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- 저자 : Mardoukhi, Yousof (검색결과 2 건)
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Fluctuations of random walks in critical random environments
Mardoukhi, Yousof,Jeon, Jae-Hyung,Chechkin, Aleksei V.,Metzler, Ralf The Royal Society of Chemistry 2018 Physical chemistry chemical physics Vol.20 No.31
<P>Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed.</P>
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Mardoukhi, Yousof,Jeon, Jae-Hyung,Metzler, Ralf The Royal Society of Chemistry 2015 Physical chemistry chemical physics Vol.17 No.44
<P>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 <I>T</I> for ergodic diffusion processes, follows a power-law ∼<I>T</I><SUP>−<I>h</I></SUP> with <I>h</I> < 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.</P> <P>Graphic Abstract</P><P>We study the strongly non-ergodic effects of a random walk on a percolation geometry below, at, and above the percolation threshold. <IMG SRC='http://pubs.rsc.org/services/images/RSCpubs.ePlatform.Service.FreeContent.ImageService.svc/ImageService/image/GA?id=c5cp03548a'> </P>
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