Negative drag route to anomalous diffusion in active fluids

Various experiments have reported that a tracer immersed in a baterial bath initially exhibits a non-ballistic superdiffusion. The origin of this behavior has been variously ascribed to hydrodynamic interactions among the bacteria or to the direct contact between each bacterium particle and the tracer. In this study, we present a simple model of the tracer dynamics in one dimension which shows that the non-ballistic superdiffusion can be induced entirely without the hydrodynamic effects. We find that the swimming bacteria can exert a "negative" drag force on the tracer in the sense that the drag force applies in the direction of the tracer's velocity. This implies that the bacteria can impart their motility to symmetric tracers through a spontaneous symmetry-breaking mechanism. At the onset of this symmetry-breaking tracer motility, the critical slowdown effects give rise to a non-ballistic superdiffusion with a power-law exponent close to those reported in experiments. While our arguments are based on a mean-field analysis of the one-dimensional model, we also numerically check their validity in more general situations, including the two-dimensional case typically encountered in experiments.