Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence $D\left(x\right)\sim {\left|x\right|}^{\alpha }$ of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function $P(x,t)$, we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus $\left|\alpha \right|$ of the scaling exponent. The fluctuations appear to diverge when the critical value $\alpha =2$ is approached, while for even larger $\alpha$ the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.

Figure 1

Schematic for the spatially varying diffusivity due to temperature or mobility gradients, shown here for the case of subdiffusion with $D\left(x\right)=D_0/(x_{\mathrm{off}}^2+x^2)$. The diffusivity of the random walker is coupled to the spatial variation of the temperature of the environment, as symbolized by the speed of the runner.

Figure 2

MSD of the heterogeneous diffusion process with power-law diffusivity (4). The MSD $\langle x^2\left(t\right)\rangle$ is represented by the thick blue curves, whose scaling exponent (7) varies with the power-law exponent $\alpha$ of $D\left(x\right)$. The individual time-averaged MSD traces $\overline{{\delta }^2}$ appear as thin red curves, and the mean time-averaged MSD $\langle \overline{{\delta }^2}\rangle$ as the thick blue curves, all of which have unit slope, apart from the critical case $\alpha =2$, where the functional MSD dependence is exponential. The theoretical asymptotes (6) for $\langle x^2\left(t\right)\rangle$ and (11) for $\langle \overline{{\delta }^2}\rangle$ correspond to the dashed black lines. The horizontal axis in the plots has the meaning of the diffusion time for the MSD and the lag time $\Delta$ for the time-averaged MSD. The analytical values for $\langle x{\left(\Delta \right)}^2\rangle$ and $\langle \overline{\delta {\left(\Delta \right)}^2}\rangle$ coincide in the limit $\Delta =T$. For $\alpha =6$ or $p=-1/2$ the theoretical asymptote (6) does not hold. We used the following parameters: for each $\alpha$ we show $N={10}^3$ traces of length $T={10}^4$, we use the offset $|x_{\mathrm{off}}{|}^{\alpha }=0.001$, $D_0=0.01$, and the starting point $x_0=0.1$.

Figure 3

Dependence of the ergodicity breaking parameters (a) EB and (b) $\mathcal{EB}$ on the lag time $\Delta$. The Brownian asymptote ${\mathrm{EB}}_{\mathrm{BM}}\left(\Delta \right)$ is shown as the dashed curve in panel (a). The initial position of the particle was $x_0=0.1$ and $N=3\times {10}^3$ traces were used for each value of $\alpha$, all other parameters are the same as those used in Fig. 2. The symbols in both panels correspond to the same parameters. The dashed lines in the corresponding color in panel (b) correspond to Eq. (17).

Figure 4

Dependence on the scaling exponent $p$, Eq. (7), of $\mathrm{EB}(\Delta =1)$ in panel (a) and the magnification in panel (b) as well as $\mathcal{EB}(\Delta =1)$ in panel (c). The theoretical predictions from Ref. [46] correspond to the dashed black curve in panels (a) and (b). The ergodicity breaking parameter of the subdiffusive CTRW given by Eq. (18) is shown by the dot-dashed red curve in panel (b). To compute each point in the graphs for $N={10}^3$ traces of $T={10}^4$ steps takes some 2 h on a standard 3-GHz core workstation. Different sets of the shown points correspond to varying particle initial conditions $x_0$ indicated by different colors. Note the alternative scale for the power exponent $\alpha =2-2/p$ on the top axes. Parameters are the same as in Fig. 2.

Figure 5

(a) Non-Gaussianity parameter $G\left(\Delta \right)$ as function of the lag time, plotted for the parameters of Fig. 3 with $N={10}^3$ trajectories for each $\alpha$ value. The approximately zero-valued trace $G\left(\Delta \right)$ for Brownian motion ($\alpha =0$) is only shown partially in panel (a). Color coding is the same as in Fig. 3. (b) The value of $G(\Delta =1)$ evaluated for varying exponent $\alpha$ and initial position $x_0$. The color scheme is the same as in Fig. 4.

Figure 6

Spreading of diffusing particles for 10 equidistantly placed initial positions. The dashed curves are the theoretical PDFs given by Eq. (8) in the diffusion limit. Parameters: $T={10}^4$, averaged over $N=200$ trajectories for each initial position $x_0$, and 5000 bins were used in the range $-100<x<100$.

Figure 7

Time evolution of the particles MSD for sub- and superdiffusive HDPs with $p=1/2$ and $p=2$ for varying initial position $x_0$, as indicated in the plots. We used $N=300$ traces for averaging.