Correlation function (statistical mechanics)

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 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.

The most common definition of a correlation function is the canonical ensemble (thermal) average of the scalar product of two random variables, ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$, at positions ${\displaystyle R}$ and ${\displaystyle R+r}$ and times ${\displaystyle t}$ and ${\displaystyle t+\tau }$: $${\displaystyle C(r,\tau )=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t+\tau )\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t+\tau )\rangle \,.}$$

Here the brackets, ${\displaystyle \langle \cdot \rangle }$, 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 ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$, ${\displaystyle \langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t+\tau )\rangle }$ from the correlated product, ${\displaystyle \langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t+\tau )\rangle }$, with the convention differing among fields. The most common uses of correlation functions are when ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$ 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 functions. 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 functions, as the random variables ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$ represent the average variations in density as a function position for two distinct elements.

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, ${\displaystyle \tau }$. In this case, we neglect the time evolution of the system, so the above definition is re-written with ${\displaystyle \tau =0}$. This defines the equal-time correlation function, ${\displaystyle C(r,0)}$. It is written as: $${\displaystyle C(r,0)=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R+r,t)\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R+r,t)\rangle \,.}$$

Often, one omits the reference time, ${\displaystyle t}$, and reference radius, ${\displaystyle R}$, by assuming equilibrium (and thus time invariance of the ensemble) and averaging over all sample positions, yielding: $${\displaystyle C(r)=\langle \mathbf {s_{1}} (0)\cdot \mathbf {s_{2}} (r)\rangle \ -\langle \mathbf {s_{1}} (0)\rangle \langle \mathbf {s_{2}} (r)\rangle }$$ where, again, the choice of whether to subtract the uncorrelated variables differs among fields. The Radial distribution function is an example of an equal-time correlation function where the uncorrelated reference is generally not subtracted. Other equal-time spin-spin correlation functions are shown on this page for a variety of materials and conditions.

One might also be interested in the temporal evolution of microscopic variables. In other words, how the value of a microscopic variable at a given position and time, ${\displaystyle R}$ and ${\displaystyle t}$, influences the value of the same microscopic variable at a later time, ${\displaystyle t+\tau }$ (and usually at the same position). Such temporal correlations are quantified via equal-position correlation functions, ${\displaystyle C(0,\tau )}$. They are defined analogously to above equal-time correlation functions, but we now neglect spatial dependencies by setting ${\displaystyle r=0}$, yielding: $${\displaystyle C(0,\tau )=\langle \mathbf {s_{1}} (R,t)\cdot \mathbf {s_{2}} (R,t+\tau )\rangle \ -\langle \mathbf {s_{1}} (R,t)\rangle \langle \mathbf {s_{2}} (R,t+\tau )\rangle \,.}$$

Assuming equilibrium (and thus time invariance of the ensemble) and averaging over all sites in the sample gives a simpler expression for the equal-position correlation function as for the equal-time correlation function: $${\displaystyle C(\tau )=\langle \mathbf {s_{1}} (0)\cdot \mathbf {s_{2}} (\tau )\rangle \ -\langle \mathbf {s_{1}} (0)\rangle \langle \mathbf {s_{2}} (\tau )\rangle \,.}$$