Thermodynamics of heat flux implied by reducing the phonon and the cattaneo heat conduction theories to the fourier theory

Abstract

We investigate heat conduction in the multiscale setting involving the Fourier, the Cattaneo, and the Peierls theories. Thermodynamics emerges in the passages among the scales. The time evolution appearing in the more microscopic theory (i.e. the theory involving more microscopic details) is split into the fast (reducing) and the slow (reduced) time evolution. During the reducing time evolution the extra state variables chosen to characterize the details become completely subjugated to the state variables used in the reduced time evolution. The reducing time evolution is generated by a potential that, if evaluated at the state reached asymptotically in the reducing time evolution, becomes a novel thermodynamic potential on the reduced level.