Heat Transport between Heat Reservoirs Mediated by Quantum Systems
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Heat Transport between Heat Reservoirs Mediated by Quantum Systems George Y. Panasyuk1, George A. Levin1, and Kirk L. Yerkes1 1 Aerospace System Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, U.S.A. ABSTRACT We explore a model of heat transport between two heat reservoirs mediated by a quantum particle. The reservoirs are modeled as ensembles of harmonic modes linearly coupled to the mediator. The steady state heat current, as well as the thermal conductance are obtained for arbitrary coupling strength and will be analyzed for the cases of weak and strong coupling regimes. It is shown that the violation of the virial theorem – the imbalance between the average potential and kinetic energy of the mediator – can be considered as a measure of the coupling strength that takes into account all the relevant factors. The dependence of the thermal conductance on the coupling strength is non-monotonic and displays a maximum. Temperature dependence of the heat conductance may reach a plateau at intermediate temperatures, similar to the classical plateau at high temperatures. We will discuss the origin of Fourier’s law in a chain of macroscopically large, but finite subsystems coupled by the quantum mediators. We will also address the origin of the anomalously large heat current between the scanning tunneling microscope tip and the substrate in deep vacuum which was found in recent experiments. INTRODUCTION A study of heat transfer through microscopic systems, such as molecules, nanotubes, and quantum dots, is one of central pursuits in modern physics contributing to both fundamental research and technological applications [1-2]. Our approach to study heat transport is based on the quantum Langevin equation [3-5] and the Drude-Ullersma model [4]. Dependencies of the derived expressions for the heat current Jth and heat conductance K on temperature T and coupling strength Ȗ are analyzed. The derived quantities are used in discussion of Fourier’s law origin and the interfacial heat transfer studied experimentally in [6]. This research is based on [7] with the following modifications: (i) major results (5)-(7) are presented here in a more general case when the couplings γ 1 and γ 2 can be different; (ii) figures 1 and 2 are drawn for different values of parameters than in [7] in order to show a wider scope of applicability of the presented theory; (iii) a possible deviation from Fourier’s law is described (including Fig. 4); (iv) a better and, in our opinion, more convincing explanation of the results [6] is provided. THEORY Our study is based on the total Hamiltonian H tot = H + H B1 + H B 2 + V1 + V2 , where ª p2 Cν2i m ω 2 x2 º p 2 kx 2 + , H Bν = ¦ « νi + νi νi νi » , and Vν = − x ¦ Cνi xνi + x 2 ¦ 2 2m 2 2 i i 2mνi ωνi i ¬ 2mνi ¼ are the Hamiltonians of the quantum system (the mediator), thermal reservoirs, and the interaction between the mediator and thermal reservoirs, respectively. Here x and p are the coordinate and momentum operators and m and k are the mass and spring constant
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