Revisiting the Boltzmann Derivation of the Stefan Law

Abstract

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature [Formula: see text] as [Formula: see text], with [Formula: see text][Formula: see text]W[Formula: see text]m[Formula: see text][Formula: see text]K[Formula: see text] the SB constant, first estimated by Stefan to within [Formula: see text] of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation 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 black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant [Formula: see text]. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.