The Study Of Heat Conduction In Low-dimensional Lattice Systems

Study of heat conduction in low-dimensional systems is a subject of both theoretical significance and potential applications. It is shown that behaviors of thermal transport in some low-dimensional systems are qualitativly different from that in 3D systems due to the constraint of space dimension and the Fourier's law is violated. This phenomenon is called anomalous heat conduction. Over the past decade, remarkable progress has been made in the study of heat conduction in classical 1D lattice systems. Meanwhile, benefited from the results of theoretical studies of heat conduction, some promising low-dimensional thermal device models have been constructed. We noticed that some low-dimensional miniature thermal devices concerned in this field are almost micro- or meso-scopic systems, and thus the quantum-mechanical effects should be taken into consideration. However, most works dealing with heat conduction in low-dimensional systems are performed in the framework of classical mechanics, and the quantum-mechanical effects are completely neglected. Generally, the study of heat conduction needs to process long-time dynamical behaviors of a large system. And the standard time-dependent quantum mechanical computation is only applicable for systems with a few degrees of freedom because of the consumption of the computation exponentially increasing with the number of degrees of freedom. Hence, we have to resort to some kind of approximate methods. Then we propose a"quantum-mechanically variational method"which is suitable for solving problems of dynamical evolution of quantum nonlinear lattices approximately. By the method, we study the heat conduction in two typical 1D quantum nonlinear lattice models. In this dissertation, our main works relating with two aspects: classical heat conduction and quantum heat conduction, which are introduced and summarized as follows.In the study of heat conduction in classical framework, firstly, we analytically investigate the heat conduction in the harmonic crystal with self-consistent stochastic reservoirs. The result shows that there is no thermal rectification in the model even with mass-gradient and linear on-site potential. Secondly, we investigate the heat conduction in the mass-graded harmonic model with harmonic/anharmonic on-site potential by nonequilibrium molecular dynamics simulation. The necessary condition for the appearance of thermal rectification is discussed in detail. Thirdly, we investigate the heat conduction in coupled inhomogeneous chains model. And the dependence of the interface thermal resistance on impurity atom is given.In the study of heat conduction in quantum-mechanical framework, we propose a"quantum-mechanically variational method", in which a wave packet in Jackiw-Kerman form is adopted to represent the wave function of a single crystal lattice, since the variational parameters contained in the Jackiw-Kerman wave function is of transparently physical significance. With the help of the Dirac's time-dependent variational principle, a set of equations for the variational parameters is obtained. They describe the dynamical evolution of the quantum-mechanical nonlinear lattices approximately, and the quantum fluctuation contained in the equations of variational parameters can be viewed as quantum corrections to the basically classical Hamiltonian equations of motion. When the quantum fluctuation vanish the classical Hamiltonian equations of motion is retrieved. Employing the"quantum-mechanically variational method", we study the heat conduction in 1D quantum FPU model and 1D quantum FK model. Influence of quantum fluctuation on the behaviors of heat conduction in the two models is discussed in great details, and some novel results are presented.