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' potential $\Phi (r)$, the two particle distribution function is defined as

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

Exact convolution equation for $g(r)$

See Eq. 5.10 of Ref. 1:

$\ln g(r_{12})+{\frac {\Phi (r_{12})}{kT}}-E(r_{12})=n\int \left(g(r_{13})-1-\ln g(r_{13})-{\frac {\Phi (r_{13})}{kT}}-E(r_{13})\right)(g(r_{23})-1)~{\rm {d}}r_{3}$

See also

• Compressibility equation
• Pressure equation
• Energy equation

References

1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)