Between equilibrium and fluctuation: Einstein's heuristic argument and Boltzmann's principle

TL;DR

The paper investigates Einstein's 1905 heuristic for light quanta, focusing on whether the argument should be read as a fluctuation or as a comparison between equilibrium states, and on how Boltzmann's principle was used and interpreted in that period. It surveys influential analyses (e.g., Dorling, Norton, Duncan & Janssen, Irons) that question the interpretation of $S = \kappa \ln W$ for radiation given nonconserved quanta, and argues that the core issue is the entropy change of monochromatic radiation, which is best read as a comparison of equilibrium entropies rather than a genuine fluctuation. The narrative emphasizes the limits of the gas–radiation analogy when applied to non-equilibrium situations and shows how Einstein's thinking evolved toward adiabatic reasoning and the broader adoption of quantum statistics. It then connects these historical debates to the modern framework in which occupancy number, not frequency, governs the emergence of classical or quantum behavior across the spectrum, culminating in the view of photons as field excitations with occupation-driven statistics and experimental support from antibunching and related phenomena. Overall, the work highlights how early debates over Boltzmann's principle and statistical reasoning in radiation helped shape the development of quantum theory and the contemporary understanding of light quanta across the electromagnetic spectrum.

Abstract

We critically revisit Einstein's 1905 heuristic argument for lightquanta, considering its internal coherence and the scope of its applicability. We argue that Einstein's reasoning, often celebrated for its originality, is ambiguous because it can be understood as a fluctuation or as a comparison between equilibrium states. A historical and conceptual analysis of Einstein's use of Boltzmann's principle in those years reveals his evolving stance on its meaning and the role of probability, as well as his persistent doubts about the nature of radiation. We use our analysis to examine the limitations of extending the notion of Einstein's lightquanta across the electromagnetic spectrum: the relevant parameter is not the frequency, but the occupancy number.

Figures (4)

• Figure 1: Diagram of the volumes considered by Einstein (this figure does not appear in his paper).
• Figure 2: Drawings of Einstein of the system of a suspended particle. In kleinkox, p. 234
• Figure 3: Drawing that appears in Dällenbach's notes. See footnote \ref{['d']}
• Figure 4: Schematic dependence of statistical weights $G(E_{\nu}, \nu)$ corresponding to the Wien, Planck and Rayleigh-Jeans laws.