Ergodic criterion of a random diffusivity model

The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function $\mathcal{C}(t_1,t_2)=\langle D\left(t_1\right)D\left(t_2\right)\rangle$ of random diffusivity $D\left(t\right)$, we quantitatively propose a general criterion of determining the ergodic property of the Langevin equation with the arbitrary random diffusivity $D\left(t\right)$. Due to the critical role of correlation function $\mathcal{C}(t_1,t_2)$, we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function $\mathcal{C}(t_1,t_2)$ of random diffusivity $D\left(t\right)$. Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function $\mathcal{C}(t_1,t_2)$.

Figure 1

EAMSD $\langle x^2\left(t\right)\rangle$ and EATAMSD $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ (red solid lines) as well as five individual time traces $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$ (blue markers) for the random diffusivity model in Eq. (2), plotted vs (lag) time. The physical time $t$ corresponds to EAMSD $\langle x^2\left(t\right)\rangle$, while the lag time $\mathrm{\Delta }$ corresponds to the time averages $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ and $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$. The theoretical results for $\langle x^2\left(t\right)\rangle$ in Eq. (6) and $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ in Eq. (9) are shown by black dashed lines. Both of the theoretical results coincide with the simulated markers when the diffusivity $D\left(t\right)$ is taken as the square of the OU process $Y^2\left(t\right)$. Other parameters: the measurement time is $T={10}^3$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 2

Amplitude scatter PDF $\varphi \left(\eta \right)$ for the random diffusivity $D\left(t\right)=Y^2\left(t\right)$, where $Y\left(t\right)$ is an OU process. The markers (circle, square, star) denote the simulations for $\mathrm{\Delta }$ (= 1, 10, 100), respectively. The solid lines are obtained by making simulations directly on the trajectories of diffusivity $D\left(t\right)$ based on the theoretical result $\eta \simeq \overline{D\left(t\right)}/\langle \overline{D\left(t\right)}\rangle$ in Eq. (12). Due to the condition $\mathrm{\Delta }\ll T$, the circle markers ($\mathrm{\Delta }=1$) are more consistent than the star markers ($\mathrm{\Delta }=100$) with the solid lines. Other parameters: the measurement time is $T={10}^3$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 3

EAMSD $\langle x^2\left(t\right)\rangle$ and EATAMSD $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ (red solid lines) as well as five individual time traces $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$ (blue markers) for the random diffusivity model in Eq. (2), plotted vs (lag) time. The physical time $t$ corresponds to EAMSD $\langle x^2\left(t\right)\rangle$, while the lag time $\mathrm{\Delta }$ corresponds to the time averages $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ and $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$. The theoretical results for $\langle x^2\left(t\right)\rangle$ in Eq. (6) and $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ in Eq. (9) are shown by black dashed lines. When the diffusivity $D\left(t\right)$ is taken as the square of the fractional OU process $Y^2(s\left(t\right))$, the theoretical results of $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ coincide with the simulated markers. The long-term consistency between the theoretical value of $\langle x^2\left(t\right)\rangle$ and $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ corresponds to the tendency to a constant of the mean diffusivity $\langle D\left(t\right)\rangle$ in Eq. (53). Other parameters: the measurement time is $T={10}^4$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 4

Amplitude scatter PDF $\varphi \left(\eta \right)$ for diffusing diffusivity: $D\left(t\right)=Y^2(s\left(t\right))$, where $Y(s\left(t\right))$ is the fractional OU process. The markers (circle, square, star) denote the simulations for $\mathrm{\Delta }$ (= 1, 10, 100), respectively. The solid lines are obtained by making simulations directly on the trajectories of diffusivity $D\left(t\right)$ based on the theoretical result $\eta \simeq \overline{D\left(t\right)}/\langle \overline{D\left(t\right)}\rangle$ in Eq. (12). Due to the condition $\mathrm{\Delta }\ll T$, the circle markers ($\mathrm{\Delta }=1$) are more consistent than the star markers ($\mathrm{\Delta }=100$) with the solid lines. Other parameters: the measurement time is $T={10}^3$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 5

Amplitude scatter PDF $\varphi \left(\eta \right)$ for diffusing diffusivity: $D\left(t\right)=Y^2(s\left(t\right))$, where $Y(s\left(t\right))$ is the fractional OU process. The markers (triangle, cross, rhombus) denote the simulations for $T$ (= 100, 10 000, 10 000), respectively. The solid lines are obtained by making simulations directly on the trajectories of diffusivity $D\left(t\right)$ based on the theoretical result $\eta \simeq \overline{D\left(t\right)}/\langle \overline{D\left(t\right)}\rangle$ in Eq. (12). Other parameters: the lag time is $\mathrm{\Delta }=1$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 6

EB parameter vs entire observation time $T$ with diffusivity $D\left(t\right)=Y^2(s\left(t\right))$. The star markers converge to a nonzero constant for a long time, and this constant agrees with the theoretical result in Eq. (55). Other parameters: the lag time is $\mathrm{\Delta }=1$, the number of trajectories used for ensemble is ${10}^4$, and the initial position is $x_0=0$.

Figure 7

EAMSD $\langle x^2\left(t\right)\rangle$ and EATAMSD $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ (red solid lines) as well as five individual time traces $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$ (blue markers) for the random diffusivity model in Eq. (2), plotted vs (lag) time. The physical time $t$ corresponds to EAMSD $\langle x^2\left(t\right)\rangle$, while the lag time $\mathrm{\Delta }$ corresponds to the time averages $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ and $\overline{{\delta }^2\left(\mathrm{\Delta }\right)}$. The theoretical results for $\langle x^2\left(t\right)\rangle$ in Eq. (6) and $\langle \overline{{\delta }^2\left(\mathrm{\Delta }\right)}\rangle$ in Eq. (9) are shown by black dashed lines. Both of these theoretical results coincide with the simulated values when the diffusivity $D\left(t\right)$ is taken as $\left|B\right(t\left)\right|$ in (a) and $B^2\left(t\right)$ in (b), where $B\left(t\right)$ is the Brownian motion. Other parameters: the measurement time is $T={10}^3$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 8

Amplitude scatter PDF $\varphi \left(\eta \right)$ for two kinds of diffusivity: the diffusivity $D\left(t\right)$ is taken as $\left|B\right(t\left)\right|$ in (a) and $B^2\left(t\right)$ in (b), where $B\left(t\right)$ is the Brownian motion. The markers (circle, square, star) denote the simulations for $\mathrm{\Delta }$ (= 1, 10, 100), respectively. The solid lines are obtained by making simulations directly on the trajectories of diffusivity $D\left(t\right)$ based on the theoretical result $\eta \simeq \overline{D\left(t\right)}/\langle \overline{D\left(t\right)}\rangle$ in Eq. (12). Due to the condition $\mathrm{\Delta }\ll T$, the circle markers ($\mathrm{\Delta }=1$) are more consistent than the star markers ($\mathrm{\Delta }=100$) with the solid lines. Other parameters: the measurement time is $T={10}^3$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 9

Amplitude scatter PDF $\varphi \left(\eta \right)$ for two kinds of diffusivity: the diffusivity $D\left(t\right)$ is taken as $\left|B\right(t\left)\right|$ in (a) and $B^2\left(t\right)$ in (b), where $B\left(t\right)$ is the Brownian motion. The markers (triangle, cross, rhombus) denote the simulations for $T$ (= 100, 10 000, 10 000), respectively. The solid lines are obtained by making simulations directly on the trajectories of diffusivity $D\left(t\right)$ based on the theoretical result $\eta \simeq \overline{D\left(t\right)}/\langle \overline{D\left(t\right)}\rangle$ in Eq. (12). Other parameters: the lag time is $\mathrm{\Delta }=1$, the number of trajectories used for ensemble is ${10}^3$, and the initial position is $x_0=0$.

Figure 10

EB parameter vs entire observation time $T$ with two kinds of diffusivity: $D\left(t\right)=\left|B\right(t\left)\right|$ in (a) and $D\left(t\right)=B^2\left(t\right)$ in (b). Instead of decaying to zero, the star markers converge to a constant for a long time. This constant agrees with the theoretical result in Eq. (63) in (a) and Eq. (68) in (b). Other parameters: the lag time is $\mathrm{\Delta }=1$, the number of trajectories used for ensemble is ${10}^4$, and the initial position is $x_0=0$.

Figure 11

Sample trajectories with different diffusivity $D\left(t\right)$, which is $Y^2\left(t\right)$ in (a), $Y^2(s\left(t\right))$ in (b), $\left|B\right(t\left)\right|$ in (c), and $B^2\left(t\right)$ in (d). Here, $Y\left(t\right)$ is the OU process, $Y(s\left(t\right))$ is the fractional OU process, and $B\left(t\right)$ is the Brownian motion. Other parameters: the measurement time is $T={10}^3$, the number of trajectories in each graph is 30, and the initial position is $x_0=0$.