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Friction coefficient within the AR model
The friction coefficient is defined as the integral over the memory function. In the discrete case we write
\begin{displaymath}
\xi_0 := \sum_{n=0}^{\infty}\Delta t\:\xi(n) =
\Delta t \Xi_{>}(1).
\end{displaymath} (4.96)

As shown in [57], the AR model allows us to express ${\Psi_{>}(z)}$ as
\begin{displaymath}
\mathrm{VACF}^{(AR)}_{>}(z)=\sum_{n=0}^{\infty}VACF^{(AR)}(n) z^{-n} = \sum_{j=1}^{P}\beta_j \frac{z_j}{z-z_j},
\end{displaymath} (4.97)

where the coefficients $\beta_j$ are given by eq. (4.76), and the roots $z_j$ must fulfill the stability criterion (4.77). Inserting (4.97) into (4.81) yields $\Xi^{(AR)}_{>}(z)$, the z-transform of the discrete memory function within the AR model,
\begin{displaymath}
\Xi^{(AR)}_{>}(z) = \frac{1}{\Delta t^2}
\left(\frac{z}{\mathrm{VACF}^{(AR)}_{>}(z)} + 1 - z\right).
\end{displaymath} (4.98)

Using (4.98) we obtain thus within the AR model
\begin{displaymath}
\xi^{(AR)}_0 = \frac{1}{\Delta t}
\frac{1}{\sum_{j=1}^{P}\beta_j\frac{1}{1-z_j}}.
\end{displaymath} (4.99)

This shows that $\xi_0$ can be obtained from the zeros $z_j$ of the characteristic polynomial $p(z)$, defined in (4.74).

In the framework of the autoregressive model, nMOLDYN allows one to calculate the VACF, the VACF memory function, the DOS, the MSD, and the AR coefficients $a_n^{(P)}$ of the velocity trajectory, averaged over all selected atoms and three Cartesian coordinates.


next up previous contents
Next: Parameters Up: Theory and implementation Previous: MSD within the AR   Contents
pellegrini eric 2009-10-06