Subdiffusion in an external force field

The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time-dependent force, we study the subdiffusion described by the subordinated Langevin equation with white Gaussian noise or, equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as the ensemble- and time-averaged mean squared displacements, position autocorrelation function, correlation coefficient, and generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are nonstationary, nonergodic, and aging.

Figure 1

Simulation results of the MSD of stochastic process (16) for different $\alpha$. Color symbols represent the simulation results of MSD with parameters $\sigma =1$ and $\gamma =0.2$ averaging over 2000 trajectories. The dot-dashed red line ($\alpha =0.4$), dotted blue line ($\alpha =0.6$), and dashed green line ($\alpha =0.8$) represent the asymptotic theoretical values of MSD for short times [see (20)]; the solid black line represents the one for long times [see (21)].

Figure 2

Simulation results of the MSD of the stochastic process described by the Langevin equation (29) for different $\alpha$. The parameters are, respectively, taken as $\sigma =1$, $\gamma =0.1$, (a) $\alpha =0.7$ or (b) $\alpha =0.3$, and the initial position $y_0=0$. Dashed blue lines and dot-dashed red lines represent the asymptotic theoretical values in Eqs. (35) and (36), respectively, and the solid black lines signify the theoretical values (34). Red circles are the simulation results.

Figure 3

Simulation results of the ensemble-time-averaged MSD of the stochastic process described by the Langevin equation (29) for different $\alpha$. The parameters are taken as $\sigma =1$, $\gamma =0.3$, and the initial position $y_0=0$. The measurement time $T=1000$. Dashed black lines (with $\alpha =0.8,0.6,0.4$ from top to bottom) and dot-dashed red lines (with $\alpha =0.8,0.6,0.4$ from top to bottom) represent the asymptotic theoretical results for short lag time and large lag time in Eq. (13) and (43), respectively. Color symbols are the simulation results about the ensemble-time-averaged MSD over 500 trajectories.

Figure 4

Simulation results of the mean and MSD of the stochastic process described by Langevin equation (45) for different $\alpha$. The parameters are, respectively, taken as $\sigma =1$, $F=1$, (a) $\alpha =0.7$ or (b) $\alpha =0.5$, and the initial position $y_0=0$. Solid black lines and blue squares represent the theoretical value in Eq. (47) and the simulation result of the first moment. Besides, the dashed black lines and the red circles are, respectively, the theoretical value in Eq. (47) and the simulation result of the ensemble-averaged MSD.

Figure 5

Ensemble-time-averaged MSD of the stochastic process described by Langevin equation (45). The solid black line shows the theoretical result (51), which coincides with the simulation result of the ensemble-time-averaged MSD, represented by the blue circles. In addition, the dot-dashed red line and the dashed blue line are the asymptotic expressions of the ensemble-time-averaged MSD for long and short lag times, scaled as ${\mathrm{\Delta }}^{\alpha +1}$ and $\mathrm{\Delta }$, respectively. Parameter values: $T={10}^5$, $F=1$, $\sigma =1$, and $\alpha =0.6$. Here we choose a big $\alpha$ to see the transition of the ensemble-time-averaged MSD more clearly.

Figure 6

First moment and MSD of the stochastic process described by Langevin equation (55) for different $\alpha$. The parameters are, respectively, taken as $\sigma =1$, $F=2$, and (a) $\alpha =0.7$ or (b) $\alpha =0.3$. The solid black lines and the blue squares represent the theoretical value $Ft$ and the simulation result of the first moment. Besides, the dashed black lines and the red circles represent the theoretical value (57) and the simulation result of the ensemble-averaged MSD.

Figure 7

Time-averaged MSD of the stochastic process described by the Langevin equation (55). The blue circles represent the simulation results of the mean value of the time-averaged MSD and the solid red lines are the time-averaged MSDs of individual particle trajectories. Parameters are $T=1000$, $\sigma =1$, and (a) $\alpha =0.7$ and $F=3$ or (b) $\alpha =0.3$ and $F=2$. The dashed black lines show the theoretical results (59), which coincide with the simulation results of the ensemble-time-averaged MSDs over 100 trajectories.

Figure 8

Fluctuation term of the second moment of the stochastic process described by Langevin equation (60). The solid black line represents the analytical result $D_1\left(t\right)$ in Eq. (67), which coincides with the simulation result averaging over ${10}^4$ trajectories, represented by red circles. Parameter values: $\alpha =0.7$, $\omega =1$, and $\sigma =1$.

Figure 9

Simulation results of the time-averaged MSD of the stochastic process described by (60) for different $\alpha$. The parameters are, respectively, taken as $\sigma =1$, $\omega =1$, $f_0=1$, and (a) $T=1000$, $\alpha =0.7$ and (b) $T=100$, $\alpha =0.3$. The solid red lines represent the simulation results of the time-averaged MSDs of individual trajectories and the blue circles are the ensemble-time-averaged MSDs over 100 trajectories, which coincide with the theoretical results (72) denoted by dashed black lines.

Figure 10

Simulation results of the time-averaged MSD of the stochastic process described by (73) for different $\alpha$. The parameters are, respectively, taken as $\sigma =1$, $\omega =1$, $f_0=1$, $T=1000$, and (a) $\alpha =0.7$ and (b) $\alpha =0.4$. The solid red lines represent the simulation results of the time-averaged MSDs of individual trajectories and the blue circles are the ensemble-time-averaged MSDs over 100 trajectories, which coincide with the theoretical results represented by dashed black lines.