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. Generalization beyond equilibrium correlation functions All of the above correlation functions have been defined in the context of equilibrium statistical mechanics. However, it is possible to define correlation functions for systems away from equilibrium. Examining the general definition of , it is clear that one can define the random variables used in these correlation functions, such as atomic positions and spins, away from equilibrium. As such, their scalar product is well-defined away from equilibrium. The operation which is no longer well-defined away from equilibrium is the average over the equilibrium ensemble. This averaging process for non-equilibrium system is typically replaced by averaging the scalar product across the entire sample. This is typical in scattering experiments and computer simulations, and is often used to measure the radial distribution functions of glasses. One can also define averages over states for systems perturbed slightly from equilibrium. See,