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Single particle tracking has become a standard tool to investigate diffusiveproperties, especially in small systems such as biological cells. Usually theresulting time series are analyzed in terms of time averages over individualtrajectories. Here we study confined normal as well as anomalous diffusionmodeled by fractional Brownian motion and the fractional Langevin equation, andshow that even for such ergodic systems time-averaged quantities behavedifferently from their ensemble averaged counterparts, irrespective of how longthe measurement time becomes. Knowledge of the exact behavior of time averagesis therefore fundamental for the proper physical interpretation of measuredtime series, in particular, for extraction of the relaxation time scale fromdata.

Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement