Subject: summary: diffusion coefficient from MSD

From: Hannes Loeffler <[email-masked]>
Date: Thu, 30 May 2002 09:21:02 +0200

Hello,

I recently asked about details concerning the calculation of the diffusion
coefficient from the mean square displacement.  I wanted to know how long I
should simulate to obtain meaningful results and why the MSD curve might be
non-linear.  In the following I will briefly summarize from the answers I
received and my own investigations.

General remarks:
1) If simulations are performed under PBCs the trajectory must be properly
   unfolded (revert PBCs) before any further calculation may be done.
2) Statistics is considerably improved if multiple time origins are taken into
   account.

"Long" time:
There seems to be some agreement that at least several hundred picoseconds are
necessary to obtain meaningful results.  The simulation time depends, of course,
also on the ensemble size (single molecules naturally need longer simulation times).

Non-linearity:
1) One cause may be insufficient statistics at long time.  In that case a
   weighted linear fit may be helpful using the standard deviation as the
   weights.
2) Another problem may arise from translational and/or rotational movement of
   the total system.

Error estimation:
1) The covariance matrix may be derived from the slope which yields a measure
   of the error bar.
2) Another method seems to be to plot ln(MSD) vs ln(t).  The slope should be
   close to 1.
3) I found the paper from G. C. Lie and E. Clementi [Phys. Rev. A, 33(1986),
   2679] very usful to interpret statical and dynamical properties in solution.
   These authors calculate the MSD for a water simulation separately for H's
   and O's. At short time (a few picoseconds) both MSDs develop into a straight
   line but different slope.  However, since water must move as a single
   entity, the curves must have the same slope after some time.

I want to thank everybody for help.  I hope nobody minds if I do not copy all
the response here to avoid wasting band width.  People sending me valuable
answers: Michael Brunsteiner, Kenneth Geisshirt, Richard Gillilan, Scott Feller,
James Kubicki,  David van der Spoel, Brian K. Peterson, Chas Simpson.

Thanks again for all the help,
Hannes.

-- 
Hannes Loeffler | tel: +43-(0)512-507-5166 | fax: +43-(0)512-507-2714
http://www-c724.uibk.ac.at/staff/loeffler/ | mailto:[email-masked]
Institute for General, Inorganic and Theoretical Chemistry
University of Innsbruck, Austria