Further Generalization of the Generalized Master Equation

Abstract

For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.