The fluctuation dissipation theorem (FDT) is the basis for a microscopic description of the interaction between electromagnetic radiation and matter. By assuming the electromagnetic radiation in thermal equilibrium and the interaction in the linear-response regime, the theorem interrelates the macroscopic spontaneous fluctuations of an observable with the kinetic coefficients that are responsible for energy dissipation in the linear response to an applied perturbation. In the quantum form provided by Callen and Welton in their pioneering paper of 1951 for the case of conductors [H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev.83 (1951) 34], electrical noise in terms of the spectral density of voltage fluctuations, SV(ω), detected at the terminals of a conductor was related to the real part of its impedance, Re[Z(ω)], by the simple relation
SV(ω)=2ℏωcoth(ℏω2KBT)Re[Z(ω)],
where KB is the Boltzmann constant, T is the absolute temperature, ℏ is the reduced Planck constant and ω=2πf is the angular frequency. The drawbacks of this relation concern with: (i) the appearance of a zero-point contribution which implies a divergence of the spectrum at increasing frequencies; (ii) the lack of detailing the appropriate equivalent-circuit of the impedance, (iii) the neglect of the Casimir effect associated with the quantum interaction between zero-point energy and boundaries of the considered physical system; (iv) the lack of identification of the microscopic noise sources beyond the temperature model. These drawbacks do not allow to validate the relation with experiments, apart from the limiting conditions when ℏω≪KBT. By revisiting the FDT within a brief historical survey of its formulation, since the announcement of Stefan–Boltzmann law dated in the period 1879–1884, we shed new light on the existing drawbacks by providing further properties of the theorem with particular attention to problems related with the electrical noise of a two-terminals sample under equilibrium conditions. Accordingly, among others, we will discuss the duality and reciprocity properties of the theorem, the role played by different statistical ensembles, its applications to the ballistic transport-regime, to the case of vacuum and to the case of a photon gas.
