Aging scaled Brownian motion

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

Figure 1

Ensemble and time averaged MSD for SBM with $\alpha =1/2$. Thin lines: Time averaged MSD for 20 individual trajectories from simulations of the SBM Langevin equation ($2$) with trajectory length $t={10}^5$. Circles: Averages over those 20 trajectories. Black thin line: Theory result (12). Thick green line: Ensemble averaged MSD (5). Three different aging times were considered (top to bottom): (a) nonaged case $t_a=0$, (b) weak aging case $t_a={10}^3$, and (c) strong aging case $t_a={10}^6$. In all simulations $K_{\alpha }^{*}=1/2$ in the following, unless otherwise indicated.

Figure 2

Ensemble and time averaged MSD for SBM ($2$) with $\alpha =3/2$ in the strong aging case, $t_a={10}^6$. The observation time is $t={10}^5$. The spread of the 20 single trajectory time averages is fairly small. As before, the ensemble and time averaged MSDs coincide, apparently restoring ergodicity.

Figure 3

Ensemble averaged MSD $\langle x^2\left(t\right)\rangle$ for confined aging SBM in the subdiffusive case with $\alpha =0.5$ and $K_{\alpha }=1$ at different aging times: (i) nonaged ($t_a=0$) denoted by the gray lines; (ii) weakly aged ($t_a=0.1$) denoted by the blue lines; and (iii) strongly aged ($t_a={10}^6$) denoted by the orange lines. In all cases, the full lines correspond to the force constant $k=0.1$, while the dashed lines stand for $k=0.01$. The green line at the bottom of the graph is a blowup [${\langle x^2\left(t\right)\rangle }^4$ of the case $t_a={10}^6$ and $k=0.1$] in which the crossover between the two plateaux is more visible.

Figure 4

Ensemble averaged MSD $\langle x^2\left(t\right)\rangle$ for confined aging SBM in the superdiffusive case with $\alpha =1.5$ and $K_{\alpha }=1$ at different aging times: (i) nonaged ($t_a=0$) denoted by the gray lines; (ii) weakly aged ($t_a=0.1$) denoted by the blue lines; and (iii) strongly aged ($t_a={10}^6$) denoted by the orange lines. In all cases, the full lines correspond to the force constant $k=0.1$, while the dashed lines stand for $k=0.01$. In all cases the terminal scaling $\simeq t^{\alpha -1}$ is reached.

Figure 5

Ensemble and time averaged MSD for confined SBM for $\alpha =1/2$. From top to bottom, the panels represent the nonaged ($t_a=0$) case, the case of weak aging ($t_a={10}^{-1}$), and the case of strong aging ($t_a={10}^6$), where the observation time is chosen as $t=5\times {10}^4$. The lines represent Eqs. (17) and (29). The force constants $k$ are indicated in the panels. Note that the time averaged MSD indeed converges to the ensemble MSD in the limit $\Delta \to t$, compare the discussion in Ref. [32] and the zoom-in provided in Fig. 7.

Figure 6

Ensemble and time averaged MSD for confined SBM with $\alpha =3/2$ in the strong aging case with $t_a={10}^6$. The observation time is $t=5\times {10}^4$.

Figure 7

Full behavior of aging confined SBM for $t_a={10}^6$ with $\alpha =1/2$ (left) and $\alpha =3/2$ (right), demonstrating the convergence of the time averaged MSD to the ensemble averaged MSD in the limit $\Delta \to t$, shown for two different potential strengths, as indicated in the panels. The observation time is $t=5\times {10}^4$. In accordance with result (35) the difference between ensemble and time averaged MSDs vanishes for growing aging time.

Figure 8

First passage time density $\wp \left(t\right)$ for $\alpha =1/2$, with $x_0=1$, and aging time $t_a={10}^6$. As shown by the dashed line, the crossover between the aging dominated slope $-1/2$ to the slope $-5/4$ is distinct.