Advanced liquid state theories for fluid criticality

One of the most basic goals of liquid state theory is the prediction of the experimentally observable thermodynamic and structural properties of macroscopic systems at given temperature and density from the forces acting between the constituent particles alone. For classical fluids, integral equation theories provide such a link between the microscopic and macroscopic realms that yields satisfactory results for fluids under a wide variety of conditions. Close to the critical point - a special state where the natural lengthscale of the system becomes infinite and the distinction between liquid and vapor becomes meaningless -, however, they develop severe problems. In order to overcome these problems, two advanced theories have been developed that extend the integral equation concept by also taking into account additional information in the form of exact results derived from statistical physics: both the "Self Consistent Ornstein-Zernike Approximation" (SCOZA) and the "Hierarchical Reference Theory" (HRT) effectively give rise to nonlinear partial differential equations that yield vastly improved descriptions of the vicinity of the critical region as well as of the liquid vapor phase transition. Unfortunately, neither SCOZA nor HRT are entirely satisfactory in their description of phase separation and the approach to the critical point. In order to improve on the current state of the art we will therefore formulate a theory that combines the essential features of both theories, investigate its mathematical and numerical properties, and assess the quality and accuracy of its results relative to those of its ancestors; of course, its representation of criticality and phase coexistence will be of special interest. This research will be carried out in several stages, corresponding to different selections of the exact relations employed. Furthermore, in addition to liquids we will also consider the mathematically more convenient models of interacting spins, the critical behavior of some of which is known to fall into the same universality class as that of liquids. In a second branch of this project, we will concentrate on modifications to SCOZA designed to extend the class of physical systems accessible to the theory as well as to reduce the complexity of, and the mathematical and numerical effort required for, its practical application. The latter involves adoption of a novel, simplified version of SCOZA that has so far shown great promise and, furthermore, is related to the unification with HRT detailed before on a technical level. In this part, too, we will consider both classical fluids and spin systems: of particular interest are the rather general continuous spin models on the one hand, and ionic fluids with their still hotly disputed critical properties on the other hand.