Beyond Fourier’s law: viscous heat equations

Heat hydrodynamics is undergoing a renewed wave of interest, after it has been theoretically proposed to take place in 2D materials and recent experimental observations in graphite. In particular, heat hydrodynamics emerges when the momentum carried by phonons is weakly dissipated, making it possible, for example, to observe second-sound, a wave-like propagation of heat. The macroscopic description of this regime, however, is still object of investigations, as the commonly used Fourier’s law isn’t capable of describing hydrodynamic transport.
In this talk, we will first introduce a first-principles formalism to compute the thermal viscosity, the transport coefficient determining the diffusion of momentum (as opposed to thermal conductivity, which describes energy diffusion). Using an exact solution of the phonon Boltzmann transport equation (BTE) based on relaxons, the eigenvectors of the scattering matrix, we show how the thermal viscosity is fully determined by the even part of the relaxon spectrum, whereas the odd part describes thermal conductivity.
Next, we show that thermal conductivity and viscosity parametrize a set of viscous heat equations, the thermal counterpart of the Navier-Stokes equations in the linear and laminar regime, with heat transport being described in terms of temperature and drift-velocity fields. These equations reduce to Fourier’s law in the limit of strong momentum dissipation, yet, if momentum dissipation is weak, can also describe second sound. These equations can be parametrized with first-principles calculations, and then solved at a similar complexity of Fourier’s law. We test the formalism on silicon, diamond and graphite, replicating for graphite the experimental observations of a hydrodynamic window and suggesting the existence of hydrodynamic transport in diamond. These equations pave the way to the description of heat transport in devices at the mesoscale.