Integral equations for equilibrium distribution functions

By explicit separation of the contribution of one, two,..., $s$ particles to the total potential energy of $N$ particles, a large number of systems of integral equations are obtained for the equilibrium distribution functions of simple liquids and gases, including as a special case the Kirkwood–Salzburg and Mayer–MontrolI systems of equations. Differentiation of the resulting equations with respect to the coordinate of any particle always leads to the BBGKY system.