Based on the functional dependence of entropy on energy, and on Wien's distribution for black-body radiation, Max Planck obtained a formula for this radiation by an interpolation relation that fitted the experimental measurements of thermal radiation at the Physikalisch Technishe Reichanstalt (PTR) in Berlin in the late 19th century. Surprisingly, his purely phenomenological result turned out to be not just an approximation, as would have been expected, but an exact relation. To obtain a physical interpretation for his formula, Planck then turned to Boltzmann's 1877 paper on the statistical interpretation of entropy, which led him to introduce the fundamental concept of energy discreteness into physics. A novel aspect of our account that has been missed in previous historical studies of Planck's discovery is to show that Planck could have found his phenomenological formula partially derived in Boltzmann's paper in terms of a variational parameter. But the dependence of this parameter on temperature is not contained in this paper, and it was first derived by Planck.
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In the limit of vanishing ϵ, Boltzmann's discrete model is only applicable to the motion of molecules in two dimension (See Ref. 13, p. 190).
Historians of science have followed Klein's statement that “it seems most likely that Planck was guided by the form Wien's distribution law,” when if fact there was not any other alternative.
It should be pointed out that it was Planck's very good fortune that Boltzmann's first treatment of his molecular gas model was in two dimensions. For each degree of freedom, the classical mean energy of a molecule is U = kT. Hence the physical three dimensional case with U = kT would not have been a useful model for Planck's electrical oscillators.
But this constraint on p does not apply in the Stirling approximation for the factorials in WB. In his book (Ref. 5, p. 49), Kuhn sets and states that “standard variational techniques lead directly to the conclusion that for ,” nj is given by Boltzmann's classical expression, Eq. (A18). But this claim is incorrect, because to obtain the distribution in the classical limit, Boltzmann set and .
Boltzmann also considered p finite, but the case of interest related to Planck's formula corresponds to .
In the Stirling approximation the resulting values of nj, Eq. (A7), are not integers. In this case λ and n are also infinite, but the ratio is fixed and the ratio is finite corresponding to the fraction of molecules with energy .
