Levy Fluctuations and Tracer Diffusion in Dilute Suspensions of Algae and Bacteria

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.

💡 Research Summary

The paper addresses the puzzling observation that passive tracer particles immersed in dilute suspensions of motile microorganisms, such as the biflagellate alga Chlamydomonas reinhardtii and various bacteria, display anomalous, non‑Gaussian diffusion. Experiments have shown that while the mean‑square displacement (MSD) eventually grows linearly with time, the probability distribution of tracer displacements retains heavy tails far beyond what a simple Brownian model would predict. The authors set out to construct a quantitative theoretical framework that links these heavy‑tailed statistics to the underlying self‑propulsion mechanisms of the swimmers.

Model construction – The authors begin by idealising the swimmer‑tracer interaction as a distance‑dependent flow field generated by a single swimmer. In the far field, the velocity induced by a swimmer decays algebraically as (u(r)\propto r^{-2}) (force‑dipole, “pusher/puller”) or (r^{-3}) (source‑dipole), capturing the essential physics of low‑Reynolds‑number propulsion. A tracer moving through this flow experiences an instantaneous “kick” whose magnitude depends on the encounter distance and swimmer orientation. To describe the statistics of these kicks, the authors adopt a truncated Lévy distribution: a power‑law tail (P(v)\sim v^{-(1+\mu)}) for large velocities, cut off at a finite value set by the system size and swimmer speed. This truncation ensures physical realism while preserving the heavy‑tail character that can generate large, rare displacements.

Temporal correlations – Unlike a Poissonian sequence of independent kicks, the fluid velocity field generated by swimmers exhibits long‑lived correlations. The authors argue that the autocorrelation of the velocity decays algebraically, (C(t)\sim t^{-\alpha}) with (\alpha\approx1), reflecting the persistence of the swimmer‑induced flow after a tracer has been kicked. This algebraic decay means that a large velocity event can influence tracer motion over a finite, but non‑negligible, time window, further enhancing the probability of extreme displacements.

Analytical results – Using stochastic calculus, Laplace transforms, and a path‑integral formulation, the authors derive the asymptotic form of the tracer position probability density function (PDF). The central part of the PDF converges to a Gaussian as expected for normal diffusion, but the tails retain a Lévy‑type power law: \