Perspectives on the mode-coupling approximation for the dynamics of interacting Brownian particles

Abstract

We analyse the field theory for the dissipative dynamics of interacting inertialess particles, using a dynamic density-wave model (Bagchi 1987Physica A 145 273), including thermal fluctuations (Dean 1996 J. Phys. A: Math. Gen. 29 L619; Fuchizaki and Kawasaki 1999 Physica A 266 400). The symmetries of the problem lead to Ward identities relating vertex functions of different orders. We show that in this theory self-consistent one-loop expansions must involve three-point functions in addition to two-point functions. We obtain correlation and response functions in a one-loop self-consistent approximation similar to, but distinct from, standard mode-coupling theories, and discuss the role of vertex renormalizations. We present a resolution of the conflict between the one-loop approximation and the fluctuation-dissipation theorem, compare our results with those of other approaches, and obtain a modified equation for the non-ergodicity parameter for this model.