Correlation functions#
Theory of correlation functions#
The theory of correlation functions is well established 108.
We describe here the implementation of the various
correlation function flavors in the GROMACS code. The definition of the
autocorrelation function (ACF)
where the notation on the right hand side indicates averaging over
however, in GROMACS there is no standard mechanism to do this
(note: you can use the xmgr
program to compute cross correlations).
The integral of the correlation function over time is the correlation
time
In practice, correlation functions are calculated based on data points
with discrete time intervals
where
Here of course
However, one needs very long simulations to get good accuracy this way, because there are many fewer points that contribute to the ACF.
Using FFT for computation of the ACF#
The computational cost for calculating an ACF according to
(442) is proportional to
Special forms of the ACF#
There are some important varieties on the ACF, e.g. the ACF of a
vector
where
In order to study torsion angle dynamics, we define a dihedral autocorrelation function as 159:
Note that this is not a product of two functions as is generally used for correlation functions, but it may be rewritten as the sum of two products:
Some Applications#
The program gmx velacc calculates the velocity autocorrelation function.
The self diffusion coefficient can be calculated using the Green-Kubo relation 108:
which is just the integral of the velocity autocorrelation function. There is a widely-held belief that the velocity ACF converges faster than the mean square displacement (sec. Mean Square Displacement), which can also be used for the computation of diffusion constants. However, Allen & Tildesley 108 warn us that the long-time contribution to the velocity ACF can not be ignored, so care must be taken.
Another important quantity is the dipole correlation time. The dipole
correlation function for particles of type
with
The viscosity of a liquid can be related to the correlation time of the
Pressure tensor