Pair distribution function
For a fluid of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N} particles, enclosed in a volume at a given temperature Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T} (canonical ensemble) interacting via the `central' intermolecular pair potential , the two particle distribution function is defined as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 1/(k_BT)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant.
Exact convolution equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r)}
See Eq. 5.10 of Ref. 1:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{12} = |{\mathbf r}_2 - {\mathbf r}_1|} .