Emergence of periodic density patterns

Abstract

Two recent statistical-mechanical theories for the periodic patterns exhibited by the local density ${\rho }_1\left(\overrightarrow{\mathrm{r}}\right)$ as a uniform fluid crystallizes are analyzed. One theory is based on the direct correlation function, the other on the pair correlation function. In the absence of tractable equations for the angular dependence of these functions in an ordered phase, similar assumptions are made in both theories which lead to similar nonlinear integral equations. The differences between investigating the local stability of the fluid in contrast to the global stability of the predicted ordered phase are considered. Calculations are given for the hard-sphere interaction in one and three dimensions and the global predictions for ${\rho }_1\left(\overrightarrow{\mathrm{r}}\right)$ and the thermodynamic properties are discussed in terms of the underlying structure of both the exact and the approximate equations. An exact treatment of either theory for an infinite system with no constraints excludes bifurcation. In light of this, the significance of bifurcation in studying phase transitions is discussed.