The Ornstein–Zernike equation and critical phenomena in fluids

Abstract

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-eta=delta(1+eta), where eta and delta 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 existence of analytical and nonanalytical solutions are indicated.