Kirkwood superposition approximation

Kirkwood 1935 (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)

$g_{N}^{(3)}(r_{1},r_{2},r_{3})=g_{N}^{(2)}(r_{1},r_{2})g_{N}^{(2)}(r_{2},r_{3})g_{N}^{(2)}(r_{3},r_{1})$

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 $g(r)$.

References

1. [JCP_1935_03_00300]
2. [PRSA_1946_188_0010]
3. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics 23 pp. 1003-1027 (1960)
4. [PR_1952_085_000777]
5. [PRSA_1953_216_0203]