Size Effects in Quasi-Static Energy Transport for Microscopic Quantum Systems
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Size Effects in Quasi-Static Energy Transport for Microscopic Quantum Systems George Y. Panasyuk1, Timothy J. Haugan1, and Kirk L. Yerkes1 1 Aerospace System Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, U.S.A.
ABSTRACT We consider finite size effects on energy transfer between nanoparticles mediated by quantum systems. The nanoparticles are considered as heat reservoirs with a finite number of modes. An expression for the quasi-static energy transport between the heat reservoirs having a finite mode frequency spacing Δ is derived. The resulting equations describing long-term (t 1/Δ) relaxation for the mode temperatures and the average temperatures of the nanoparticles are solved. The solutions depend on small number of measurable parameters and show unusual peculiarities in their temporal variations. As is shown, Fourier’s law in a chain of identical subsystems (nanoparticles) can be validated only on a short time scale. For a larger times, when t ~ 1/Δ, the temperatures of different modes deviate from each other, thus preventing thermal equilibrium in each subsystem, and the validity of Fourier’s law cannot be established.
INTRODUCTION A study of size effects in nanostructured materials makes an important part of modern research. One fundamental question here is related to the applicability of macroscopic theories when a particle is only a few nanometers in size. While the study of finite size effects on linear and nonlinear responses to the application of electromagnetic fields has a rather long history [15], the role of these effects on thermal properties of small bodies has been investigated only recently. In [6-8], static thermodynamic properties of nanostructures were explored. In [9], the authors revealed the critical role of the on-site pinning potential in establishing steady-state conditions of heat transport in finite systems. The presented research is a continuation of our study on size effects in long-term quasi-static heat transport [10] and its extension to a description incorporating more realistic parameter values, thus increasing the scope of applicability of the proposed model. MODEL Our study is based on the total Hamiltonian H tot H H B1 H B 2 V1 V2 , where
H
p2 m 2 x2 p 2 kx2 x2 C2 , H B i i i i , and V x Ci xi i 2 2 i mii 2 2m 2 i i 2mi
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 of the
(1)
mediator, xi and pi are the coordinates and momentum operators, whereas mi and i are the masses and frequencies of the oscillators for the ith mode that belongs to the νth bath (ν = 1, 2). In addition, we employ the Drude-Ullersma model which assumes that in the absence of the interaction with the quantum system, each bath consists of uniformly spaced modes and introduces the following ω
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