# Pair distribution function

For a fluid of ${\displaystyle N}$ particles, enclosed in a volume ${\displaystyle V}$ at a given temperature ${\displaystyle T}$ (canonical ensemble) interacting via the `central' intermolecular pair potential ${\displaystyle \Phi (r)}$, the two particle distribution function is defined as

${\displaystyle {\rm {g}}_{N}^{(2)}({\mathbf {r} }_{1},{\mathbf {r} }_{2})=V^{2}{\frac {\int ...\int e^{-\beta \Phi ({\mathbf {r} }_{1},...,{\mathbf {r} }_{N})}{\rm {d}}{\mathbf {r} }_{3}...{\rm {d}}{\mathbf {r} }_{N}}{\int e^{-\beta \Phi ({\mathbf {r} }_{1},...,{\mathbf {r} }_{N})}{\rm {d}}{\mathbf {r} }_{1}...{\rm {d}}{\mathbf {r} }_{N}}}}$

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

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

See Eq. 5.10 of Ref. 1:

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

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