Spectral Analysis of Brownian Motion with its Rheological Analogues

TL;DR

This paper develops a rheological analogue model to analyze the power spectrum of Brownian motion in viscoelastic materials, linking it to the real part of the complex dynamic fluidity and simplifying calculations for various complex fluids.

Contribution

It introduces a novel rheological analogue involving an inerter to simplify the spectral analysis of Brownian motion in viscoelastic media.

Findings

Power spectrum proportional to real part of complex fluidity.

Simplifies analysis for Maxwell, Jeffreys, and subdiffusive fluids.

Provides a unified framework for diverse viscoelastic materials.

Abstract

The power spectrum of the Brownian motion of probe microparticles with mass m and radius R immersed in a viscoelastic material reveals valuable information about repetitive patterns and correlation structures that manifest in the frequency domain. In this paper, we employ a viscous viscoelastic correspondence principle for Brownian motion and we show that the power spectrum of Brownian motion in any linear, isotropic viscoelastic material is proportional to the real part of the complex dynamic fluidity of a linear rheological network that is a parallel connection of the linear viscoelastic material within which the Brownian particles are immersed and an inerter, with distributed intrance with mass mR. The synthesis of this rheological analogue simplifies appreciably the calculation of the power spectrum for Brownian motion within viscoelastic materials such as Maxwell fluids, Jeffreys fluids, subdiffusive materials, or in dense viscous fluids that give rise to hydrodynamic memory.