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Correlation Length

The correlation length is a measure of the spatial extent over which the order parameter is correlated, indicating the range of influence of neighboring molecules.
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The statement of the theorem

Let ψ(x)\psi(\mathbf{x}) be the order parameter field. The two-point correlation function is defined as ⟨ψ(x)ψ(x′)⟩\langle \psi(\mathbf{x}) \psi(\mathbf{x}') \rangle. The correlation length ξ\xi is defined by the exponential decay of this function with separation ∣x−x′∣|\mathbf{x} - \mathbf{x}'|: \nlim⁡∣x−x′∣→∞ln⁡⟨ψ(x)ψ(x′)⟩∣x−x′∣=−1ξ\lim_{|\mathbf{x} - \mathbf{x}'| \to \infty} \frac{\ln \langle \psi(\mathbf{x}) \psi(\mathbf{x}') \rangle}{\left|\mathbf{x} - \mathbf{x}'\right|} = -\frac{1}{\xi} \nAlternatively, in Fourier space, ξ−2\xi^{-2} is related to the coefficient of the quadratic term in the inverse susceptibility χ−1(k)\chi^{-1}(\mathbf{k}):\nχ−1(k)≈1ξ2+c2k2+…\chi^{-1}(\mathbf{k}) \approx \frac{1}{\xi^2} + c_2 k^2 + \dots