Heat conduction in the Frenkel-Kontorova model

Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a first-principle 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 finite-size 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.