Rapid Fickian Yet Non-Gaussian Diffusion after Subdiffusion

The recently discovered Fickian yet non-Gaussian diffusion (FnGD) is here finely tuned and investigated over a wide range of probabilities and timescales using a quasi-2D suspension of colloidal beads under the action of a static and spatially random optical force field. This experimental model allows one to demonstrate that a “rapid” FnGD regime with a diffusivity close to that of free suspension can originate from earlier subdiffusion. We show that these two regimes are strictly tangled: as subdiffusion deepens upon increasing the optical force, deviations from Gaussianity in the FnGD regime become larger and more persistent in time. In addition, the distinctive exponential tails of FnGD are quickly built up in the subdiffusive regime. Our results shed new light on previous experimental observations and suggest that FnGD may generally be a memory effect of earlier subdiffusive processes.

Figure 1

At optical power $\mathrm{\Psi }=0.67\mathrm{W}$, (a) and (b) show two snapshots of the speckle field: “empty” and in the presence of the colloidal tracers, respectively.

Figure 2

At optical power $\mathrm{\Psi }=0.67\mathrm{W}$, (a) MSD normalized by the square grain size as a function time. The dashed line $⟨\mathrm{\Delta }{x}^2⟩\propto t$ corresponds to a Fickian behavior. (b) Displacement distribution $p(|\mathrm{\Delta }x|,\mathrm{\Delta }t)$ for time $\mathrm{\Delta }t=400\mathrm{s}$. The dashed line corresponds to the Gaussian distribution for a standard Brownian motion. The data in both panels cover a limited portion of the investigated range.

Figure 3

At optical power $\mathrm{\Psi }=0.67\mathrm{W}$, (a) MSD over the whole investigated time span. For a direct comparison, the dataset of Fig. 2 is also reported (crosses). (b),(c) Distributions of particle displacements after performing the rescaling discussed in the text at different lag-times $\mathrm{\Delta }t$ within the time spans of the subdiffusive and FnGD regimes, respectively. The data cover the two decades in probability below those presented in Fig. 2. The universal Gaussian distribution $G(|X|)$ is reported as a continuous line. The dashed lines in both panels are the same exponential law $A\exp [-(|X|/\mathrm{\Lambda })]$ obtained by fit to the data at the Fickian time $\mathrm{\Delta }t=t_f$.

Figure 4

(a) MSD and (b) its logarithmic derivative $u(\mathrm{\Delta }t)=(d\log ⟨\mathrm{\Delta }{x}^2(\mathrm{\Delta }t)⟩/d\log \mathrm{\Delta }t)$ as a function of time for the different optical powers $\mathrm{\Psi }$, as indicated. The dashed lines are fits to the long-time Fickian regime $⟨\mathrm{\Delta }{x}^2(\mathrm{\Delta }t)⟩=2D\mathrm{\Delta }t$, with diffusivities $D=0.031$, 0.059, 0.066, $0.106\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ for $\mathrm{\Psi }=0.67$, 0.61, 0.43, 0 W, respectively. Tails of the distributions of particle displacements after performing the rescaling discussed in the text, (c), at lag-time $\mathrm{\Delta }t=t_f(\mathrm{\Psi })$ and for different optical powers $\mathrm{\Psi }$ and (d) for the intermediate optical power $\mathrm{\Psi }=0.61$ W and different times $\mathrm{\Delta }t$ ranging in the subdiffusive and FnGD regimes. In both panels (c) and (d), the universal Gaussian distribution $G(|X|)$ is reported as a continuous line. The dashed lines are exponentials $A\exp [-(|X|/\mathrm{\Lambda })]$ obtained by fitting the tails of $P(|X|,\mathrm{\Delta }t=t_f)$ at the different powers.