Critical Point Correlations of the Yvon-Born-Green Equation

The critical behavior of the Yvon-Born-Green integral equation for fluids is analyzed by a moment expansion which yields a nonlinear differential equation accurately describing the long-range correlations. Phase plane analyses show that for dimensions $d<~4$ a critical point is characterized by $\eta =4-d$ with $g\left(\mathcal{r}\right)-1\mathrm{}$ negative for large distances, $\mathcal{r}$, in contrast to normal expectations. For $d>\mathrm{}4\mathrm{}$ the differential equation allows $\left[g\right(\mathcal{r})-1]>\mathrm{}0\mathrm{}$ and $\eta =0\mathrm{} or 4-d$. The compressibility never diverges if $d=1\mathrm{}$.