The Fourier Law at the Macro and Nanoscale Available to Purchase

The Fourier law implicitly assumes transient thermal disturbances are carried throughout the solid at an infinite velocity while not defining the carrier mechanism. Paradoxically, the phonon and electron carriers on which the Fourier law is based are limited to acoustic velocities. At the macroscale, the paradox is resolved by the thermal BB photons of QM that carry the Planck energy E = kT of the atoms in the disturbance throughout the solid at the speed of light. BB stands for blackbody and QM for quantum mechanics. The traditional Fourier equation in lattice temperature is expressed in terms of the Planck energy E of the atoms to show infinite carrier velocity is reasonably approximated by BB photons at the speed of light, thereby avoiding the unphysical alternative that absent BB photons the Fourier law is required to rely on thermal disturbances travelling at infinite velocity. Practically, the effect of BB photons on the accuracy of the Fourier solution is insignificant as the BB transient response of the semi-infinite solid is shown identical to that which includes the lag time caused by the speed of light. Fourier’s law is not applicable at the nanoscale as by QM the Planck energy of the atom is not available to be carried through the solid by the BB photon.