for example, http://xbeams.chem.yale.edu/~batista/vaa/node56.html Measuring correlation functions Correlation functions are typically measured with scattering experiments. For example, x-ray scattering experiments directly measure electron-electron equal-time correlations. From knowledge of elemental structure factors, one can also measure elemental pair correlation functions. See Radial distribution function for further information. Equal-time spin–spin correlation functions are measured with neutron scattering as opposed to x-ray scattering. Neutron scattering can also yield information on pair correlations as well. 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. Optical microscopy is thus common for colloidal suspensions, especially in two dimensions. Time evolution of 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. This is known as the Onsager regression hypothesis. As the values of microscopic variables separated by large timescales, , 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. There is actually an intuitive connection between the time evolution of correlation functions and the time evolution of macroscopic systems: on average, the correlation function evolves in time in the same manner as if a system was prepared in the conditions specified by the correlation function's initial value and allowed to evolve. Equilibrium fluctuations of the system can be related to its response to external perturbations via the Fluctuation-dissipation theorem. The connection between phase transitions and correlation functions Continuous phase transitions, such as order-disorder transitions in metallic alloys and ferromagnetic-paramagnetic transitions, involve a transition from an ordered to a disordered state. In terms of correlation functions, the equal-time correlation function is non-zero for all lattice points below the critical temperature, and is non-negligible for only a fairly small radius above the critical temperature. As the phase transition is continuous, the length over which the microscopic variables are correlated, , must transition continuously from being infinite to finite when the material is heated through its critical temperature. This gives rise to a power-law dependence of the correlation function as a function of distance at the critical point. This is shown in the figure in the left for the case of a ferromagnetic material, with