Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist

Single particle trajectories are investigated assuming the coexistence of two subdiffusive processes: diffusion on a fractal structure modeling spatial constraints on motion and heavy-tailed continuous time random walks representing energetic or chemical traps. The particles’ mean squared displacement is found to depend on the way the mean is taken: temporal averaging over single-particle trajectories differs from averaging over an ensemble of particles. This is shown to stem from subordinating an ergodic anomalous process to a nonergodic one. The result is easily generalized to the subordination of any other ergodic process (i.e., fractional Brownian motion) to a nonergodic one. For certain parameters the ergodic diffusion on the underlying fractal structure dominates the transport yet displaying ergodicity breaking and aging.

Figure 1

(Color online) Ensemble and temporal averaging of MSD of diffusion on a percolation cluster at criticality. The MSDs are plotted on a log-log scale, where the slope is equivalent to the time exponent $\beta$. The plots of the ensemble and the temporal averaging coincide displaying ergodicity. The percolation cluster was generated using site percolation on a square lattice of size $250\times 250$, with periodic boundary conditions, at critical concentration $p_c=0.592746$. Note that due to the overlap of the curves it is difficult to distinguish between the ensemble and temporal averages plots. The theoretical slope with $\beta =0.697$ appears as a guide.

Figure 2

(Color online) Heavy tailed CTRW on a Sierpinski gasket. Plot of the sum of coordinates of the RWs position $x+y$ as a function of time. In the inset a schematic representation of the process is shown, where circles represent different waiting times. In our simulations we used a Sierpinski gasket of 11th generation in two dimensions.

Figure 3

(Color online) A log-log plot of the temporal and ensemble MSDs of a CTRW $\left(\alpha =0.3\right)$ on a percolation cluster at criticality. The distribution of diffusion constants is apparent for the temporal MSDs of different trajectories. The theoretical slopes are given as a guide for the eye. The numerically calculated ensemble average, at the top, fits the given theoretical slope. The same can be said for the different numerically calculated temporal MSDs, shown at the bottom part of the plot.

Figure 4

(Color online) Schematic picture of events for the construction of the forward waiting time distribution, Eq. (11) and the averaging in Eq. (9).