Generalized Ornstein-Zernike Approach to Critical Phenomena

Abstract

A generalization of the Ornstein‐Zernike integral equation is derived and suggestions are made about a possible application to an improved theory of critical phenomena. A fundamental maximum principle of statistical mechanics is used to place the generalized equation in the context of phase transitions and critical points. The equation is a relationship between a generalized correlation matrix by means of which the average fluctuation product of any two sum functions may be expressed and a generalized direct‐correlation matrix which is the second functional derivative of the functional in the maximum principle. The existence of a critical eigenvector of the direct‐correlation matrix is proposed and three physical meanings of this vector are given. An explicit formula for the direct‐correlation matrix is given and is used to derive two asymptotic properties. This formula exhibits an unexpected relationship between the generalized Ornstein‐Zernike equation and the Percus‐Yevick equation.