Consider a system of  particles in a volume  (for an average number density ) and at a temperature  (let us also define ;  is Boltzmann’s constant). The particle coordinates are , with . The potential energy due to the interaction between particles is  and we do not consider the case of an externally applied field.
The appropriate averages are taken in the canonical ensemble , with  the configurational integral, taken over all possible combinations of particle positions. The probability of an elementary configuration, namely finding particle 1 in , particle 2 in , etc. is given by
The total number of particles is huge, so that  in itself is not very useful. However, one can also obtain the probability of a reduced configuration, where the positions of only  particles are fixed, in , with no constraints on the remaining  particles. To this end, one has to integrate () over the remaining coordinates :
If the particles are non-interacting, in the sense that the potential energy of each particle does not depend on any of the other particles, , then the partition function factorizes, and the probability of an elementary configuration decomposes with independent arguments to a product of single particle probabilities,
Note how for non-interacting particles the probability is symmetric in its arguments. This is not true in general, and the order in which the positions occupy the argument slots of matters. Given a set of positions, the way that the  particles can occupy those positions is  The probability that those positions ARE occupied is found by summing over all configurations in which a particle is at each of those locations. This can be done by taking every permutation, , in the symmetric group on  objects, , to write . For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting,
In fact, for this special case every n-particle density is independent of coordinates, and can be computed explicitlywhere it is clear that the n-point correlation function is dimensionless.