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The origin of the tricritical point (tcp) in ionic solutions and its dependence on the short-range effective free energies (SR) are studied within a Landau–Ginzburg–Wilson approach. The tcp is associated with a continuous transition to a charge-ordered phase, in which the charge density oscillates in space. SR interactions affect significantly the charge ordering and the position of the tcp depends sensitively on the strength, shape and state dependence of the SR potentials. The tcp can be located close to the coulombic critical point or close to the solvophobic critical point. For particular forms of the SR interactions either the tricritical or the critical point can be preempted by the occurrence of the other transition.

The critical behavior of a self-consistent Ornstein–Zernike approach (SCOZA) that describes the pair correlation function and thermodynamics of a classical fluid, lattice gas, or Ising model is analyzed in three dimensions below the critical temperature, complementing our earlier analysis of the supercritical behavior. The SCOZA subcritical exponents describing the coexistence curve, susceptibility (compressibility), and specific heat are obtained analytically ( β=7/20, γ′=7/5, α′=−1/10 ). These are in remarkable agreement with the exact values ( β≈0.326, γ′≈1.24, α′≈0.11 ) considering that the SCOZA uses no renormalization group concepts. The scaling behavior that describes the singular parts of the thermodynamic functions as the critical point is approached is also analyzed. The subcritical scaling behavior in the SCOZA is somewhat less simple than that expected in an exact theory, involving two scaling functions rather than one.

Our earlier work applying the self-consistent Ornstein-Zernike approach (SCOZA) to the D-vector model is extended to a more general class of continuous-spin models for which the constraint of a single-spin magnitude is relaxed.

The self-consistent Ornstein-Zernike approach (SCOZA) developed earlier by the authors for the lattice gas and Ising model as well as simple continuum fluids is extended to the D-vector spin model that includes the classical rotator model (D = 2), the classical Heisenberg model (D = 3), and the spherical-model limit (D → ∞), in which the SCOZA becomes exact. The approach entails first formulating an exact treatment of the model that includes its response to the presence of a field H with one primary component H|) and a transverse component H⊥.

The development of a general theory of clustering is sketched. The primary focus is on the use of the pair connectedness function in problems of correlated clustering and on the Ornstein-Zernike formalism that has been developed to evaluate this function and relate it to observed mesoscopic and macroscopic properties. The treatment here stresses percolation, which has already been found to be directly relevant to complex-matter problems. It goes on to show the way the general theory becomes a theory of chemical association when applied to chemically associating fluids. In the complete-association limit, it becomes a site-site theory of molecular fluids.

We derive the replica Ornstein-Zernike equations (ROZ) satisfied by the two-point correlation functions for a partly quenched system, i.e., a system in which some of the particles are quenched, or frozen in place, and some are annealed, or allowed to equilibrate. A formally exact closure for these equations is given. A set of integral equations given by Madden and Glandt are recovered as an approximation to the exact ROZ equations. We discuss the fundamental status of this approximation. We also present new exact relations that express the thermodynamic quantities of partly quenched media using the canonical ensemble in terms of the correlation functions in such media.

Recently, integral equations have been developed for continuum percolation models by relating them to the one-state limit of a continuum Potts model (CPM). In this paper, we use the CPM formalism to develop series expansions for the main quantities of continuum percolation theory. This provides necessary background for the work with integral equations and also sheds light on the basic structure of percolation theory. We develop virial series for the mean number of clusters 〈nc〉 and the mean cluster size S( ρ ). A basic relation between these series is developed using a correlation function new to percolation theory which we denote h′(x). We develop the virial theorem for continuum percolation, which relates the mean number of clusters to the two-point blocking function. A geometric interpretation of this theorem is offered in the case of random sphere percolation. For a general, correlated percolation model, the virial theorem involves our new correlation function h′(x). We use the CPM formalism to give computationally efficient expressions for the general coefficient in the virial expansion of all of these quantities. The combinatorial weighting factors that appear in these expansions are discussed in detail. For completeness, we prove the equivalence of our formulas to the others that have appeared in the literature. We do this in the hope of providing a useful reference for further work in continuum percolation.

Some properties of the Boltzmann equation (BE) for a spatially uniform system of Maxwellian molecules are considered, including the explicit evaluation of the energy-space transition probability P ( y , z ; x ), the evaluation of the constants λ n and μ n,i which enter in the moment equations, the form of the energy BE, the linearization of the moment equations, and a symmetry property of the BE itself.