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 $\beta :=1/(k_{B}T)$ , where $k_{B}$ is the Boltzmann constant.

Exact convolution equation for ${\mathrm {g} }(r)$

See Eq. 5.10 of Ref. 1:

\ln {\mathrm {g} }(r_{12})+{\frac {\Phi (r_{12})}{k_{B}T}}-E(r_{12})=n\int \left({\mathrm {g} }(r_{13})-1-\ln {\mathrm {g} }(r_{13})-{\frac {\Phi (r_{13})}{k_{B}T}}-E(r_{13})\right)({\mathrm {g} }(r_{23})-1)~{\rm {d}}{\mathbf {r} }_{3}

where, i.e. $r_{12}=|{\mathbf {r} }_{2}-{\mathbf {r} }_{1}|$ .