Subdiffusion in time-averaged, confined random walks

Certain techniques characterizing diffusive processes, such as single-particle tracking or molecular dynamics simulation, provide time averages rather than ensemble averages. Whereas the ensemble-averaged mean-squared displacement (MSD) of an unbounded continuous time random walk (CTRW) with a broad distribution of waiting times exhibits subdiffusion, the time-averaged MSD, $\overline{{\delta }^2}$, does not. We demonstrate that, in contrast to the unbounded CTRW, in which $\overline{{\delta }^2}$ is linear in the lag time $\Delta$, the time-averaged MSD of the CTRW of a walker confined to a finite volume is sublinear in $\Delta$, i.e., for long lag times $\overline{{\delta }^2}\sim {\Delta }^{1-\alpha }$. The present results permit the application of CTRW to interpret time-averaged experimental quantities.

Figure 1

(Color online) CTRW simulations. The results in Figs. 1, 2, 3, 4 were obtained as follows. All CTRW simulations were started at $x=0$ with an initial jump at $\tau =0$. The jump lengths were taken from a Gaussian distribution with variance $⟨\delta x^2⟩=1$. The walker moves in the spatial interval $\left[-10,10\right]$, i.e., $L=20$, with reflecting boundaries. Uniformly distributed random numbers $r∊\left(0,1\right)$ were generated with the long-period random number generator of L’Ecuyer with Bays-Durham shuffle and added safeguards [16]. A random variable ${\tau }_w$ with the distribution given in Eq. (2) can be obtained from the uniformly distributed $r$ from the transformation ${\tau }_w=r^{-1/\alpha }-1$ [the time unit in Eq. (2) is ${\tau }_0=1$ for all simulations]. Simulations were performed for $\alpha =0.3$, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 on various time scales ranging from $t={10}^7$ up to ${10}^{14}$. Averages were taken over 1000 realizations for $t={10}^7–{10}^{11}$ time units and over 100 realizations from ${10}^{12}$ to ${10}^{14}$ time units. In the figure above, the dotted lines are time-averaged MSD, $\overline{{\delta }^2}\left(\Delta ,t\right)$, of individual realizations of CTRWs, all with $\alpha =0.9$, $t={10}^9$, and a reflecting boundary $\left(L=20\right)$, such that ${\tau }_c=1.3\times {10}^3{\tau }_0$. Full line: mean $⟨\overline{{\delta }^2}\left(\Delta ,t\right)⟩$ of 1000 of the above simulations.

Figure 2

(Color online) Simulation length $\left(t\right)$ dependence of $\overline{{\delta }^2}$ of a CTRW with exponent $\alpha =0.9$ and reflecting boundary $\left(L=20\right)$. The averages are shown over 1000 CTRWs for total lengths, $t$, varying between ${10}^7$ and ${10}^{11}$ and 100 CTRWs for ${10}^{12}$ time units ($t$ increases from the left to the right). In order that each time series should contain the same number of points $\left({10}^7\right)$, the time resolution of the longer simulations was reduced. Therefore, the short-$\Delta$ behavior of the MSDs of larger $t$ is absent. The quasifree linear-$\Delta$ behavior breaks down when the boundary becomes sensed at ${\tau }_c=1.3\times {10}^3{\tau }_0$. A $\Delta$-sublinear MSD, i.e., $\sim {\Delta }^{1-\alpha }$, is seen on longer time scales. $⟨\overline{{\delta }^2}⟩$ is bounded by the value $L^2/6$. For longer $t$, $⟨\overline{{\delta }^2}⟩$ is shifted to smaller values but does not reach a constant plateau.

Figure 3

(Color online) (A) Scaled, TA MSD, $t^{1-\alpha }⟨\overline{{\delta }^2}⟩$ of CTRWs with different $\alpha =0.5,0.7,0.9$ (top down) and reflecting boundaries $\left(L=20\right)$ such that ${\tau }_c$ has the values $1.4\times {10}^4{\tau }_0$, $1.9\times {10}^3{\tau }_0$, and $1.3\times {10}^3{\tau }_0$, respectively. For each $t={10}^7,{10}^8,\dots$ up to ${10}^{11}$ and each $\alpha$, 1000 simulations, (and for $t={10}^{12}$ to ${10}^{14}$, 100 simulations) were performed. The figure displays the average over the different time-averaged MSDs. (B) Fit of Eq. (12) to data of A (dotted), for the same and further exponents, $\alpha =0.3$, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 (top down). The good fit to the simulation data of the analytical curve (dashed) demonstrates the validity of the approximations made in the derivation of Eqs. (10, 11) and the usefulness of interpolation formula (12).

Figure 4

(Color online) Ergodicity breaking parameter EB as a function of the observation time $t$ (logarithmic scale) for various $\alpha$ from 0.3 to 0.9 (top down, see legend). The parameter was calculated for $\Delta ={10}^5$ (open markers) and $\Delta ={10}^6$ (filled markers; if only filled marker is visible, both values coincide). The full lines indicate the prediction of Eq. (13). The ensemble average was performed over 1000 realizations for $t={10}^7,\dots ,{10}^{11}$ and 100 realizations for $t={10}^{12}$.