A generalization of the Nyquist formula for an oscillating electrical circuit with slowly varying parameters is given. Using the momentum method and the time multiscale technique, it is shown that not only the dissipation parameter (resistance), but also the time derivatives of the dispersive parameters (inductance and capacity) determine the spectral properties of the noise in the LC -circuit. This additional contribution is characterized by a new non-local dispersive term which is not related to Joule dissipation and which results from an additional phase shift between the force and the response of the system. The influence of the dispersion contributions on the quality factor of the system is discussed.
The famous Callen-Welton formula [1] is generalized to the systems with slowly varying parameters. Using the momentum method and the time multiscale technique, developed for a nonlocal plasma in [2] it is shown that not only the dissipation but also the time derivatives of the dispersion determine the amplitude and the width of the spectrum lines of the fluctuations. In the general case, the contribution of the second one may be comparable with the first one. This contribution is affected by a new nonlocal dispersive term which is not related to Joule dissipation and appears because of an additional phase shift between the force and response of the system. The general formalism is illustrated by applications to several particular types of system. The influence of the dispersion contributions on the quality factor of the system is discussed.
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