Investigation of invariants of a many-particle system by the method of projection operators

The projection operator method is used to look for equations that determine invariants of stationary and equilibrium states of a statistical system. The method is based on Bogolyubov's fundamental idea advanced in 1945–1946 that a statistical process of high dimension can be reduced to a sequence of processes of lower dimension. A matrix representation of the Liouville operator with respect to four projection operators is used to split the Liouville equation into equations for the invariants in subspaces of lower dimension, in the derivation of the operator equation for the invariants of the equilibrium states of the system a concrete scheme of projection operators is proposed that employs another of Bogolyubov's ideas: that of successive allowance for a hierarchy of interactions in the system. From a known invariant – the equilibrium distribution function of the canonical ensemble – an integrodifferential equation is obtafned for the radial distribution function of particles in a homogeneous classical fluict, this generalizing Bogolyubov's well-known equation.