next up previous

5.2 Pair Correlation Function


a(\mbox{$\bf r$};\mbox{$\bf\Gamma$}_{c}) = \sum_{i}\delta(\mbox{$\bf r$}_{i}-\mbox{$\bf r$})

An average of this quantity represents the relative frequency, or probability densitiy of some particle being situated near $\mbox{$\bf r$}$. In other words, $\langle a(\mbox{$\bf r$})\rangle=\rho(\mbox{$\bf r$})$ is simply the mean fluid density at position $\mbox{$\bf r$}$:

\rho(\mbox{$\bf r$})=\langle \sum_{i}\delta(\mbox{$\bf r$}_{i}-\mbox{$\bf r$})\rangle

In a fluid we usually have $\rho(\mbox{$\bf r$})=const$; only in the presence of external fields or near surfaces $\rho(\mbox{$\bf r$})$ varies in a non-trivial manner.

Let us proceed to the ``pair correlation function'' (PCF)
g(r)=\frac{V}{4\pi r^{2}N^{2}}
\langle\sum_{i}\sum_{j\neq i} \delta (r-r_{ij}) \rangle
\end{displaymath} (5.1)

This is the conditional probability density of finding a particle at $\mbox{$\bf r$}$, given that there is a particle at the coordinate origin. Thus $g(r)$ provides a measure of local spatial ordering in a fluid.

To determine $g(r)$, proceed like this: The typical shape of the PCF at liquid densities is depicted in Fig. 5.1.

Figure 5.1: Pair correlation function of the Lennard-Jones liquid

Significance of $g(r)$ in fluid physics:

PROJECT MD/MC (LENNARD-JONES): Augment your Lennard-Jones MD (or MC) program by a routine that computes the pair correlation function $g(r)$ according to 5.2; remember to apply the nearest image convention when computing the pair distances. As the subroutine ENERGY already contains a loop over all particle pairs $(i,j)$, it is best to increment the $g(r)$ histogram within that loop.

Plot the PCF and see whether it resembles the one given in Figure 5.1.

next up previous
F. J. Vesely / University of Vienna