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Soviet Physics Uspekhi

Fluctuation-dissipation relations. Role of the finiteness of the correlation time. Quantum generalization of Nyquist's formula

Yu. L. Klimontovich

Published Feb 28, 1987• 17 citations
Publisher

Abstract

The fluctuation-dissipation relations (FDR) in physical systems 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, and 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, is not well-founded and leads to unphysical consequences. A different expression, used in the literature, for the quantum Nyquist formula is examined. It is not universal, but holds in many important cases. Its region of applicability is determined by the corresponding quantum kinetic equations. The consequences of the two forms of the quantum Nyquist formula, which can be checked experimentally, are studied. The question of the formulation of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source and the corresponding diffusion coefficient in quantum systems. It arises, in particular, also in quantum electrodynamics in the calculation of the Lamb shift (Sec. 12). In this connection two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the subtraction formalism of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula. The exposition is illustrated with many specific examples.

DOI
10.1070/pu1987v030n02abeh002812

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The fluctuation-dissipation relations (FDR) in physical systems 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, and 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, is not well-founded and leads to unphysical consequences. A different expression, used in the literature, for the quantum Nyquist formula is examined. It is not universal, but holds in many important cases. Its region of applicability is determined by the corresponding quantum kinetic equations. The consequences of the two forms of the quantum Nyquist formula, which can be checked experimentally, are studied. The question of the formulation of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source and the corresponding diffusion coefficient in quantum systems. It arises, in particular, also in quantum electrodynamics in the calculation of the Lamb shift (Sec. 12). In this connection two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the subtraction formalism of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula. The exposition is illustrated with many specific examples.
The generalized Langevin equation and its attendant fluctuation–dissipation relation (FDR) for both classical and quantum systems is explictly derived for a large class of system-bath interaction potentials. We demonstrate for this class of potentials that the classical FDR involving only the temperature of the bath is satisfied, and that in general the decay times of the dissipative processes and of the system are temperature dependent. We also demonstrate that the quantum FDR depends in detail on the nature of the bath and on the specific system-bath interaction. Thus we conclude that while the classical Langevin equation is phenomenologically useful, its quantum counterpart is much more limited.
The quantum fluctuation dissipation theorem (QFDT) in the Callen–Welton [ Phys. Rev.83 (1951) 34] form is critically revisited. We show that the role of the system eigenvalues is in general not correctly accounted for by the accepted form of the QFDT. As a consequence, a series of quantum results claimed in the literature, like the presence of zero point fluctuations, the violation of the quantum regression hypothesis, the non-white spectrum of the Langevin force, etc. emerge as a consequence of an incorrect application of the theorem. In this context the case of the single harmonic oscillator is illustrated as a typical example where the accepted form of the QFDT is proven to fail.
An operator is constructed for the spectral density of a fluctuating dynamical variable. The formulation is based on an analysis of method of measurement. An essential feature of the operator is the replacement of classical dynamical variables by time-dependent Heisenberg operators. The expectation value of the spectral density includes fluctuations of statistical origin, as well as fluctuations of quantum mechanical origin. It can be computed for any physical system when the initial state has been specified. Two applications are given: Nyquist's law is rigorously deduced for a physical system that is similar to the one discussed by Callen and Welton, and the shot effect is calculated for free uncorrelated electrons. A quantum correction is obtained whose origin lies in the wavelike character of the electron.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.