# 12.4: General Properties of Time Correlation Functions

• • Mark Tuckerman
• New York University
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Define a time correlation function between two quantities $$A (x)$$ and $$B (x)$$ by

\begin{align*} C_{AB} (t) &= \langle A(0)B(t)\rangle \\[4pt] &= \int d{\rm x}f({\rm x})A({\rm x})e^{iLt}B({\rm x}) \end{align*}

The following properties follow immediately from the above definition:

## Property 1

$\langle A(0)B(t)\rangle = \langle A(-t)B(0)\rangle \nonumber$

## Property 2

$C_{AB}(0) = \langle A({\rm x})B({\rm x})\rangle \nonumber$ Thus, if $$A = B$$, then

$C_{AA}(t) = \langle A(0)A(t)\rangle \nonumber$

known as the autocorrelation function of $$A$$, and

$C_{AA}(0) = \langle A^2\rangle \nonumber$

If we define $$\delta A = A - \langle A \rangle$$, then

$C_{\delta A\delta A}(0) = \langle (\delta A)^2\rangle =\langle ( A - \langle A \rangle )^2 \rangle = \langle A^2\rangle - \langle A\rangle^2 \nonumber$

which just measures the fluctuations in the quantity $$A$$.

## Property 3

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

$C_{AB}(t) = \lim_{T\rightarrow\infty}{1 \over T-t}\int_0^{T-t}ds A({\rm x}(s))B({\rm x}(t+s)) \nonumber$

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

$C_{AB}(k\Delta t) = {1 \over N-k}\sum_{j=1}^{N-k}A({\rm x}_k)B({\rm x}_{k+j})\;\;\;\;\;\;\;\;\;\;k=0,1,2,...,N_c \nonumber$

where $$N$$ is the total number of time steps, $$\Delta t$$ is the time step and $$N_c << N$$.

## Property 4: Onsager regression hypothesis

In the long time limit, $$A$$ and $$B$$ eventually become uncorrelated from each other so that the time correlation function becomes

$C_{AB}(t) = \langle A(0)B(t)\rangle \rightarrow \langle A\rangle\langle B\rangle \nonumber$

For the autocorrelation function of $$A$$, this becomes

$C_{AA}(t)\rightarrow \langle A\rangle^2 \nonumber$

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 Figure $$\PageIndex{1}$$. 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 $$\PageIndex{1}$$

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

This page titled 12.4: General Properties of Time Correlation Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mark Tuckerman.