Role of the Wiener-Khinchin theorem in statistical mechanics

Equilibrium fluctuations of dynamic quantities and the associated spectral amplitudes can be regarded as stationary functions and orthogonal measures, respectively, with values in a Hilbert space. The mathematical basis of this description and of the equivalent probabilistic description is re-examined. The latter is in terms of stationary random functions and orthogonal random measures, with real or complex values in the classical case, and values in the Hilbert space of state vectors in quantum mechanics. In either case, the Wiener-Khinchin theorem assures that self-correlation functions can be represented as Fourier transforms of non-negative measures—the squared norms of the amplitudes. The role of the theorem in linear response theory is discussed.