The fluctuation-dissipation theorem

A system in equilibrium will in general exhibit microscopic fluctuations about the equilibrium state. The fluctuation-dissipation theorem relates the spectrum of these fluctuations to a solution of the macroscopic equation describing the approach to equilibrium from a non-equilibrium state. The aim here is to show exactly what the theorem is and how it is to be used. An account of the quantum version of the theorem is given in three parts, depending on the solution of the macroscopic equation used to express the fluctuations: the relaxation function, the response function or the Green function for continuous systems. Each part is illustrated with an example: charge fluctuations in an RLC circuit for the first two and electric field fluctuations in vacuum for the last.