An elementary example of temporal correlations of the form
is the velocity autocorrelation in fluids
Simple kinetic theory, which takes into account only binary
. 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]
Velocity autocorrelation function of the Lennard-Jones fluid
Procedure for calculating autocorrelation functions
At regular intervals of time steps, mark starting values
Since in the further process only the
starting values are required, it is best to store
them in registers that are cyclically overwritten.
At each time , compute the products
and relate them to the (discrete) time displacements
; a particular defines
a channel number
indicating the particular histogram channel to be incremented by .
To simplify the final normalization it is recommended to count the number
of times each channel is incremented.
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 (LENNARD-JONES):
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