We show that there are two classes of the closure equations for the Ornstein-Zernike equation: The analytical equations for the bridge functional $B=B^{\left(\mathit{an}\right)}$ like hypernetted-chain approximation, Percus-Yevick approximation, etc., and nonanalytic equation $B=B^{\left(\mathit{nan}\right)}$, where $B^{\left(\mathit{nan}\right)}=B^{\left(\mathit{rg}\right)}+B^{\left(\mathit{cr}\right)}$ and $B^{\left(\mathit{rg}\right)}$ is the regular (analytical) component of the bridge functional, and $B^{\left(\mathit{cr}\right)}$ is the critical (nonanalytical) component of $B^{\left(\mathit{nan}\right)}$. The closure equation $B^{\left(\mathit{an}\right)}$ defines coordinates of the critical point and other individual features of critical phenomena, and $B^{\left(\mathit{nan}\right)}$ defines all the known relations between critical exponents. It is shown, that the necessary condition for existence of the nonanalytic solution of the OZ equation is the equality $5-\eta =\delta (1+\eta )$, where $\eta ,\delta$ are the critical exponents, values of which can change in a narrow interval. We also show that the transition from the analytical solution to the nonanalytic one is accompanied by a break of the pressure derivative. The boundaries between the areas, where each of these solutions exists, are indicated on the phase diagram.

• Received 21 September 2008

DOI:https://doi.org/10.1103/PhysRevE.79.031119