Instability of the liquid state at high density: a detailed numerical analysis

Abstract

Many, but not all, liquid state theories show a ‘transition’ to infinite-range oscillatory solutions at a density in the region of a true freezing transition. The physical significance of such solutions to liquid-state integral equations has long been a source of controversy. This paper considers the case of hard sphere correlations, as predicted by modern density functional theory. From a careful finite-element numerical analysis we are able to conclude: (i) in the limit of arbitrarily small amplitude, such an instability would be a true bifurcation into two non-decaying oscillatory solution branches of different wavelength, (ii) the position of this ‘instability’ scales precisely with the square of the numerical mesh size, (iii) full nonlinear solutions at the ‘transition’ to non-decaying oscillatory profiles show no bifurcation, confirmed by an eigenvalue analysis, and (iv) the observed nonlinear solution at the linear ‘instability’ is fully consistent with an extended ‘asymptotic’ theory. Finally, we contrast our numerical results at even higher densities with those reported by previous authors.