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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
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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The above assumption may seem non-intuitive at first: how can an ensemble which is time-invariant have a non-uniform temporal correlation function? Temporal correlations remain relevant to talk about in equilibrium systems because a time-invariant, ''macroscopic'' ensemble can still have non-trivial temporal dynamics ''microscopically''. One example is in diffusion. A single-phase system at equilibrium has a homogeneous composition macroscopically. However, if one watches the microscopic movement of each atom, fluctuations in composition are constantly occurring due to the quasi-random walks taken by the individual atoms. Statistical mechanics allows one to make insightful statements about the temporal behavior of such fluctuations of equilibrium systems. This is discussed below in the section on the [[#Time evolution of correlation functions|temporal evolution of correlation functions and Onsager's regression hypothesis]].
===Time correlation function===
Time correlation function plays a significant role in nonequilibrium statistical mechanics as partition function does in equilibrium statistical mechanics.<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
| first = Robert
| date = 1965
| title = Time-Correlation Functions and Transport Coefficients in Statistical Mechanics
| journal = Annual Review of Physical Chemistry
| volume = 16
| 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 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>
and this definition applies for both classical and quantum version. For stationary (equilibrium) system, the time origin is irrelevant, and <math>C_{AB}(\tau)=C_{AB} (t_1, t_2)</math>, with <math>\tau = t_2 - t_1 </math> as the time difference.
The explicit expression of classical time correlation function is,
<math display="block"> C_{AB}(t) = \int d^N \mathbf{r} d^N \mathbf{p} f(\mathbf{r}_0, \mathbf{p}_0) A(\mathbf{r}_0, \mathbf{p}_0) B(\mathbf{r}_t, \mathbf{p}_t)</math>
where <math>A(\mathbf{r}_0, \mathbf{p}_0)</math> is the value of <math>A</math> at time <math>t=0</math>, <math>B(\mathbf{r}_t, \mathbf{p}_t)</math> is the value of <math>B</math> at time <math>t</math> given the initial state <math> (\mathbf{r}_0, \mathbf{p}_0) </math>, and <math>f(\mathbf{r}_0, \mathbf{p}_0) </math> is the phase space distribution function for the initial state. If the [[ergodicity]] is assumed, then the ensemble average is the same as time average in a long time; mathematically,
<math display="block">C_{AB}(\tau) = \langle A(\tau) B(0) \rangle = \lim_{T \to \infty} \frac{1}{T} \int_0^{T-\tau} dt \, A(t+\tau) B(t)</math>
scanning different time window <math>\tau</math> gives the time correlation function. As <math>t \to 0</math>, the correlation function <math>C_{AB}(0) = \langle A B \rangle</math>, while as <math>t \to \infty</math>, we may assume the correlation vanishes and <math>\lim_{t \to \infty} C_{AB}(t) = \langle A \rangle \langle B \rangle</math>.
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>
{{cite journal
| last = Miller
| first = William H
| date = 2001
| title = The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations
| journal = Journal of Physical Chemistry A
| volume = 105
| issue = 13
| pages = 2942–2955
| doi = 10.1021/jp003712k
}}</ref> is a family to evaluate the quantum time correlation function from the definition.
Additionally, there are two alternative quantum time correlations,<ref name= Tuckerman>{{cite book |first= Mark E. |last = Tuckerman |title = Statistical Mechanics: Theory and Molecular Simulation |publisher = Oxford Graduate Texts |year= 2023 |isbn=978-0198825562}}</ref> and they both related to the definition of quantum time correlation function in the Fourier space. The first symmetrized correlation function <math>G_{AB}(t)</math> is defined by,
<math display="block">G_{AB}(t) = \frac{1}{Q(N, V, T)} \text{Tr} \left[\hat{A} e^{i\hat{H}\tau_c^*/\hbar} \hat{B} e^{-i\hat{H}\tau_c/\hbar}\right]</math>
with <math>\tau_c \equiv t - i\beta \hbar / 2</math> as a complex time variable. <math>G_{AB}(t)</math> is related with the definition of quantum time correlation function by,
<math display="block">\tilde{C}_{AB}(\omega) = e^{\beta \hbar \omega / 2} \tilde{G}_{AB}(\omega)</math>
The second symmetrized (Kubo transformed) correlation function is,
<math display="block">K_{AB}(t) = \frac{1}{\beta Q(N, V, T)} \int_0^{\beta} d\lambda \operatorname{Tr} \left[e^{-(\beta-\lambda)\hat{H}} \hat{A} e^{-\lambda \hat{H}} e^{i\hat{H}t/\hbar} \hat{B} e^{-i\hat{H}t/\hbar}\right]</math>
and <math>K_{AB}(t)</math> reduces to its classical counterpart both in the high temperature and harmonic limit. <math>K_{AB}(t)</math> is related with the definition of quantum time correlation function by,
<math display="block">\tilde{C}_{AB}(\omega) = \left[\frac{\beta \hbar \omega}{1-e^{-\beta \hbar \omega}}\right] \tilde{K}_{AB}(\omega)</math>
The symmetrized quantum time correlation function are easier to evaluate, and the Fourier transformed relation makes them applicable in calculating spectrum, transport coefficients, etc. Quantum time correlation function can be approximated using the [[path integral molecular dynamics]].
===Generalization beyond equilibrium correlation functions===
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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=
==Time evolution of correlation functions==
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:<math>C (r) \approx \frac{1}{r^{(d-2+\eta)}}\,,</math>
where <math>\eta</math> is a [[critical exponent]], which does not have any simple relation with the non-critical exponent <math>\vartheta</math> introduced above.
For example, the exact solution of the two-dimensional Ising model (with short-ranged ferromagnetic interactions) gives precisely at criticality <math>\eta = \frac{1}{4}</math>, but above criticality <math>\vartheta = \frac{1}{2}</math> and below criticality <math>\vartheta = 2</math>.<ref>B.M. McCoy and T.T. Wu
As the temperature is lowered, thermal disordering is lowered, and in a continuous [[phase transition]] the correlation length diverges, as the correlation length must transition continuously from a finite value above the phase transition, to infinite below the phase transition:
:<math>\xi\propto |T-T_c|^{-\nu}\,,</math>
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However, such higher order correlation functions are relatively difficult to interpret and measure. For example, in order to measure the higher-order analogues of pair distribution functions, coherent x-ray sources are needed. Both the theory of such analysis<ref>{{Cite journal | doi = 10.1103/PhysRevB.82.104207| title = X-ray cross-correlation analysis and local symmetries of disordered systems: General theory| journal = Physical Review B| volume = 82| issue = 10| pages = 104207| year = 2010| last1 = Altarelli | first1 = M.| last2 = Kurta | first2 = R. P.| last3 = Vartanyants | first3 = I. A.|arxiv = 1006.5382 |bibcode = 2010PhRvB..82j4207A | s2cid = 119243898}}</ref><ref>{{Cite journal | doi = 10.1107/S1600576714012424| title = Detecting orientational order in model systems by X-ray cross-correlation methods| journal = Journal of Applied Crystallography| volume = 47| issue = 4| pages = 1315| year = 2014| last1 = Lehmkühler | first1 = F. | last2 = Grübel | first2 = G. | last3 = Gutt | first3 = C. | arxiv = 1402.1432| s2cid = 97097937}}</ref> and the experimental measurement of the needed X-ray cross-correlation functions<ref>{{Cite journal | last1 = Wochner | first1 = P. | last2 = Gutt | first2 = C. | last3 = Autenrieth | first3 = T. | last4 = Demmer | first4 = T. | last5 = Bugaev | first5 = V. | last6 = Ortiz | first6 = A. D. | last7 = Duri | first7 = A. | last8 = Zontone | first8 = F. | last9 = Grubel | first9 = G. | doi = 10.1073/pnas.0905337106 | last10 = Dosch | first10 = H. | title = X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter | journal = Proceedings of the National Academy of Sciences | volume = 106 | issue = 28 | pages = 11511–4 | year = 2009 | pmid = 20716512| pmc = 2703671|bibcode = 2009PNAS..10611511W | doi-access = free }}</ref> are areas of active research.
== See also ==
* [[Ornstein–Zernike equation]]
==References==
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