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 (we call it the Planck variable) whose distribution is just the Bose distribution yielding Planck's law of black-body radiation. The fractional part is the 'dark part' represented by the 'dark variable' with a continuous distribution, which is, of course, not observed in the experiments. It is proved that the Bose distribution is infinitely divisible, and the irreducible decomposition of it is given. This means that the Planck variable can be decomposed into an infinite sum of independent binary random variables representing the 'binary photons' (more accurately photo-molecules or photo-multiplets) of energies 2shν with s=0, 1, 2, .... These binary photons follow Fermi statistics. According to our present analysis, the black-body radiation can be viewed as a mixture of statistically and thermodynamically independent fermion gases consisting of 'binary photons'. The binary photons give a natural tool for the dyadic expansion of 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