A Rheological Analogue for Brownian Motion with Hydrodynamic Memory

TL;DR

This paper introduces a rheological analogue model for Brownian motion with hydrodynamic memory, simplifying calculations of particle displacement by linking fractional derivatives to a mechanical network.

Contribution

It presents a novel mechanical network model using a fractional Scott-Blair element and an inerter to represent hydrodynamic memory in Brownian motion.

Findings

The model captures long-range force correlations in Brownian particles.

Mean-square displacement calculations are simplified using the network.

Results are expressed with Mittag-Leffler functions.

Abstract

When the density of the fluid surrounding suspended Brownian particles is appreciable, in addition to the forces appearing in the traditional Ornstein and Uhlenbeck theory of Brownian motion, additional forces emerge as the displaced fluid in the vicinity of the randomly moving Brownian particle acts back on the particle giving rise to long-range force correlations which manifest as a ``long-time tail'' in the decay of the velocity autocorrelation function known as hydrodynamic memory. In this paper, after recognizing that for Brownian particles immersed in a Newtonian, viscous fluid, the hydrodynamic memory term in the generalized Langevin equation is essentially the 1/2 fractional derivative of the velocity of the Brownian particle, we present a rheological analogue for Brownian motion with hydrodynamic memory which consists of a linear dashpot of a fractional Scott-Blair element and an inerter. The synthesis of the proposed mechanical network that is suggested from the structure of the generalized Langevin equation simplifies appreciably the calculations of the mean-square displacement and its time-derivatives which can also be expressed in terms of the two-parameter Mittag--Leffler function.