Classical and Quantum Chaos and Control of Heat Flow

We discuss the problem of heat conduction in classical and quantum low dimensional systems from a microscopic point of view. At the classical level we provide convincing numerical evidence for the validity of Fourier law of heat conduction in linear mixing systems, i.e. in systems without exponential instability. At the quantum level, where motion is characterized by the lack of exponential dynamical instability, we show that the validity of Fourier law is in direct relation with the onset of quantum chaos. We then study the phenomenon of thermal rectification and briefly discuss the different types of microscopic mechanisms that lead to the rectification of heat flow. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.