  We continue our study of the statistical properties of particles in
equilibrium obeying Smoluchowski dynamics. We show that the system is governed
by a kinetic equation of the memory function form and that the memory function
is given by one of the self-energies available via perturbation theory as
introduced in previous work. We determine the memory function explicitly to
second-order in an expansion in a pseudo-potential. The method we use allows
for a straightforward computation of corrections via a formal expansion and we
therefore view it as an improvement over the conventional mode-coupling theory
(MCT) formalism where it is not clear how to make systematic corrections. In
addition, the formalism we have introduced is flexible enough to allow for a
wide array of different approximation schemes, including density expansions.
The convergence criteria for our formal series are not worked out here, but the
second order equation that we derive is promising in the sense that it leads to
analytic and numerical results consistent with expectations from computer
simulations of the hard sphere system in addition to replicating the desired
features from conventional MCT (e.g., a two-step decay). These particular
solutions will be discussed in forthcoming work.
