Dynamical Evolution of a Gas with Short-Range Repulsive Forces

To develop from first principles the kinetic theory for a dense gas with short-range potentials, Boltzmann's collision integral must be generalized to include interactions among more than two bodies. This generalization requires knowledge of the behavior of the $s$-particle distribution functions for times long compared with the duration of a collision. For a spatially homogeneous gas, we obtain the asymptotic formula for the $s$-particle distribution functions expanded in powers of density. The functions are obtained for arbitrary order $\mu$ in the expansion. Mild restrictions are imposed on the initial distributions, but strong restrictions are needed on the repulsive interparticle potential. This expansion is in many respects the nonequilibrium analog of the virial expansion of Ursell and Mayer for equilibrium. We prove that the asymptotic behavior of the $\mu \mathrm{th}$-order distribution function is given by a polynomial in time whose leading power is $\mu$. The constant term in the asymptotic expression can be interpreted in terms of reversible evolution in the gas. The growing (or "secular") terms of the polynomial are directly responsible for the irreversible evolution, since the secular terms describe changes that occur during the relaxation to equilibrium. The lowest secularity for the one-particle distribution function coincides with a result of Bogoliubov and generates the Boltzmann collision integral. This secular term corresponds to the phase mixing of two particles that have undergone a completed collision. The coefficient of time can be expressed as a scattering cross section averaged over the velocity distribution. We show that the higher secularities describe the phase mixing of two particles that have undergone a number of interactions, and their coefficients can also be expressed as averaged cross sections. A major technical result that follows is a complete classification of the secularities that arise in any distribution function and their physical interpretation. The collisional contributions that are required for the construction of the higher-order kinetic equations are thereby isolated.