Revisiting the Coupling of Thermodynamics and Electromagnetics
We revisit the coupling of continuum thermodynamics and electromagnetic theory for polarisable and magnetisable matter in motion. Two routes are followed and then compared. The first route is the axiomatic bulk theory of Dreyer, Guhlke and M\"uller, in which universal balance laws are closed by an entropy principle. We show that the source of the internal energy balance must be built with the non-convective electric current, that the polarisation current and the Lorentz magnetisation enter through one single identity, which Dreyer et al.\ do not write down, and that this identity fixes both the admissible entropy variables and the signs of the bound-current ansatz. The second route is the statistical-mechanical one of Mazur, in which the macroscopic Maxwell equations are obtained by ensemble averaging over a system of atoms with internal charge carriers. Mazur stops before the conservation laws, so we derive them, and we estimate the size of the mass-correction terms that appear. The comparison shows that after a redefinition of polarisation and magnetisation the two sets of equations agree structurally. The only irreducible difference is a momentum contribution from microscopic field fluctuations, which can not be reproduced in a purely macroscopic theory. We further show that the electromotive intensity $\mathcal{E}$ and the Lorentz magnetisation $\mathcal{M}$ are not modelling choices but appear by themselves, and that the asymmetric look of the entropy function is a consequence of the chosen energy variable and not a defect of the theory.