Underlying probability distributions of Planck's radiation law

Abstract

The derivation of Planck's radiation law can be considered as a transformation of a thermodynamic relation for black-body radiation into a fundamental relation in which the error law is the negative binomial distribution. In both limiting frequency ranges it transforms into Poisson distributions; in the Wien limit, it is the distribution of the number of photons, whose most probable value is given by Boltzmann's expression, while in the Rayleigh-Jeans limit, it is the distribution of the number of Planck oscillators. In the general case, they are Bernoullian random variables. In the Rayleigh-Jeans limit, the probability of determining the number of oscillators in a given frequency interval for a fixed value of the energy can be inverted to determining the probability of the energy for a fixed number of oscillators. The probability density is that of the canonical ensemble.