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