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In this paper, we present a Hamiltonian and thermodynamic theory of heattransport on various levels of description. Transport of heat is formulatedwithin kinetic theory of polarized phonons, kinetic theory of unpolarizedphonons, hydrodynamics of polarized phonons, and hydrodynamics of unpolarizedphonons. These various levels of description are linked by Poisson reductions,where no linearizations are made. Consequently, we obtain a new phononhydrodynamics that contains convective terms dependent on vorticity of the heatflux, which are missing in the standard theories of phonon hydrodynamics.Moreover, the equations are hyperbolic and Galilean invariant, unlike currenttheories for beyond-Fourier heat transport. The vorticity-dependent termsviolate the alignment of the heat flux with the temperature gradient even inthe stationary state, which is expressed by a Fourier-Crocco equation. The newterms also cause that temperature plays in heat transport a similar role aspressure in aerodynamics.

Multiscale heat transport with inertia and thermal vortices