# General properties of time correlation functions

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

 $$C_{AB} (t)$$ $$\langle A(0)B(t)\rangle$$ $$\underline {\int d{\rm x}f({\rm x})A({\rm x})e^{iLt}B({\rm x})}$$

The following properties follow immediately from the above definition:

1.

$\langle A(0)B(t)\rangle = \langle A(-t)B(0)\rangle$
2.

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

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

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 $$N_c << N$$.
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
$C_{AB}(t) = \langle A(0)B(t)\rangle \rightarrow \langle A\rangle\langle B\rangle$

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:

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