Critical behavior of the hypernetted-chain equation

We present numerical solutions in the singular (‘‘critical’’) region of the hypernetted-chain (HNC) equation for a model fluid system described by a pair potential consisting of a highly repulsive core and an attractive well. In contrast to the behavior of real systems, we find that the isothermal compressibility ${\kappa }_T$ does not actually diverge on the critical isochore ρ=${\rho }_c$ as the temperature T is lowered toward the critical temperature $T_c$. Rather, there exists a locus of temperatures $T_s$(ρ), parametrized by density ρ [with $T_s$(${\rho }_c$)=$T_c$], on which ${\kappa }_T$ remains finite, but below which no physical solutions to the HNC equation exist. This behavior breaks down at densities near triple-point conditions, where ${\kappa }_T$ has a true divergence at a low enough temperature. There is a striking similarity between certain aspects of this critical behavior and that of the Percus-Yevick equation, for which comparison can be made to the analytic solution available for the case of the sticky hard-sphere potential.