Dancing to the swimmers' beat: Loopy Lévy flights enhance tracer diffusion in active suspensions

The diffusion process followed by a tracer in a medium out of equilibrium typically exhibits anomalous diffusion that cannot be modelled by Brownian motion. Prototypical active media are suspensions of swimming microorganisms like algae and bacteria, where the tracer is dragged by the hydrodynamic flow generated by the swimmers. Several experiments have characterised the tracer diffusion in dilute conditions by a greatly enhanced diffusion coefficient, non-Gaussian tails of the displacement statistics, and crossover scaling phenomena. Despite the abundant experimental results, there is so far no comprehensive theory that can describe all these features. Here we present a theoretical framework of the enhanced tracer diffusion from first-principles, by coarse-graining the microscopic tracer-swimmer interactions as a coloured Lévy Poisson process. This theory not only provides the toolkit necessary to characterise theoretically the tracer diffusion but also paves the way to the study of its stochastic thermodynamics.