The Puzzling of Stefan-Boltzmann Law: Classical or Quantum Physics

TL;DR

This paper examines the longstanding debate over the classical versus quantum foundations of the Stefan-Boltzmann law, proposing a resolution to the classical-quantum compatibility puzzle through quantum statistical justification.

Contribution

It demonstrates that Boltzmann's classical assumptions are justified by quantum statistics, clarifying the classical-quantum relationship in black-body radiation.

Findings

Classical assumptions are compatible with quantum statistics.

Boltzmann's derivation aligns with Planck's quantum approach.

Recent classical simulations of black-body radiation support the classical-quantum compatibility.

Abstract

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.