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6.4.2 Autocorrelation Functions

An elementary example of temporal correlations of the form

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

is the velocity autocorrelation in fluids
\begin{displaymath}
C(t)\equiv \langle \mbox{$\bf v$}_{i}(0)\cdot\mbox{$\bf v$}_{i}(t)\rangle
\end{displaymath} (6.10)

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

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

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 6.7: Velocity autocorrelation function of the Lennard-Jones fluid
\begin{figure}\includegraphics[width=300pt]{figures/f6acf.ps}\end{figure}


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 6.7.


next up previous
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," Kluwer-Plenum 2001