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.