Lévy walking droplets

Foraging and other natural phenomena show superdiffusive transport that are described by Lévy walks. We present a model experimental system, an octanoic acid (OA) drop on aqueous unsaturated OA solutions, for studying Lévy walks in controlled environments. The drop spontaneously moves due to unbalanced Marangoni forces, in a manner that mimics biological propulsion in living systems. The drop motion is shown to be a Lévy walk by demonstrating superdiffusive transport, Lévy-stable distributed step lengths, and power-law decay in the velocity autocorrelation function, with consistent exponents.

Figure 1

The variation of the surface tension ${\sigma }_s$ of the OA-water solution versus $\lambda$. The drop sizes used in the measurements were the same for all measurements. For the given drop size, the different regimes seen in the experiments are also indicated.

Figure 2

Instantaneous positions and images of OA drop during one experiment on an aqueous solution with $\lambda =0.6$. The drop trajectory (from top-left corner to lower-right corner) is created by merging frames that are 5 s apart in time. The vectors indicate the velocity of the drop centroid. Note the occasional but large steps in the trajectory, which are a characteristic of a Lévy walk. $5\times$ magnified images of the triple lines at four different instances are shown in green.

Figure 3

Trajectories of the drop for (a) $\lambda =0.6$ (b) $\lambda =0.7$, and (c) $\lambda =1$. Colors indicate different trials. All trajectories have been terminated after the same elapsed time of 180 s. As $\lambda$ increases, the range of exploration decreases due to smaller Marangoni force. Note the oscillatory behavior in the zoomed-in view of panel (b) at large times (see Supplemental Material for movie [30]).

Figure 4

The variation of MSD ($\langle \mathrm{\Delta }{r}^2\rangle$) with time $\tau$ of an OA drop at (a) $\lambda =0.6$ and (b) $\lambda =0.7$, obtained by averaging over all trajectories. The dash-dotted line is a power law ${\tau }^{1.6}$, where the exponent is the maximum likelihood estimate for $\tau \in [12,90]$. The solid black lines correspond to ballistic and diffusive transport. Inset shows MSD for individual trajectories for $\lambda =0.6$, as evidence of reproducibility in different trials.

Figure 5

Cumulative distribution functions $C(\ell ,t)$ for $\lambda =0.6$ and $\lambda =0.7$. The data for different times collapse onto a single curve when $C(\ell ,t)$ and step lengths $\ell$ are scaled as in Eq. (1) with $\alpha =1.4$. The dash-dotted line represents the expected asymptotic scaling of $C(\ell ,t)$ for large $z$, where $z=\ell t^{-1/\alpha }$. Inset shows probability distribution function $P(x,0.04s)$ for $\lambda =0.6$, where $x$ is step lengths in $x$ or $y$ directions over $t=0.04\text{s}$. The solid line is a Gaussian fit. The distribution clearly shows heavy tails.

Figure 6

Simulation results for a model Lévy walker with $\alpha =1.4$. (a) Mean squared displacement (MSD) vs $\tau$. As expected, $\text{MSD}\propto {\tau }^{\nu }$ with $\nu =3-\alpha =1.6$. (b) Cumulative distribution functions $C(\ell ,t)$ for different times $t$. The data for different times collapse onto a single curve when scaled as in Eq. (1). The dash-dotted line is the expected behavior for large distances i.e., $z^{-1.4}$, where $z$ is the argument of the function $g$ in Eq. (1). The results are similar to the experimental results in Fig. 5.

Figure 7

The variation of the velocity autocorrelation function $c\left(\tau \right)$ with $\tau$ for $\lambda =0.6$, 0.7. For both values of $\lambda$, $c\left(\tau \right)\sim {\tau }^{-0.4}$ for large times ($\tau >10\text{s}$).

Figure 8

(a), (b) Top views of rigid and deformable drops and the corresponding surface tension force calculations, respectively.