Authors: Y C Kamala
Abstract: In 1924 Satyendra Nath Bose, then at the University of Dacca, derived Planck’s radiation law without appeal to classical electrodynamics. He divided the phase space of light quanta into cells of volume h³ and counted the ways in which quanta could be distributed among them. Einstein translated the paper and extended its method to material particles, and the result became Bose–Einstein statistics. This paper re-reads Bose’s derivation as an exercise in combinatorics. It shows that Bose’s own counting, in terms of the numbers of cells that contain a given number of quanta, is exactly equivalent to the “stars and bars” count of indistinguishable objects in distinguishable boxes, and it proves the equivalence with a generating function. Exact enumerations confirm the equivalence for small systems and show how sharply it departs from the count for distinguishable quanta. Numerical maximisation of Bose’s weight recovers the geometric distribution of quanta per cell that yields Planck’s law, and exact finite-system calculations show the convergence of the single-cell distribution and of the fluctuations to their thermodynamic limits, including Einstein’s fluctuation formula with its particle and wave terms. The paper relates Bose’s counting to Planck’s combinatorial argument of 1900, to Natanson’s analysis of 1911 and to the Ehrenfest–Kamerlingh Onnes derivation of 1915, and argues that the novelty of Bose’s work lay less in a new formula than in a new choice of the objects to be counted.
DOI: https://doi.org/10.5281/zenodo.23473406
