Revisiting the Boltzmann derivation of the Stefan law

The Stefan-Boltzmann (SB) law relates the radiant emittance of an ideal black-bodycavity at thermal equilibrium to the fourth power of the absolute temperatureTasq=σT4, withσ= 5.67×10−8Wm−2K−4the SB constant, firstly estimated by Stefanto within 11 per cent of the present theoretical value. The law is an important achieve-ment of modern physics since, following Planck (1901), its microscopicderivation impliesthe quantization of the energy related to the electromagnetic field spectrum. Somewhatastonishing, Boltzmann presented his derivation in 1878 making use only of electro-dynamic and thermodynamic classical concepts, apparently without introducing anyquantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmannderivation implies two assumptions not justified within a classical approach, namely:(i) the zero value of the chemical potential and, (ii) the internal energy of the blackbody with a finite value and dependent from both temperature and volume. By usingPlanck (1901) quantization of the radiation field in terms of a gas of photons, the SBlaw received a microscopic interpretation free from the above assumptions that also pro-vides the value of the SB constant on the basis of a set of universal constants includingthe quantum action constanth. However, the successive consideration by Planck (1912)concerning the zero-point energy contribution was found to be responsible of anotherdivergence of the internal energy for the single photon mode at high frequencies. Thisdivergence is of pure quantum origin and is responsible for a vacuum-catastrophe, tokeep the analogy with the well-known ultraviolet catastrophe of the classical black-bodyradiation spectrum, given by the Rayleigh-Jeans law in 1900. As a consequence, froma rigorous quantum-mechanical derivation we would expect the divergence of the SBlaw (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes arerevised by accounting for both the quantum-relativistic photon gasproperties, and the Casimir force.