Homogeneous generalized master equation for an arbitrary initial state of a many-particle classical system

A special projection operator is introduced, which exactly transforms the Nakajima–Zwanzig inhomogeneous Generalized Master Equation (GME) with an irrelevant initial condition term (a source) into a completely closed homogeneous GME (with no initial correlations term) for the relevant part of an N-particle distribution function FN(t) for an arbitrary initial distribution function FN(0) of a system of classical particles. It resolves the problem of the derivation of a closed linear evolution equation from the Liouville equation with no “molecular chaos” type approximation. The obtained equation is equivalent to the completely closed equation for s-particle (s≤N) distribution function Fs(t). The initial correlations are accounted for in this equation in the modified by the introduced projection operator kernel governing the evolution of Fs(t). This equation is further simplified by presenting its kernel in the linear in the particles’ density n approximation. In this approximation the equations for one-particle F1(t) and two-particle F2(t) distribution functions are derived. Special attention is given to analyzing the equation for F1(t) and showed that this equation in the Markov approximation can be written as the linear Boltzmann or Landau (for a small interparticle interaction) equations but with additional terms caused by initial correlations.