Distribution of particle displacements due to swimming microorganisms

TL;DR

This paper models how swimming microorganisms cause non-Gaussian particle displacements that exhibit diffusive scaling, explaining experimental observations through a simple stochastic model and analyzing the transition to Gaussian behavior.

Contribution

It introduces a minimal model for particle displacements due to microorganisms, linking the shape of the distribution to swimmer volume fraction and providing criteria for Gaussian convergence.

Findings

Displacements follow a non-Gaussian distribution with diffusive scaling.

The model explains the transient nature of the non-Gaussian distribution.

Analytical solutions are provided for specific drift functions.

Abstract

The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively -- the 'diffusive scaling.' We use a simple model where the particles undergo repeated 'kicks' due to the swimmers to explain the shape of the distribution as a function of the volume fraction of swimmers. The net displacement is determined by the inverse Fourier transform of a single-swimmer characteristic function. The only adjustable parameter is the strength of the stresslet term in our spherical squirmer model. We give a criterion for convergence to a Gaussian distribution in terms of moments of the drift function, and show that the experimentally-observed diffusive scaling is a transient related to the slow crossover of the fourth moment from a ballistic to a linear regime with path length. We also present a simple model, with logarithmic drift function,that can be solved analytically.