Linear Response Theory of Scale-Dependent Viscoelasticity for Overdamped Brownian Particle Systems

TL;DR

This paper develops a linear response framework for understanding how the relaxation properties of overdamped Brownian particles depend on spatial scale, using stress tensor correlations and Fourier analysis.

Contribution

It introduces a scale-dependent relaxation modulus tensor expressed via stress correlations, providing explicit formulas for simple Brownian models.

Findings

Derived explicit expressions for relaxation moduli in simple models

Connected stress tensor correlations to relaxation behavior

Identified shear and bulk relaxation moduli as key components

Abstract

We show the linear response theory of spatial-scale-dependent relaxation moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood stress tensor field as the microscopic stress tensor field. We show that the scale-dependent relaxation modulus tensor, which characterizes the response of the stress tensor field to the applied velocity gradient field, can be expressed by using the correlation function of the Irving-Kirkwood stress tensor field. The spatial Fourier transform of the relaxation modulus tensor gives the wavenumber-dependent relaxation modulus. For isotropic and homogeneous systems, the relaxation modulus tensor has only two independent components. The transverse and longitudinal deformation modes give the wavenumber-dependent shear relaxation modulus and the wavenumber-dependent bulk relaxation modulus. As simple examples, we derive the explicit expressions for the relaxation moduli for two simple models the non-interacting Brownian particles and the harmonic dumbbell model.