  The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show
that the displacements of small particles affected by swimming microorganisms
achieve a non-Gaussian distribution, which nevertheless scales diffusively --
the 'diffusive scaling.' We use a simple model where the particles undergo
repeated 'kicks' due to the swimmers to explain the shape of the distribution
as a function of the volume fraction of swimmers. The net displacement is
determined by the inverse Fourier transform of a single-swimmer characteristic
function. The only adjustable parameter is the strength of the stresslet term
in our spherical squirmer model. We give a criterion for convergence to a
Gaussian distribution in terms of moments of the drift function, and show that
the experimentally-observed diffusive scaling is a transient related to the
slow crossover of the fourth moment from a ballistic to a linear regime with
path length. We also present a simple model, with logarithmic drift
function,that can be solved analytically.
