L\'evy Fluctuations and Tracer Diffusion in Dilute Suspensions of Algae and Bacteria

TL;DR

This paper develops a theoretical model explaining the non-Gaussian, anomalous diffusion of passive tracers in dilute suspensions of swimming microorganisms, linking flow field statistics to self-propulsion mechanisms.

Contribution

It introduces a simplified tracer-swimmer interaction model that captures the flow field scaling and explains the origin of non-Gaussian tracer distribution tails.

Findings

Non-Gaussian tails arise from truncated Lévy velocity statistics.

Algebraic decay of fluid velocity correlations explains anomalous diffusion.

Simulations confirm the theoretical predictions.

Abstract

Swimming microorganisms rely on effective mixing strategies to achieve efficient nutrient influx. Recent experiments, probing the mixing capability of unicellular biflagellates, revealed that passive tracer particles exhibit anomalous non-Gaussian diffusion when immersed in a dilute suspension of self-motile Chlamydomonas reinhardtii algae. Qualitatively, this observation can be explained by the fact that the algae induce a fluid flow that may occasionally accelerate the colloidal tracers to relatively large velocities. A satisfactory quantitative theory of enhanced mixing in dilute active suspensions, however, is lacking at present. In particular, it is unclear how non-Gaussian signatures in the tracers' position distribution are linked to the self-propulsion mechanism of a microorganism. Here, we develop a systematic theoretical description of anomalous tracer diffusion in active suspensions, based on a simplified tracer-swimmer interaction model that captures the typical distance scaling of a microswimmer's flow field. We show that the experimentally observed non-Gaussian tails are generic and arise due to a combination of truncated L\'evy statistics for the velocity field and algebraically decaying time correlations in the fluid. Our analytical considerations are illustrated through extensive simulations, implemented on graphics processing units to achieve the large sample sizes required for analyzing the tails of the tracer distributions.