Ornstein–Zernike equation

In statistical mechanics the Ornstein–Zernike equation (named after Leonard Salomon 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 r₁₂ away with g(r₁₂) as the radial distribution function. In 1914 Ornstein and Zernike proposed 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(r₁₂). 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. The OZ equation has the interesting property that if one multiplies the equation by $e^{i\mathbf{k \cdot r_{12}}}$ with $\mathbf{r_{12}}\equiv |\mathbf{r}_{2}-\mathbf{r}_{1}|$ and integrate with respect to $d \mathbf{r}_{1}$ and $d \mathbf{r}_{2}$ one obtains:

$\int d \mathbf{r}_{1} d \mathbf{r}_{2} h(r_{12})e^{i\mathbf{k \cdot r_{12}}}=\int d \mathbf{r}_{1} d \mathbf{r}_{2} c(r_{12})e^{i\mathbf{k \cdot r_{12}}} + \rho \int d \mathbf{r}_{1} d \mathbf{r}_{2} d \mathbf{r}_{3} c(r_{13})e^{i\mathbf{k \cdot r_{12}}}h(r_{23}). \,$

If we then denote the Fourier transforms of h(r) and c(r) by $\hat{H}(\mathbf{k})$ and $\hat{C}(\mathbf{k})$ this rearranges to

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

from which we obtain that

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

In order to solve this equation a closure relation must be found. One commonly used closure relation is the Percus–Yevick approximation.

More information can be found in^[2].

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. ^ 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
• OzOS, (a Linux distribution named after Leonard Salomon Ornstein and Frederik Zernike)