Relationship of equilibrium correlations to power spectrum of dynamics: Weiner-Khinchin theorem

Let be system variables for a system with stationary ergodic dynamics. The correlation function of and , using the equilibrium time-dependent distribution , the probability that and Y(t)=y (independent of t because of stationarity), is given by
 
Because of ergodicity, this correlation may be written as
 
Now, the Fourier transform of the process X(t) is given by

with inverse
 
and similarly for Y. Substituting equation 10.4 into equation 10.2, and considering that X and Y are real, the correlation may be written
 
which shows that the time-dependent correlation is the inverse Fourier transform of a product of the amplitude spectra. This proves the Weiner-Khinchin theorem, equation 10.5. Equations 10.1 and 10.5 are two ways that the time dependent correlation function can be written as an inner product.

Correlations at zero time difference are the objects that may be calculated from the equilibrium distribution. Setting to zero in equation 10.5 we see that these correlations are related to the amplitude spectra by

Thus, the equilibrium correlations give directly the inner product of the amplitude spectra, and in the case that X=Y, the correlation gives the total power in the spectrum, i.e.

so that the details of the time progression and spectrum of a single system variable are not available from the equilibrium distribution, but the total power in the spectrum of the system variable is available.

As a final comment on a different use of equilibrium correlation functions, when a linear response for the system function to (small) perturbations, and a dissipative stochastic relaxation dynamics of the system function are assumed, the time response coefficient of the system function to changes in the system perturbed from equilibrium is related to the temperature and the equilibrium fluctuation strength D (correlation of the fluctuations with themselves) by, [69, 30]. This relationship is called the fluctuation-dissipation theorem. It is a relationship between the equilibrium fluctuations of a system variable and the response of the system variable to perturbations from equilibrium.

In the next section the derivatives of the distribution of states, entropies, and moments with respect to the external field, coupling constant, and temperature in the coupled spin system are discussed.