Correlation function (statistical mechanics)

Correlation functions describe how microscopic variables, such as spin and density, at different positions are related.

More specifically, correlation functions measure quantitatively the extent to which microscopic variables fluctuate together, on average, across space and/or time.

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.

This could be purely coincidental or due to other underlying factors, known as confounding variables, which cause both points to covary (statistically).

In these materials, atomic spins tend to align in parallel and antiparallel configurations with their adjacent counterparts, respectively.

The most common definition of a correlation function is the canonical ensemble (thermal) average of the scalar product of two random variables,

For example, in multicomponent condensed phases, the pair correlation function between different elements is often of interest.

Often, one is interested in solely the spatial influence of a given random variable, say the direction of a spin, on its local environment, without considering later times,

Other equal-time spin-spin correlation functions are shown on this page for a variety of materials and conditions.

They are defined analogously to above equal-time correlation functions, but we now neglect spatial dependencies by setting

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 temporal evolution of correlation functions and Onsager's regression hypothesis.

[1] For instance, transport coefficients [2] are closely related to time correlation functions through the Fourier transform; and the Green-Kubo relations,[3] used to calculate relaxation and dissipation processes in a system, are expressed in terms of equilibrium time correlation functions.

If evaluating the (non-symmetrized) quantum time correlation function by expanding the trace to the eigenstates,

Nevertheless, semiclassical initial value representation (SC-IVR) [5] is a family to evaluate the quantum time correlation function from the definition.

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.

, it is clear that one can define the random variables used in these correlation functions, such as atomic positions and spins, away from equilibrium.

This is typical in scattering experiments and computer simulations, and is often used to measure the radial distribution functions of glasses.

See, for example, http://xbeams.chem.yale.edu/~batista/vaa/node56.html Archived 2018-12-25 at the Wayback Machine Correlation functions are typically measured with scattering experiments.

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.

In 1931, Lars Onsager proposed that the regression of microscopic thermal fluctuations at equilibrium follows the macroscopic law of relaxation of small non-equilibrium disturbances.

, should be uncorrelated beyond what we would expect from thermodynamic equilibrium, the evolution in time of a correlation function can be viewed from a physical standpoint as the system gradually 'forgetting' the initial conditions placed upon it via the specification of some microscopic variable.

This is shown in the figure in the left for the case of a ferromagnetic material, with the quantitative details listed in the section on magnetism.

It describes the canonical ensemble (thermal) average of the scalar product of the spins at two lattice points over all possible orderings:

Even in a magnetically disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale

For example, the exact solution of the two-dimensional Ising model (with short-ranged ferromagnetic interactions) gives precisely at criticality

In an isotropic XY model, time and temperature correlations were evaluated by Its, Korepin, Izergin & Slavnov.

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[12][13] and the experimental measurement of the needed X-ray cross-correlation functions[14] are areas of active research.

Schematic equal-time spin correlation functions for ferromagnetic and antiferromagnetic materials both above and below versus the distance normalized by the correlation length, . 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 , 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.
The caption is very descriptive.
Equal-time correlation functions, , as a function of radius for a ferromagnetic spin system above, at, and below at its critical temperature, . Above , exhibits a combined exponential and power-law dependence on distance: . The power-law dependence dominates at distances short relative to the correlation length, , while the exponential dependence dominates at distances large relative to . At , the correlation length diverges, , resulting in solely power-law behavior: . is distinguished by the extreme non-locality of the spatial correlations between microscopic values of the relevant order parameter without long-range order. Below , the spins exhibit spontaneous ordering, i.e. long-range order, and infinite correlation length. Continuous order-disorder transitions can be understood as the process of the correlation length, , transitioning from being infinite in the low-temperature, ordered state, to infinite at the critical point, and then finite in a high-temperature, disordered state.