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5.3 Autocorrelation Functions

An elementary example of temporal correlations of the form

C_{a}(t)\equiv\langle a(0) a(t)\rangle

is the velocity autocorrelation in fluids
C(t)\equiv \langle \mbox{$\bf v$}_{i}(0)\cdot\mbox{$\bf v$}_{i}(t)\rangle
\end{displaymath} (5.2)

Simple kinetic theory, which takes into account only binary collisions, predicts $C(t)\propto e^{-\lambda t}$. At fluid densities a different behavior is to be expected. Nevertheless the first results on $C(t)$ obtained by Alder [ALDER 67] provided some surprises. It turned out that at intermediate fluid densities and long times $C(t) \propto t^{-3/2}$ instead of showing an exponential decay.

This has profound consequences. The diffusion constant $D$ of a liquid is given by

D=\frac{1}{3}\int\limits_{0}^{\infty}C(t)  dt

Due to the long time tail of $C(t)$ the simulation result for $D$ is about $30$ percent larger than its kinetic estimate.

The reason for the long time tail in $C(t)$ 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 5.2: Velocity autocorrelation function of the Lennard-Jones fluid

Procedure for calculating autocorrelation functions $<a(0)   a(t)>$:

EXERCISE: Run your MD program for $2000$ 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 5.2.

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F. J. Vesely / University of Vienna