Maximum entropy formulation of the Kirkwood superposition approximation.

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g(1234)=g(123)g(124)g(134)g(234)/[g(12)g(13)g(14)g(23)g(24)g(34)] used in the Born-Green-Yvon 2 equations which are a pair of integral equations for the triplet and pair correlation functions.