Stress correlation function and linear response of Brownian particles

TL;DR

This paper derives the nonlocal stress autocorrelation tensor for Brownian particles, connecting microscopic dynamics to macroscopic viscoelastic responses, and recovers elastic and viscous behavior in different limits.

Contribution

It introduces a comprehensive framework that includes transverse contributions and extends existing theories to relate stress correlations to particle displacements in viscoelastic states.

Findings

Derived the stress autocorrelation tensor including elastic terms.

Connected the memory function to shear and bulk viscosities.

Formulated hydrodynamic equations from microscopic stress correlations.

Abstract

We determine the nonlocal stress autocorrelation tensor in an homogeneous and isotropic system of interacting Brownian particles starting from the Smoluchowski equation of the configurational probability density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we go beyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection operator formalism by introducing the proper irreducible dynamics following Cichocki and Hess, and Kawasaki. Differently from these authors, we include transverse contributions as well. This recovers the expression for the stress autocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, in case that those are evaluated in the overdamped limit. Finally, we argue that the found memory function reduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations and derive the corresponding hydrodynamic equations.