Hydrodynamic, electronic and optic analogies with heat transport in extended thermodynamics

Over the last twenty-five years, the search for generalized equations that allow us to better understand the phenomenon of heat conduction has become an active frontier both in transport theory, and in non-equilibrium thermodynamics, due to the growing interest in nanotechnologies, thermal metamaterials and fast devices. Here we review how some mathematical analogies between generalized heat-transport equations and well-known equations in hydrodynamics, electronics and optics have been helpful to infer new forms of heat transfer arising in extended thermodynamics and to inspire the consideration of new phenomena. We also examine in each case the thermodynamic basis of the respective formulation.