Non-Gaussian limit fluctuations in active swimmer suspensions

We investigate the hydrodynamic fluctuations in suspensions of swimming microorganisms (Chlamydomonas) by observing the probe particles dispersed in the media. Short-term fluctuations of probe particles were superdiffusive and displayed heavily tailed non-Gaussian distributions. The analytical theory that explains the observed distribution was derived by summing the power-law-decaying hydrodynamic interactions from spatially distributed field sources (here, swimming microorganisms). The summing procedure, which we refer to as the physical limit operation, is applicable to a variety of physical fluctuations to which the classical central limiting theory does not apply. Extending the analytical formula to compare to experiments in active swimmer suspensions, we show that the non-Gaussian shape of the observed distribution obeys the analytic theory concomitantly with independently determined parameters such as the strength of force generations and the concentration of Chlamydomonas. Time evolution of the distributions collapsed to a single master curve, except for their extreme tails, for which our theory presents a qualitative explanation. Investigations thereof and the complete agreement with theoretical predictions revealed broad applicability of the formula to dispersions of active sources of fluctuations.

Figure 1

(a) Chlamydomonas were dispersed and sealed in a glass-bottom dish together with microstirrer bars. (b) A single Chlamydomonas was randomly inserted into a spherical volume, and a probe was placed at the origin. A pair of balanced forces $\mathit{F}$ and $-\mathit{F}$ were exerted on the surrounding medium and form a force dipole. The swimming direction is represented by $\widehat{\mathit{n}}$.

Figure 2

(a) Theoretical curves of normalized $W\left(v_x\right)$ versus $v_x/\sigma$. Plots with the same $c{R}_{\min }^3$ are shown in the same color and collapse to the same curve. The predicted curves for $c{R}_{\min }^3\ll 1$ approach a heavily tailed Lévy distribution that is characterized with its power-law tail $\propto u_x^{-5/2}$ (broken curve), whereas those corresponding to $c{R}_{\min }^3\gg 1$ converge to Gaussian distributions (dash-dotted curve). (b) Schematic illustration of $c{R}_{\min }^3$ that indicates the numbers of Chlamydomonas in the $R_{\min }^3$ volumes, which are indicated by the dashed circles.

Figure 3

(a) Velocity distribution of Chlamydomonas. (b) MSD of probes $(2a=5\mu \mathrm{m})$ in Chlamydomonas suspensions. Superdiffusive motion at small $\mathrm{\Delta }t$ tends to become diffusive (MSD $\propto \mathrm{\Delta }t$) at longer time periods. (c), (d) Displacement distributions of probes [ $2a=10$ and 5 μm for (c) and (d), respectively] in Chlamydomonas suspensions prepared with different number densities $c$. The curves are fits of the inverse Fourier transform in Eq. (3) obtained using $R_{\min }$ and $\gamma$ as fit parameters. The broken line is the power law $\propto u_x^{-5/2}$ expected for the corresponding Lévy distribution. (e), (f) Fit values for $R_{\min }$ and $\gamma$. Open and filled symbols correspond to data for probes with $2a=10$ and 5 μm, respectively. Solid and dashed lines in (e) indicate the averages for probes with $2a=10$ and 5 μm that exactly match to the sum of the probe radius and that of Chalmydomonas $(\sim 5\mu \mathrm{m})$. The solid line in (f) indicates the average of $\gamma$ that agrees with independent estimation [30].

Figure 4

(a) The lag-time dependence of $P_{\mathrm{Ch}}\left(u_x;\mathrm{\Delta }t\right)$, which was normalized by $\sigma$, for $c=1.7\times 10^{13}{\mathrm{m}}^{-3}$ and a probe diameter of $10\mathrm{m}$. For these data which have $\mathrm{\Delta }t\le 1\mathrm{s}$, the central part of each distribution in the range $|u_x/\sigma |<5$ has collapsed onto a single curve. (b) Time evolution of the NGP calculated from the same data used in (a). (c) When there are no Chlamydomonas close to the probe (left), the complex flow close to a Chlamydomonas is mostly negligible. A single Chlamydomonas appearing by chance close to a probe (right) causes large, unpredictable displacements. These displacements affect the extreme tails of the distributions and are rarely observable with sufficient statistics.