Ornstein-Zernike equation

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In statistical mechanics the Ornstein-Zernike equation (named after Leonard Salomon Ornstein and Frederik 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.

[edit] 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 postions 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]

[edit] 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
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