Abstract
We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T(-h) with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.
Highlights
We investigate the ergodic properties of a random walker performing diffusion on a random fractal geometry
For the evaluation of physical observables Nz random percolation geometries were generated at a given percolation density p, and N tracer trajectories for each percolation geometry were recorded
We study the functional behaviour of the fluctuations of the time averaged mean squared displacement (MSD) as the observation time T is increased
Summary
Mean squared displacement (MSD) grows non-linearly with time in the form hr2(t)i C ta (1). To identify a specific anomalous diffusion process behind observed data allows one to learn more about the physical nature of the system and predict secondary quantities such as the first passage behaviour responsible for (bio)chemical reactions.[29,31,56,57,58,59] Weakly non-ergodic processes in Bouchaud’s sense exhibit the above-mentioned disparity between the ensemble averaged MSD (1) and the time averaged MSD defined below, even in the limit of long measurement times These processes exhibit ageing, the dependence of observables on the time span between initiation of the process and start of the measurement, and the amplitude of individual time averages fluctuates significantly.
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