Critical Correlations and the Square-Gradient Theory

A nonlinear differential equation for the asymptotic decay of the pair correlation function of a fluid at its critical point is obtained from the square-gradient theory (and its extension to fourth order), and analyzed when the critical exponent $\eta$ is either zero or nonzero. Its solutions are shown to be consistent with the correct power-law decay if and only if the ordinary scaling relations together with hyperscaling (for $\eta >0\mathrm{}$) are valid.