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.
Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.
R. Metzler,Jae-Hyung Jeon,Andrey G. Cherstvy,E. Barkai
Published 2014 in Physical Chemistry, Chemical Physics - PCCP
ABSTRACT
PUBLICATION RECORD
- Publication year
2014
- Venue
Physical Chemistry, Chemical Physics - PCCP
- Publication date
2014-09-22
- Fields of study
Medicine, Physics
- Identifiers
- External record
- Source metadata
Semantic Scholar, PubMed
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CLAIMS
CONCEPTS
- anomalous diffusion models
A family of stochastic models used to describe diffusion that departs from standard Brownian scaling or stationarity assumptions.
Aliases: anomalous diffusion model
- anomalous dynamics
Particle motion whose statistical behavior departs from standard Brownian diffusion.
Aliases: anomalous diffusion dynamics
- anomalous stochastic processes
Random processes that generate anomalous diffusion behavior rather than standard Brownian transport.
Aliases: anomalous processes
- brownian motion
The reference random-walk diffusion process used as the conventional baseline for comparison in the abstract.
Aliases: Brownian-like motion
- ensemble-averaged mean squared displacement
The mean squared displacement obtained by averaging over many realizations of a stochastic process at fixed time.
Aliases: ensemble averaged MSD, EAMSD
- ergodicity breaking parameter
A statistic used to quantify deviations from ergodic behavior across trajectories or time windows.
Aliases: EB parameter, ergodicity-breaking parameter
- ergodic properties
The properties describing whether time averages and ensemble averages coincide for a stochastic process.
Aliases: ergodicity
- physical origins
The mechanistic sources or system-level causes responsible for the different anomalous diffusion behaviors.
Aliases: underlying physical origins
- single-particle tracking
An experimental tracking approach that records the path of an individual labeled particle over time.
Aliases: SPT, single particle tracking
- time-averaged mean squared displacement
A trajectory-level displacement statistic computed by averaging squared displacements over lag times along a single path.
Aliases: time averaged MSD, TAMSD