Generalized Ornstein-Zernike Approach to Critical Phenomena. III. The Critical Null Space

The generalized Ornstein-Zernike formalism developed in previous papers is extended to systems specified by an arbitrary number of spatially varying extensive variables. This extension exhibits the critical eigenvector previously introduced as the first member of a more general structure of nested null spaces of the $T$ matrix which is invariant under an affine transformation of the extensive variables. A $k$-dependent version of Tisza's method of diagonalization of the stiffness and compliance matrices by completion of squares is shown to provide a natural coordinate system for critical fluctuations. This coordinate system defines a sequence of nested null spaces of the stiffness matrix which leads to the structure of null spaces of the $T$ matrix referred to above. The nature of critical fluctuations revealed in this way is used to derive an asymptotic expression for the $G\left(\right[m],[n\left]\right)$ matrix when [$m$] and [$n$] are far apart, as well as an expression for the short-range critical behavior of the molecular distribution functions. The conclusions of the paper are illustrated for simple and binary fluids. The fundamental invariance theorem for the structure of nested null spaces of the stiffness matrix is proved in an appendix.