Taking the detail balance relations into account in the construction of approximations of kinetic equations

We propose an approach to the construction of approximations of kinetic equations of a lattice gas in the context of cluster approaches, taking account of the detail balance relations. This approach is applied to construct approximations for a one-component lattice gas on a homogeneous one-dimensional lattice, in which it is shown that in the absence of migration the additional requirement of unique solutions of the approximate equations leads to the well-known Kirkwood approximations in the context of the quasi-chemical approximations. We give the equations for a one-component lattice gas on a homogeneous rectangular lattice, describing a family of approximations satisfying these requirements and conditions on the additional relations needed to close the system. Bibliography: 5 titles.