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6.4.2 Autocorrelation Functions
An elementary example of temporal correlations of the form
is the velocity autocorrelation in fluids

(6.10) 
Simple kinetic theory, which takes into account only binary
collisions, predicts
. At fluid densities
a different behavior is to be expected. Nevertheless the first results on
obtained by Alder [ALDER 67] provided some surprises.
It turned out that at intermediate fluid densities and long times
instead of showing an exponential decay.
This has profound consequences. The diffusion constant of a liquid
is given by
Due to the long time tail of the simulation result for
is about percent larger than its kinetic estimate.
The reason for the long time tail in was later explained as a
collective dynamical effect: part of the momentum of a particle
is stored in a microscopic vortex that dies off very slowly.[DORFMAN 72]
Figure 6.7:
Velocity autocorrelation function of the LennardJones fluid

Procedure for calculating autocorrelation functions
:
EXERCISE:
Run your MD program for time steps and store the velocity vector
of a certain particle (say, no. 1) at each time step. Write and test a
program that evaluates the autocorrelation function of this vector.
PROJECT MD (LENNARDJONES):
Using the experience gathered in the above exercise, write a procedure
that computes the velocity ACF, averaged over all particles, during
an MD simulation run.
Plot the ACF and see whether it resembles the one given in
Figure 6.7.
Next: 6.5 Particles and Fields
Up: 6.4 Evaluation of Simulation
Previous: 6.4.1 Pair Correlation Function
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001