---
tag: 29T4NC87AR
kind: example
name: Ornstein–Zernike equation
msc: [82D15, 82B21, "45E10"]
instance_of: [EY75YD988Z]
sources:
  - url: https://en.wikipedia.org/wiki/Ornstein%E2%80%93Zernike_equation
same_as:
  wikidata: Q3277494
  wikipedia: Ornstein–Zernike equation
---

## Example

Consider a homogeneous and isotropic classical fluid in $\R^3$ with number density $\rho > 0$.
Let $g(r)$ be its radial distribution function and $h(r) = g(r) - 1$ the total correlation function, where $r$ is the distance between two particles.
The Ornstein–Zernike equation defines the direct correlation function $c$ by
$$h(\abs{x - y}) = c(\abs{x - y}) + \rho \int_{\R^3} c(\abs{x - z}) \, h(\abs{z - y}) \, dz, \quad x, y \in \R^3.$$
For known $h$ it is a [[convolution integral equation|EY75YD988Z]] of the second kind for $c$ on $\R^3$, and for known $c$ one for $h$.

Check by the Fourier transform: the integral is the convolution $c * h$, so the equation becomes $\hat h(q) = \hat c(q) + \rho \, \hat c(q) \hat h(q)$.
Hence $\hat h = \frac{\hat c}{1 - \rho \hat c}$ and $\hat c = \frac{\hat h}{1 + \rho \hat h}$, wherever the denominators do not vanish.

## Remarks

The equation splits the correlation between two particles into a direct part $c$ and an indirect part carried through a chain of other particles.
By itself it has two unknown functions, so it needs a second relation, a closure, such as the [[Percus–Yevick approximation|K3E536PSRJ]] or the [[hypernetted-chain equation|MRJ2FHX6K1]].

## Explanations

### L1 human

In a liquid, the position of one molecule influences where its neighbours are likely to be.
The equation says that this influence is the sum of a direct effect and of effects passed on through a third molecule, then a fourth, and so on.
Because the indirect part is a convolution, it is simple in Fourier space.