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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.

Abstract. Fractional dynamics has experienced a firm upswing during the last years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional FokkerPlanck equation [R. Metzler and J. Klafter, Phys. Rep. 339 (2000) 1-77]. It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub-and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.

The irreproducibility of time-averaged observables in living cells poses fundamental questions for statistical mechanics and reshapes our views on cell biology.

We introduce a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlinear force and a thermal heat bath. For the forcefree case, a subdiffusive behavior is recovered. The equation is shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution. The relaxation of single modes is shown to follow a Mittag-Leffler decay. We discuss the example of a particle in a harmonic potential.

Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion. of larger molecules or tracers in living cells [3,[6][7][8][9][10][11][12][13][14][15][16]. While normal diffusion, by virtue of the central limit theorem, is characterized by the universal Gaussian probability density function and therefore uniquely determined by the first and second moments [17], anomalous diffusion of the form (1) is non-universal and may be caused by different stochastic mechanisms. These would give rise to vastly different behavior for diffusional mixing, diffusionlimited reactions, signaling, or regulatory processes. To better understand cellular dynamics, knowledge of the underlying stochastic mechanism is thus imperative.Here we report experimental evidence from extensive single trajectory time series of lipid granule motion in Schizosaccharomyces pombe (S. pombe) fission yeast cells obtained from tracking with optical tweezers (resolving 10 −6 sec to 1 sec) and video microscopy (10 −2 sec to 100 sec). Using complementary analysis tools we demonstrate that at short times the data are described best by continuous time random walk (CTRW) subdiffusion, revealing pronounced features of weak ergodicity breaking in the time averaged mean squared displacement. At longer times the stochastic mechanism is closest to subdiffusive fractional Brownian motion (FBM). The time scales over which this anomalous behavior persists is relevant for biological processes occurring in the cell. Anomalous diffusion may indeed be a good strategy for cellular signaling and reactions [7,8].CTRW and FBM both effect anomalous diffusion of the type (1) [18]. Subdiffusive CTRWs are random walks with finite variance δx 2 of jump lengths, while the waiting times between successive jumps are drawn from a density ψ(t) ≃ τ α /t 1+α with diverging characteristic time [4,17]. Such scale-free behavior results from multiple trapping events in, e.g., comb-like structures [19] or random energy landscapes [20]. Power-law waiting time distributions were also identified for tracer motion in reconstituted actin networks [21]. Subdiffusive FBM is a random process driven by Gaussian noise ξ with longrange correlations, ξ(0)ξ(t) ≃ α(α − 1)t α−2 [22], and it is related to fractional Langevin equations [23]. The finite characteristic time scales associated with FBM contrast the ageing property of the subdiffusive CTRW processes [Supplementary Material (SM)].Single particle tracking microscopy has become a standard tool to probe the motion of individual tracers, espe...

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