Abstract
Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.
Highlights
Following the three groundbreaking papers on the theory of Brownian motion[1] by Albert Einstein,[2] Marian Smoluchowski,[3] and Paul Langevin,[4] in 1908 Jean Perrin reported the first systematic single particle tracking results in his seminal paper on diffusion
When we require that the fractional Gaussian noise (fGn) is internal and should fulfil the Kubo generalised fluctuation-dissipation theorem, the resulting particle motion in the overdamped limit is described by the fractional Langevin equation (FLE)[30] γ t
While for the free Fractional Brownian motion (FBM) and FLE motion ergodic behaviour is found the crossover to the stationary plateau turns out to be transiently non-ergodic
Summary
Following the three groundbreaking papers on the theory of Brownian motion[1] by Albert Einstein,[2] Marian Smoluchowski,[3] and Paul Langevin,[4] in 1908 Jean Perrin reported the first systematic single particle tracking results in his seminal paper on diffusion. Perrin used microscopic diffusion measurements of small putty particles to determine Avogadro’s number via the Einstein-Stokes-Smoluchowski relation.[5] Due to the relatively short trajectories, Perrin used the ensemble information of many measured, not completely identical particles in his analysis.[5] Only six years after Perrin’s first publication and exactly hundred years ago, in 1914 Ivar Nordlund conceived an experimental setup, that allowed him to record long time traces of. This is an Open Access article published by World Scientific Publishing Company. At a fixed lag time ∆, the width of this distribution decreases with increasing measurement time T ,29 and sufficiently long individual trajectories are in that sense reproducible
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