Levy 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)(T)> alpha 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) (D) similar to e(-D/D0). In the present study, we model the diffusivity of a particle as a Levy flight process so that D has a power-law tailed distribution, viz., pi(e) (D) similar to D-1-alpha with 0 < alpha < 1. We find that in the short time limit, the width of displacement distribution is proportional to root 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.