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.
Definitions
The most common definition of a correlation function is the canonical ensemble (thermal) average of the scalar product of two random variables, and , at positions and and times and :
Here the brackets, , indicate the above-mentioned thermal average. It is a matter of convention whether one subtracts the uncorrelated average product of and , from the correlated product, , with the convention differing among fields. The most common uses of correlation functions are when and 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 and represent the average variations in density as a function position for two distinct elements.
Equilibrium equal-time (spatial) correlation functions
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, . In this case, we neglect the time evolution of the system, so the above definition is re-written with . This defines the equal-time correlation function, . It is written as:
Often, one omits the reference time, , and reference radius, , by assuming equilibrium (and thus time invariance of the ensemble) and averaging over all sample positions, yielding:
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.
Equilibrium equal-position (temporal) correlation functions
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, and , influences the value of the same microscopic variable at a later time, (and usually at the same position). Such temporal correlations are quantified via equal-position correlation functions, . They are defined analogously to above equal-time correlation functions, but we now neglect spatial dependencies by setting , yielding:
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:
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 temporal evolution of correlation functions and Onsager's regression hypothesis.