Born-Green equation

$k_{B}T{\frac {\partial \ln {\rm {g}}(r_{12})}{\partial {\mathbf {r} }_{1}}}={\frac {-\partial \Phi (r_{12})}{\partial {\mathbf {r} }_{1}}}-\rho \int \left[{\frac {\partial \Phi (r_{13})}{\partial {\mathbf {r} }_{1}}}\right]{\rm {g}}(r_{13}){\rm {g}}(r_{23})~{\rm {d}}{\mathbf {r} }_{3}$

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 \Phi(r_{nm})} is the intermolecular pair potential, T is the temperature, 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 k_B} is the Boltzmann constant.

References

1. M. Born and Herbert Sydney Green "A General Kinetic Theory of Liquids I: The Molecular Distribution Functions", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 188 pp. 10-18 (1946)