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Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster

  • 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 theWe 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 similar to 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.show moreshow less

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Metadaten
Author:Yousof Mardoukhi, Jae-Hyung Jeon, Ralf MetzlerORCiDGND
DOI:https://doi.org/10.1039/c5cp03548a
ISSN:1463-9076 (print)
ISSN:1463-9084 (online)
Pubmed Id:http://www.ncbi.nlm.nih.gov/pubmed?term=26503611
Parent Title (English):Physical chemistry, chemical physics : a journal of European Chemical Societies
Publisher:Royal Society of Chemistry
Place of publication:Cambridge
Document Type:Article
Language:English
Year of first Publication:2015
Year of Completion:2015
Release Date:2017/03/27
Volume:17
Issue:44
Pagenumber:14
First Page:30134
Last Page:30147
Funder:Academy of Finland (Suomen Akatemia) within the Finland Distinguished Professor (FiDiPro) programme
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert