Fractional kinetics emerging from ergodicity breaking in random media

We present a modeling approach for diffusion in a complex medium characterized by a random length scale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as ergodicity breaking, $p$ variation, and aging. In particular, this approach recapitulates characteristic features previously described in part by the fractional Brownian motion and in part by the continuous-time random walk. Moreover, for a proper distribution of the length scale, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from nonfractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking.

Figure 1

Time-averaged mean-square displacement $\overline{{\delta }^2}$ as a function of the time lag $\mathrm{\Delta }$ calculated for several trajectories (thin red lines) performing the fBm in a random medium, according to the ggBm (7) with $\beta =H=0.3$ and $T={10}^4$. The dashed line corresponds to the time- and ensemble-averaged mean-square displacement. The continuous thick line is a guide to the eye.

Figure 2

Plot of $E_B\left(T\right)$ for the stochastic process (7) with $\beta =0.5$ and $H=0.3$ at various time lags $\mathrm{\Delta }$ as a function of the measurement time $T$. Larger $\mathrm{\Delta }$ produces an increase of $E_B\left(T\right)$ at short time $T$. The $E_B\left(T\right)$ values at large time $T$ shown are in agreement with the theoretical expectation (16) (dashed line). (Inset) Results of the the $p$-variation test with $p=2$ for the stochastic process (7) with $\beta =0.5$ and $H=0.3$, showing the same trend as the pure fBm.

Figure 3

Plot of the time- and ensemble-averaged mean-square displacement $\langle \overline{{\delta }^2}\left(T\right)\rangle$ at various time lags $\mathrm{\Delta }$ and as a function of the measurement time $T$ for the process (19) with $\beta =H=0.3$ and $\alpha =-0.3$. The curves asymptotically show a power-law decay $T^{\alpha }$ (dashed lines), demonstrating the presence of aging in the process.

Figure 4

Contour plot of the aging exponent $\alpha$ as a function of the exponents controlling the power-law growth of the time- ($2H$) and ensemble-averaged ($\alpha +2H$) mean-square displacement for the process (19). The continuous black line corresponds to the absence of aging ($\alpha =0$). Dashed green lines separate sub- and superdiffusive regions, characterized by exponent values $<1$ and $>1$, respectively.