The temperature-dependent thermal scalar and vector potentials in quantum Boltzmann equation

To explore the thermal transport procedure driven by temperature gradient in terms of linear response theory, Luttinger and Tatara proposed the thermal scalar and vector potentials, respectively. In this manuscript, we try to address the microscopic origin of these phenomenological thermal potentials. Based on the temperature-dependent damping force derived from quantum Boltzmann equation (QBE), we express the thermal scalar and vector potentials by the distribution function in damping force, which originates from the scattering of conduction electrons. We illustrate this by the scattering of electron-impurity interaction in a transport system. The temperature and temperature gradient will appear in the thermal potentials, as done in previous works (Luttinger J. M., Phys. Rev., 135 (1964) A1505; Tatara G. G., Phys. Rev. Lett., 114 (2015) 196601). The influence from quantum correction terms of QBE is also considered, which contributes not only to the damping force, but also to the anomalous velocity in the velocity term. An approximated solution for the QBE is given, the numerical results for the damping force, thermal current density as well as other physical observables are shown in figures.