Statistical mechanics, structure of matter Definition

Correlation function

Definition

Let $d \ge 1$ , and let $\mu$ be an infinite-volume Gibbs measure for a spin system on $\Z^d$ with bounded real spins $\sigma_x$ , $x \in \Z^d$ . Write $\langle \cdot \rangle$ for the expectation under $\mu$ .

For $x_1, \dots, x_n \in \Z^d$ , the $n$ -point correlation function is $\langle \sigma_{x_1} \cdots \sigma_{x_n} \rangle$ .
The truncated, or connected, two-point function is $G(x, y) = \langle \sigma_x \sigma_y \rangle - \langle \sigma_x \rangle \langle \sigma_y \rangle$ .
If $\mu$ is invariant under translations, $G(x, y)$ depends only on $x - y$ and is written $G(x - y)$ .

The correlation length in the direction of the unit vector $e_1$ is $\xi = \big( \lim_{n \to \infty} -\frac{1}{n} \log \abs{G(n e_1)} \big)^{-1}$ , when this limit exists.

Remarks

Example: the one-dimensional Ising model with coupling $J > 0$ at zero field has a unique Gibbs measure, with $\langle \sigma_x \rangle = 0$ and $G(n) = \langle \sigma_0 \sigma_n \rangle = (\tanh \beta J)^n$ . Check: write $e^{\beta J \sigma_i \sigma_{i+1}} = \cosh(\beta J) (1 + \sigma_i \sigma_{i+1} \tanh \beta J)$ and expand; only the term containing all the bonds between $0$ and $n$ survives the sum over spins. So $\xi = -1 / \log \tanh(\beta J)$ , which is finite for every $\beta$ .

Non-example: in a finite box with boundary conditions, $\langle \sigma_x \sigma_y \rangle$ depends on the position of $x$ and $y$ in the box, not only on $x - y$ . So clause 3 fails there, which is one reason to define $G(x - y)$ in infinite volume.

For a fluid, the corresponding object is the pair correlation of the particle density, encoded by the radial distribution function.

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Gibbs measure concept
Ising model example
Correlation function definition

Tag	`FANDMJWCV9`
URI	`m2://FANDMJWCV9`
Kind	definition
Writing	W0 stub
MSC	82B05, 82B20, 60G60
Defines	Correlation function
Uses	Gibbs measure, Ising model
Changed	3 November 2026

Correlation function

Definition

Remarks

Reading path

Further reading

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