Various functions, by means of which the distribution of molecules in a system may be described, are defined, and the relations between them are derived. Equations, by means of which these functions may be calculated from a knowledge of the mutual potential energy between pairs of molecules, are also derived. The equations applicable to gaseous systems, which are derived by a development previously used in the theory of condensing systems, enable the calculation of the distribution functions by means of a power series in either the fugacity or the inverse volume. The coefficients of these power series are direct integrals involving the pair potentials between molecules. In condensed systems an alternative procedure is required. The equations then appear in the form of integral equations involving the unknown functions under the integral signs. These equations are not readily solved, due primarily to the difficulties of performing multiple coordinate integrations. By the use of functions involving the probabilities of population of virtual cells, the cell equation frequently used for the calculation of the thermodynamic properties of liquids is derived as a consequence of the method used here. The direct connection between the equations derived in the theory of condensing systems, and in the usual cell approximations, is thus established. The equations derived here, however, permit consistent use of the cell method to arrive at higher approximations. This development may possibly be of value in making numerical evaluations of the thermodynamic functions of liquids.

Mayer, J. E., & Montroll, E. (1941). Molecular Distribution. The Journal of Chemical Physics, 9(1), 2–16.