An average of this quantity represents the relative frequency,
or probability densitiy of some
particle being situated near
. In other words,
is simply the mean
fluid density at position
:
In a fluid we usually have
; only in the presence
of external fields or near surfaces
varies in a
non-trivial manner.
Let us proceed to the ``pair correlation function'' (PCF)
(5.1)
This is the
conditional probability density of finding a particle at
,
given that there is a particle at the coordinate origin.
Thus provides a measure of local spatial ordering in a fluid.
To determine , proceed like this:
Divide the range of -values (at most , where is the side
length of the basic cell) into intervals of length .
A given configuration
is scanned to
determine, for each pair , a ``distance channel''number
In a histogram table the corresponding value is then incremented by
. This procedure is repeated every, say, MD steps (or MC steps).
At the end of the simulation run the histogram is normalized according to
5.1.
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 in fluid physics:
The average of any quantity that depends on the pair
potential may be expressed as an integral over .
Example: pressure (see also 1.2)
Theory: analytical approximations to for a given pair potential
.[KOHLER 72],[HANSEN 86]
Experiment: the Fourier transform of , the
``scattering law''
is just the relative intensity of neutron or X-ray scattering at
a scattering angle
with
PROJECT MD/MC (LENNARD-JONES):
Augment your Lennard-Jones MD (or MC) program by a routine that
computes the pair correlation function 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 , it is best to increment the histogram
within that loop.
Plot the PCF and see whether it resembles the one given in
Figure 5.1.