Ballistic-diffusive heat conduction at nanoscale: GENERIC approach

Heat conduction in nanosize systems has to be studied in settings involving microscopic details that are not seen in the classical Fourier theory. G. Chen has suggested [Phys. Rev. Lett. 86 (2001) 2297] a combination of the Cattaneo setting in which the velocity of the heat propagation is finite and the kinetic theory setting in which phonons are seen as heat carriers. In this Letter we show that if the Cattaneo and the kinetic theories are combined in a way that preserves the structure expressing their compatibility with thermodynamics (GENERIC structure) then both the Cattaneo and the kinetic equations become modified. The modified Cattaneo equations involve the term introduced by Chen and, in addition, new terms that are nonlinear in quantities that disappear at equilibrium. The kinetic equation is modified by new terms involving gradients of the heat flux and the local temperature.