Of particular importance is the case n=2, or the correlation function
known as the pair correlation function.
The explicit expression for is
In general, for homogeneous systems in equilibrium, there are no special
points in space, so that 
should depend only on the
relative position of the particles or the difference 
.
In this case, it proves useful to introduce the change of variables
Then, we obtain a new function 
, a function of 
and 
:
In general, we are only interested in the dependence on . Thus, we
integrate this expression over and obtain a new correlation function
defined by
For an isotropic system such as a liquid or gas, where there is no preferred
direction in space, only the maginitude or , 
is
of relevance. Thus, we seek a choice of coordinates that involves
r explicitly. The spherical-polar coordinates of the vector
is the most natural choice. If 
then the spherical
polar coordinates are
where 
and 
are the polar and azimuthal angles, respectively.
Also, note that
where
Thus, the function g(r) that depends only on the distance r between two particles is defined to be
Integrating g(r) over the radial dependence, one finds that
The function g(r) is important for many reasons. It tells us about the
structure of complex, isotropic systems, as we will see below, it determines
the thermodynamic quantities at the level of the pair potential approximation,
and it can be measured in neutron and X-ray diffraction experiments.
In such experiments, one observes the scattering of neutrons or X-rays
from a particular sample. If a detector is placed at an angle
from the wave-vector direction of an incident beam of particles, then
the intensity 
that one observes is proportional to the
structure factor
where 
is the vector difference in the wave vector between the incident
and scattered neutrons or X-rays (since neutrons and X-rays are
quantum mechanical particles, they must be represented by plane
waves of the form 
). By computing the ensemble average
(see problem 4 of problem set #5), one finds that 
and
S(k) is given by
Thus, if one can measure S(k), g(r) can be determined by Fourier transformation.