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 are related. More specifically, correlation functions quantify how microscopic variables co-vary with one another on average across space and time. A classic example of such spatial correlations is in ferro- and antiferromagnetic materials, where the spins prefer to align parallel and antiparallel with their nearest neighbors, respectively. The spatial correlation between spins in such materials is shown in the figure to the right.

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 ${\displaystyle t+\tau }$:

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