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
