A numerical bifurcation analysis of the Ornstein-Zernike equation

Robert E Beardmore
Dept. of Mathematics, Imperial College London
South Kensington Campus, London SW7 2AZ UK
r.beardmore@ic.ac.uk
A Peplow, F Bresme

The isotropic Ornstein-Zernike (OZ) equation

$\displaystyle \qquad \qquad \qquad h(r) = c(r) + \rho \int_{\mathbb{R}^{3}} h(\Vert{\bf x}-{\bf y}\Vert)c(\Vert{\bf y}\Vert)d{\bf y}, \qquad \qquad \qquad (1) $

that is the subject of this paper was presented almost a century ago to model the molecular structure of a fluid at varying densities. In order to form a well-posed mathematical system of equations from (1) that can be solved, at least in principle, we assume the existence of a closure relationship. This is an algebraic equation that augments (1) with a pointwise constraint that is deemed to hold throughout the fluid and it forces a relationship between the total and direct correlation functions ($ h$ and $ c$ respectively).

Some closures have a mathematically appealing structure in the sense that the total correlation function is posed as a perturbation of the Mayer f-function given by

$\displaystyle f(r)=-1+e^{-\beta u(r)}.$

This perturbation depends on the potential $ u$, temperature (essentially $ 1/\beta$) and the indirect correlation function through a nonlinear function that we denote $ G$:

$\displaystyle \qquad \qquad \qquad h = f(r) + e^{-\beta u(r)}G(h-c), \qquad (G(0)= 0), \qquad \qquad \qquad (2) $

so that (1-2) are solved together with $ \beta$ and $ \rho$ as bifurcation parameters. There are many closures in use and if we write $ \exp_1(z) = -1+e^z$ then the hyper-netted chain (HNC) closure

$\displaystyle \qquad \qquad \qquad \qquad \qquad G(\gamma) = \exp_1(\gamma) \qquad \qquad \qquad \qquad \qquad (3) $

has the form of (2) and is popular in the physics and chemistry literature.

The purpose of the talk is show that any reasonable discretisation method applied to (1-2) suffers from an inherent defect if the HNC closure is used that can be summarised as follows: phase transitions lead to fold bifurcations. The existence of a phase transition is characterised by the existence of a bifurcation at infinity with respect to $ h$ in an $ L^1$ norm at a certain density, such that boundedness of $ h$ is maintained in a certain $ L^p$ norm. This behaviour is difficult to mimic computationally by projecting onto a space of fixed and finite dimension and, as a result, projections of (1-2) can be shown to undergo at least one fold bifurcation if such a bifurcation at infinity is present. However, other popular closure relations do not necessarily suffer from the same defect.