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Physical Review Letters

There is No Quantum Regression Theorem

G. W. Ford, Robert F. O’Connell

Published Jul 29, 1996• 107 citations
Publisher

Abstract

The Onsager regression hypothesis states that the regression of fluctuations is governed by macroscopic equations describing the approach to equilibrium. It is here asserted that this hypothesis fails in the quantum case. This is shown first by explicit calculation for the example of quantum Brownian motion of an oscillator and then in general from the fluctuation-dissipation theorem. It is asserted that the correct generalization of the Onsager hypothesis is the fluctuation-dissipation theorem.

DOI
10.1103/physrevlett.77.798

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The Onsager regression hypothesis states that the regression of fluctuations is governed by macroscopic equations describing the approach to equilibrium. It is here asserted that this hypothesis fails in the quantum case. This is shown first by explicit calculation for the example of quantum Brownian motion of an oscillator and then in general from the fluctuation-dissipation theorem. It is asserted that the correct generalization of the Onsager hypothesis is the fluctuation-dissipation theorem.
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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.
In the attempt to derive the regression theorem from the fluctuation dissipation theorem several authors claim the violation of the former theorem in the quantum case. Here we pose the question: does it exisist a quantum fluctuation dissipation theorem (QFDT) in its conventional interpretation ? It is shown that the relation usually called as the QFDT is the condition of detailed macroscopic energetic balance. Following this interpretation the existing conflict between the two theorems in the quantum case is removed.
Some 93 years ago, Onsager postulated his so-called regression hypothesis, which relates the time-dependent relaxation of a close-to-equilibrium system to its temporal correlations at thermal equilibrium. As far as quantum systems are concerned, it is long-known that this hypothesis cannot be quantitatively correct. Here, the authors analytically derive a suitably adjusted version of Onsager's original formula, provide a wide variety of numerical illustrations, and compare it with some pertinent previous works.