  Electric, thermal and thermoelectric transport in correlated electron systems
probe different aspects of the many-body dynamics, and thus provide
complementary information. These are well studied in the low- and
high-temperature limits, while the experimentally important intermediate
regime, in which elastic and inelastic scattering are both important, is less
understood. To fill this gap, we provide comprehensive solutions of the
Boltzmann equation in the presence of an electric field and a temperature
gradient for two different cases: First, when electron-electron collisions are
treated within the relaxation-time approximation while the full momentum
dependence of electron-impurity scattering is included and, second, when the
electron-impurity scattering is momentum-independent, but the electron-electron
collisions give rise to a momentum-dependent inelastic scattering rate of the
Fermi-liquid type. We find that for Fermi-liquid as well as for Coulomb
interactions, both methods give the same results for the leading temperature
dependence of the transport coefficients. Moreover, the inelastic relaxation
rate enters the electric conductivity and the Seebeck coefficient only when the
momentum dependence of the electron-impurity collisions, analytical or
non-analytical, is included. Specifically, we show that inelastic processes
only mildly affect the electric conductivity, but can generate a non-monotonic
dependence of the Seebeck coefficient on temperature and even a change of sign.
Thermal conductivity, by contrast, always depends on the inelastic scattering
rate even for a constant elastic relaxation rate.
