A bifurcation analysis of the Ornstein–Zernike equation with hypernetted chain closure

Motivated by the large number of solutions obtained when applying bifurcation algorithms to the Ornstein–Zernike (OZ) equation with the hypernetted chain (HNC) closure from liquid state theory, we provide existence and bifurcation results for a computationally-motivated version of the problem.We first establish the natural result that if the potential satisfies a short-range condition then a low-density branch of smooth solutions exists. We then consider the so-called truncated OZ HNC equation that is obtained when truncating the region occupied by the fluid in the original OZ equation to a finite ball, as is often done in the physics literature before applying a numerical technique.On physical grounds one expects to find one or two solution branches corresponding to vapour and liquid phases of the fluid. However, we are able to demonstrate the existence of infinitely many solution branches and bifurcation points at very low temperatures for the truncated one-dimensional problem provided that the potential is purely repulsive and homogeneous.