For dilute systems (e.g. gases), the correlations in the positions of the particles that  accounts for are only due to the potential  engendered by the reference particle, neglecting indirect effects. In the first approximation, it is thus simply given by the Boltzmann distribution law:
If  were zero for all  &ndash; i.e., if the particles did not exert any influence on each other, then  for all  and the mean local density would be equal to the mean density : the presence of a particle at O would not influence the particle distribution around it and the gas would be ideal. For distances  such that  is significant, the mean local density will differ from the mean density , depending on the sign of  (higher for negative interaction energy and lower for positive ).
As the density of the gas increases, the low-density limit becomes less and less accurate since a particle situated in    experiences not only the interaction with the particle in O but also with the other neighbours, themselves influenced by the reference particle. This mediated interaction increases with the density, since there are more neighbours to interact with: it makes physical sense to write a density expansion of , which resembles the virial equation:
This similarity is not accidental; indeed, substituting () in the relations above for the thermodynamic parameters (Equations ,  and ) yields the corresponding virial expansions. The auxiliary function  is known as the cavity distribution function. It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given  under equilibrium is unique up to an additive constant, if it exists.
In recent years, some attention has been given to develop pair correlation functions for spatially-discrete data such as lattices or networks.