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Starting from the memory function equation of the VACF (Eq. 4.72), the first step towards a
numerical computation of the memory function consists in discretizing Eq. 4.72
 |
(4.79) |
Eq. (4.79) is now subjected to a one-sided z-transform. Using that
 |
(4.80) |
for any discrete function
whose one-sided z-transform exists, one obtains from (4.79)
 |
(4.81) |
using that
. The one-sided z-transform of an arbitrary discrete function
is defined as
.
Here it has been used that the one-sided z-transform of the discrete convolution integral is just the product
. Inserting the definition of the one-sided z-transform for
and
, this equation can be rewritten as
![\begin{displaymath}
\sum_{j=0}^\infty \xi(j) z^{-j} = \frac{1}{\Delta t^2}...
...ight] z^{-j}}
{ \sum_{j=0}^\infty VACF(j) z^{-j} }.
\end{displaymath}](img270.gif) |
(4.82) |
Note that the term proportional to z cancels out. The time dependent memory function is, in principle, obtained by
comparing the coefficients of the series on the lhs and the rhs of Eq. (4.82). To construct a numerical method,
we replace the series by polynomials of order N, where
defines the time window for the memory function to be computed.
After this first step a polynomial division is performed on the rhs of Eq. (4.82), and after a subsequent multiplication
of both sides with
one obtains the time dependent memory function,
, by comparison of coefficients,
Within nMOLDYN
is replaced by the autocorrelation function calculated in the framework of the autoregressive model,
, as in Eqs. 4.75 and 4.76. The coefficients
are obtained by polynomial division and
is a rest which does not contain information on the memory function within the time interval
. The discrete
memory function is therefore given by
.
A remark concerning the discretization scheme (4.79) is in place here. The discrete convolution
sum is effectively a first order approximation of the convolution integral. More sophisticated approximations could be used,
but they would lead to less convenient expressions upon z-transformation. Correspondingly, we have chosen a first order
approximation for the differentiation on the left-hand side of (4.72). In this way the first order
(integro-)differential equation (4.72) is transformed into the first order difference equation
(4.79).
However, this simple discretization scheme together with the use of the one-sided z-transform leads to a significant
error in
. It is clear from Eq. (4.72) that due to the symmetry of the autocorrelation function
(
), the derivative
should vanish at
However, in the discretized version it is
approximated by a forward difference that is always negative. A higher-order calculation shows that the estimate for
that results from the procedure described above should be doubled.
Next: MSD within the AR
Up: Theory and implementation
Previous: Density of states within
  Contents
pellegrini eric
2009-10-06