Non-equilibrium field theory with a temperature gradient: Thermal current in a nanowire

A perturbative framework is developed within the standard non-equilibrium field theory techniques to incorporate a temperature gradient across a thermoelectric device. The framework uses a temperature-dependent pseudo-Hamiltonian generated from the exact density matrix in the presence of non-uniform temperatures. We develop a perturbation theory for small temperature gradients in long wires and obtain the non-linear thermal conductance as a function of temperature difference. The framework should be adaptable to more general cases of temperature inhomogeneities in either fermionic or bosonic systems.