Abstract
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). 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.
Highlights
Random walks are one of the most central concepts in modern physics
The study of stochastic trajectories starts with an analysis of the mean squared displacement (MSD), which for Brownian motion reads 〈x2〉 = 2Ddt, with D being the diffusion coefficient and d the spatial dimension
This scaling of distance with time is typical for continuous-time Brownian motion and serves to distinguish it from other random processes, whose scaling ∼t α deviates from the standard value α = 1
Summary
Random walks are one of the most central concepts in modern physics. In particular, they are the key concept to describe transport processes in complex physical systems, ranging from electrons in solids and colloids in fluids to tracer particles in the geosciences and aerosol particles in the atmosphere of the earth. The study of stochastic trajectories starts with an analysis of the mean squared displacement (MSD), which for Brownian motion reads 〈x2〉 = 2Ddt, with D being the diffusion coefficient and d the spatial dimension. For Brownian motion, it has been established early that a sufficiently long single trajectory can be used to extract the ensemble value for the diffusion coefficient [8].
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