Criticality in the Yvon–Born–Green and similar integral equations

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Abstract
The critical behavior implied by the Yvon–Born–Green integral equation for the pair correlation function of a fluid, and by a class of similar integral equations, is considered. By using the inverse correlation length κ, which vanishes at a true critical point, as a small parameter the integral equations are reduced, via a moment expansion, to a nonlinear differential equation which describes the critical behavior of the correlation function outside the (finite) range of the potential. Analysis then shows that for spatial dimensionalities d≳4 any critical behavior is, asymptotically, precisely of classical, Ornstein–Zernike scaling form. However, for d≤4 either there is no true criticality, in the sense that both the correlation length and the compressibility KT always remain finite, or the behavior becomes anomalous in that the net correlation function h(r)≡g(r)−1 must, in general, become negative for intermediate and large distances r as κ→0. In the latter case, asymptotic and universal scaling behavior occurs, with a critical point decay exponent η=4−d, and KT→+∞ for d≳d0≂2.2, although KT→−∞ for 2≤d<d0. For d<2 the compressibility always remains finite.