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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Comment on ‘‘Energy balance for a dissipative system’’

I. R. Senitzky

Published May 1, 1995• 9 citations
Publisher

Abstract

In a recent paper [Phys. Rev. E 48, 1547 (1993)] Li, Ford, and O'Connel argue that the power lost through dissipation by a system in equilibrium with a thermal reservoir is compensated by the action of a fluctuation force that the reservoir exerts on the system. Their analysis is quantum mechanical and their expression for the power supplied by the reservoir remains positive definite even at absolute zero. This Comment outlines an analysis of a harmonic oscillator coupled to a loss mechanism and shows how the quantum-mechanical formalism should be interpreted to avoid unphysical conclusions. An explicit distinction is made between thermal and zero-point fluctuations, and the physical significance of the latter is discussed.

DOI
10.1103/physreve.51.5166

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The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.

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In a recent paper Croxson [Phys. Rev. A $49,$ 588 (1994)] reported dissipation-induced transition probabilities from the ground state to higher states for a quantum harmonic oscillator. A partial approximate result was given for the oscillator to remain in its ground state, while for higher transitions the situation became complicated. Our approach provides an exact simple form closed expression for any order of transition. In addition, we supply the evolution of the work done by the system on its environment.

Physical Review A•Apr 1, 1999•4 citations
In a recent paper [Phys. Rev. E 48, 1547 (1993)] Li, Ford, and O'Connel argue that the power lost through dissipation by a system in equilibrium with a thermal reservoir is compensated by the action of a fluctuation force that the reservoir exerts on the system. Their analysis is quantum mechanical and their expression for the power supplied by the reservoir remains positive definite even at absolute zero. This Comment outlines an analysis of a harmonic oscillator coupled to a loss mechanism and shows how the quantum-mechanical formalism should be interpreted to avoid unphysical conclusions. An explicit distinction is made between thermal and zero-point fluctuations, and the physical significance of the latter is discussed.
The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.
We discuss how various parts of a quantum many-body system exchange energies at thermal equilibrium. To show this, we assume a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. We use the Quantum Langevin equation to show such energy exchange at equilibrium. We conclude that different parts of a physical system can exchange energies even at absolute zero temperature.
The role of random force in maintaining equilibrium in a dissipative quantum system is studied here. We compute the instantaneous power supplied by the fluctuating (random) force, which provides information about the work done by the random force on the quantum subsystem of interest. The quantum Langevin equation formalism is used here to verify that, at equilibrium, the work done by the fluctuating force balances the energy lost by the quantum subsystem to the heat bath. The quantum subsystem we choose to couple to the heat bath is the charged oscillator in a magnetic field. We perform the calculations using the Drude regularized spectral density of bath oscillators instead of using a strict ohmic spectral density that gives memoryless damping. We also discuss the energy balance for our dissipative quantum system and in this regard it is to be understood that the physical system is the charged magneto-oscillator coupled to the heat bath, not the uncoupled charged magneto-oscillator.
In a recent paper Croxson [Phys. Rev. A 49,49,49, 588 (1994)] reported dissipation-induced transition probabilities from the ground state to higher states for a quantum harmonic oscillator. A partial approximate result was given for the oscillator to remain in its ground state, while for higher transitions the situation became complicated. Our approach provides an exact simple form closed expression for any order of transition. In addition, we supply the evolution of the work done by the system on its environment.