Pair distribution function
For a fluid of \(N\) particles, enclosed in a volume \(V\) at a given temperature \(T\) (canonical ensemble) interacting via the `central' intermolecular pair potential \(\Phi(r)\), the two particle distribution function is defined as
\[{\rm g}_N^{(2)}({\mathbf r}_1,{\mathbf r}_2)= V^2 \frac{\int ... \int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_3...{\rm d}{\mathbf r}_N}{\int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_1...{\rm d}{\mathbf r}_N}\]
where \(\beta := 1/(k_BT)\), where \(k_B\) is the Boltzmann constant.
[edit] Exact convolution equation for \({\mathrm g}(r)\)
See Eq. 5.10 of Ref. 1:
\[\ln {\mathrm g}(r_{12}) + \frac{\Phi(r_{12})}{k_BT} - E(r_{12}) = n \int \left({\mathrm g}(r_{13}) -1 - \ln {\mathrm g}(r_{13}) - \frac{\Phi(r_{13})}{k_BT} - E(r_{13}) \right)({\mathrm g}(r_{23}) -1) ~{\rm d}{\mathbf r}_3\]
where, i.e. \(r_{12} = |{\mathbf r}_2 - {\mathbf r}_1|\).
