Author: Nicolae Mazilu

Published on Tuesday, March 2nd, 2010 in category ProtoQuant


ABSTRACT. The black-body radiation is an open problem, and in our opinion it will always be, because its foundations lack something essential. It is determined by two equilibrium conditions: one of its elements with those of the surrounding matter, the other of internal equilibrium. There is much uncertainty in describing both equilibriums due the lack of physical model for the fundamental unit of the statistical ensemble representing the radiation. The classical solution of the problem points to the fact that in this case the temperature is not connected to a sufficient statistic like in the case of ideal gas. This shows that the overall thermodynamic equilibrium might not be as important as the equilibrium determined by interactions at a lower level. Based on classical approach to the problem, these interactions can be described as measurements. However, a statistical element is needed at variance with Planck’s fit. It is shown here such an alternative and illustrate it on COBE-FIRAS data, pointing out to a special tensor nature of the cosmic background radiation. 



One usually tries not to identify the spectral density of radiation with a probability density, because of insurmountable mathematical problems arising with such a consideration. In general the Fourier analysis of the complete light signal has no closed form solution (Houstoun, 1914a,b). In the cases where another function can be found in order to approximate the spectrum, usually that approximation is either obliterated by the impeccability of Planck’s spectrum or leads to physically unsound results, or both. One such case is the work of G. W. Walker (Walker, 1914a,b). In keeping with the oscillator structure of the matter, Walker found a very close approximation of the black-body spectrum, having all of the required physical characterisics. However, Walker’s function is way under Planck’s when it comes to fitting quality to the data. Moreover, it suggests that the physical structure explaining the radiation from mechanical point of view is a critically damped harmonic oscillator – an aperiodic structure. This is not in accordance with the idea of quantization of the energy levels in an atom – the assumed physical structure generating the radiation – and with the fact that the concept of frequency closely connected to radiation. There were, of course, in history, many cases explaining one side or another of the problem, however not explicitly addressed to considering the spectral density as a probability density for frequency in the spectrum – the natural inspiration brought about by the Boltzmann factor. For example in the classical case of Einstein and Hopf calculations (Einstein, Hopf, 1910) the probability density of the spectral components in the complete signal representing radiation was assumed to be a Gaussian, and the conclusion of calculations was that Planck spectrum is physically unsustainable. As known, this raised immediate criticism on the validity of such a statistical assumption (Varró, 2006).


Transformation to Gaussian

However, we can say that in a certain way it is indeed sound enough to asume normality of the spectrum when compared to data, and we try now to explain this in detail. It is quite a regular statistical practice to use transformations of the data in order to bring their distribution to a Gaussian form and thus unveil their clean statistical content. Quite obviously – at least from a mathematical point of view – the black-body radiation spectrum can be considered as a probability density of finding a certain frequency at a given temperature in the radiation spectrum, or perhaps vice versa. It should come then as no surprise that the spectrum can be reduced to a Gaussian form by the transformation




This fact was discovered in 1919 by Irwin G. Priest, who showed that the spectral density of black-body radiation can be closely represented this way as a Gaussian (Priest, 1919a,b).

The present author is not aware of any other mention of this exquisite representation in the literature he surveyed. Probably, not having any physical interpretation, the representation remained at the level of a statistical trick. Thus, the first thing to do was to check the truth of Priest’s claim using, of course, the old original data (Lummer, 1900). The result was, indeed, astounding: just as Priest claims, the Gaussian fit is visibly better than the Planck function when referred to those data. But, for the present purpose we illustrate the issue with fresher, readily available data.

Priest’s basic equation is written in the old manner of experimentalists, i.e. using the wavelength instead of frequency:




Dividing by the maximum spectral density at a given temperature – which is more convenient for curve-fitting purposes – we have




The notations are as follows: Eλ is the spectral density of radiation energy al wavelength λ, which has a maximum Em; further on, A and D1,2 are constants depending on the absolute temperature T. In order to bring a present-day example of the quality of this function in fitting the data, we used here the equation (3) in fitting the COBE-FIRAS official data with it (Fixsen et al., 1996). In Figure 1 it is shown what Priest means by the fact that data represent a Gaussian curve: simply plotting the data points versus cubic root of frequency gives a symmetric curve. The line is the Gaussian of best fit using EXCEL’s SOLVER. Even the dipole data seem to respect this symmetry (Figure 2). In Figure 3, the result of curve fitting is given for the monopole data, in terms of frequency. The two curves – Planck’s and Priest’s are, at least visibly, not too far away from each other.

A Few Speculations…

Just how much the classical conclusion of Einstein and Hopf, that the harmonic oscillator cannot lead to Planck spectrum, would have been changed, with no further assumption about the distribution of the spectral components of radiation, except for this direct approach to the data, can be judged by the following sketchy calculations. First of all, the emphasis in the physical problem would be radically changed. Specifically, knowing the spectral density as a function of frequency in the complete signal, one can only ask what is the physical structure sustaining the thermodynamic equilibrium. If we assume the classical relation between the signal and its Fourier transform, then Priest’s function is actually the square of the Fourier transform of the signal. This means that the complex function representing the signal had to be estimated by an integral of the form




where t is the time, ν is the frequency and A, a, b are three constants depending on the equilibrium temperature. This integral has obviously a close connection with an Airy function, whose representation is




Indeed, the exponent in (4) can be put in the form




with the following identifications




With these, the equation in (4) becomes




These rough results show that the black-body radiation is certainly in the natural classical order of things, at least from a measurement point of view. First of all the Airy function has everything to do with the light intensity measurements in the neighborhood of the caustic (Airy, 1838, 1848). We can thus “foresee” that the Airy packets would have been introduced perhaps way earlier than it was done (Berry, Balasz, 1979) and probably we would have discussed today in some other terms about the coherent states and about the correlation between the Schrödinger equation and the heat equation (Albeverio, Høegh-Krohn, 1974), to mention only those subjects having an explicit connection with the Airy function. The heat equation itself has to be of third order in space variables (Widder, 1979), and probably the Schrödinger equation would too. Anyway, the issue would have certainly very much impacted the study of stochastic Airy processes (for a recent review of such processes see (Majumdar, Comtet, 2004)) and probably we would have discussed the problem of the cosmic background radiation in some other terms.

But the things remained apparently with no consequence. As mentioned the present author is not aware of the fact that Priest’s observations have raised more than a momentary curiosity when they were communicated in 1919. It seems however that Priest himself thought otherwise. We quote:

Equation (1) (our equation (2) here, n.a.) has precisely the form of the well-known equation of the “probability curve” which suggests that the proposed equation may have some significance, other than a mere empiric relation (Priest, 1919a; our italics)


The author has no theoretical basis for proposing this equation; but recalling that the Planck equation was first an empiric relation to which theory was later forced to conform, he has thought that it might be worthy of notice and perhaps of some theoretical consideration by others. (Priest, 1919b; italics in the original)

Obviously, Priest attached a great physical significance to the fact that there is a transformation which reduces the black-body radiation spectrum to a Gaussian. Perhaps he even tried to force some physical consideration based upon this, in the good old manner of Planck, but apparently with no success. “Others” did not seem to be interested either, so that the present author found the works of Priest quite accidentally. Calculations as above, and the awareness of the importance of Airy function in optics, quantum mechanics, stochastic processes etc., forced him to pay a closer attention to the moments related to the development of the theory of black-body radiation. Priest’s Gaussian certainly gives an experimental “Boltzmann factor” allowing us to repeat the previous line of reasoning for the radiation. However, the main problem here still remains: because the spectrum is given by measurements, one needs to describe the physical structure based on it. In the case of classical Boltzmann factor there is the statistical idea of ensemble of given energy at a certain temperature. Here, however, the things get more complicated: because the “Priest factor” is revealed experimentally, one needs its physical explanation, inasmuch as the statistical one is already at hand. Priest’s Gaussian comes as a probability related to frequency, and reminds us that such a statistic is indeed necessary. In this direction the analysis reveals some suggestions to be added to the already existing ones.

… and An Explanation

Coming back to our main subject – the “Priest factor” – the presence of the cubic root of frequency in Priest’s Gaussian seems to point out in entirely another direction even from the realm of the stochastic Airy processes or K-distributed noise. If one considers the equation of a damped harmonic oscillator as merely a mechanical model of a physical structure which represents what is experimentally recorded, then that equation should be taken a little bit differently. Specifically, whatever we take into consideration as frequency is given, in the good classical manner, by the two roots of the characteristic equation of the damped harmonic oscillator. If l is the damping and ω0 is the basic frequency, these are of the form



We don’t know how to define the theoretical frequency otherwise. However, what we are actually looking for is the energy of an electromagnetic field, viz. a quantity somehow related to the eigenvalues of the corresponding Maxwell tensor. Usually, one assumes that the component fields to be directly measured (Wilhelm, 1985, 1993) in such instances. However, at least for the case of thermal radiation, Priest’s result suggests that the complex frequencies from equation (9) serve in construction of something else which is measured in connection with those real eigenvalues. This “something” can still be theoretically described by a tensor, whose eigenvalues are given, for instance, by the following algebraic recipe:



where ε is the complex cubic root of unity. The data then seem to show simply that a certain algebraic combination of these eigenvalues is normally distributed, which is quite a regular condition in statistical physics. Which combination gives that statistic – can actually be shown by the following procedure of Novozhilov, who was referring originally to the stress tensor of continuum mechanics (Novozhilov, 1952), but certainly can be applied to any tensor.

When we measure a physical tensor, we actually measure its components on a plane in space, and normal to that plane. If the process represented by that tensor is isotropic, as the cosmic background radiation appears to be, then we can safely assume that the measurement in a point in space yields actually two averages: one of the normal components of the tensor on all planes through that point, the other of the corresponding plane components of that tensor. If the eigenvalues of our tensor are denoted by r1,2,3 then the two averages are



Applying this recipe to our eigenvalues from equation (10), the average of normal components is zero, while the average in-plane components is



According to this scenario the data for cosmic background radiation – and perhaps for the thermal radiation in general – tell us that this radiation is an electromagnetic field mechanically represented by a slightly damped harmonic oscillator – in the way we just have shown above – whose equilibrium thermodynamics is dictated by a “superstatistic”, i.e. Priest’s Gaussian for the isotropic average from equation (12). The mean and variance of this distribution are determined by the equilibrium temperature, still according to Wien’s displacement law, as shown by Priest. Mention should be made, however, that such a result can very well be specifically related to the way the measurements of the radiation are usually made, i.e. with a bolometer. A refined spectral and directional measurement would probably show other results.


It is hard nowadays to find experimental data on thermal radiation unbiased by the idea that the spectrum is a Planck one. For instance in engineering applications there are practically no other alternatives. This situation is, of course, economically justifiable: the measurements are costly, and the Planck spectrum is quite sufficient for technological purposes. However, we can see no reasons to cling on it in researches of purely scientific interest, especially today, after the notable successes in electromagnetism, quantum mechanics, theory of stochastic processes etc.

The problem of Planck’s spectrum is constantly reiterated in the literature, and analyzed from all the possible angles with the declared purpose of finding its right physical interpretation. This fact seems to indicate that the fundamental concepts of the physics lying in the background of this problem are not sufficiently well defined. In our opinion the key point is that, when it comes to black-body radiation, the absolute temperature does not reflect a sufficient statistic. We showed here explicitly how the natural “bi-partition” in temperature worked for Planck’s case. It seems to work even for the more sophisticated stochastic structures describing the radiation (Irons, 2000; Varró, 2006a). However, this key point could be noticed even in the early days of the quantum theory, because it was actually quite explicit. Let’s recall that when Planck’s function was applied to the calculation of specific heats of solids, it gave no full satisfaction. Its remedy suggested the zero point energy, which eventually led Timothy Boyer to the hypothesis of a statistic of radiation independent of temperature (Boyer, 1969), in order to obtain the right Planck spectrum. As a matter of fact a statistic independent of the temperature is necessary even for the ideal gas defining the absolute temperature. In a broader sense, Boyer made here a step equivalent to that of Copernicus in the field of astronomy. He realized that there is not a unitary scale for the measured radiation, as there was not a unitary scale for the measurement of distances in the geocentric image of the planetary system. However, exactly as in the classical case with the solar system, we are lucky enough, because Planck’s constant proved actually to be an explicit expression of such a scale, which is what Boyer duly noticed and explicitly used.

From Gaussian statistics point of view, Planck’s reasoning was based on the clear characteristic of the spectrum of thermal equilibrium radiation of showing two distinct parts, of low and high frequency. In this case, we have shown that the “quantum” should be connected with the correlation coefficient – among others – of the two replicas of the statistical process representing the equilibrium radiation. More importantly, in view of the Wien’s displacement law, this fact can be seen as related to frequency as a statistic, thus supplying the missing statistic characterizing the thermal equilibrium. In the limit of weak correlation, the correlation coefficient exclusively dominates the statistics, and the “quantum” depends strictly on temperature, as in the classical counterpart of the system – the ideal gas. Paraphrasing the conclusion of Einstein’s from 1905, one might say that the “ideal gas behaves like radiation of low temperature”. However, in the general case of arbitrary temperature the interpretation is no more so precise, and proves to depend on two distinct parameters, as classically known from the work of Gilbert N. Lewis, the godfather of photons.

The black-body radiation spectrum seems however to reveal a proper statistic specific to frequencies, and this is the Gaussian statistic, that can be taken instead of the a priori Boltzmann factor. Of course this is not a distribution of the frequency per se, but of a cubic-root transform of the frequency. In theoretical statistics such a transformation is known as Box-Cox transformation to normality (Box, Cox, 1964), and this specific cubic-root transformation is known to normalize the Poisson distribution for instance (Efron, 1982). However it seems that there is no physical reason to accept such a transformation as being in the natural order, except for the fact that the cubic root of frequency represents the eigenvalue of a 3×3 tensor somehow related to the frequency of the black-body radiation. In other words, in the classical Rayleigh-Jeans procedure of counting the modes, the frequency should be considered not as a vector, but as a tensor. Then we find quite naturally that, if the radiation is represented by an electromagnetic field, we cannot actually measure but the eigenvalues of its energy tensor – the Maxwell tensor – and the Gaussian can be the statistical expression of this situation. The eigenvalues we are measuring are real numbers and they should be somehow related to the cubic root of the frequency, for instance by a sort of constitutive law. As long as we take theoretically the frequency as being a parameter related to a mechanical harmonic oscillator, which is what gives it physical meaning, this conclusion seems unavoidable. Thus, quite probably, the Gaussian distribution revealed by Irwin G. Priest represents this situation.

The most important conclusion though regards the cosmic background radiation, for which one can hardly say – and with good reasons – that represents a blackbody radiation in the classical sense (Robitaille, 2009). Priest’s results seem however to indicate, on one hand that the experimental data are dominated by the electromagnetic nature of the field and, on the other hand that the global thermal equilibrium – therefore Kirchhoff’s universality – is not a necessary condition in defining the spectrum.



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Figure 1. Priest’s Gaussian representation of CMBR monopole data


Figure 2. Priest’s Gaussian representation of CMBR dipole data


Figure 3. Priest’s and Planck’s CMBR Curves

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