Correlation function (statistical mechanics): Difference between revisions

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==The connection between phase transitions and correlation functions==

[[File:Ferromagnetic correlation functions around Tc.svg|thumb|left|alt=The caption is very descriptive.|Equal-time correlation functions, <math>C(r,\tau =0)</math>, as a function of radius for a ferromagnetic spin system above, at, and below at its critical temperature, <math>T_ C</math>. Above <math>T_ C</math>, <math>C(r,\tau =0)</math> exhibits a combined exponential and power-law dependence on distance: <math>C (r,\tau = 0)\propto r^{-(d-2+\eta )} e^{-r/\xi (T)} </math>. The power-law dependence dominates at distances short relative to the correlation length, <math>\xi</math>, while the exponential dependence dominates at distances large relative to <math>\xi</math>. At <math>T_ C</math>, the correlation length diverges, <math>\xi (T_C)=\infty</math>, resulting in solely power-law behavior: <math>C(r,\tau =0) \propto r^{-(d-2+\eta)}</math>. <math>T_ C</math> is distinguished by the extreme non-locality of the spatial correlations between microscopic values of the relevant order parameter without long-range order. Below <math>T_ C</math>, the spins exhibit spontaneous ordering andbut thus infinitefinite correlation length again. Continuous order-disorder transitions can be understood as the process of the correlation length, <math>\xi</math>, transitioning from being infinitefinite in the low-temperature, ordered state, to infinite at the critical point, and then again finite in a high-temperature, disordered state.]]

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, <math>\xi</math>, 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 the quantitative details listed in the section on magnetism.