The Joint Energy Distributions of the Bose-Einstein and of the Fermi-Dirac Particles

Abstract

Vincze (1959, 1961) derived the Planck-Bose-Einstein (PBE) probability density function for the energy distribution of black body radiation. This derivation was based on an expression for the information measure (negentropy) belonging to a continuous random variable and on the Bose-Einstein statistics; the quantum hypothesis of Planck was not needed. The present note considers the joint distribution of several variables. In earlier works, the method for solving the extreme value problem happened with a method due to Kullback and Leibler (1951) and the result did not agree with the formula given by Planck (1900), unless certain element was neglected. In this paper, the authors consider again the extreme value problem but through Lagrange method of the theory of variation which yields the PBE formula exactly. For the Fermi-Dirac case, the procedure goes on the same line. As a consequence, one has the property “indistinguishability of the particles” as a tool for arriving at the results, but the dependence of the random, variables is essential. This suggests that our procedure is appropriate even for gases which are neither bosons nor fermions, but the particles influence each other and their distributions are not independent [see Wilczek (1991)].