Some Critical Properties of Ornstein-Zernike Systems and Spherical Models

Abstract

A recent work on the critical properties of Ornstein-Zernike (OZ) systems in which the direct correlation function $C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ behaves as $\ensuremath{-}{(\mathrm{kT})}^{\ensuremath{-}1}w(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}g0$, where $w(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is the finite part of the pair potential between particles of a lattice gas with point hard cores, is extended to long-range interactions $w(r)\ensuremath{\propto}{r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$, for $\ensuremath{\sigma}g0$, where $r=|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|$ and $d$ is the dimensionality. The relationship between OZ systems, the spherical model, and the mean spherical model is established, noting that the conditions defining an OZ system are precisely those of the mean spherical model. Explicit expressions for the asymptotic correlation functions in the critical region are obtained, and it is shown that, despite the long-range nature of the interaction, the relation between the exponent $\ensuremath{\eta}$ and the shape of the critical isotherm coincides with the relation predicted by the scaling theory, within a certain range of $\ensuremath{\sigma}$ for fixed $d$. The equation of state is considered both above and below the critical temperature ${T}_{c}$, and a breakdown of Widom's homogeneity argument is exhibited. The equation of state for $Tl{T}_{c}$ makes possible the determination of the coexistence curve for all values of $d$ and $\ensuremath{\sigma}$ for which the system has a critical point; and it is shown that, when applied to short-range interactions, the difficulties found previously by Stell no longer appear.