The power of a single trajectory - Schnellbächer, Nikolas David; Schwarz, Ulrich S. (New journal of physics, 2018)
Random walks are often evaluated in terms of their mean squared displacements, either for a large number of trajectories or for one very long trajectory. An alternative evaluation is based on the power spectral density, but here it is less clear which information can be extracted from a single trajectory. For continuous-time Brownian motion, Krapf et al now have mathematically proven that the one property that can be reliably extracted from a single trajectory is the frequency dependence of the ensemble-averaged power spectral density (Krapf et al 2018 New J. Phys. 20 023029 [http://iopscience.iop.org/article/10.1088/1367-2630/aaa67c] ). Their mathematical analysis also identifies the appropriate frequency window for this procedure and shows that the diffusion coefficient can be extracted by averaging over a small number of trajectories. The authors have verified their analytical results both by computer simulations and experiments.