Puzzles in the theory of heat conduction in low-dimensional systems

Fourier's law of heat conduction predicts that heat propagation in a solid is diffusive. Fourier's law is a phenomenological macroscopic law and its derivation for a microscopic model with Hamiltonian dynamics is an open problem. A large number of studies over the last few decades suggest that this law is in fact not generally valid in low-dimensional systems. It appears that the heat carriers perform super-diffusive Levy walks rather than simple random walks. I will discuss what we presently know about this problem and some of the interesting open questions in this field.