Abstract
During the physical foundation of his radiation formula in his December 1900 talk and subsequent 1901 article, Planck refers to Boltzmann’s 1877 combinatorial-probabilistic treatment and obtains his quantum distribution function, while Boltzmann did not. For this, Boltzmann’s memoirs are usually ascribed to classical statistical mechanics. Agreeing with Bach, it is shown that Boltzmann’s 1868 and 1877 calculations can lead to a Planckian distribution function, where those of 1868 are even closer to Planck than that of 1877. Boltzmann’s and Planck’s calculations are compared based on Bach’s three-level scheme ‘configuration–occupation–occupancy’. Special attention is paid to the concepts of interchangeability and the indistinguishability of particles and states. In contrast to Bach, the level of exposition is most elementary. I hope to make Boltzmann’s work better known in English and to remove misunderstandings in the literature.
Highlights
During the physical foundation of his radiation formula in his December 1900 talk and subsequent 1901 article, Planck refers to Boltzmann’s 1877 combinatorial-probabilistic treatment and obtains his quantum distribution function, while Boltzmann did not
But not least, Boltzmann’s statistical definition of entropy is the first one that applies to non-equilibrium states, “opening the door to the statistical mechanics of non-equilibrium states and irreversible processes.” [7] (p. 1974)
“First, we find how often in all J complexions a molecule has kinetic energy 0, how often the kinetic energy is ε, 2ε, etc., and say that the ratios of these numbers should provide the probabilities that a molecule has kinetic energy 0, ε, 2ε, etc. at thermal equilibrium
Summary
In the relationship between statistical mechanics and thermodynamics, Ludwig Boltzmann’s and Max Planck’s works play a leading role. In his combinatorial calculations for founding his radiation law, Planck [1] refers to Boltzmann’s 1877 [2] definition and counting of states and the definition of entropy based thereon. In agreement with Bach but at an elementary level, I will show that Boltzmann’s 1868 [5] and 1877 [2] definitions and counting of states can lead to a Planckian distribution law, where the 1868 memoir [5] is even much closer to Planck’s 1901 treatment [6] than the 1877 memoir [2]. Some of the formulae below will be derived from this fundamental result
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