Correlation function (statistical mechanics): Difference between revisions

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[[File:Ferro antiferro spatial corrs png.png|thumb|right|Schematic equal-time spin correlation functions for ferromagnetic and antiferromagnetic materials both above and below <math>T_\text{Curie}</math> versus the distance normalized by the correlation length, <math>\xi</math>. In all cases, correlations are strongest nearest to the origin, indicating that a spin has the strongest influence on its nearest neighbors. All correlations gradually decay as the distance from the spin at the origin increases. Above the Curie temperature, the correlation between spins tends to zero as the distance between the spins gets very large. In contrast, below <math>T_\text{Curie}</math>, the correlation between the spins does not tend toward zero at large distances, but instead decays to a level consistent with the long-range order of the system. The difference in these decay behaviors, where correlations between microscopic random variables become zero versus non-zero at large distances, is one way of defining short- versus long-range order.]]

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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===Time correlation function===

Time correlation function plays a significant role in nonequilibrium statistical mechanics as partition function does in equilibrium statistical mechancismechanics.<ref name= McQuarrie>{{cite book |first= Donald A. |last = McQuarrie |title = Statistical Mechanics |publisher = University Science Books |year= 2000 |isbn=978-1891389153}}</ref> For instance, transport coefficients <ref>

{{cite journal

| last = Zwanzig

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| pages = 67–102

| doi = 10.1146/annurev.pc.16.100165.000435

}}</ref> are closely related to time correlation functions through the [[Fourier transform]]; and the [[Green-Kubo]] relations,<ref name= Zwangzig>{{cite book |first=Robert |last= Zwanzig |title = Nonequilibrium Statistical Mechanics |year= 2001 |isbn=978-0195140187}}</ref> used to calculate relaxation and dissipation processes in a system, are expressed in terms of equilibrium fluctuations time correlation functions. The time correlation function of two observable <math>A</math> and <math>B</math> is defined as,<ref name= Nitzen>{{cite book |first=Abraham |last= Nitzen |title=Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems |year= 2014 |isbn=978-0199686681

}}</ref>

<math display="block">C_{AB} (t_1, t_2) = \langle A(t_1) B(t_2) \rangle</math>

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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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==Measuring correlation functions==

Correlation functions are typically measured with scattering experiments. For example, x-ray scattering experiments directly measure electron-electron equal-time correlations.<ref name=Sethna>{{cite book |first=James P. |last= Sethna |title=Statistical Mechanics: Entropy, Order Parameters, and Complexity |publisher=Oxford University Press |year=2006 |isbn=978-0198566779 |chapter= Chapter 10: Correlations, response, and dissipation|chapter-url= httphttps://pages.physics.cornell.edu/~sethna/StatMech/}}</ref> From knowledge of elemental structure factors, one can also measure elemental pair correlation functions. See [[Radial distribution function]] for further information. Equal-time spin–spin correlation functions are measured with [[neutron scattering]] as opposed to x-ray scattering. Neutron scattering can also yield information on pair correlations as well. For systems composed of particles larger than about one micrometer, optical microscopy can be used to measure both equal-time and equal-position correlation functions. Optical microscopy is thus common for colloidal suspensions, especially in two dimensions.

==Time evolution of correlation functions==