The second-order correlation function  is of special importance, as it is directly related (via a Fourier transform) to the structure factor of the system and can thus be determined experimentally using X-ray diffraction or neutron diffraction.
If the system consists of spherically symmetric particles,  depends only on the relative distance between them, . We will drop the sub- and superscript: . Taking particle 0 as fixed at the origin of the coordinates,  is the average number of particles (among the remaining ) to be found in the volume  around the position .
We can formally count these particles and take the average via the expression , with  the ensemble average, yielding:
where the second equality requires the equivalence of particles . The formula above is useful for relating  to the static structure factor , defined by , since we have:
and thus:
, proving the Fourier relation alluded to above.
This equation is only valid in the sense of distributions, since  is not normalized: , so that  diverges as the volume , leading to a Dirac peak at the origin for the structure factor. Since this contribution is inaccessible experimentally we can subtract it from the equation above and redefine the structure factor as a regular function:
Finally, we rename  and, if the system is a liquid, we can invoke its isotropy: