Decay of Pair Correlations

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Abstract
It is shown that when the second moment of the direct correlation function is negative, the usual Ornstein-Zernike analysis is inadequate. The calculation is extended and it is shown that when both the second and fourth moments are negative, the decay of correlations is oscillatory. It is then argued, following Fisher and Widom, that there will be a locus in the density-temperature plane such that at low temperatures and densities the second moment will be positive and the decay of correlations will be monotonic, while at high temperatures and densities the second and fourth moments will be negative, and the decay of correlations oscillatory.

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