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We discuss the problem of heat conduction in classical and quantum lowdimensional systems from a microscopic point of view. At the classical level weprovide convincing numerical evidence for the validity of Fourier law of heatconduction in linear mixing systems, i.e. in systems without exponentialinstability. At the quantum level, where motion is characterized by the lack ofexponential dynamical instability, we show that the validity of Fourier law isin direct relation with the onset of quantum chaos. We then study thephenomenon of thermal rectification and briefly discuss the different types ofmicroscopic mechanisms that lead to the rectification of heat flow. The controlof heat conduction by nonlinearity opens the possibility to propose new devicessuch as a thermal rectifier.

Classical and Quantum Chaos and Control of Heat Flow