Statistical equilibrium can only be established if there is exchange of energy between the resonators, otherwise each resonator would retain its initial energy, which is arbitrary, indefinitely, and the final distribution would obey no law. This exchange could not be done by radiation if the resonators were fixed and locked in a fixed enclosure. Indeed, each resonator could emit or absorb only light of a specific wavelength, so it could send energy to resonators of the same period.
It is not the same if one supposes that the enclosure is deformable or contains moving bodies. And indeed the light reflecting on a mobile mirror changes wavelength by virtue of the famous principle of Döppler-Fizeau. And this is the first mode of radiation exchange.
There is a second; the resonators can react mechanically on each other, either directly or rather via moving atoms and electrons that flow from one to the other and collide them. It is the exchange by collision. This is the one I studied recently, finding and confirming the results of Mr. Planck.
As I explained above, it is necessary that all modes of exchange of energy lead to the same conditions of statistical equilibrium, otherwise the Carnot principle would be in breach. This is necessary in order to give an account of the experience, but it is necessary that we should be able to give this surprising concordance a satisfactory explanation, that we should not be forced to attribute it to a sort of providential chance. In the old Mechanics, this explanation was all found, it was the universality of Hamilton’s equations; are we going to find something analogous here?
I have not yet finished the study of the exchange by radiation, and I do not know yet if we know all the conditions of equilibrium which leads this mode of exchange; I would not be surprised to discover new ones that could cause us some embarrassment.
For the moment, there is one that has been revealed to us by the work of Mr. Wien; this is called Wien’s law, according to which the product of the energy of the radiation by the fifth power of the wavelength depends only on the temperature multiplied by the wavelength.
We see immediately that, for this law of Wien to be compatible with the statistical equilibrium due to the exchange by collisions, it is necessary that, in this exchange by collisions, the energy can vary only by quanta inversely proportional to the wavelength. This is a mechanical property of the resonators, which is obviously quite independent of the Döppler-Fizeau principle, and it is difficult to understand by what mysterious preestablished harmony these resonators were endowed with the only mechanical property which could be suitable. . If the statistical equilibrium is invariable, it is no longer for a single and universal reason, it is by the concurrence of multiple and independent circumstances.
In Mr. Planck’s mode of exposition, this duality of the modes of exchange does not appear, but it is only concealed and I thought it necessary to draw attention to this point.
This difficulty is not the only one; a resonator can yield energy to another only by integer multiples of its quantum; this one can only receive by integer multiples of its quantum to him; as these two quanta will generally be incommensurable, this is enough to exclude the possibility of a direct exchange, but the exchange can be done through the atoms, supposing that the energy of these atoms can vary in a continuous way.
This is not the most serious; the resonators must lose or gain each quantum abruptly or rather they must earn all their entire quantum or they gain nothing. But they need some time to win or lose; this is what the phenomenon of interference requires. Two quanta emitted by the same resonator at different times can not interfere with each other. Both emissions should indeed be regarded as two independent phenomena and there would be no reason for the time interval between them to be constant. This is even impossible; this interval must be greater if the light is weak than if it is intense; unless it is assumed that the interval is constant, that each emission may consist of several quanta and that the intensity depends on the number of quanta emitted at a time. But that can not go either; the interval must be small relative to a period to fit with the interference observations; the value of the quantum results from Planck’s own formula; therefore, we would have a minimum of the possible intensity of the light, and we have observed light emissions below this minimum.
It is therefore every quantum that interferes with itself; it is therefore necessary that, once in the form of luminous vibrations of the ether, it divides itself into several parts, that some parts lag behind the others of several wavelengths and consequently that they have not been issued at the same time.
It seems that there is a contradiction; perhaps it is not insoluble. Imagine a system consisting of a number of Hertz exciters, all identical; each of them is charged by a source of electricity and as soon as its charge has reached a certain value, the spark bursts, the emission begins and nothing can stop it anymore, until the exciter is fully discharged; it must either lose its quantum entirely, or lose nothing (the quantum is the amount of energy that corresponds to the explosive potential). But this quantum is not lost abruptly, each emission lasts a certain time and the waves emitted are susceptible of regular interference.
Planck hypothesized that the relation between the energy of a resonator and its radiation was the same as in Maxwell ‘s Electrodynamics; we could give up this hypothesis, and suppose that the mechanical collisions are made according to ancient laws. The distribution of the energy between the resonators would then be done according to the law of the equipartition, but the resonators with short period would radiate less with equal energy. We could then account for the law of radiation, but we would not explain the anomalies of the specific heats at low temperatures, unless we admit that the exchange by collision is no longer possible for very cold solids, and that their molecules exchange more heat only by radiation at a short distance.
We could go further, assuming that there is never any collision, that all the so-called mechanical forces are of electromagnetic origin; that they are due to actions at a distance, which can themselves be explained by radiation. It would then be necessary to leave only the mode of exchange by radiation and by the play of the Döppler-Fizeau principle; perhaps then we would be led to hypotheses very different from that of quanta.