Duality and reciprocity of fluctuation-dissipation relations in conductors

By analogy with linear response, we formulate the duality and reciprocity properties of current and voltage fluctuations expressed by Nyquist relations, including the intrinsic bandwidths of the respective fluctuations. For this purpose, we individuate total-number and drift-velocity fluctuations of carriers inside a conductor as the microscopic sources of noise. The spectral densities at low frequency of the current and voltage fluctuations and the respective conductance and resistance are related in a mutually exclusive way to the corresponding noise source. The macroscopic variances of current and voltage fluctuations are found to display a dual property via a plasma conductance that admits a reciprocal plasma resistance. Analogously, the microscopic noise sources are found to obey a dual property and a reciprocity relation. The formulation is carried out in the frame of the grand canonical (for current noise) and canonical (for voltage noise) ensembles, and results are derived that are valid for classical as well as degenerate statistics, including fractional exclusion statistics. The unifying theory so developed sheds new light on the microscopic interpretation of dissipation and fluctuation phenomena in conductors. In particular, it is proven that for fermions, as a consequence of the Pauli principle, nonvanishing single-carrier velocity fluctuations at zero temperature are responsible for diffusion but not for current noise, which vanishes in this limit.