Kinetic Equations Governing Smoluchowski Dynamics in Equilibrium

TL;DR

This paper derives a systematic kinetic equation for particles obeying Smoluchowski dynamics in equilibrium, improving upon mode-coupling theory by enabling controlled corrections and broader approximation schemes.

Contribution

It introduces a flexible formalism for kinetic equations with explicit second-order solutions, enhancing the systematic correction capability over traditional mode-coupling theory.

Findings

Second-order equations match simulation results

Formalism allows systematic corrections

Reproduces key features of MCT, like two-step decay

Abstract

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.