Born-Green equation

$kT{\frac {\partial \ln g(r_{12})}{\partial r_{1}}}={\frac {-\partial U(r_{12})}{\partial r_{1}}}-\rho \int \left[{\frac {\partial U(r_{13})}{\partial r_{1}}}\right]g(r_{13})g(r_{23})~dr_{3}$

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, volume 188 p. 10" (1946)