Kinetic equations governing Smoluchowski dynamics in equilibrium

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 pseudopotential. 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.