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VACF within the AR model
The autocorrelation function $r(t)$ introduced in the previous Section is here the normalized VACF
\begin{displaymath}
VACF(t) := \frac{\langle v(t)v(0)\rangle}{\langle v^2(0)\rangle}
\end{displaymath} (4.71)

hence $r(0)=VACF(0)=1$. Here $v(t)$ is the x-, y-, or z-component of the velocity of a `tagged' atom. The memory function $\xi(t)$ of $\psi(t)$ is defined by the relation
\begin{displaymath}
\frac{d}{dt}\VACF(t) = -\int_{0}^{t}d\tau  \xi(t-\tau)\VACF(\tau).
\end{displaymath} (4.72)

Eq. (4.72) is called the memory function equation.

Within the AR-model the z-transform of the VACF has the form

\begin{displaymath}
\mathrm{VACF}^{(AR)}(z) = \frac{1}{a^{(P)}_P}\frac{-z^P\sigma_P^2}
{\prod_{k=1}^P (z-z_k) \prod_{l=1}^P(z-z^{-1}_l)}.
\end{displaymath} (4.73)

Here the $\{z_k\}$ are the zeros of
\begin{displaymath}
p(z) = z^P-\sum_{k=1}^{P}a^{(P)}_k z^{P-k}.
\end{displaymath} (4.74)

We recall that the z-transform of an arbitrary discrete function $f(n)$ is given by $F(z) = \sum_{n=-\infty}^{+\infty} f(n) z^{-n}$, and the inverse transform by $f(n) = \frac{1}{2\pi i}\oint_C dz z^{n-1}
F(z)$. Applying the inverse z-transform to (4.73) yields
\begin{displaymath}
VACF^{(AR)}(n) = \sum_{j=1}^{P}\beta_jz_j^{\vert n\vert},
\end{displaymath} (4.75)

where the coefficients $\beta_j$ are given by
\begin{displaymath}
\beta_j = \frac{1}{a_P}\frac{-z_j^{P-1}\sigma_P^2}
{\prod_{k=1,k\ne j}^P (z_j-z_k) \prod_{l=1}^P(z_j-z^{-1}_l)}.
\end{displaymath} (4.76)

Note that $VACF^{(AR)}(n)$ has a multiexponential form, and that the stability criterion
\begin{displaymath}
\vert z_j\vert < 1,\quad j=1,\ldots,P,
\end{displaymath} (4.77)

must be fulfilled. This is guaranteed by the Burg-algorithm [60,61].


next up previous contents
Next: Density of states within Up: Theory and implementation Previous: Theory and implementation   Contents
pellegrini eric 2009-10-06