Noisy swimming at low Reynolds numbers

TL;DR

This paper analyzes how small microswimmers transition between active swimming and Brownian motion at low Reynolds numbers, deriving formulas for key behaviors and identifying three characteristic motion regimes.

Contribution

It provides analytical formulas for orientation correlation, velocity, and displacement of a simplified three-sphere swimmer, linking size to motion regimes at low Reynolds number.

Findings

Identified three regimes: Brownian, quasi-ballistic, and quasi-diffusive.

Derived formulas validated by numerical simulations.

Results applicable to bacterial foraging and artificial microswimmer design.

Abstract

Small organisms (e.g., bacteria) and artificial microswimmers move due to a combination of active swimming and passive Brownian motion. Considering a simplified linear three-sphere swimmer, we study how the swimmer size regulates the interplay between self-driven and diffusive behavior at low Reynolds number. Starting from the Kirkwood-Smoluchowski equation and its corresponding Langevin equation, we derive formulas for the orientation correlation time, the mean velocity and the mean square displacement in three space dimensions. The validity of the analytical results is illustrated through numerical simulations. Tuning the swimmer parameters to values that are typical of bacteria, we find three characteristic regimes: (i) Brownian motion at small times, (ii) quasi-ballistic behavior at intermediate time scales, and (iii) quasi-diffusive behavior at large times due to noise-induced orientation flipping. Our analytical results can be useful for a better quantitative understanding of optimal foraging strategies in bacterial systems, and they can help to construct more efficient artificial microswimmers in fluctuating fluids.