Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

By Ralf Metzler, Jae-Hyung Jeon, Andrey G. Cherstvy, Eli Barkai. Physical Chemistry, Chemical Physics - PCCP, 2014

1,370 citations recorded by the source corpus.

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

Subfields: Modeling and Simulation, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/021a8735f98c7a92ffc17f993e2c25fafc84b648

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Citations in the corpus

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “We refer interested readers to the comprehensive surveys [7, 8, 9] and the monograph [10] for an extensive list of practical applications and physical modeling in engineering, physic, and biology.”
Time-fractional diffusion equation in the fractional Sobolev spaces (2015, background): “In the very recent survey paper [21] that is devoted to the systems, which exhibit anomalous diffusion, about three hundred references to relevant works are given.”
Protein Crowding in Lipid Bilayers Gives Rise to Non-Gaussian Anomalous Lateral Diffusion of Phospholipids and Proteins (2016, methodology): “the increase of T, we evaluate the ergodicity breaking parameter [35,104,105]”
Numerical approximation of fractional powers of elliptic operators (2013, background): “A recent discussion about the properties of such models and their applications to chemistry, geosciences, and engineering can be found in [13, 19].”
Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion (2018, background): “[11, 12, 34] as well as weak ergodicity breaking and ageing [35, 36].”