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6.4.1 Pair Correlation Function
Let
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)
|
(6.9) |
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:
The typical shape of the PCF at liquid densities is depicted in
Fig. 6.6.
Figure 6.6:
Pair correlation function of the Lennard-Jones liquid
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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 6.2)
- 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
6.4.1; 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 6.6.
Next: 6.4.2 Autocorrelation Functions
Up: 6.4 Evaluation of Simulation
Previous: 6.4 Evaluation of Simulation
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001