Non-normal tracer diffusion from stirring by swimming microorganisms

The experimentally observed non-Gaussian form of passive tracer distributions in media stirred by active swimmers (Leptos et al., Phys. Rev. Lett. 103, 198103 (2009)) are analyzed in terms of continuous time random walks. The walks are characterized by a trapping time distribution ψ(τ) with long time behaviour ψ(τ) is proportional to τ (-1-α) and a step size distribution p(Δx) is proportional to (Δx)(-2-β). The experimentally observed behaviour that 〈x (2)〉is proportional to t is obtained for a one-parameter family of exponents with β = 2α. However, the distribution function for this case is non-Gaussian and shows exponential tails. The shape of the distributions agrees rather well with the experimental observations from Leptos et al. and allows for the determination of the exponents.