Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation

It is shown that the energy of a mode of a classical chaotic field, following the continuous exponential distribution as a classical random variable, can be uniquely decomposed into a sum of its fractional part and of its integer part. The integer part is a discrete random variable whose distribution is just the Bose distribution yielding the Planck’s law of black-body radiation. The fractional part – as we call it – is the “dark part” with a continuous distribution, which is, of course, not observed in the experiments. It is proved that the Planck-Bose distribution is infinitely divisible, and the irreducible decomposition of it is given. The Planck variable can be decomposed into an infinite sum of independent binary random variables representing the “binary photons” – more accurately photo-molecules or multiplets – of energy 2hν with s = 0, 1, 2, ... . These binary photons follow the Fermi statistics. Consequently the black-body radiation can be viewed as a mixture of thermodynamically independent fermion gases consisting of “binary photons”. The binary photons give a natural Sándor Varró : Irreducible Decomposition of Gaussian Distributions 2 tool for the dyadic expansion of an arbitrary (but not coherent) ordinary photon excitations. It is shown that the binary photons have wave-particle fluctuations of fermions. These fluctuations combine to give the wave-particle fluctuations of the original bosonic photons, expressed by Einstein’s fluctuation formula.