# 10.2: Correlation Function from a Discrete Trajectory

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In practice classical correlation functions in molecular dynamics simulations or single molecule experiments are determined from a time-average over a long trajectory at discretely sampled data points. Let’s evaluate $$C _ {A A}$$for a discrete and finite trajectory in which we are given a series of $$N$$ observations of the dynamical variable $$A$$ at equally separated time points ti. The separation between time points is $$t _ {i + 1} - t _ {i} = \Delta t$$, and the length of the trajectory is $$T = N \Delta t$$. Then we have ,

$C _ {A A} = \frac {1} {T} \sum _ {i , j = 1}^{N} \Delta t A \left( t _ {i} \right) A \left( t _ {j} \right) = \frac {1} {N} \sum _ {i , j = 1}^{N} A _ {i} A _ {j} \label{9.16}$

where $$A _ {i} = A \left( t _ {i} \right)$$. To make this more useful we want to express it as the time interval between points $$\tau = t _ {j} - t _ {i} = ( j - i ) \Delta t$$, and average over all possible pairwise products of $$A$$ separated by $$\tau$$. Defining a new count integer $$n = j -i$$, we can express the delay as $$\tau = n \Delta t$$. For a finite data set there are a different number of observations to average over at each time interval (n). We have the most pairwise products—$$N$$ to be precise—when the time points are equal (ti=tj). We only have one data pair for the maximum delay $$\tau = T$$. Therefore, the number of pairwise products for a given delay $$\tau$$ is $$N-n$$. So we can write Equation \ref{9.16} as

$C _ {A A} ( \tau ) = C ( n ) = \frac {1} {N - n} \sum _ {i = 1}^{N - n} A _ {i + n} A _ {i} \label{9.17}$

Note that this expression will only be calculated for positive values of $$n$$, for which $$t_j≥t_i$$. As an example consider the following calculation for fluctuations in a vibrational frequency $$\omega(t)$$, which consists of 32000 consecutive frequencies in units of $$cm^{-1}$$ for points separated by 10 femtoseconds, and has a mean value of $$\omega _ {0} = 3244 \mathrm {cm}^{- 1}$$. This trajectory illustrates that there are fast fluctuations on femtosecond time scales, but the behavior is seemingly random on 100 picosecond time scales

After determining the variation from the mean $$\delta \omega \left( t _ {i} \right) = \omega \left( t _ {i} \right) - \omega 0$$, the frequency correlation function is determined from Equation \ref{9.17}, with the substitution $$\delta \omega \left( t _ {i} \right) \rightarrow A _ {i}$$.

We can see that the correlation function reveals no frequency correlation on the time scale of 104 –105 fs, however a decay of the correlation function is observed for short delays signifying the loss of memory in the fluctuating frequency on the 103 fs time scale. From Equation \ref{9.15}, we find that the correlation time is $$\tau_C = 785\, fs$$.

10.2: Correlation Function from a Discrete Trajectory is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.