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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 $$r_{12}$$ 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. 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

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

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].

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

References

^ Frisch, H. & Lebowitz J.L. The Equilibrium Theory of Classical Fluids (New York: Benjamin, 1964)
^ 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
^ D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976