Lévy flight with absorption: A model for diffusing diffusivity with long tails

We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of “diffusing diffusivity” [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)], it was shown that the mean square displacement of particle remains Fickian, i.e., $〈x^2\left(T\right)〉\propto T$ at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of $D$ in that model has a steady state limit which is exponential, i.e., ${\pi }_e\left(D\right)\sim e^{-D/D_0}$. In the present study, we model the diffusivity of a particle as a Lévy flight process so that $D$ has a power-law tailed distribution, viz., ${\pi }_e\left(D\right)\sim D^{-1-\alpha }$ with $0<\alpha <1$. We find that in the short time limit, the width of displacement distribution is proportional to $\sqrt{T}$, implying that the diffusion is Fickian. But for long times, the width is proportional to $T^{1/2\alpha }$ which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index $2\alpha$ which preserves its shape at all times.

Figure 1

Width of the distribution as a function of time for different values of $\alpha$. (a) Linear plots showing that smaller is the value of $\alpha$, faster is the growth of width in time. (b) Log-log plots of width versus time. For short times the curves for different values of $\alpha$ are parallel with a slope of $1/2$, but at larger times the slope becomes $1/2\alpha$. The insets show the variation of width with time for smaller values of $\alpha$ (0.2 and 0.3).