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arXiv:0905.4254

Fluctuation-dissipation theorem and harmonic oscillators

Vincenzo Branchina, Marco Fedele Di Liberto, Ivano Lodato

Published May 26, 2009• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.0905.4254

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The fluctuation-dissipation theorem (FDT) is very general and applies to a broad variety of different physical phenomena in condensed matter physics. With the help of the FDT and following the famous work of Caldeira and Leggett, we show that, whenever linear response theory applies, any generic bosonic or fermionic system at finite temperature T can be mapped onto a fictitious system of free-harmonic oscillators. To the best of our knowledge, this is the first time that such a mapping is explicitly worked out. This finding provides further theoretical support to the phenomenological harmonic oscillator models commonly used in condensed matter. Moreover, our result helps in clarifying an interpretation issue related to the presence and physical origin of the Bose-Einstein factor in the FDT.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.
This chapter provides a summary of all results concerning the fluctuation-dissipation theorem needed to understand many aspects of noisy systems. The proof is a succinct one in density matrix language, with a review of the latter provided for those who wish to follow the proof. Callen and Welton (1951) were the first to emphasise the universal nature of the relation between fluctuations and dissipation. They provided a quantum-mechanical derivation that involved detailed sums over the eigenstates. In this chapter, density operators are introduced to deal with an ensemble of systems rather than a single system. The response function, equilibrium theorems, hermiticity and time reversal, and application of the fluctuation-dissipation theorem to a harmonic oscillator are discussed. To make some of the abstract concepts of dissipative motion become meaningful, they are illustrated by using a reservoir consisting of a continuum of harmonic oscillators.
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Abstract One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the mean kinetic energy E k equals , where is the Boltzmann constant and T is the temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years, still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and consider the simplest system, i.e. the Caldeira–Leggett model for a free quantum Brownian particle in contact with a thermostat consisting of an infinite number of harmonic oscillators. We prove that the mean kinetic energy E k of the Brownian particle equals the mean kinetic energy per one degree of freedom of the thermostat oscillators, i.e. . We show that this relation can be obtained from the fluctuation–dissipation theorem derived within the linear response theory and is universal in the sense that it holds true for any linear and non-linear systems in contact with a bosonic thermostat.