Fluctuation-dissipation theorem and harmonic oscillators

Vincenzo Branchina, Marco Di Liberto, Ivano Lodato

Published 2009-05-26Version 1

The question of the "physical meaning" and "origin" of the Bose-Einstein (BE) factor in the fluctuation-dissipation theorem (FDT) is often raised and this term is sometimes interpreted as originating from a real harmonic oscillator composition of the physical system. Such an interpretation, however, is not really founded. Inspired by the famous work of Caldeira and Leggett, we have been able to show that, whenever linear response theory is applicable, which is the main hypothesis under which the FDT is established, any generic bosonic and/or fermionic system at temperature $T$ can be mapped onto a fictitious system of harmonic oscillators so that the suscettivity and the mean square of the fluctuating observable of the real system coincide with the corresponding quantities of the fictitious one. We claim that it is in this sense, and only in this sense, that the BE factor can be interpreted in terms of harmonic oscillators, no other physical meaning can be superimposed to it. At the best of our knowledge, this is the first time that such a mapping is explicitly worked out.