Boltzmann approach to high-order transport: the non-linear and non-local responses

The phenomenological textbook equations for the charge and heat transport are extensively used in a number of fields ranging from semiconductor devices to thermoelectricity. We provide a rigorous derivation of transport equations by solving the Boltzmann equation in the relaxation time approximation and show that the currents can be rigorously represented by an expansion in terms of the 'driving forces'. Besides the linear and non-linear response to the electric field, the gradient of the chemical potential and temperature, there are also terms that give the response to the higher-order derivatives of the potentials. These new, non-local responses, which have not been discussed before, might play an important role for some materials and/or in certain conditions, like extreme miniaturization. Our solution provides the general solution of the Boltzmann equation in the relaxation time approximation (or equivalently the particular solution for the specific boundary conditions). It differs from the Hilbert expansion which provides only one of infinitely many solutions which may or may not satisfy the required boundary conditions.