2002, Vol.5, No.4, pp.372-379

The fluctuation-dissipation relations (FDR) are studied at all levels of the statistical description. The most general FDR are the relations for the fluctuations of many-body distribution functions.
It is pointed out the problem of formulation of FDR is related to the problem of deriving irreversible equations based on the reversible equations of classical and quantum mechanics.
The FDR are divided into two classes: 1) FDR for fluctuations with infinite correlation times ("collisionless approximation"), which correspond to infinitely narrow resonances; 2). FDR for fluctuations with finite correlation times ("collisional approximation"). The corresponding spectral densities have finite widths, determined by the "collision integrals".
The fundamental questions about which different viewpoints have been published in the literature are critically analyzed: 1) the limits of applicability of the Callen--Welton formula and 2) the quantum generalization of Nyquist's formula for the intensity of a Langevin source of oscillatory systems. It is shown that the traditional form of the quantum Nyquist formula leads to unphysical consequences. The different expression for the quantum Nyquist formula is examinated.
The consequences of the two forms of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source in quantum systems.
It arises, in particular, in quantum electrodynamics in the calculation of the Lamb shift. Two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the "substruction formalism" of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula.
Key words: fluctuation-dissipation relations, quantum fluctuations, quantum Nyquist formula

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