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In [[statistical mechanics]], the '''correlation function''' is a measure of the order in a system, as characterized by a mathematical [[correlation function]]. Correlation functions describe how microscopic variables, such as spin and density, at different positions or times are related. More specifically, correlation functions measure quantitatively the extent to which microscopic variables fluctuate together, on average, across space and/or time. Keep in mind that correlation doesn’t automatically equate to causation. So, even if there’s a non-zero correlation between two points in space or time, it doesn’t mean there is a direct causal link between them. Sometimes, a correlation can exist without any causal relationship. This could be purely coincidental or due to other underlying factors, known as confounding variables, which cause both points to covary (statistically).
A classic example of spatial correlation can be seen in [[Ferromagnetism|ferromagnetic]] and antiferromagnetic materials. In these materials, atomic spins tend to align in parallel and antiparallel configurations with their adjacent counterparts, respectively. The figure on the right visually represents this spatial correlation between spins in such materials.
==Definitions==
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<math display="block">C (r,\tau) = \langle \mathbf{s_1}(R,t) \cdot \mathbf{s_2}(R+r,t+\tau)\rangle\ - \langle \mathbf{s_1}(R,t) \rangle\langle \mathbf{s_2}(R+r,t+\tau) \rangle\,.</math>
Here the brackets, <math>\langle \cdot \rangle </math>, indicate the above-mentioned thermal average. It is important to note here, however, that while the brackets are called an average, they are calculated as an [[expected value]], not an average value. It is a matter of convention whether one subtracts the uncorrelated average product of <math>s_1</math> and <math>s_2</math>, <math>\langle \mathbf{s_1}(R,t) \rangle\langle \mathbf{s_2}(R+r,t+\tau) \rangle </math> from the correlated product, <math>\langle \mathbf{s_1}(R,t) \cdot \mathbf{s_2}(R+r,t+\tau)\rangle</math>, with the convention differing among fields. The most common uses of correlation functions are when <math>s_1</math> and <math>s_2</math> describe the same variable, such as a spin-spin correlation function, or a particle position-position correlation function in an elemental liquid or a solid (often called a [[Radial distribution function]] or a pair correlation function). Correlation functions between the same [[random variable]] are [[autocorrelation function]]s. However, in statistical mechanics, not all correlation functions are autocorrelation functions. For example, in multicomponent condensed phases, the pair correlation function between different elements is often of interest. Such mixed-element pair correlation functions are an example of [[Cross-correlation|cross-correlation functions]], as the random variables <math>s_1</math> and <math>s_2</math> represent the average variations in density as a function position for two distinct elements.
===Equilibrium equal-time (spatial) correlation functions===
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Correspondingly, the quantum time correlation function is, in the canonical ensemble,<ref name=Nitzen/>
<math display="block"> C_{AB}(t) = \frac{1}{Q(N, V, T)} \text{Tr}\left[e^{-\beta \hat{H}} \hat{A} e^{i\hat{H}t/\hbar} \hat{B} e^{-i\hat{H}t/\hbar}\right]</math>
where <math>\hat{A}</math> and <math>\hat{B}</math> are the quantum operator, and <math>\hat{B}(t) = e^{i\hat{H}t/\hbar} \hat{B}(0) e^{-i\hat{H}t/\hbar}</math> in the [[Heisenberg picture]]. If evaluating the (non-symmetrized) quantum time correlation function by expanding the trace to the eigenstates,
<math display="block"> C_{AB}(t)=\frac{1}{Q(N, V, T)} \sum_{j, k} e^{\beta E_j} e^{i(E_k-E_j)t/\hbar} A_{jk} B_{kj} </math>
Evaluating quantum time correlation function quantum mechanically is very expensive, and this cannot be applied to a large system with many degrees of freedom. Nevertheless, semiclassical initial value representation (SC-IVR) <ref>
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