On the stability of infinite dimensional fluid of hard hyperspheres: a statistical mechanical estimate of the density of closest packing of simple hypercubic lattices in spaces of large dimensionality

Abstract

We report an analysis of the bifurcation of the solution to the nonlinear equation for the inhomogeneous singlet density in a system of hard hyperspheres; the instability examined corresponds to the liquid-to-simple hypercubic lattice transition. We propose that in the limit d→∞ the continuous bifurcation which occurs is at the maximum achievable density in a simple hypercubic lattice. Extension of this result to 1<d<∞ leads to estimates of the closest packing densities of simple hypercubic lattices in d dimensions. An examination of the liquid-to-simple hypercubic lattice transition for particles with a Gaussian pair repulsion leads to the identification of that transition with the onset of absolute instability, i.e., the spinodal of the liquid.