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.

Contents

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}) %2B \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}}} %2B \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}) %2B \rho \hat{H}(\mathbf{k})\hat{C}(\mathbf{k}) \,

from which we obtain that

\hat{C}(\mathbf{k})=\frac{\hat{H}(\mathbf{k})}{1 %2B\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[3].

See also

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