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' intermolecular pair potential $\Phi (r)$, 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 $\beta =1/(k_{B}T)$, 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 (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 \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}{\mathbf r}_3}

where, i.e. 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|} .

See also

• Radial distribution function
• 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)