Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Figure 1

MSD of the aging UDSBM process: analytical results [solid lines, with the full expression of Eq. (23) and with the asymptotes (27) and (28) shown] and the results of computer simulations (data points), plotted for $\alpha =1$ (a) and $\alpha =1.4$ (b) and different aging times $t_a$. Parameters: ${\tau }_0=100,{\gamma }_0=0.02,m=1,D_0=1$, and the observation time is $T={10}^4$. The initial temperature is ${\mathcal{T}}_0=m{\gamma }_0{D}_0={\gamma }_0$. In (a) we have $t_{\text{min}}=50$, for (b) at $t_a={10}^6,{10}^5,{10}^4$ we have, respectively, $t_{\text{min}}\approx 1.26,3.16,7.93$. (c) Shows the MSD for the case of strong aging $t_a={10}^6\gg T$ and different $\alpha$. For superdiffusive UDSBM, the ballistic MSD regime shrinks to shorter times $t\ll 1/{\gamma }_0$ for larger $\alpha$ values, in accord with the analytical prediction (31). In (c) at $t_a={10}^6$ we have $t_{\text{min}}\approx 1.26,7.93,50$ for $\alpha =1.4,1.2,1$, correspondingly.

Figure 2

Delay of the overdamping transition for aging UDSBM. (a) Theoretical (solid lines) and computer simulation (data points) results for the MSD, obtained for $\alpha =0.5$. The initial ballistic regime is the dashed line following Eq. (27). The intermediate normal diffusion asymptote is the dashed line (28). The trace length is $T={10}^5$ and the aging times $t_a$ are as indicated. Parameters: $D_0=1,{\gamma }_0=1,{\tau }_0=30$. For (a) at $t_a={10}^6,{10}^3,0$ we obtain $t_{\text{min}}\approx 183,5.77,1,$ respectively. (b) Theoretical results for the MSD, plotted for different aging times and for $\alpha =0.5,$ $D_0=1,{\gamma }_0=0.02,{\tau }_0={10}^3,T={10}^7$, show three different scaling regimes. Note that the values of ${\tau }_0,{\gamma }_0$, and $T$ in (b) differ from those in (a). For (b), at $t_a={10}^8,{10}^6$, and 0 we get $t_{\min }\approx 15,800,1,580,50$, respectively. The dashed asymptotes are according to Eqs. (27), (28), and (35).

Figure 3

Analytical results for the MSD (solid lines) and time-averaged MSD (data points) of aging superdiffusive UDSBM at $\alpha =1.5$. The asymptotes for the initial ballistic, intermediate linear, and long time anomalous behavior are according to Eqs. (27), (28), and (25), respectively. Note that long aging times diminish and eventually remove the weak ergodicity breaking. Parameters: $T={10}^5,{\tau }_0={10}^3$, and ${\gamma }_0=0.02$. For these parameters, at $t_a={10}^6,{10}^4$, and 0 the ballistic regime is expected to range in $t<t_{\min }\approx 1.58,15.8$, and 50, respectively. Also note the almost fully superimposing time-averaged MSD data points at aging times $t_a=0$ and $t_a={10}^4$.

Figure 4

Theoretical [solid curves, Eqs. (39) and (40)] and computer simulation results (data points) for the time-averaged MSD of aging subdiffusive UDSBM processes. The asymptotes shown as the dashed lines are according to Eqs. (46) and Eqs. (43) and (45) for short and intermediate lag times, respectively. The findings are plotted for $\alpha =\frac{1}{6}$ (a) and $\alpha =\frac{1}{2}$ (b), with $T={10}^5,{\tau }_0={10}^2$, and ${\gamma }_0=0.1$.

Figure 5

MSD (blue points and dashed lines) and time-averaged MSD (green lines and solid black line) for the aging subdiffusive UDSBM processes, for $\alpha =\frac{1}{2}$ and two different aging times $t_a$ as indicated in (a) and (b). The analytical expressions for the MSD and time-averaged MSD are given by Eqs. (23) and (38), respectively. The red curves represent individual time-averaged MSD realizations. Other parameters are the same as in Fig. 2.

Figure 6

Analytical results for the time-averaged MSD of aging UDSBM, plotted for $\alpha =\frac{1}{6}$ and varying aging times. The asymptotes are according to Eqs. (46) and (45). Parameters: $T={10}^7,{\tau }_0={10}^3$, and ${\gamma }_0=0.02$.

Figure 7

Ratio of the aging versus nonaging time-averaged MSDs of UDSBM, ${\mathrm{\Lambda }}_{\alpha }\left(t_a\right)$, as obtained from our computer simulations, plotted as a function of the aging time $t_a$ for $\mathrm{\Delta }=1$ and different values of the scaling exponents. Parameters: $T={10}^4,{\tau }_0=30,N={10}^3$. The analytical asymptotes in the limit of long aging times (53) are shown by the dashed lines.

Figure 8

(a), (b) Distribution $\varphi \left(\xi \right)$ of the relative amplitude of the time-averaged MSD for nonaging UDSBM at $\alpha =\frac{1}{2}$ and ultraslow UDSBM, computed for different lag times, as indicated in the panels. For longer lag times, the distribution gets progressively wider and becomes asymmetric. (c) Comparison between $\varphi \left(\xi \right)$ for aging and nonaging UDSBM processes for $\alpha =0.5$ and lag time $\mathrm{\Delta }=1$. In the strong aging regime, the distribution becomes slightly wider. (d) Distributions $\varphi \left(\xi \right)$ for the nonaging UDSBM process for different $\alpha$ exponents, as indicated in the plot, and for $\mathrm{\Delta }=1$. Parameters for all the plots: $T={10}^4$ and $N={10}^4$.

Figure 9

EB parameter obtained from computer simulations versus the aging time $t_a$ for the aging UDSBM process, computed for traces with $T={10}^4$ steps and averaged over $N={10}^3$ trajectories, for different $\alpha$ values (as denoted in the plot) and $\mathrm{\Delta }=1$. The black dashed lines show the phenomenological strong aging scaling relation (56). The parameters used here are $T={10}^4,{\tau }_0=30,{\tau }_v=1$, and $D_0=1$.

Figure 10

Ergodicity breaking parameter variation with the trace length $T$, computed for different scaling exponents $\alpha$ in the absence of aging (a) and for rather strong aging (b), after averaging over $N={10}^3$ trajectories. The value of the lag time is $\mathrm{\Delta }=1$ for both panels so that $\mathrm{\Delta }/T\ll 1$. The asymptotes shown in (a) and (b) are Eqs. (57) and (58), in the limit of short and long aging times, respectively.

Figure 11

MSD (a) and time-averaged MSD (b) as obtained from computer simulations (data points) and theoretically [solid lines; Eq. (61) for the MSD and Eqs. (68) and (69) for the time-averaged MSD] for the aging ultraslow UDSBM process at $\alpha =0$. The short time ballistic asymptote and the long time logarithmic behavior of the MSD are shown as dashed lines, plotted according to Eqs. (65) and (67), correspondingly. The ballistic asymptotes for the time-averaged MSD are the dashed lines [Eq. (70)]. The values of the aging time $t_a$ are as indicated in the plots. Other parameters are $D_0=1,{\gamma }_0=1$, and ${\tau }_0=30$.