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##### VACF within the AR model

The autocorrelation function introduced in the previous Section is here the normalized VACF

| (4.71)

hence . Here is the x-, y-, or z-component of the velocity of a `tagged' atom. The memory function of is defined by the relation

| (4.72)

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

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

| (4.73)

Here the are the zeros of

| (4.74)

We recall that the z-transform of an arbitrary discrete function is given by , and the inverse transform by . Applying the inverse z-transform to (4.73) yields

| (4.75)

where the coefficients are given by

| (4.76)

Note that has a multiexponential form, and that the stability criterion

| (4.77)

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

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