Some Critical Properties of Ornstein-Zernike Systems

Abstract

A study is made of some critical properties of systems that satisfy the Ornstein-Zernike (OZ) condition that the direct correlation function $\mathrm{c}\left(\overrightarrow{\mathrm{r}}\right)$ behaves like $-{\mathrm{}\left(\mathrm{kT}\right)\mathrm{}}^{-1\mathrm{}}$ times the pair potential $V\left(\overrightarrow{\mathrm{r}}\right)$ for $\overrightarrow{\mathrm{r}}$ such that $V\left(\overrightarrow{\mathrm{r}}\right)\ll \mathrm{kT}$, even at the critical point. It is pointed out that a number of models of interest that satisfy this condition exist. The relationship (and great difference) between the implication of this condition and the results of the van der Waals-Bragg-Williams-Weiss approach is clarified, and it is noted that in systems satisfying the OZ condition both Widom's homogeneity condition and Kadanoff's scaling hypothesis can be violated, although a self-similarity condition is in general satisfied. The importance of the subtle interplay between small $\left|\overrightarrow{\mathrm{r}}\right|$ and large $\left|\overrightarrow{\mathrm{r}}\right|$ correlations, which is lost in both the mean field and scaling picture, is discussed.