Summary and Conclusions

The Berlin spectroscopists did not let Planck rejoice for long about his fundamental derivation of Wien's law. In the very year Planck completed his program, 1899, they began to observe systematic deviations from Wien's law in the infrared part of the blackbody spectrum. This helped Planck realize that, contrary to his earlier conviction, there were an infinite number of expressions for the resonator entropy compatible with his electromagnetic H -theorem, and thus an infinity of corresponding blackbody laws. In fact, in order for the total entropy to increase, the only constraint on the expression for the resonator entropy was that its second derivative (with respect to energy) should be negative. Then, on the basis of a new independent argument, Planck imposed an additional constraint on this derivative and recovered Wien's law, to the experimenters' great disbelief.

[100] See n. 90. The absence of a quantum discontinuity in Planck's early derivations of his blackbody law was first observed by Kuhn 1978 against a long historical tradition that asserted the contrary.

[101] Planck 1906, 135: ". . . resonators that carry a given amount of energy (better: that fall into a given 'energy domain') . . ." The notion of Elementargebiete der Wahrscheinlichkeit was systematically developed in the second edition (1913) of Planck 1906.

The new argument was wrong, and Planck publicly withdrew it in October 1900, after the experimental violations of Wien's law had become more obvious. In the same communication he proposed an alternative blackbody law, a happy guess based on a simple modification of the expression for the second derivative of the resonator entropy corresponding to Wien's law. The new blackbody law immediately proved to fit empirical data quite well, and Planck started to think about a more fundamental derivation. This led him to consider the relation between entropy and probability which Boltzmann had introduced in 1877.

According to the relevant memoir of Boltzmann, in a dilute gas the equilibrium distribution of velocities—that is, Maxwell's distribution—was also the most "probable"; and the entropy (or the function -H ) was given by the logarithm of the (unnormalized) "probability." Calling (according to modern terminology) the exact microscopic configuration of the molecular model a microstate, and the distribution of velocities a macrostate, Boltzmann's (unnormalized) "probability" was defined as the number of microstates compatible with a given macrostate. Of course, this definition has problems since there is a continuous infinity of microstates corresponding to every macrostate. To solve this difficulty, Boltzmann divided up the configuration space of a molecule into cells and regarded all configurations belonging to a given cell as one single configuration. For instance, in a simple model for which the configuration of a molecule is completely determined by its energy, the energy axis is cut up into equal intervals or energy elements, and a microstate is obtained by assigning to each molecule one of these intervals. Boltzmann's subsequent calculations required the energy elements to be finite (so that the number of molecules in an energy interval could be very large) but small enough not to blur the definition of macrostates, to which the quantities of physical importance pertained. On this condition the energy elements disappeared from the end results; and Maxwell's distribution and the corresponding entropy were recovered. In other words, Boltzmann employed the energy elements as a mathematical artifice, for the purpose of giving a definite meaning to the "probability" of a macrostate. They did not belong to the microscopic model, nor could they enter macroscopic laws; for these could be reached independently of the relation between entropy and "probability," through the H -theorem or the ergodic hypothesis.

The relation between entropy and probability played only a minor role in Boltzmann's subsequent work. For instance, in his Gastheorie it appeared only as a "mathematical illustration" of the expression for the H -function. Boltzmann (rightly) believed that derivations of thermodynamic

quantities and laws through the H -theorem or through the ensemble technique were more fundamental. In 1900 Planck faced a different situation: his electromagnetic H -theorem had proved useless in determining the entropy of a resonator, so that the relation between entropy and probability, far from being superfluous, seemed to be the only available access to the blackbody law. Planck accepted the relation but not its original context, which was a probabilistic interpretation of the irreversibility theorem. Instead he reinterpreted Boltzmann's "probability" as a quantitative measure of elementary disorder, a notion that was at the core of his (Planck's) non-probabilistic conception of irreversibility. Such reinterpretation also had a practical advantage: it provided some guidance about how to extend the analogy between gas theory and radiation theory.

Planck first discussed the type of disorder to be found in a resonator, knowledge gleaned from the requirements of derivation of the electromagnetic H -theorem. In this way he determined what played the roles of microstates and macrostates, as the states of the system respectively in the detailed and the physical levels of description. Next, following Boltzmann, he introduced finite energy elements in order to obtain a definite value for the "probability," that is, the number of microstates in a given macro-state. The logarithm of this "probability" gave him the entropy of a resonator, which leads to Planck's new blackbody law—if only the energy elements can be taken to be proportional to the frequency of the resonator.

Contrary to Boltzmann's case the energy elements now appeared in the final thermodynamic expressions. Planck attributed this peculiarity to a difference in the type of disorder. Indeed, his understanding of the disorder in a resonator led to a notion of macrostate (characterized by the total energy of an ensemble of resonators) that was insensitive to the introduction of energy elements; therefore, Boltzmann's condition that the energy elements should be small enough not to blur the definition of macrostates had no counterpart in Planck's case, and nothing seemed to forbid the appearance of the energy element in the final entropy formula.

In this situation Planck had no reason to question the continuity of the resonator energy. Moreover, such a step would have contradicted, among other things, his derivation of the "fundamental equation," which was necessary for his proof of the blackbody law. In his mind the energy elements were something like the gauge of elementary disorder; they therefore pertained to the indeterminate internal structure of resonators, and they did not contradict his electrodynamic reasonings, which were independent of this structure. In short, Planck relaxed Boltzmann's connection between microworld and macroworld by leaving part of the micromodel

indeterminate. This allowed him to maintain strict irreversibility in the macroworld, by adjusting the indeterminate part of the micromodel (introduction of elementary disorder). In turn, this adjustment permitted his derivation of the blackbody law, without contradicting the determinate part of the micromodel.

As is well known, a few years ago Thomas Kuhn published an in-depth study of blackbody theory at the turn of the century. I will briefly indicate how my account may differ from his. Kuhn concludes, as I do, that Planck did not restrict the energy of his resonators to discontinuous values. His reasoning may be summarized as follows: Boltzmann introduced finite energy elements with no intention of jettisoning the continuity of molecular dynamics; Planck reached his expression for the resonator entropy working in close analogy with Boltzmann's method; therefore, despite some delusive formal manipulations, he did not quantize the energy of the resonators. As convincing as it might be, this argument does not say why Planck did not feel compelled, within the framework of his own thermodynamics, to imitate Boltzmann's procedure even more closely, which would have led to an absurd blackbody law (the so-called Rayleigh-Jeans law). My explanation for this rests on the idiosyncratic nature of Planck's conception of the microscopic foundations of thermodynamics. Kuhn describes Planck's conversion to Boltzmann's views and methods as quasi-complete (as starting with the introduction of "natural radiation"). In fact, as Allan Needell first demonstrated, Planck did not renounce his nonstatistical conception of irreversibility until much later (around 1914). This in turn explains the role elementary disorder played in orienting Planck's use of analogies in his derivation of the blackbody law in 1900. It also explains why Planck's early readers (and a good number of later ones) found his derivation either obscure or implicitly based on an intrinsic quantization of resonators: they were wearing Boltzmann's spectacles.^[102]

During the first ten years of this century, Planck's theory of radiation, and more generally the problem of thermal radiation, became the object of critical investigations by unusually penetrating minds, among whom were two young physicists, Ehrenfest and Einstein, and the venerated H. A. Lorentz. Some of Planck's results survived: the electromagnetic H -theorem (so named by Ehrenfest) proving the spatial uniformizing effect of resonators, and the blackbody law with its characteristic energy elements and the new fundamental constant h .^[103]

[102] Kuhn 1978; Needell 1980, 1988.

[103] See Kuhn 1978; Klein 1963b, 1967; Darrigol 1988b.

However, the central concept of Planck's theory, namely his notion of elementary chaos, appeared to be untenable. According to Einstein, no coherent conception of microscopic dynamics was able to provide a strict and indefinite increase of entropy. On the contrary, microscopic disorder implied observable effects like the perpetual agitation of Brownian particles and mirrors. Within Boltzmannian orthodoxy, Planck's assumption of finite energy elements proved to be incompatible with the foundation of electrodynamics. No interpretation of the blackbody law could be given without emancipating the resonators from their classical (even secular) behavior.

In 1906 Einstein reinterpreted the formal skeleton of Planck's derivation of the blackbody law on the basis of a discrete quantization of resonators. In other words, he turned Boltzmann's "fiction" into a reality, interpreting the energy unit as the minimal amount of energy that resonators could exchange with radiation. This idea of a radical quantum discontinuity was certainly paradoxical, for no one (not even Einstein) could imagine a satisfactory mechanism of the quantum jumps. Nevertheless, it quickly led Einstein to a successful theory of specific heat. By the Solvay congress of 1911 an increasing number of specialists (but not Planck) were convinced that the energy of atomic entities could only take discrete values. The ground was ready for even sharper departures from classical theory, which Bohr soon brought with his atomic theory.^[104]

To conclude, the retrospective successes and defects of Planck's program can be largely understood as deriving from certain powerful analogies with Boltzmann's theory, these analogies being constrained by a belief in the absolute validity of the entropy principle (which was not Boltzmann's). One of these successes, the electromagnetic H -theorem, depended upon a formal analogy between the notions of natural radiation and of molecular chaos. Further, the conception of disorder bound to this analogy guided Planck in his exploitation of another analogy, that between Boltzmann's combinatorics and resonator combinatorics. The resulting derivation of the blackbody law happened to be formally meaningful, even though its conservative interpretation would not survive the quantum revolution initiated by Einstein.

[104] Einstein 1906. See Kuhn 1978, and Klein 1965.