Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion

Wang et al. [Proc. Natl. Acad. Sci. U.S.A. 106, 15160 (2009)] have found that in several systems the linear time dependence of the mean-square displacement (MSD) of diffusing colloidal particles, typical of normal diffusion, is accompanied by a non-Gaussian displacement distribution $G(x,t)$, with roughly exponential tails at short times, a situation they termed “anomalous yet Brownian” diffusion. The diversity of systems in which this is observed calls for a generic model. We present such a model where there is diffusivity memory but no direction memory in the particle trajectory, and we show that it leads to both a linear MSD and a non-Gaussian $G(x,t)$ at short times. In our model, the diffusivity is undergoing a (perhaps biased) random walk, hence the expression “diffusing diffusivity”. $G(x,t)$ is predicted to be exactly exponential at short times if the distribution of diffusivities is itself exponential, but an exponential remains a good fit for a variety of diffusivity distributions. Moreover, our generic model can be modified to produce subdiffusion.

Figure 1

The displacement distributions after different numbers of steps $N$ for the diffusing diffusivity model [Eqs. (3), (6), and (7)] with $d=0.0025$ and $s=0.01$ simulated as described in the text. The solid line fits are exponential for the three smallest values of $N$, Gaussian for $N=2000$, and the interpolating function $G(x)=A\exp (-B\sqrt{1+(x/X_0{)}^2})$ for $N=700$. The inset shows the MSD and a linear fit.

Figure 2

The displacement distributions after different numbers of steps $N$ for the diffusing diffusivity model with coupling to particle displacement with $d_0=5\times {10}^{-5}$ and $f={10}^{-3}$ in Eq. (11), simulated as described in the text. The solid line fits are exponential for the four smallest values of $N$, Gaussian for $N=5000$, and the interpolating function $G(x)=A\exp (-B\sqrt{1+(x/X_0{)}^2})$ for $N=2000$. The inset shows the MSD and a linear fit.