For a fluid of
particles, enclosed in a volume
at a given temperature
(canonical ensemble) interacting via the `central' intermolecular pair potential
, 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450354cc005edaa0aa2e735687a7aa0c1683d410)
where
, where
is the Boltzmann constant.
Exact convolution equation for ![{\displaystyle {\mathrm {g} }(r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3c0f8778d0ec69f18edb83dd2a6cda35764847)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac2e772321710ad1c836aa45d037c16ae47a6d9)
where, i.e.
.
See also
References
- J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)