  Energy transport can be influenced by the presence of other conserved
quantities. We consider here diffusive systems where energy and the other
conserved quantities evolve macroscopically on the same diffusive space-time
scale. In these situations the Fourier law depends also from the gradient of
the other conserved quantities. The rotor chain is a classical example of such
systems, where energy and angular momentum are conserved. We review here some
recent mathematical results about diffusive transport of energy and other
conserved quantities, in particular for systems where the bulk Hamiltonian
dynamics is perturbed by conservative stochastic terms. The presence of the
stochastic dynamics allows to define the transport coefficients (thermal
conductivity) and in some cases to prove the local equilibrium and the linear
response argument necessary to obtain the diffusive equations governing the
macroscopic evolution of the conserved quantities. Temperature profiles and
other conserved quantities profiles in the non-equilibrium stationary states
can be then understood from the non-stationary diffusive behaviour. We also
review some results and open problems on the two step approach (by weak
coupling or kinetic limits) to the heat equation, starting from mechanical
models with only energy conserved.
