## Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity

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**Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity.** / Cherstvy, Andrey G.; Metzler, Ralf.

Research output: Contribution to journal › Article › Scientific › peer-review

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*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2015, no. 5, P05010. https://doi.org/10.1088/1742-5468/2015/05/P05010

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*Journal of Statistical Mechanics: Theory and Experiment*,

*2015*(5), [P05010]. https://doi.org/10.1088/1742-5468/2015/05/P05010

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TY - JOUR

T1 - Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity

AU - Cherstvy, Andrey G.

AU - Metzler, Ralf

PY - 2015/5/15

Y1 - 2015/5/15

N2 - We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) ∼ D0|x|αtβ depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.

AB - We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) ∼ D0|x|αtβ depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.

KW - diffusion

UR - http://www.scopus.com/inward/record.url?scp=84930653082&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2015/05/P05010

DO - 10.1088/1742-5468/2015/05/P05010

M3 - Article

VL - 2015

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 5

M1 - P05010

ER -