[3] Universal PDF of 2D Brownian and Persistent Trajectories

I am grateful to Martin Reichelsdorfer, who has programmed and applied the CRFSA-method (center, rotate, flip, scale and average), as described in post [2], to various synthetic and experimental trajectories in 2D.

In contrast to the original method of Gonzalez et al., his raw data trajectories consisted of real points in 2D. Cells were only introduced in the final averaging step. Therefore, the flipping operation had to be redefined: It was done in such a way that the last point of the trajectory was always located right of the first point.

He first tested the method for a standard Brownian Random Walk. For a given time period dt, the PDF of a Random Walk is known to be a Gaussian, centered at the origin, with a variance proportional to dt. This is also what one naively expects to see as the ensemble averaged spatial distribution. However, after CRFSA one obtaines a double-peak structure:


Second, Martin applied CRFSA to measured trajectories of micro-beads bound to living MEVO cells. On long time scales, those trajectories show superdiffusive behaviour with fractional powerlaw exponents in the mean squared displeacement (MSD), corresponding to directional persistence in the bead motion. This persistence is a qualitative difference to the Brownian Random Walk and one might expect new features for the ensemble averaged spatial distribution function (SDF). However, the results for the Persistent Walks are not much different from before.


No significant changes are observed when the ensemble of trajectories is divided into groups of similar powerlaw exponent.

These results lead to a couple of questions:

• Why is the CRFSA-SDF of a Random Walk not a Gaussian ?
  
• What exactly is the origin of the double-peak ?
• What makes the trajectories of mobile phone users so qualitatively different from Brownian or peristent walks ?