Analysis of Single Particle Trajectories: from Normal to Anomalous Diffusion∗

With modern experimental tools it is possible to track the motion of single nanoparticles in real time, even in complex environments such as biological cells. The quest is then to reliably evaluate the time series of individual trajectories. While this is straightforward for particles performing normal Brownian motion, interesting subtleties occur in the case of anomalously diffusing particles: it is no longer granted that the long time average equals the ensemble average. We here discuss for two different models of anomalous diffusion the detailed behaviour of time averaged mean squared displacement and related quantities, and present possible criteria to analyse single particle trajectories. An important finding is that although the time average may suggest normal diffusion the actual process may in fact be subdiffusive.