Correlation function (statistical mechanics)

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The correlation function in statistical mechanics is measure of the order in a system. Usually is is called \xi\,. It tells us how microscopic variables at different positions are correlated. In a spin system, it is the thermal average of the scalar product of the spins at two lattice points over all possible orderings. The correlation function is hence,

G (r) = \langle \mathbf{s}(R) \cdot \mathbf{s}(R+r)\rangle\,.

Here the brackets mean the above-mentioned thermal average.

Even in a disordered phase, spins at different positions are correlated, i.e., if the distance r is \ll \xi, the spins are almost parallel. The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures one sees an exponential correlation with the correlation function being given asymptotically by

G (r) \approx \frac{1}{r^{d-2+\eta}}\exp{\left(\frac{-r}{\xi}\right)}\,,

where r is the distance between spins and d is the dimension of the system. Futhermore, η is a critical exponent.

As the temperature is lowered, thermal disordering is lowered, and in a continuous phase transition the correlation length diverges, namely

\xi\propto |T-T_c|^{-\nu}\,,

with another exponent ν.

This power law correlation is responsible for the scaling, seen in these transitions. All exponents mentioned are independent of temperature. They are in fact universal, i.e found to be the same in a wide variety of systems.

One very important correlation function is the radial distribution function which is seen often in statistical mechanics.

[edit] Literature

  • Phase Transitions and Critical Phenomena, vol. 1-20 (1972-2001), Academic Press, Ed.: C. Domb, M.S. Green, J.L. Lebowitz
  • M.E. Fisher, Renormalization Group in Theory of Critical Behavior, Reviews of Modern Physics, vol. 46, p. 597-616 (1974)
  • J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford Science Publications, 1992) ISBN 0198517300
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