Dynamics of interacting Brownian particles: a diagrammatic formulation

TL;DR

This paper develops a diagrammatic theoretical framework for analyzing the time evolution of density fluctuations in equilibrium interacting Brownian particles, connecting it with mode-coupling theory.

Contribution

It introduces a diagrammatic formulation using orthogonalized density fluctuations and derives a self-consistent approximation equivalent to mode-coupling theory.

Findings

Derived an exact hierarchy of equations of motion.

Provided a diagrammatic interpretation of memory functions.

Showed the one-loop approximation matches mode-coupling results.

Abstract

We present a diagrammatic formulation of a theory for the time dependence of density fluctuations in equilibrium systems of interacting Brownian particles. To facilitate derivation of the diagrammatic expansion we introduce a basis that consists of orthogonalized many-particle density fluctuations. We obtain an exact hierarchy of equations of motion for time-dependent correlations of orthogonalized density fluctuations. To simplify this hierarchy we neglect contributions to the vertices from higher-order cluster expansion terms. An iterative solution of the resulting equations can be represented by diagrams with three and four-leg vertices. We analyze the structure of the diagrammatic series for the time-dependent density correlation function and obtain a diagrammatic interpretation of reducible and irreducible memory functions. The one-loop self-consistent approximation for the latter function coincides with mode-coupling approximation for Brownian systems that was derived previously using a projection operator approach.