General properties of time correlation functions

Define a time correlation function between two quantities and by

The following properties follow immediately from the above definition:

1.

2.

Thus, if A=B, then

known as the autocorrelation function of A, and

If we define , then

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

which is valid for t;SPMlt;;SPMlt;T. In molecular dynamics simulations, where the phase space trajectory is determined at discrete time steps, the integral is expressed as a sum

where N is the total number of time steps, is the time step and .

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

For the autocorrelation function of A, this becomes

Thus, decays from at t=0 to as .

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.