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IV. Second Thing Left Behind

By accepting the Boltzmann’s and Gibbs’ view regarding the relation between probability and entropy, Physics placed itself into the hands of the so-called class of exponential families of distributions. In the case of one statistical variable X such a family, having the elementary probability given by

(25)

is known as natural exponential family. Here ξ is the parameter scanning the family, while ν(dx) is a Stieltjes measure of the domain of statistical variable X. The parameter ξ, usually related to the measurement of the variable X, is connected to the mean of the ensembles characterized by (25) through equation

(26)

a relation used explicitly before in order to interpret u as a mean over an ensemble of oscillators.

Now, the previous stride of reasoning may be flawed by the fact that the differential equation (9) looks much like an interpolation equation, having no substance of principle involved in its derivation. Its formal deduction as based on Gaussian distribution in the previous section might thus be tarnished as being too particular, if not very approximate. Fact is that one can get the equation for the fluctuations starting from physical considerations upon the field sustaining the fluctuations, and the result is a quadratic polynomial for the variance. This fact proves to be essential from a certain point of view. Indeed, if we limit the considerations to natural exponentials, we then have necessarily

(27)

where V(ξ) is the variance function of the family of exponentials. If this function depends on the parameter in such a way that it can be arranged as a quadratic polynomial in the mean of the family of distributions, then we have a particular case of exponentials, modernly termed as quadratic variance distribution functions (Morris, 1982). These distributions cover just about everything we use today in the realm of Physics and Engineering: Gaussian, Binomial, Negative Binomial, Poisson, Gamma, Generalized Hyperbolic Secant (Morris, 1982).

Then, one can notice the advantage of the Born’s approach to Planck’s problem even if we do not use the Gaussian approach: in principle, the statistics of radiation can be characterized by such a quadratic relationship between the mean and the variance of an ensemble representing the radiation. The corresponding distribution is not necessarily Gaussian. In this case the whole problem of radiation can be treated in its utmost generality, for the two limiting processes are quite different in nature, albeit both quadratic variance processes. Namely, the equation (9) which is the characteristic of Born’s line of reasoning can, by a simple metamorphosis, be put in the Morris’ canonical form for a quadratic variance distribution function (Morris, 1982, Table 1)

(28)

Here we denoted

(29)

where u₀ is an arbitrary constant energy density. The parameter r is not a correlation coefficient anymore. Then, according to Morris’ scheme, Planck’s result is a Negative Binomial distribution (NB(r, p)) with the density of probability given by

(30)

where the variable X is discrete, x = 1, 2, … , p ≡ e^θ is a probability (the “Boltzmann factor”) and q = 1 - p. The mean of this distribution is given by

(31)

which is Planck’s formula, written however in such a way as to make explicit the occurrence of unit u₀: u = mu₀. Let us try to find some limit distributions for NB(r, p).

First of all, we are interested in those probabilities p close to one. As we have

(32)

and by definition

(33)

one can directly write

(34)

According to Morris’ classification, this represents a Gamma distribution having the density

(35)

for x real and positive. For r = 1, i.e. ε₀ = u₀, this distribution is the classical exponential. In general, for finite r, this limit distribution of radiation refers to the case where the mean energy of the ensemble characterizing the heat radiation is high (m → ∞). The temperature of this state cannot be but high too, in order to make p ~ 1. At arbitrary but finite densities of radiation energy this limit distribution is also characteristic for r → 0, which comes down to ε₀ << u₀, no matter of the relationship between ε₀ and kT, i.e. no matter of the value of the mean m. In other words, the process can be a Gamma process in classical as well as in quantum case, depending on the unit we choose for the measurement of the radiation spectrum.

Another limiting case of the general Negative Binomial distribution characterizing the Planck process, is that where the probability p is very small. According to the definition of p this happens for low temperatures, so that θ → ─ ∞, and is realized by an ensemble of very low energy density. In equation (32) the linear term prevails so that, according to Morris’ classification the process is a Poisson one as characterized by the probability density

(36)

for the variable x = 0, 1, 2, … . For arbitrary energy densities, the Poisson limit distribution can also be realized when r → ∞, i.e. when ε₀ >> u₀, no matter of the relationship between ε₀ and kT. Again, just as before, the process can be a Poisson process in classical as well as in quantum case, depending on the unit we choose for the measurement of the radiation spectrum.

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