Pair distribution function

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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 $Φ(r)$ , 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 $β: = 1 / (k B T)$ , where $k B$ is the Boltzmann constant.

[edit] Exact convolution equation for $g(r)$

See Eq. 5.10 of Ref. 1:

$\ln {\mathrm g}(r_{12}) + \frac{\Phi(r_{12})}{k_BT} - E(r_{12}) = n \int \left({\mathrm g}(r_{13}) -1 - \ln {\mathrm g}(r_{13}) - \frac{\Phi(r_{13})}{k_BT} - E(r_{13}) \right)({\mathrm g}(r_{23}) -1) ~{\rm d}{\mathbf r}_3$