The probability distribution of a trapped Brownian particle in plane shear flows

TL;DR

This paper analyzes the probability distribution of a trapped Brownian particle in shear flows, providing analytical solutions for linear shear and approximate solutions for Poiseuille flow, validated by numerical results.

Contribution

It offers the first analytical solution for the distribution in linear shear flow and develops perturbation approximations for Poiseuille flow, supported by numerical validation.

Findings

Analytical distribution for linear shear flow obtained.

Perturbation approximations for Poiseuille flow validated numerically.

Good agreement between analytical and numerical results across parameters.

Abstract

We investigate the statistical properties of an over-damped Brownian particle that is trapped by a harmonic potential and simultaneously exposed to a linear shear flow or to a plane Poiseuille flow. Its probability distribution is determined via the corresponding Smoluchowski equation, which is solved analytically for a linear shear flow. In the case of a plane Poiseuille flow, analytical approximations for the distribution are obtained by a perturbation analysis and they are substantiated by numerical results. There is a good agreement between the two approaches for a wide range of parameters.