We have seen before that pure PL velocity correlations can produce positive kurtosis, however so far only in a very unrealistic way. Truely convincing would be a PL trajectory with an exponential (in stead of Gaussian) step width distribution (SWD). Can the velocity phases be tuned in such a way ?

To answer that question, en ensemble of 1000 trajectories was generated, with velocity phases randomly distributed between -pi an +pi. For each trajectory, the SWD was computed and least-square-fitted to an exponential distribution. The resulting error was high for all trajectories and fluctuated between about 20 and 40 (a good fit would have an error smaller than 1).

The trajectory with the lowest error was nr. 558. However, even this guy had still an extremely Gaussian SWD:

The kurtosis-versus-lagtime was also computed for 558 and compared to that of an artificial trajectory with truely exponential steps:

Obviosly, 558 is still far away from our goal.

In conclusion, it seems very probable that pure PL trajectories cannot have exponential SWDs. Or is the island of exponential-SWD concentrated in a very tiny region of the huge phase space, so that even among 1000 random probes not a single one comes close ?

Of course, PL trajectories might be compatible with other types of realistic distributions which positive kurtosis, like Gaussian peaks with abnormally fat tails.

## No comments:

## Post a Comment