Mayer Cluster Expansion

Introduction

For many classical problems, no truly analytical answer has yet been developed; a great many theories resemble leaping sparks that flare for a moment in history and then give way to silence and echo-like ripples. The main subject of this article, the Ornstein-Zernike Form (or equation), is an integral equation describing the particle density distribution function in an isotropic liquid. It was proposed by Leonard Ornstein and Frits Zernike in 1914 — a remarkable date. [PDF]

The Ornstein-Zernike Form describes a scheme that focuses on two-body properties such as the distribution function.^[1] Given the definition of the total correlation function $h (r_{a c}) = g (r_{a c}) - 1$ , the Ornstein-Zernike Form reads:

h (r_{a c}) = c (r_{a c}) + ρ \int c (r_{a b}) h (r_{b c}) d^{3} r_{b}

The distribution function is split into a direct correlation function $c_{a c}$ and an indirect correlation part $c (r_{a b}) h (r_{b c})$ . This is a perturbation theory. It can be conveniently extended to multicomponent systems:

h_{α γ} (r_{a c}) = c_{α γ} (r_{a c}) + \sum_{λ} λ ρ \int c_{α γ} (r_{a b}) h_{α γ} (r_{b c}) d^{3} r_{b}

Transformation

The Ornstein-Zernike Equation is especially well suited to solution systems. This is partly because, compared with the Virial Expansion and the Mayer Cluster Expansion, the property it focuses on — the correlation function — is eminently suitable for studying liquids; and partly because, for interfacial problems, the interface can be treated as an infinitely large, infinitely dilute solution component and incorporated into the Ornstein-Zernike equation.

Closure / How to Solve

Since two unknown terms, $c (r)$ and $h (r)$ , appear, an assumption must be introduced to complete the system of equations: two assumptions are commonly used.

	HNC Approximation	PY Approximation
Closure condition	$c (r) = h (r) - \ln g (r) - β u (r)$	$c (r) = [1 - e^{β u (r)}] g (r)$
Applicable potential	Long-range potentials (e.g. Coulomb forces), soft repulsive potentials	Short-range hard-core potentials (e.g. hard-sphere fluids)
Treatment of many-body correlations	Includes more chain-like correlation diagrams	Neglects higher-order chain diagrams, retaining only lower-order terms