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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]

Procedure for calculating autocorrelation functions :

• At regular intervals of time steps, mark starting values . Since in the further process only the preceding 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.

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 (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 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