Integral Equation Method in the Theory of Liquids

The integral equation method in the treatment of the equation of state, particularly for liquids, can be regarded as the problem of the solution of the reciprocal of a known matrix having the activity as a parameter, and dependent on the pair potential divided by $\mathrm{kT}$. The Kirkwood and Yvon-Born-Green integral equations are shown to be special cases of the general operation by which the equations can be derived. The solutions are usually made by a closure which can be called the Kirkwood approximation. If this approximation is consistently used a considerable number of different equations can be derived, which, however, need not all lead to identical answers, due to the error in the approximation. The simplest method of correction of the approximation by successive steps leads to a divergent procedure. A very simple integral equation can be derived by an approximation somewhat analogous to that of Kirkwood. The solution of this equation does show phase transitions, but with little resemblance to those actually observed. A somewhat more complicated equation, of the same order of complexity as that of Kirkwood, is also derived in which the type of approximation used is not that of Kirkwood.