Anomalous Diffusion and Lévy Walks Distinguish Active from Inertial Turbulence

Bacterial swarms display intriguing dynamical states like active turbulence. Now, using a hydrodynamic model, we show that such dense active suspensions manifest superdiffusion, via Lévy walks, which masquerades as a crossover from ballistic to diffusive scaling in measurements of mean-squared displacements, and is tied to the emergence of hitherto undetected oscillatory streaks in the flow. Thus, while laying the theoretical framework of an emergent advantageous strategy in the collective behavior of microorganisms, our Letter underlines the essential differences between active and inertial turbulence.

Figure 1

MSDs of Lagrangian trajectories (a) for mild activity and (b) with increasing levels of activity (vertically staggered for clarity). Local slopes [(b), inset] of $\mathrm{\Delta }{x}^2$: While $\xi$ continuously decreases from 2 to 1 for $\alpha \le -4$, it plateaus (between the vertical lines) at $\xi \approx 4/3$ for $\alpha =-6$. Representative trajectories for (a) $\alpha =-1$ (upper inset) and (c) $\alpha =-6$ (see text) reflecting the change from diffusive to anomalous behavior and, also, seen in snapshots of particle positions (initially localized within the small white disk) at $t\approx 10$ for $\alpha =-1$ [panel (a), lower inset] and (d) $\alpha =-6$; the brightness of the colors reflects the particle density (see movie in the Supplemental Material [33]).

Figure 2

For highly active suspensions (case G), probability distributions $p(\tau )$ of the waiting time $\tau$ with a ${\tau }^{-8/3}$ scaling (dashed line), as a guide to the eye, for different ${\theta }_c$ [33]. (Lower inset) Local slope analysis of $p(\tau )$ and $p(d)$ ($x$ axis rescaled for visualization) yields $\gamma =1.7\pm 0.3$ and $\gamma =1.6\pm 0.2$, respectively, consistent with Lévy walk estimates, showing a scaling range (between dashed [$p(\tau )$] and dotted [$p(d)$] vertical lines) for about a decade. (Upper inset) Joint probability distribution of $d$ and $\tau$ confirming a finite velocity through the near linear relationship between the two.

Figure 3

Representative snapshots of vorticity fields (and their magnified sections with velocity vectors as arrows and instantaneous streamlines as solid lines) for (a) case D ($\alpha =-1$), and (b) case G ($\alpha =-6$). In the latter, highly active suspensions, a new feature shows up: The vorticity field is now populated by oscillatory streaks (see, also, Fig. 4 and Ref. [33]).

Figure 4

Extremely active suspensions show that (a) the vorticity field is dominated by streaks interspersed with fewer vortical spots [magnified view in panel (b) with velocity vectors as arrows and instantaneous streamlines as solid lines] and is associated with (c) robust anomalous diffusion $\mathrm{\Delta }{x}^2\sim t^{4/3}$, reflected in a representative trajectory (inset).