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Additional info for From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory (California Studies in the History of Science)

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70] [69][70] ― 51 ― Planck Versus Boltzmann From a systematic comparison of Planck's reasonings with those which led Boltzmann to the H -theorem we may surmise that Planck benefited at various steps from suggestive analogies. Table 1 summarizes the correspondence between Planck's and Boltzmann's arguments. " The third and fifth sections respectively give the equations ruling the microscopic evolution of the system and those ruling the directly observable quantities. The transition between these two types of equations is expressed in the fourth section, with the idea of disorder, and the corresponding simplification of the interaction within the system.

40] Wien's argument, or more exactly Planck's version of it,[41] starts with the consideration of a cavity with perfectly reflecting walls, one of which is part of a mobile piston (fig. 5). At the initial time this cavity contains isotropic electromagnetic radiation with the spectral density rv . Under a slow displacement of the piston at a constant speed v , the light of frequency [40][41] ― 28 ― v in an incident beam is Doppler-shifted by where q is the angle of incidence, and c the velocity of light.

Then the energetic coupling between the variable charge distribution around O and the external field can be determined without knowing the internal structure of this distribution . To this end Planck cleverly considered the energy flux, F , through a sphere E centered on O, much larger than the charge distribution but small enough to make the variations of the enclosed fields negligible . Assuming that the charge distribution exchanges energy only with the electromagnetic field (there is no Joule effect and no internal electromotive force), this flux must be equal to the diminution rate —dY/dt of the energy of the charge distribution plus the diminution rate —dW e/dt of the energy of the external field enclosed by E in the absence of a charge distribution: Another expression of this flux results from Poynting's formula: This integral may be split into four terms according to the development of the vector product.

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