  Swimming bacteria create long-range velocity fields that stir a large volume
of fluid and move around passive particles dispersed in the fluid. Recent
experiments and simulations have shown that long-time mean-squared displacement
of passive particles in a bath of swimming bacteria exhibits diffusive
behaviour with the effective diffusion coefficient significantly larger than
its thermal counterpart. Comprehensive theoretical prediction of this effective
diffusion coefficient and understanding of the enhancement mechanism remain a
challenge. Here, we adapt the kinetic theory by Lin et al., J. Fluid Mech. 669,
167 (2011) developed for 'squirmers' to the bacterial case to quantitatively
predict enhanced diffusivity of tracer particles in dilute two- and
three-dimensional suspensions of swimming bacteria. We demonstrate that the
effective diffusion coefficient is a product of the bacterial number density,
their swimming speed, a geometric factor characterising the velocity field
created by a single bacterium, and a numerical factor. We show that the
numerical factor is, in fact, a rather strong function of the system
parameters, most notably the run length of the bacteria, and that these
dependencies have to be taken into account to quantitatively predict the
enhanced diffusivity. We perform molecular-dynamics-type simulations to confirm
the conclusions of the kinetic theory. Our results are in a good agreement with
the values of enhanced diffusivity measured in recent two- and
three-dimensional experiments.
