  We demonstrate that the well-known expression for the charge magnetization of
a sample with a non-zero Berry curvature can be obtained by demanding that the
Einstein relation holds for the electric transport current. We extend this
formalism to the transport energy current and show that the energy
magnetization must satisfy a particular condition. We provide a physical
interpretation of this condition, and relate the energy magnetization to
circulating energy currents in Chern insulators due to chiral edge states. We
further recover the expression for the energy magnetization with this
alternative formalism. We also solve the Boltzmann Transport Equation for the
non-equilibrium distribution function in 2D for systems with a non-zero Berry
curvature in a magnetic field. This distribution function can be used to obtain
the regular Hall response in time-reversal invariant samples with a non-zero
Berry curvature, for which there is no anomalous Hall response.
