s\mathrm{}\left(t\right)\mathrm{}$ which appear in the kernel of the equation of evolution derived in the preceding article. The clusters ${\beta }_1\mathrm{}\left(t\right)\mathrm{}$ and ${\beta }_2\mathrm{}\left(t\right)\mathrm{}$—which correspond to binary and ternary collisions, respectively—are calculated in detail. They are each found to divide into the two following parts: (1) a "completed" collision part which corresponds to collisions which are eventually completed (scattering processes) and (2) an "incompleted" part which corresponds to those collisions not completed by time $t$. The incompleted collision parts contribute to the "memory" of the equation of evolution and are shown to be relatively small when $t$ is large. The completed collision parts, which play a central role in the theory of transport coefficients, are time-independent scattering operators in momentum space and do not contribute to the memory. By means of the "binary-collision expansion" a systematic method is presented for the calculation of the three-body scattering operator [${\lim }_{t\to \infty }{t}^{-1\mathrm{}}{\beta }_2\mathrm{}\left(t\right)\mathrm{}$] which is directly applicable to interaction forces with infinite repulsions. An approximate formula is then derived for this scattering operator in a form which can be readily used to calculate the density correction to transport coefficients which arise from ternary collisions.