Structure of the BBGKY hierarchy near phase transition

A nonperturbative method, as opposed to diagrammatic expansions, is used to study critical phenomena in a fluid with a small hard core and a weak, long‐ranged attractive potential. Using the natural small parameter related to the inverse of the range of the attractive potential, spatially uniformly valid asymptotic estimates are made for the magnitudes of all correlations (which are defined as the excess from the generalized superposition approximation) in a region near phase transition in arbitrary number of dimensions. It is shown that if the dimension of the space is larger than four, the correlation hierarchy truncates at the three‐body level. The pair correlation satisfies a linear equation. The solution is precisely of Ornstein‐Zernike form. For dimensions smaller than four, the hierarchy is still an infinite chain, but considerably simpler than the BBGKY hierarchy. In this case, at the critical point, the correlations are shown to satisfy a scaling law which is the same as that for S 4 spin systems.