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arXiv:0801.4057 · Chinese Physics Letters

The Third Law of Quantum Thermodynamics in the Presence of Anomalous Couplings

Wang Chun-Yang, Bao Jing-Dong

Published Feb 1, 2008• 18 citations
arXivHTMLPDF

Abstract

The quantum thermodynamic functions of a harmonic oscillator coupled to a heat bath through velocity-dependent coupling are obtained analytically. It is shown that both the free energy and the entropy decay fast with the temperature in relation to that of the usual coupling from. This implies that the velocity-dependent coupling helps to ensure the third law of thermodynamics.

DOI
10.1088/0256-307x/25/2/021

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The quantum thermodynamic functions of a harmonic oscillator coupled to a heat bath through velocity-dependent coupling are obtained analytically. It is shown that both the free energy and the entropy decay fast with the temperature in relation to that of the usual coupling from. This implies that the velocity-dependent coupling helps to ensure the third law of thermodynamics.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
We investigate the low-temperature statistical properties of a harmonic oscillator coupled to a heat bath, where the low-frequency spectrum vanishes. We obtain the exact result of the zero point energy. Due to the low frequency shortage of environmental oscillators' spectral density, the coordinate and momentum correlation functions decay as τ−4 and τ−6 respectively at zero temperature, where τ is the correlation time. The low-temperature behavior of the mean energy does not violate the third law of thermodynamics, but differs largely from the Ohmic spectrum case.
We have studied the specific heat of the (NS + NB) model for an NS-body harmonic oscillator (HO) system which is strongly coupled to an NB-body HO bath without dissipation. The system specific heat of CS(T) becomes NSkB at T → ∞ and vanishes at T = 0 in accordance with the third law of thermodynamics. The calculated CS(T) at low temperatures is not proportional to NS and shows an anomalous temperature dependence, strongly depending on NS, NB, and the system-bath coupling. In particular at very low (but finite) temperatures, it may become negative for a strong system-bath coupling, which is in contrast with non-negative specific heat of a HO system with NS = 1 reported by Ingold, Hänggi, and Talkner [Phys. Rev. E 79, 061105 (2005)]. Our calculation indicates an importance of taking account of finite NS in studying open quantum systems which may include an arbitrary number of particles in general.
Entropy of a quantum oscillator in the presence of a quantum environment (or heat bath) is studied here. The spectral density, which comprises the environmental influences, here corresponds to a quasi-monochromatic thermal harmonic noise. This further implies that the environment is a structured one compared to the usually employed unstructured strictly ohmic environment. The entropy is found to be vanishing as the temperature approaches its absolute zero value, which thereby guarantees the third law of thermodynamics. At very high temperatures the classical behavior is observed. Similar to the usually employed ohmic-Drude bath model, the coefficient of the leading temperature dependent term in the entropy (at low temperatures) is proportional to the damping strength but not on other parameters of the spectral density characterising the bath.
The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to absolute zero. The third law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent ζ of the cooling process dT(t)/dt∼-T^{ζ} when approaching absolute zero, T→0. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled two-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat-driven refrigerator (absorption refrigerator) is compared to a power-driven refrigerator. When optimized, both cases lead to the same exponent ζ, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat bath models are considered: a bath composed of harmonic oscillators and a bath composed of ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the third law are discussed. In the Appendices, the theory of periodically driven open systems and its implication for thermodynamics are outlined.