Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients $D\left(x\right)$ are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, $D\left(x\right)$ is considered to be centerd at the origin and the entire $x$ axis is characterized by a single scaling exponent $\alpha$. In this work we consider a spatially random scenario: in periodic intervals (“layers”) in space $D\left(x\right)$ is centerd to the midpoint of each interval. In each interval the scaling exponent $\alpha$ is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains (“layer thickness”) of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Figure 1

Diffusion coefficient $D\left(x\right)$ for a stratified layered medium with local particle diffusion governed by HDPs. Panels (a) and (b) correspond to the cases of, respectively, ${\alpha }_0=1$ and ${\alpha }_0=-2$. Other parameters are $\sigma =0.05$, 0.5, and 1 (corresponding to very weak, intermediate, and strong disorder) and $\delta x=1$ (meaning that $2\delta x$ is the domain periodicity). These values of parameters ${\alpha }_0\mathrm{and}\sigma$ are often used in the analysis below.

Figure 2

MSD, individual TAMSDs, and mean TAMSD for the unconfined HDPs with constant $\alpha$-exponents driven by GCN, plotted for different values of the scaling exponent of the diffusion coefficient (values provided in the plots) as function of diffusion time. Here, as well as in other plots, the MSDs $\langle x^2\left(t\right)\rangle$ are the thick green curves, the individual TAMSDs $\overline{{\delta }_k^2\left(\mathrm{\Delta }\right)}$ are the red lines, and the mean TAMSDs $〈\overline{{\delta }^2\left(\mathrm{\Delta }\right)}〉$ are the thick blue curves. The MSD asymptotes at short and long times as well as the TAMSD asymptote are the dashed and solid lines, respectively. The black dot-dashed line is the Brownian asymptote, $\langle x^2\left(t\right)\rangle =2D_{0}t$. In panels (a)–(c), the initial position of the OU process is taken as a fixed constant ${\xi }_0=0.$ In panels (d)–(f), the initial position ${\xi }_0$ is a random number satisfying the equilibrium distribution of the OU process Eq. (5), i.e., ${\xi }_0\sim N\left(0,1/\sqrt{2\tau }\right)$. In panels (a), (d), to make the slope of MSD clear, we assume $x_0=0.7$, while in panels (b), (c), (e), (f), $x_0=0.1.$ In all the figures, the relaxation time is $\tau =20$.

Figure 3

Dynamic characteristics of particles obeying stratified HDPs with random exponents diffusing in a medium with a quenched disorder. Panels (a)–(c) show stratified HDPs with mean diffusivity exponent ${\alpha }_0=0,1,-2$, respectively. The graphs are plotted by systematically varying $\tau$, as indicated in the plots. The black dot-dashed line is the Brownian asymptote, $\langle x^2\left(t\right)\rangle =2D_{0}t$. Other parameters: variance of the HDP exponent variation $\sigma =1$, domain periodicity of the medium $\delta x$ = 1, initial particle position $x_0=0.1+\delta x$ (as in Refs. [47]), and ${\xi }_0=0$. We fix the constants $D_0=0.01$ and $D_{\mathrm{off}}=0.001$ throughout the paper. The notations for curves and lines are the same as described in the caption of Fig. 2.

Figure 4

The same as described in the caption of Fig. 3, with the same notations for different curves and asymptotes, plotted for three values of the scaling exponents of $D\left(x\right)$ and for varying layer thickness, $\delta x=1,10,\mathrm{and}100$. Note that in Figs. 3(c1)–3(c3) the initial position is taken as $x_0=0.1$ and $D_0=0.2$. The gray lines correspond to Eq. (A2) or Eq. (27). In panel (c3) the gray line is not given because the corresponding mean first exit time is too large to compute its exact value. Parameters: $\sigma$ = 0.05, $\tau$ = 20, and ${\xi }_0$ = 0.

Figure 5

Particle-diffusion characteristics for HDPs with GCN in a quenched-disorder medium plotted for different variance $\sigma$ of the scaling-exponent distribution $p\left(\alpha \right)$; see Eq. (4). Parameters: $\delta x=1,\tau =20,$ and ${\xi }_0=0$. The notations for curves and lines are the same as described in the caption of Fig. 3.

Figure 6

Ergodicity breaking parameter, normalized to the reference behavior for Brownian motion, $E_{\mathrm{EB}}/E_0$, plotted as function of lag time, $\mathrm{\Delta }$. The colors of symbols are specified in the legend. The line types from thin to thick correspond to $\tau =1$, 10, and 100, respectively. Other parameters are $\sigma =0.05,\delta x=10,$ and ${\xi }_0=0$.

Figure 7

Complementary ergodicity parameter, $E_{\mathrm{NE}}$, as function of the lag time. The green, blue, and pink curves represent HDPs with GCN for ${\alpha }_0=0,1, -2$ (the color scheme is similar to that in Fig. 6). Parameter: $\sigma =0.05,\delta x=1,$ and ${\xi }_0=0$. (a)${\alpha }_0=0$; (b) ${\alpha }_0=-2$; (c) ${\alpha }_0=1$.

Figure 8

Variation of $E_{\mathrm{EB}}$ as function of the correlation time $\tau$ of GCN. The red, green, and blue curves and corresponding symbols indicate the cases ${\alpha }_0=-2$, 0, and 1, respectively. Parameters: ${\xi }_0=0$, and (a) $\mathrm{\Delta }=1,\delta x=1,\sigma =0.05$; (b) $\mathrm{\Delta }=1,\delta x=1,\sigma =1$; (c) ${\alpha }_0=-2,\mathrm{\Delta }=1,\sigma =0.05$. The lines from thin to thick in panel (c) indicate the situations with $\delta x=1$, 10, and 100, respectively.

Figure 9

Variation of the non-Gaussianity parameter $G\left(\mathrm{\Delta }\right)$ versus lag time $\mathrm{\Delta }$, for the values of ${\alpha }_0$ given in the legends. Parameters: $\tau =20,\delta x=10,{\xi }_0=0$ and (a) $\sigma =0.05$ and (b) $\sigma =1$.

Figure 10

Non-Gaussianity parameter as function of ${\alpha }_0$, plotted for noise correlation times $\tau =1$, 10, and 100 (respectively, the red, blue, and green symbols). Parameters: medium periodicity $\delta x=10$, lag time $\mathrm{\Delta }=1, {\xi }_0=0$ and for panels (a) and (b) the variance of $p \left(\alpha \right)$ distribution is $\sigma =0.05$ and 1, correspondingly. A Gaussian distribution, for comparison, has $G=0$ for all $\mathrm{\Delta }$.

Figure 11

When $\left|\alpha \right|\le 1$ in panels (a)–(d), the blue line is the theoretical result given above. The theoretical results fit well with numerical simulations for large $\delta x$. When $\left|\alpha \right|>1$ in panels (e) and (f), $\lambda$ is obtained by numerically calculating the mean first exit time from the Langevin equation. The blue line is $\langle x^2\left(t\right)\rangle =4\lambda \delta x^{2}t$, which fits well with numerical simulation for small $\delta x$. The initial position is $x_0=0.1$, the gray line is the MSD and yellow line is the mean TAMSD.