Bogolyubov's abridged description of equilibrium systems and derivation of an equation for the radial distribution function in a liquid

The fundamental idea by N. N. Bogoliubov about reducing the description of equilibrium statistical systems is applied to the problem of deriving the equation for the radial disrtibution function (RDF) of particles in simple classical liquids. A projective construction of the reduced description of equilibrium state is developed which is based on Bogoliubov's idea about the successive taking into account the nierarchy of interactions in many-particle systems. Generalised integro-differential equation for the RDF is obtained which after partial linearization and different simplifications leads to the well-known equations by Bogoliubov, Kirkwood–Salzburg, Percus–Yevick, superintertwining chains, etc. New additional contributions due to correlations between particles are found in the linear and quadratic (with respect to the density) terms in the equation.