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Anomalous diffusion has been widely observed by single particle trackingmicroscopy in complex systems such as biological cells. The resulting timeseries are usually evaluated in terms of time averages. Often anomalousdiffusion is connected with non-ergodic behaviour. In such cases the timeaverages remain random variables and hence irreproducible. Here we present adetailed analysis of the time averaged mean squared displacement for systemsgoverned by anomalous diffusion, considering both unconfined and restricted(corralled) motion. We discuss the behaviour of the time averaged mean squareddisplacement for two prominent stochastic processes, namely, continuous timerandom walks and fractional Brownian motion. We also study the distribution ofthe time averaged mean squared displacement around its ensemble mean, and showthat this distribution preserves typical process characteristic even for shorttime series. Recently, velocity correlation functions were suggested todistinguish between these processes. We here present analytucal expressions forthe velocity correlation functions. Knowledge of the results presented here areexpected to be relevant for the correct interpretation of single particletrajectory data in complex systems.

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking