Distribution of particle displacements in biomixing

The experiments of Leptos et al. [1] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively. We use a simple model where the particles undergo repeated ‘kicks’ due to the swimmers to explain the shape of the distribution. Leptos et al. study the microscopic algae Chlamydomonas reinhardthii. They measure experimentally the probability density function (PDF) of tracer displacements, ρXt(x). Thus, ρXt(x) dx is the probability of observing a particle displacement Xt ∈ [x, x+ dx] after waiting a time t. The range of t is chosen small enough that the swimmers are ‘ballistic,’ so their velocity is roughly constant. At zero volume fraction (φ = 0), the distribution ρXt(x) is Gaussian, due solely to molecular diffusivity. For higher number densities, exponential tails appear and the Gaussian core broadens. Leptos et al. fit their the distribution to the sum of a Gaussian and an exponential: