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The relation between the VACF and the MSD reads
![\begin{displaymath}
{MSD}(t)=\langle[x(t)-x(0)]^2\rangle=2 \int_{0}^{t}d\tau (t - \tau)
C_{vv}(t)
\end{displaymath}](img286.gif) |
(4.84) |
By discretizing the above equation one obtains
 |
(4.85) |
Using
 |
|
|
(4.86) |
into Eq. 4.85, gives
 |
(4.87) |
Making use of the one-side z-transform (equivalent to the Laplace transform for discrete functions), we obtain
 |
(4.88) |
where
 |
(4.89) |
Introducing Eq. 4.89 into Eq. 4.88 yields
 |
(4.90) |
and its inverse z-transform reads
 |
(4.91) |
Using the expression of the non-normalized
obtained in the framework of the AR model
 |
(4.92) |
one finds the expression of MSD within the same AR model
 |
(4.95) |
which allows one to compute the MSD within the AR model from the poles and the
coefficients of the non-normalized VACF.
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pellegrini eric
2009-10-06