A projection operator and self-consistent field equations for reduced nonequilibrium distribution functions

Abstract

A projection operator is constructed which, when applied to the Liouville equation, yields self-consistent field equations for products of reduced nonequilibrium distribution functions for interacting particles. The self-consistent field equations superficially appear as the evolution equations for fictitious independent subunits making up the system of interest. They, in fact, represent closures of the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy at various levels of reduced description. When the integral kernel is suitably approximated, they yield the well-known Boltzmann equation and kinetic equations for reduced distribution functions for uncorrelated subsystems comprising the system. On the basis of the self-consistent field equations, some deductions are made for kinetic equations that may be used for constructing thermodynamic theories of irreversible processes consistent with the laws of thermodynamics.