Figure 2.

Velocity probability density function (PDF) of a tracer particle in the flow generated by different concentrations of swimmers. The solid curves are based on approximation (3.3), using the exact second and fourth moments of the velocity PDFs, as shown in the appendices. (a) For the co-oriented model (2.4) with long-range hydrodynamics n = 1, the velocity PDFs from simulations (symbols) converge rapidly to the Gaussian distribution predicted by the central limit theorem (solid curves), even at low volume fractions φ ≃ 0.4%. (b) By contrast, for the co-oriented model with n = 2, the central limit theorem convergence is very slow and the velocity distribution exhibits strongly non-Gaussian features at volume fractions similar to those realized by recent experiments [6]. (c) The velocity PDF for the dipolar swimmer model looks very similar to that of our co-oriented model (b), which means the angular dependence does not play an important role for the velocity distribution. Simulation parameters are κ = 0.4875, V = 50 µm s⁻¹, and ε = 5 µm. The sample size is 4 × 10⁶ throughout. Red circles or solid line, ϕ = 0.4%; orange squares or solid line, ϕ = 0.8%; green triangles or solid line, ϕ = 1.6%; cyan inverted traingles or solid line, ϕ = 3.2%. In figure 7, the one-dimensional data of (a) and (b) are displayed as a semilog plot.