  We present a general quantum kinetic theory that accounts for the interplay
between a temperature gradient, momentum-space Berry curvatures of Bloch
electrons, and Bloch-state scattering. Using a theory that incorporates the
presence of a temperature gradient by introducing a "thermal vector potential",
we derive a quantum kinetic equation for Bloch electrons in the presence of
disorder and a temperature gradient. Taking also into account the presence of
electric and magnetic fields, the quantum kinetic equation we derive makes it
possible to compute transport coefficients at arbitrary orders of
electric-field $\vec{E}$, magnetic-field $\vec{B}$, and temperature-gradient
$\nabla T$ strengths $|\vec{E}|^a |\vec{B}|^b |\nabla T|^c$. Our theory enables
a systematic calculation of magnetothermoelectric and magnetothermal
conductivities of systems with momentum-space Berry curvatures. As an
illustration, we derive from a general microscopic electron model a general
expression for the rate of pumping of electrons between valleys in parallel
temperature gradient and magnetic field. From this expression we find a
relation, which is analogous to the Mott relation, between the rate of pumping
due to a temperature gradient and that due to an electric field. We also apply
our theory to a two-band model for Weyl semimetals to study thermoelectric and
thermal transport in a magnetic field. We show that the Mott relation is
satisfied in the chiral-anomaly induced thermoelectric conductivity, and that
the Wiedemann-Franz law is violated in the chiral-anomaly induced thermal
conductivity, which are both consistent with the results obtained by invoking
semiclassical wave-packet dynamics.
