Ageing effects in single particle trajectory averages

We study time averages of single particle trajectories in scale free anomalous diffusion processes, in which the measurement starts at some time ta > 0 after initiation of the process at the time origin, t = 0. Using ageing renewal theory we show that for such non-stationary processes a large class of observables are affected by a unique ageing function, which is independent of boundary conditions or the external forces. We quantify the weakly non-ergodic nature of this process in terms of the distribution of time averages and the ergodicity breaking parameter which both explicitly depend on the ageing time ta. Consequences for the interpretation of single particle tracking data are discussed.