Diffusing diffusivity: a new derivation and comparison with simulations

Abstract

Many experiments are now available where it has been shown that the probability distribution function (pdf) for the position of a Brownian particle diffusing in a heterogeneous medium is not Gaussian. However, in spite of this non-Gaussianity, the mean square displacement (MSD) still remains Fickian, i.e., $$\left\langle x^{2}\right\rangle \propto T$$ . One possible explanation of this non-Gaussian yet Brownian behavior is that the diffusivity of the particle itself is “diffusing”. Chubynsky and Slater (Phys. Rev. Lett. 113 098302 2014) proposed a model of “diffusing diffusivity” which they were able to solve analytically at small time scales, but simulations were performed for intermediate to large time scales. We present here a class of diffusing diffusivity models and show that the problem of calculating pdf for the position of diffusing particle is equivalent to calculating the survival probability of a particle undergoing Brownian motion in the presence of a sink. We give exact analytical results for all time scales and show that the pdf is non-Gaussian at short times which crosses over to a Gaussian at long times. The MSD is also shown to vary linearly with time at all times. We find that our results reproduce the numerical results of Chubynsky and Slater quite well. SYNOPSIS A model of diffusing diffusivity is analyzed and exact analytical results are obtained. The MSD is shown to vary linearly with time at all times yet the pdf is not Gaussian at all times. A comparison of these results is also made with the numerical results of Chubynsky and Slater.