Fluctuation-dissipation theorem and harmonic oscillators

TL;DR

This paper demonstrates that the Bose-Einstein factor in the fluctuation-dissipation theorem can be interpreted as arising from a theoretical mapping of any bosonic or fermionic system onto a fictitious harmonic oscillator system, clarifying its physical meaning.

Contribution

It explicitly constructs a mapping from generic quantum systems to harmonic oscillators, providing a new interpretation of the Bose-Einstein factor in the fluctuation-dissipation theorem.

Findings

The BE factor can be derived from a harmonic oscillator mapping.

Any bosonic or fermionic system at temperature T can be represented by a fictitious harmonic oscillator system.

This mapping preserves susceptibility and fluctuation quantities.

Abstract

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.