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MSD within the AR model
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} (4.84)

By discretizing the above equation one obtains
\begin{displaymath}
MSD(n)=2 \sum_{k=0}^n \Delta t^2(n-k)C_{vv}(k).
\end{displaymath} (4.85)

Using
$\displaystyle f_1(n)=\Theta(n)\cdot n
f_2(n)=\Theta(n)\cdot C_{vv}(n)$     (4.86)

into Eq. 4.85, gives
\begin{displaymath}
MSD(n)=2 \sum_{k=-\infty}^{+\infty} \Delta t^2f_1(n-k)f_2(k)
\end{displaymath} (4.87)

Making use of the one-side z-transform (equivalent to the Laplace transform for discrete functions), we obtain
\begin{displaymath}
{MSD}^{>}(z)= 2 F_1^{>} F_2^{>}\Delta t^2 ,
\end{displaymath} (4.88)

where
\begin{displaymath}
F_1^{>}=\frac{z}{(z-1)^2}.
\end{displaymath} (4.89)

Introducing Eq. 4.89 into Eq. 4.88 yields
\begin{displaymath}
MSD^{>}(z)= 2 \frac{z\Delta t^2}{(z-1)^2}C_{vv}^{>}(z)
\end{displaymath} (4.90)

and its inverse z-transform reads
\begin{displaymath}
{MSD}(n)= 2\Delta t^2\cdot\frac{1}{2\pi i}\oint dz z^{n-1}\cdot\frac{z}{(z-1)^2}\cdot C_{vv}^{>}(z).
\end{displaymath} (4.91)

Using the expression of the non-normalized $C_{vv}^{\rangle}(z)$ obtained in the framework of the AR model
\begin{displaymath}
C_{vv}^{>}(z)= \sum_{j=1}^P\beta_j\frac{z}{z-z_j}\cdot\langle v^2\rangle
\end{displaymath} (4.92)

one finds the expression of MSD within the same AR model
$\displaystyle MSD^{AR}(n)$ $\textstyle =$ $\displaystyle 2\Delta t^2\langle v^2\rangle \sum_{j=1}^P\beta_j \frac{1}{2\pi i}\oint dz \frac{z^{n+1}}{(z-1)^2}\cdot\frac{1}{z-z_j}$ (4.93)
  $\textstyle =$ $\displaystyle 2\Delta t^2\langle v^2\rangle \sum_{j=1}^P\beta_j \{\frac{n}{1-z_j}-\frac{z_j}{(1-z_j)^2}+\frac{z_j^{n+1}}{(1-z_j)^2}\}.$ (4.94)


\begin{displaymath}
MSD^{AR}(n)\stackrel{n\rightarrow +\infty}{\simeq} 2Dn\Delta...
...{j=1}^P\frac{\beta_j}{1-z_j}=\frac{\langle v^2\rangle}{\gamma}
\end{displaymath} (4.95)

which allows one to compute the MSD within the AR model from the poles and the $\beta_j$ coefficients of the non-normalized VACF.


next up previous contents
Next: Friction coefficient within the Up: Theory and implementation Previous: Discrete memory function of   Contents
pellegrini eric 2009-10-06