Kirkwood superposition approximation

The Kirkwood superposition approximation takes its name from John G. Kirkwood (see Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)

{\rm {g}}_{N}^{(3)}({\mathbf {r} }_{1},{\mathbf {r} }_{2},{\mathbf {r} }_{3})={\rm {g}}_{N}^{(2)}({\mathbf {r} }_{1},{\mathbf {r} }_{2}){\rm {g}}_{N}^{(2)}({\mathbf {r} }_{2},{\mathbf {r} }_{3}){\rm {g}}_{N}^{(2)}({\mathbf {r} }_{3},{\mathbf {r} }_{1})

It appears that this was used as a basis of a closure for the Kirkwood integral equation (Ref. 1) and the Yvon, and Born-Green (Ref. 2) until the work of Morita and Hiroike (Ref. 3). It was pointed out in Ref.s 4 and 5, that there is an inconsistency between the pressure and the compressibility equation if this superposition approximation is used to generate $g(r)$ . This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density. It has recently been shown that the Kirkwood superposition approximation precludes the existence of a critical point (Ref. 6).

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Kirkwood superposition approximation

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