Ornstein–Zernike equation

In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function. It basically describes how the correlation between two molecules can be calculated. Its applications can mainly be found in fluid theory.

Derivation

The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to^[1] for the full derivation.

It is convenient to define the total correlation function:

$h(r_{12})=g(r_{12})-1 \,$

which is a measure for the "influence" of molecule 1 on molecule 2 at a distance $r_{12}$ away with $g(r_{12})$ as the radial distribution function. In 1914 Ornstein and Zernike proposed ^[2] to split this influence into two contributions, a direct and indirect part. The direct contribution is defined to be given by the direct correlation function, denoted $c(r_{12})$. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as

$h(r_{12})=c(r_{12}) + \rho \int d \mathbf{r}_{3} c(r_{13})h(r_{23}) \,$

which is called the Ornstein–Zernike equation. Its interest is that, by eliminating the indirect influence, $c(r)$ is shorter-ranged than $h(r)$ and can be more easily described.

If we define the distance vector between two molecules $\mathbf{r}_{ij}:= \mathbf{r}_{i}-\mathbf{r}_{j}$ for $i,j=1,2,3$, the OZ equation can be rewritten using a convolution.

$h(\mathbf{r}_{12})= c(\mathbf{r}_{12}) + \rho \int d \mathbf{r}_{3} c(\mathbf{r}_{12}-\mathbf{r}_{32})h(-\mathbf{r}_{32}) = c(\mathbf{r}_{12}) + \rho \int d \mathbf{r}_{32} c(\mathbf{r}_{12}-\mathbf{r}_{32})h(\mathbf{r}_{32}) = c(\mathbf{r}_{12}) + \rho (c\, *\, h)(\mathbf{r}_{12}) \,$.

If we then denote the Fourier transforms of $h(\mathbf{r})$ and $c(\mathbf{r})$ by $\hat{H}(\mathbf{k})$ and $\hat{C}(\mathbf{k})$, respectively, and use the convolution theorem we obtain

$\hat{H}(\mathbf{k})=\hat{C}(\mathbf{k}) + \rho \hat{H}(\mathbf{k})\hat{C}(\mathbf{k}), \,$

which yields

$\hat{C}(\mathbf{k})=\frac{\hat{H}(\mathbf{k})}{1 +\rho \hat{H}(\mathbf{k})} \,\,\,\,\,\,\, \hat{H}(\mathbf{k})=\frac{\hat{C}(\mathbf{k})}{1 -\rho \hat{C}(\mathbf{k})}. \,$

One needs to solve for both $h(r)$ and $c(r)$ (or, equivalently, their Fourier transforms). This requires an additional equation, known as a closure relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function $c(r)$ in terms of the total correlation function $h(r)$. The details of the system under study (most notably, the shape of the interaction potential $u(r)$) are taken into account by the choice of the closure relation. Commonly used closures are the Percus–Yevick approximation, well adapted for particles with an impenetrable core, and the hypernetted-chain equation, widely used for "softer" potentials. More information can be found in.^[3]

See also

• Percus–Yevick approximation, a closure relation for solving the OZ equation
• Hypernetted-chain equation, a closure relation for solving the OZ equation

References

1. Frisch, H. & Lebowitz J.L. The Equilibrium Theory of Classical Fluids (New York: Benjamin, 1964)
2. Ornstein, L. S. and Zernike, F. Accidental deviations of density and opalescence at the critical point of a single substance. Proc. Acad. Sci. Amsterdam 1914, 17, 793-806
3. D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976

External links

• The Ornstein–Zernike equation and integral equations
• Multilevel wavelet solver for the Ornstein–Zernike equation Abstract
• Analytical solution of the Ornstein–Zernike equation for a multicomponent fluid
• The Ornstein–Zernike equation in the canonical ensemble
• Ornstein–Zernike Theory for Finite-Range Ising Models Above T_c