Interplay between Brownian and hydrodynamic tracer diffusion in suspensions of swimming microorganisms

The general problem of tracer diffusion in non-equilibrium baths is important in a wide range of systems, from the cellular level to geographical lengthscales. In this paper, we revisit the archetypical example of such a system: a collection of small passive particles immersed in a dilute suspension of non-interacting dipolar microswimmers, representing bacteria or algae. In particular, we consider the interplay between thermal (Brownian) diffusion and hydrodynamic (active) diffusion due to the persistent advection of tracers by microswimmer flow fields. Previously, it has been argued that even a moderate amount of Brownian diffusion is sufficient to significantly reduce the persistence time of tracer advection, leading to a significantly reduced value of the effective active diffusion coefficient $D_A$ compared to the non-Brownian case. Here, we show by large-scale simulations and kinetic theory that this effect is in fact only practically relevant for microswimmers that effectively remain stationary while still stirring up the surrounding fluid, so-called \emph{shakers}. In contrast, for moderate and high values of the swimming speed $v_s$, relevant for biological microswimmer suspensions, the effect of Brownian motion on $D_A$ is negligible, leading to the effects of advection by microswimmers and Brownian motion being additive. This conclusion contrasts with previous results from the literature, and encourages a reinterpretation of recent experimental measurements of $D_A$ for tracer particles of varying size in bacterial suspensions.