  We show that in the linear response approximation only entropy provides
coupling between thermal and electric phenomena. The dissipationless quantum
currents -- magnetization, superconducting, persistent and topological edge
currents -- do not produce and transfer entropy and may be excluded from final
formulas for thermomagnetic coefficients. The magnetization energy flux, eM X
E, in crossed electric and magnetic fields strongly modifies the Poynting
vector in magnetic materials and metamaterials, but do not contribute to the
heat current. Calculating entropy fluxes of fluctuating Cooper pairs, we find
the fluctuation Nernst coefficient in pure superconductors. To account electron
scattering, we generalize the gauge-invariant Kubo formalism developed for the
Hall effect to thermomagnetic entropy transfer. We also introduce the
thermomagnetic entropy per unit charge and derive the Nernst coefficient
proportional to the difference of the thermoelectric and thermomagnetic
entropies. This explains the Sondheimer cancellation and high sensitivity of
thermomagnetic phenomena to correlations. In 2D superconductors, the transport
entropy transferred by a vortex moving through the background formed by
vortex-antivortex pairs is the configuration entropy of kB ln 2, which strongly
exceeds the intrinsic entropy of vortex core. Beyond the linear response, the
non-entropic forces can lead to phenomena unexpected from thermodynamics, such
as vortex attraction to the moving hot spot. Quantum currents do not transfer
entropy and may be used as ideal connectors to quantum nanodetectors.
