Radial distribution function

Density Expansion of the radial distribution function

The radial distribution function of a compressed gas may be expanded in powers of the density (Ref. 1)

$\left.{\rm {g}}(r)\right.=e^{-\beta \Phi (r)}(1+\rho {\rm {g}}_{1}(r)+\rho ^{2}{\rm {g}}_{2}(r)+...)$

where $\rho$ is the number of molecules per unit volume. The function ${\rm {g}}(r)$ is normalized to the value 1 for large distances. As is known, ${\rm {g}}_{1}(r)$, ${\rm {g}}_{2}(r)$, ... can be expressed by cluster integrals in which the position of of two particles is kept fixed. In classical mechanics, and on the assumption of additivity of intermolecular forces, one has

${\rm {g}}_{1}(r_{12})=\int f(r_{13})f(r_{23})~{\rm {d}}r_{3}$

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}_2 (r_{12})= \frac{1}{2}({\rm g}_1 (r_{12}))^2 + \varphi (r_{12}) + 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})}

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_{ik}} is the distance 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_i -r_k|} , 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 f(r)} is the Mayer f-function

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 \left. f(r) \right. = e^{-\beta U(r)} -1}

and

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 \varphi (r_{12}) = \int f (r_{13}) f (r_{24}) f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4}

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 \psi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{24}) f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4}

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 \chi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{14}) f (r_{24}) f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4}

References

1. B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review 85 pp. 777 - 783 (1952)