How to derive macroscopic heat diffusion, governed by heat equation (or, in the stationary situation, Fourier’s law), from microscopic hamiltonian dynamics? I will give some mathematical precise statement of what we would like to prove, and what could be the ”ingredients” of the proof, in particular linear response in equilibrium. Then I will expose some mathematical results that can be obtained if are allowed some energy conserving stochastic perturbation of the hamiltonian dynamics. I will also expose some surprising numerical results about the ”thermal conductivity” of a chain of rotors, when it is really far from equilibrium.