Ageing Scher–Montroll Transport

Krüsemann, Henning; Schwarzl, Richard; Metzler, Ralf

doi:10.1007/s11242-016-0686-y

Cited by 13 publications

(7 citation statements)

References 65 publications

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“…For instance, when we measure the motion of an endogenous lipid granule in a living cell, it is not clear what the starting point for this process is. Or, we could probe the charge current in amorphous semiconductors and on purpose introduce a time shift between the creation of the charge carriers and the moment when we switch on the driving electrical field 81 , 82 . If the dynamics is stationary, a delay between system initiation and start of the measurement does not make any difference, as the correlation functions solely depend on the time difference.…”

Section: Resultsmentioning

confidence: 99%

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Schwarzl

Godec

Metzler

2017

Sci Rep

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Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.

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Section: Resultsmentioning

confidence: 99%

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Schwarzl

Godec

Metzler

2017

Sci Rep

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“…In particular, for out-ofequilibrium processes such as SBM one expects severe effects of ageing onto the particle dynamics [99,100]. Therefore, the time interval t a impacts the statistical properties [11,101,102]. Effects of ageing are observed, for instance, in glassy systems [103][104][105][106][107], homogeneously cooled granular fluids [108], for diffusion in plasma cell membranes [109], protein dynamics [110,111], in polymeric semiconductors [112], as well as for blinking statistics of quantum dots [113,114].…”

Section: Introductionmentioning

confidence: 99%

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

et al. 2017

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We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

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“…This method is based on a semi-Markov representation of the CTRW limit, where the memory is explicitly included, see also Germano et al [11]. Krüsemann, Schwarz and Metzler [17] demonstrate how the non-Markovian nature (ageing, or memory) can be observed in Scher-Montroll experiments on transient photocurrent in amorphous materials. Barkai…”

Section: Uncoupled Ctrwmentioning

confidence: 99%

Continuous time random walks and space-time fractional differential equations

Meerschaert¹,

Scheffler²

2019

Basic Theory

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The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of this model is governed by a fractional diffusion equation. If the step length of the random walk follows a power law, we get a spacefractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives.

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Ageing Scher–Montroll Transport

Cited by 13 publications

References 65 publications

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

Continuous time random walks and space-time fractional differential equations

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