  Stefan-Boltzmann law was empirically deduced by Stefan in 1874 by fitting
existing experiments and theoretically validated by Boltzmann in 1884 on the
basis of a classical model involving thermodynamics principles and the Maxwell
equations. At first sight the electromagnetic (EM) gas assumed by Boltzmann and
identifiable as an ensemble of $N$ classical normal-modes, looks like an
extension of the classical model of the massive ideal-gas. Accordingly, for
this EM gas the internal total energy, $U$, was taken to be function of volume
$V$ and temperature $T$ as $U=U(V,T)$, and the equation of state was given by
$U=3PV$, with $P$ the radiation pressure. In addition, Boltzmann implicitly
assumed that, for given values of $V$ and $T$, $U$ and $N$ would take finite
values. However, from one hand these assumptions are not justified by Maxwell
equations since, in vacuum (i.e. far from the EM sources), according to
classical statistics, the values of $U$and $N$ diverge. From another hand,
Boltzmann derivation of Stefan law is found to be macroscopically compatible
with its derivation from quantum statistics announced by Planck in 1901.
Accordingly, this letter presents a solution of this puzzling classical/quantum
compatibility by noticing that the implicit assumption made by Boltzmann is
fully justified by quantum statistics. Furthermore, we shed new light on the
interpretation of recent classical simulations of a black-body carried out by
Wang, Casati, and Benenti in 2022 who found an analogous puzzling compatibility
to induce speculations on classical physics and black-body radiation that are
claimed to require a critical reconsideration of the role of classical physics
for the understanding of quantum mechanics.
