Nonergodicity mimicking inhomogeneity in single particle tracking

Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Recent single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with an asymptotic power law distribution of waiting times lacking the mean. We show, that contrary to what is expected, the mean squared displacement obtained as a moving temporal average along the single trajectory, exhibits a simple diffusive behavior with diffusion coefficients which strongly differ from one trajectory to another. This finding mimics the inhomogeneous behavior as in an ensemble of random walkers with different diffusion coefficients. The information about the anomaly can be restored by an additional ensemble average over these diffusion coefficients, which results in an effective diffusion-coefficient which depends on the length of the trajectory T and on the exponent of the waiting time distribution.