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Temperature and Frequency

Fact is that the Planck’s quantum as used by de Broglie is an explicit production of statistics, and this seems to haunt de Broglie’s attempts to associate a frequency with particles from the very starting point. Along this line of argument, we can see for instance a statistical ensemble disguised under the concept of group of waves. The proposal in [2] goes along the same essential statistical ideas. De Broglie must have sensed this problem for, as M. Lochak shows [3], in his attempt to unify the three fundamental classical principles, he has been led to consider the energy in broader terms, in order to also broaden the area covered by the concept of frequency. Specifically, he also considered a frequency related to temperature. It might help to note though that de Broglie has been led to such ideas by considering an exponential statistic for the proper mass itself. While the formal procedure traces back to technical details in the analysis of the specific heats of solids (for instance Debye temperature and the like), de Broglie’s association bears the burden of a principle requiring us to trace its deep origins. For this we have to go back as far as Planck himself, and the very origins of quantization.

The procedure Planck used in order to come up with the idea of quantum, to be described in the very next paragraphs is, in our opinion, the first significant precursor of the current theory of quadratic variance distribution functions [9]. Let us illustrate that procedure in detail, in order to get some sense of the issue. Assume we have two Gaussian processes as representing the limit cases of blackbody radiation, Rayleigh-Jeans and Wien, and that they are in a general relationship, i.e. not statistically independent. One may further assume that the general thermal radiation is actually a linear combination between the two processes but, for the sake of argument, we limit here the line of reasoning to just the sum of the two processes. The general bivariate normal distribution is given by the probability density

(19)

In terms of the variances σ_x, σ_y of the two processes and their correlation coefficient r, the coefficients a, b, c can be written as

(20)

Now we can write the probability density of the compound process (X + Y), which is

(21)

i.e. a Gaussian with the variance (a + c – 2b)/(ac – b²) or, in terms of variances and correlation coefficient of the two component processes

(22)

Now, within this statistical theoretical environment, Planck used the basic definition equations of the Rayleigh-Jeans and Wien’s processes as limit cases, in order to interpolate between them:

(23)

where u is the density of energy of radiation, β is the inverse of temperature and a prime means derivative of function with respect to variable. The writing in equation (23) is justified by the fact that Planck found, perhaps for the first time in the history of Theoretical Statistics, the trait of what has later on become known as exponential type of distributions: u’(β) represents, up to a sign, the variance of the distribution characterizing the black radiation. What followed appeared indeed as an interpolation process as Born takes it [10], however it is a little more subtle. For, Planck considered that the left hand side of (23) is a universal characteristic of the variance function of the process characterizing the blackbody radiation. As the variance of two statistically independent processes is the sum of their variances, Planck wrote a differential equation for u’(β) by taking into consideration for it simply the sum of the two components from the right hand side of (23). If we take the same route here, the equation (22) would suggest that Planck approach can be expressed by the equation:

(24)

(Planck’s original equation has r = 0). This equation can be integrated to give

(25)

Here something is immediately obvious, which shows that, in the past, our focus might have been misplaced due to the mirage of a quick physical interpretation already at hand. First of all, it does not seem to exist a closed form solution for u. Secondly the energy ε₀, which has been introduced from dimensional considerations by Wien according to his displacement law, has to be apriori explained. As known, it has been subsequently explained as a quantum of energy, proportional to frequency, to be carried by an invented particle (the photon) and it is this frequency that has been used by de Broglie in his association wave-particle. Then our idea is that, if we want to deal with de Broglie’s frequency, we have first to deal with ε₀, for this energy is itself uncertain. One can remark, in this respect, the tremendous advantage of considering the two classical limit processes as statistically independent (r = 0), for in that situation one has directly the Planck’s case with all of the known advantages thereof: the use of Wien’s displacement law in order to infer that ε₀ is proportional to frequency and then find that u is actually a mean over a canonical ensemble of discrete oscillators.

In the present general case the things are a little out of hand, in that the question immediately pops up as to the physical meaning of ε₀. We ought to have an explanation at any rate, but unlike the classical case, do not have the ensemble of oscillators at our disposal anymore. Here is a scenario, involving the de Broglie’s line of reasoning: in the good old fashion of Statistical Mechanics, we correlate this energy with an exponential factor, which can play the role of a partition function over a certain ensemble, and which can be easily extracted from equation (25) as

(26)

The left hand side of this equation represents a thermal ensemble for the energy ε₀, having the mean β, or vice versa, of energy β having the mean ε₀. There are two odd things here. The first of them is that the right hand side of (26) also depends on ε₀. However, this dependence occurs through the intermediary of the ratio w, which allows us to say that a statistical interpretation actually depends upon a sort of ε₀-content of the density of energy of our ensemble, whatever it may be. This conclusion sounds quite normal: an experimentalist, for instance, knows exactly how to characterize the radiation depending on its density. Should this density be of the order of ε₀ then w ≈ 1, and the right hand side of equation (26) does not depend but on the correlation coefficient between the two processes:

(27)

Here it is just a matter of option to consider the quantum as it has been introduced originally by Einstein, i.e. for ε₀ proportional to frequency. Equation (27) may mean more. Three decades ago, Ioannidou [11] tried to explain the quantum through the correlation of ensembles associated with oscillators, based on the uncertainty relation. The attempt has been forgotten, probably due to the connection it suggested. Well, that connection seems sound, for here it is again, in equation (27), which explicitly puts down a relationship between the quantum and the correlation coefficient of the two processes representing the radiation. Further on, if the correlation of the two processes is faint, as in the Planck’s case, then

(28)

independently of any other consideration. Thus, in this limit, the “quantum”, is directly proportional to temperature. This is a fact that has been discussed at length by Louis de Broglie [12] who skilfully identified the action throughout the three fundamental physical principles as mentioned by M. Lochak in his note [3]. We hope to show in a forthcoming paper that this is precisely the gnoseological reason for the de Broglie’s theory of double solution, if only for the fact that is parallels the original case that led to idea of quantum, with its two extreme situations. In other words, de Broglie’s principle, inasmuch as it refers to associating a periodic process to energy, goes well beyond the quantization and relativity.

A few more words about previous considerations may be worth our while. In the general case, when the ε₀-content and the correlation of the two extreme processes are both arbitrary (double solution!), it helps noticing that the right hand side of (26) is actually the generating function of a particular class of Pollaczek polynomials [13]. Specifically, we can write (26) in the form

(29)

The orthogonality relation of polynomials involved here is given by

(30)

with the weight function ρ given by

(31)

where Γ is the Euler function of the first kind, generalization of the factorial. Should we agree to interpret ε₀ as the energy of a photon, as has historically been the case, then the formula (31) would be the source of constructions of some modern quantum states related to the coherence properties of radiation. Going beyond this point, may be indeed very beneficial for physical knowledge, and the way can be considered as already paved. Indeed the equation (24) opens the door of a new kind of probability densities. These have apparently been discovered by V. Seshadri, and they have been studied in detail by Célestin Kokonendji [14]. Perhaps the opinion of a theoretical statistician in these physical matters will allow a clarification of them.

The bottom line is that we ought to supply for ε₀ an apriori explanation; otherwise we have to face problems of aposteriori explanations, as indeed was historically the case. One of these explanations, which we would like to call de Broglie’s principle, aims the very existence of the photon, electron, or of any other ‘ons’ for that matter. In our opinion, the ongoing discussion upon the legitimacy of the quantum precepts and the physical reality for de Broglie frequency just proves this idea.

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