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General properties of time correlation functions

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    Define a time correlation function between two quantities \(A (x) \) and \(B (x) \) by

    \(C_{AB} (t) \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(\langle A(0)B(t)\rangle \)
    $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \( \underline {\int d{\rm x}f({\rm x})A({\rm x})e^{iLt}B({\rm x})} \)

    The following properties follow immediately from the above definition:


    \[\langle A(0)B(t)\rangle = \langle A(-t)B(0)\rangle \]

    \[C_{AB}(0) = \langle A({\rm x})B({\rm x})\rangle\]

    Thus, if \(A = B \), then
    \[C_{AA}(t) = \langle A(0)A(t)\rangle \]

    known as the autocorrelation function of \(A\), and
    \[C_{AA}(0) = \langle A^2\rangle\]

    If we define \(\delta A = A - \langle A \rangle \), then
    \[C_{\delta A\delta A}(0) = \langle (\delta A)^2\rangle =\langle ( A - \langle A \rangle )^2 \rangle = \langle A^2\rangle - \langle A\rangle^2 \]

    which just measures the fluctuations in the quantity \(A\).
    A time correlation function may be evaluated as a time average, assuming the system is ergodic. In this case, the phase space average may be equated to a time average, and we have
    \[C_{AB}(t) = \lim_{T\rightarrow\infty}{1 \over T-t}\int_0^{T-t}ds A({\rm x}(s))B({\rm x}(t+s)) \]

    which is valid for \( t<<T \). In molecular dynamics simulations, where the phase space trajectory is determined at discrete time steps, the integral is expressed as a sum
    \[ C_{AB}(k\Delta t) = {1 \over N-k}\sum_{j=1}^{N-k}A({\rm x}_k)B({\rm x}_{k+j})\;\;\;\;\;\;\;\;\;\;k=0,1,2,...,N_c \]

    where \(N\) is the total number of time steps, \(\Delta t \) is the time step and \(N_c << N \).
    Onsager regression hypothesis: In the long time limit, \(A \) and \(B\) eventually become uncorrelated from each other so that the time correlation function becomes
    \[ C_{AB}(t) = \langle A(0)B(t)\rangle \rightarrow \langle A\rangle\langle B\rangle \]

    For the autocorrelation function of \(A\), this becomes
    \[ C_{AA}(t)\rightarrow \langle A\rangle^2 \]

    Thus, \(C_{AA} (t) \) decays from \(\langle A^2 \rangle \) at \( t = 0\) to \( \langle A^2 \rangle \) as \(t \rightarrow \infty \).

    An example of a signal and its time correlation function appears in the figure below. In this case, the signal is the magnitude of the velocity along the bond of a diatomic molecule interacting with a Lennard-Jones bath. Its time correlation function is shown beneath the signal:

    Figure 1:

    Over time, it can be seen that the property being autocorrelated eventually becomes uncorrelated with itself.

    Contributors and Attributions

    Mark Tuckerman (New York University)

    General properties of time correlation functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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