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Unformatted text preview: Heat conduction in the Frenkel–Kontorova model Bambi Hu Department of Physics, Centre for Nonlinear Studies, and The BeijingHong KongSingapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong, China and Department of Physics, University of Houston, Houston, Texas 772045005 Lei Yang a ! Department of Physics, Centre for Nonlinear Studies, and The BeijingHong KongSingapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong, China s Received 28 October 2004; accepted 10 January 2005; published online 28 March 2005 d Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a firstprinciple derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are far from clear. In this paper we will review recent works on heat conduction in a simple nonintegrable model called the Frenkel–Kontorova model. The thermal conductivity of this model has been found to be finite. We will study the dependence of the thermal conductivity on the temperature and other parameters of the model such as the strength and the periodicity of the external potential. We will also discuss other related problems such as phase transitions and finitesize effects. The study of heat conduction is not only of theoretical interest but also of practical interest. We will show various recent designs of thermal rectifiers and thermal diodes by coupling nonlinear chains together. The study of heat conduction in low dimensions is also important to the understanding of the thermal properties of carbon nanotubes. © 2005 American Institute of Physics . f DOI: 10.1063/1.1862552 g Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a firstprinciple derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions has recently been called into question. In this work we will review recent works on heat conduction in a simple model called the Frenkel–Kontorova model. We will study the thermal conductivity, its dependence on the temperature and vari ous parameters of the model, phase transitions, and finitesize effects. As potential applications, we will show various designs of thermal rectifiers and diodes. This study will be of relevance to carbon nanotubes. I. INTRODUCTION The Fourier law describing heat conduction states that the heat flux J is proportional to the temperature gradient „ T and the coefficient of proportionality is the thermal conduc tivity k : J =  k „ T . s 1 d It has been almost 200 years since Fourier proposed this phe nomenological law that a firstprinciple derivation from sta tistical mechanics is still lacking. Worse still, even the validtistical mechanics is still lacking....
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This note was uploaded on 03/18/2012 for the course PHYSICS 303 taught by Professor Ihn during the Spring '12 term at Swiss Federal Institute of Technology Zurich.
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