The Ornstein–Zernike equation and critical phenomena in fluids

It is shown that there are two classes of closure equations for the Ornstein–Zernike (OZ) equation: the analytical equations B=B(an) type of hyper-netted-chain approximation, Percus-Yevick approximation etc., and the nonanalytical equation B=B(non), where B(nan)=B(RG)+B(cr); B(RG) is the regular (analytical) component of the bridge functional, and B(cr) is the critical (nonanalytical) component of B(nan). The closure equation B(an) defines coordinates of a critical point and other individual features of critical phenomena, and B(nan) defines known relations between critical exponents. It is shown that a necessary condition for the existence of a nonanalytical solution of the OZ equation is the equality 5−η=δ(1+η), where η and δ are critical exponents, the values of which can change in a narrow interval. It is shown that the transition from analytical solution to nonanalytical solution is accompanied by a step of derivative of pressure. On the phase diagram of fluids the boundaries dividing the area of exi...